Rival Systems Options Knowledge Base
Intermediate Guide to Options
 Select Your Level
 Beginner
 Intermediate
 Advanced
Chapters
 Intermediate Guide to Options
 Theoretical Pricing Models
 Applying Probability to Option Pricing
 The BlackScholes Model
 Volatility
 Estimating Volatility
 Types of Volatility
 Using an Option's Theoretical Value
 The End Result
 Option Sensitivities
 Defining the Sensitivities
 Volatility Spreads
 Types of Volatility Spreads
 Spread Sensitivities
 Adjustments
 Directional Strategies
 Bull and Bear Volatility Spreads
 Vertical Spreads
 Option Arbitrage
Intermediate Options
Intermediate options training represents the next phase in a trader's journey to master the world of financial derivatives. Building upon the foundational knowledge gained in basic options training, intermediatelevel education delves deeper into the intricacies of options trading.
This material aims to equip traders with more advanced tools and strategies, enabling them to take their trading skills to a higher level. In this stage, traders develop a deeper understanding of options pricing, volatility, and risk management, allowing them to make more informed and strategic decisions in the complex world of options.
One key aspect of intermediatelevel options techniques is the exploration of more sophisticated trading strategies. Traders move beyond simple calls and puts to discover strategies like vertical spreads, iron condors, and butterflies. These multileg strategies provide traders greater flexibility in structuring their positions and managing risk.
We'll also delve into the concepts of delta, gamma, theta, and vega – the Greeks – which help traders understand how an option's value changes in response to various market factors. This knowledge empowers traders to finetune their strategies and adapt to market conditions.
Risk management is another crucial component of intermediatelevel options programming. Traders learn how to use options not only for speculation but also for hedging existing positions. They gain insights into portfolio management and position sizing techniques, which are essential for protecting capital and mitigating potential losses.
Additionally, traders explore the concept of implied volatility and its impact on options pricing, enabling them to identify mispriced options and potential arbitrage opportunities. Intermediate options training equips traders with the tools and insights needed to navigate the markets confidently and precisely, whether their goal is income generation, risk reduction, or capital growth.
Disclaimer
The material in this options guide is meant for educational purposes only. The information, strategies, and examples presented are not to be construed as trading or investment advice. Rival Systems does not endorse or recommend specific trading or investment decisions and users are encouraged to exercise their own judgement and seek professional advice before making any financial decisions.
Users are urged to carefully consider their financial objective, risk tolerance, and level of experience before engaging in trading or investment activities. Rival Systems is not responsible for any inaccuracies, errors, or omissions in the educational content or for any actions taken in reliance on such content.
Chapter 02
Theoretical Pricing Models
Theoretical Pricing Models
While they do not solve all the problems with which option traders must deal, the great majority of active option traders use theoretical pricing models to aid in decisionmaking and risk management. An indepth understanding of the mathematics of pricing models is not a requirement for success in options (indeed, most option traders are not mathematicians), but the ability to interpret the numbers generated by theoretical pricing models and a basic understanding of the assumptions upon which the models are based, will enable a trader to make the best use of model generated values.
Theoretical pricing model
A mathematical model designed to evaluate a security or contract given certain prior assumptions about characteristics of the contract as well as conditions in the marketplace.
The Role of Probability
There is almost an infinite variety of strategies a trader can pursue in the marketplace, given an opinion about market conditions. Whatever the basis for the opinion, it will probably be expressed with terms such as "good chance," "highly likely," "possible," "improbable," etc.
The problem with this approach is that opinions cannot easily be expressed in numerical terms. What do we really mean by "good chance"? Or by "highly unlikely"? If we want to approach option markets logically, we will need some method of quantifying our opinions about price movements.
Since one can never be certain about future market conditions, almost all trading decisions are based on the laws of probability. Under some circumstances, if a trader believes that a strategy has a very high probability of profit and a very low probability of loss, they will be satisfied with a small potential profit since the profit is likely to be quite secure.
On the other hand, if the probability of profit is very low, the trader will demand a large profit when market conditions do develop favorably. Because of the importance of probability in the decisionmaking process, it will be worthwhile to consider some basic probability concepts, expected return, and theoretical value.
Expected Return
Consider a roulette bet of the type commonly offered in the United States. The roulette wheel has 38 slots, numbered 1 through 36, 0 and 00. Suppose a casino allows a player to choose a number. If the player's number comes up, he receives 36; if any other number comes up, they receive nothing.
If a player were allowed to pick a number an infinite number of times, on average, how much would they expect to get back each time they picked a number? There are 38 slots on the roulette wheel, each with equal probability, but only one slot will return 36 to the player. If we divide the one way to win 36 by the 38 slots on the wheel, the result is 36/38 = .9474, or about 95¢.
This is the average payback or expected return. A player who pays 95¢ for the privilege of picking a number at the roulette table can expect to break about even in the long run.
Expected return
The average amount that someone can expect to receive at the culmination of a particular wager or investment. This determination is usually based on the laws of probability.
Of course, no casino will let a player buy such a bet for 95¢. Under those conditions, the casino would make no profit. In the real world, a player who wants to purchase such a bet will have to pay more than the expected return, typically 1. The 5¢ difference between the 1 price of the bet and the 95¢ expected return represents the profit potential, or edge, to the casino. In the long run, for every dollar bet at the roulette table, the casino can expect to keep about 5¢.
Theoretical Edge or Edge
The difference between the expected return and the current price of a particular wager or investment.
Given the above conditions any player interested in making a profit would rather switch places with the casino so that they could be the house. Then they would have a 5¢ edge on their side by selling bets worth 95¢ for 1. Alternatively, the player would like to find a casino where they could purchase the bet for less than its expected return of 95¢, perhaps 88¢. Then the player would have a 7¢ edge over the casino.
The concept of expected return is important in option pricing because it tells us what the average payoff on a proposition will be based on the laws of probability. While the expected return does not tell us what will happen on any particular outcome, we know that in the long run, the average outcome will converge to the expected return.
The expected return can be calculated by multiplying the amount of the payoff by the probability of the payoff. If there is more than one possible payoff, then the expected return is the sum of the individual payoffs multiplied by each associated probability:
Formula: Expected Return
expected return = (payoff1 x probability1) +...+ (payoffn x probabilityn)
For example, suppose you are given a chance to roll a 6sided die, with faces numbered 1 through 6, and you will receive an amount equal to the number rolled (1 if you roll 1, 2 if you roll 2, ..., 6 if you roll 6). What is the expected return for one roll of the die? There is a 1/6 chance of rolling any number, so the total expected return is:
(1/6 x 1) + (1/6 x 2) + (1/6 x 3) + (1/6 x 4) + (1/6 x 5) + (1/6 x 6)
= 1/6 x (1 + 2 + 3 + 4 + 5 + 6)
= 21/6
= 3.50
The expected return is 3.50.
Theoretical Value
The theoretical value of a proposition is the price one would expect to pay in order to just break even in the long run. Thus far, the only factor we have considered in determining the value of a proposition is the expected return. We used this concept to calculate the 95¢ fair price for the roulette bet. There may, however, be other considerations.
Theoretical value
An option value is generated by a mathematical theoretical pricing model given certain prior assumptions about the terms of the option, the characteristics of the underlying contract, and prevailing interest rates.
Suppose that in our roulette example, the casino decides to change the conditions of the bet slightly. The player may now purchase the roulette bet for its expected return of 95¢ and, as before, if they lose, the casino will immediately collect their 95¢. Under the new conditions, however, if the player wins, the casino will send them their 36 winnings in two months. Will both the player and the casino still break even on the proposition?
Where did the player get the 95¢ they used to place their bet at the roulette wheel? In the immediate sense, they may have taken it out of their pocket. But a closer examination may reveal that they withdrew the money from their savings account prior to visiting the casino.
Since they won't receive their winnings for two months, they will have to take into consideration the two months' interest they would have earned had they left the 95¢ in their savings account. If interest rates are 12% annually, the interest loss over two months is 2% x 95¢, or about 2¢. If the player purchases the bet for its expected return of 95¢, they will be a 2¢ loser because of the cost of carrying a 95¢ debit for two months. The casino, on the other hand, will take the 95¢, put it in an interestbearing account, and at the end of two months, collect 2¢ in interest.
Under these new conditions, the theoretical value of the bet is the expected return of 95¢ less the 2¢ carrying cost on the bet, or about 93¢. If a player pays 93¢ for the roulette bet today and collects their winnings in two months, neither they nor the casino can expect to make any profit in the long run.
The two most common considerations in a financial investment are the expected return and carrying costs. There may, however, be other considerations. For example, suppose the casino decided to send the player a 1¢ bonus over the next two months. They could then add this additional payment to the previous theoretical value of 93¢ to get a new theoretical value of 94¢. This is like the dividend paid to owners of stock in a company.
By purchasing a proposition for less than its theoretical value or by selling a proposition for more than its theoretical value, a trader creates a positive theoretical edge.
A positive theoretical edge is no guarantee of a profit on any one occurrence. But in the long run, if a trader has correctly evaluated a proposition, consistently making trades with a positive theoretical edge will, on average, yield a profit for the trader.
The goal of option evaluation is to determine, through the use of theoretical pricing models, the theoretical value of an option. A trader can then make an intelligent decision whether the option is overpriced or underpriced in the marketplace and whether the theoretical edge is sufficient to justify going into the marketplace and making a trade.
Summary
Expected Return 

Theoretical Value 

Chapter 03
Applying Probability to Option Pricing
Applying Probability to Option Pricing
All theoretical pricing models work in essentially the same way. They propose a series of possible prices for the underlying contract, assign a probability to each of these prices, and from this, calculate the expected return for an exercise price at expiration. From the expected return, the model deducts the carrying cost, thereby yielding a theoretical value for the option.
Assigning Probabilities to Different Outcomes
Suppose an underlying contract is trading at 100 and that on a certain date in the future, which we will call expiration, the contract can take on one of five different prices: 80, 90, 100, 110, or 120. Assume, moreover, that each of the five prices is equally likely with 20% probability. The prices and probabilities might be represented as follows:
80  90  100  110  120 

20%  20%  20%  20%  20% 
If we take a long position in the underlying contract at today's price of 100, what will be the expected return from this position at expiration? Twenty percent of the time, we will lose 20 when the contract ends up at 80. Twenty percent of the time, we will lose 10 when the contract ends up at 90. Twenty percent of the time, we will break even when the contract ends up at 100. Twenty percent of the time, we will make 10 when the contract ends up at 110. And twenty percent of the time, we will make 20 when the contract ends up at 120. We can write the arithmetic:
(20% x 20)  (20% x 10) + (20% x 0) + (20% x 10) + (20% x 20) = 0
Since the profits and losses exactly offset each other, the expected return to the long position is zero. The same reasoning will show that the expected return to a short position taken at the current price of 100 is also zero. Given the prices and probabilities, if we take either a long or short position we can expect to just break even in the long run.
Now suppose that we take a long position in a 100 call. Putting aside for a moment the question of what we might pay for the call, what will be the expected return given our prices and probabilities? If the underlying contract finishes at 80, 90, or 100, the call will expire worthless. If the underlying contract finishes at 110 or 120, the call will be worth 10 and 20, respectively. The arithmetic is:
(20% x 0) + (20% x 0) + (20% x 0) + (20% x 10) + (20% x 20) = 6
The call can never be worth less than zero, so the expected return from the call position is always a nonnegative number, in this case, 6.
Choosing Appropriate Probabilities
We previously attempted to generate a theoretical value by assigning the same probability of 20% to all price outcomes in the underlying contract. In the real world, however, we know from experience that not all outcomes are equally likely. It therefore makes more sense to assign different probabilities to different prices.
For example, we may want to assign lower probabilities to outcomes that are farther away from the current price. This is shown below:
80  90  100  110  120 

10%  20%  40%  20%  10% 
Now the expected return from a 100 call is:
(10% x 0) + (20% x 0) + (40% x 0) + (20% x 10) + (10% x 20) = 4.00
Suppose also that the option has two months to expiration and that we must pay for it today. If interest rates are 12% annually (1% per month), the carrying costs are 2%. The theoretical value of the option will then be:
4.00  (2% x 4.00) = 3.92
Note that even though we changed the probabilities, the expected return to a long (or short) stock position taken at 100 is still zero:
(10% x 20) + (20% x 10) + (40% x 0) + (20% x 10) + (10% x 20) = 0
As long as the probabilities are arranged symmetrically around the current price, such that for every upward move, there is a downward move of equal probability and equal magnitude, the expected return is zero.
ArbitrageFree Markets
Up to now, even though we may have assigned different probabilities to different outcomes, all outcomes and probabilities have been arranged symmetrically. For each upward price move, there is a downward move of equal magnitude and probability. We might, however, believe that the expected return to an underlying contract is not zero, and that there is a greater chance that the contract will move in one direction rather than another. Look at the following price outcomes and probabilities:
80  90  100  110  120 

10%  20%  30%  25%  15% 
Using these new probabilities, the expected return from a long position in the underlying contract is:
(10% x 20)  (20% x 10) + (30% x 0) + (25% x 10) + (15% x 20) = 1.50
The expected return for the 100 call is:
(10% x 0) + (20% x 0) + (30% x 0) + (25% x 10) + (15% x 20) = 5.50
Note that the underlying contract now has a positive expected return, so it may seem that there is money to be made simply by purchasing the underlying contract. This would be true if there were no other considerations. But suppose the underlying contract is a stock, which we purchase at today's price of 100 and hold for some period. There is clearly a carrying cost associated with the investment.
If the carrying cost is exactly equal to the expected return of 1.50, we will just break even. For a long stock position to be profitable, the stock must appreciate by at least the amount of carrying costs over the holding period. Therefore, the expected return from the stock must be some positive number. If we assume that any stock trade will just break even, the expected return must be equal to the carrying costs.
Dividend
An amount paid by a company to the shareholders in the company. Dividends are usually paid from one to four times annually, with the quoted dividend representing the amount paid for each share of stock held.
Some stocks also pay dividends. If the dividend is paid during the holding period, it will affect the expected return. A trader who buys stock will have to pay out carrying costs, but they will receive the dividends. If we again assume that a stock trade will break even, the expected return at the end of the holding period must be identical to the carrying costs less the dividend. If the carrying cost for the stock over some period is 3.50, and a 1 dividend is expected during this period, the expected return at the end of the period must be 2.50.
A trader who purchases the stock today will incur an interest debit of 3.50 at the end of the holding period, but this will be exactly offset by the 1.00 dividend that they receive during the holding period, as well as the 2.50 expected return at the end of the period. (Of course, the trader can also earn interest on the dividend from the time they receive it until the end of the holding period. Since this will usually be a very small amount in relation to the other factors, for practical purposes we can ignore it.)
Arbitrage
The purchase and sale of the same product in different markets to take advantage of a price disparity between the two markets.
Almost all theoretical pricing models assume that the underlying market is arbitragefree; no profit can be made by either buying or selling an underlying contract, and all credits and debits, including the expected return, must exactly cancel out.
If we assume an arbitragefree market, we must necessarily assume that the average price of the contract at the end of the holding period is the current price plus an expected return, which will exactly offset all other credits and debits.
If the holding costs on a 100 stock over some period are 4, the average price at the end of the period must be 104. If the stock also pays a 1 dividend, the average price must be 103. In both cases, the credits and debits will exactly cancel out.
We can now summarize the necessary steps in developing a model:
 Propose a series of possible prices at expiration for the underlying contract
 Assign an appropriate probability to each possible price
 Maintain an arbitragefree underlying market
 From the prices and probabilities in steps 1, 2, and 3, calculate the expected return for the option
 From the option's expected return, deduct the carrying cost
If we can accomplish all this, we will finally have a theoretical value from which we can begin to trade.
Summary
 Theoretical pricing models attempt to calculate the expected return for an option by assigning probabilities to different underlying prices at expiration. By factoring in carrying costs, the model then generates a theoretical value for the option.
 Theoretical pricing models always assign probabilities to underlying prices in such a way that a trade made in the underlying contract will always break even.
Chapter 04
The BlackScholes Model
The BlackScholes Model
In its most basic form, the BlackScholes model is designed to evaluate European options where the underlying contract is a specified number of shares of stock.
The reasoning, that led to the development of the BlackScholes model, depends on the five steps we listed in the previous section. To calculate an option's theoretical value using the model, we need to know a minimum of five characteristics of the option and its underlying contract.
These are:
 The option's exercise price
 The amount of time remaining to expiration
 The current price of the underlying contract
 The riskfree interest rate over the life of the option
 The volatility of the underlying contract
The last input, volatility, may be unfamiliar to the new trader. While we will put off a detailed discussion of this input to the next section, from our previous discussion, one can reasonably infer that volatility is related to the probabilities that are assigned to different price outcomes.
If we know each of the required inputs, we can feed them into the theoretical pricing model and thereby generate a theoretical value.
The Hedge Ratio
Black and Scholes also incorporated into their model the concept of the riskless hedge. For every option position, there is a theoretically equivalent position in the underlying contract such that, for small price changes in the underlying contract, the option position will gain or lose value at exactly the same rate as the underlying position.
In theory, one can profit from a mispriced option by offsetting the option position with this theoretically equivalent underlying position. That is, whatever option position we take, we must take an opposing market position in the underlying contract. The correct proportion of underlying contracts needed to establish this riskless hedge is known as the hedge ratio option by offsetting the option position with this theoretically equivalent underlying position.
Mispriced
The determination that the price in the marketplace of a wager or investment differs from a modelgenerated theoretical value.
Hedge
A secondary position is taken to protect the value of some primary position. If the value of the primary position declines, the losses are at least partially offset by an increase in the value of the hedge position.
Hedge ratio
The theoretically correct number of underlying contracts to option contracts is required to establish a neutral hedge.
Why is it necessary to establish a riskless hedge? Recall that in our simplified approach an option's theoretical value depended on the probability of various price outcomes for the underlying contract. As the underlying contract changes in price, the probability of each outcome will also change. If the underlying price is currently 100 and we assign a 25% probability to 120, we might drop the probability for 120 to 10% if the price of the underlying contract falls to 80. By initially establishing a riskless hedge and then adjusting this hedge as market conditions change, we are taking into consideration these changing probabilities.
In this sense an option can be thought of as a substitute for a similar position in the underlying contract. A call is a substitute for a long position; a put is a substitute for a short position. Whether it is better to take the position in the option or in the underlying contract depends on the theoretical value of the option and its price in the marketplace.
If a call can be purchased (sold) for less (more) than its theoretical value, it will, in the long run, be more profitable to take a long (short) market position by purchasing (selling) calls than by purchasing (selling) the underlying contract. In the same way, if a put can be purchased (sold) for less (more) than its theoretical value, it will, in the long run, be more profitable to take a short (long) market position by purchasing (selling) puts than by selling (buying) the underlying contract.
Since the theoretical value obtained from a theoretical pricing model is no better than the inputs into the model, a few comments on each of the inputs will be worthwhile.
Model Inputs
Exercise Price as a Model Input
There should never be any doubt about the exercise price of an option, since it is fixed in the terms of the contract and does not vary over the life of the contract. An ABC March 55 call cannot suddenly turn into a March 60 call or a March 50 call. An XYZ July 80 put cannot turn into a July 75 put or a July 85 put. (It is true that an exchange may adjust the exercise price of a stock option if there is a stock split. In practical terms this is not really a change in the exercise price because the exercise price retains the same relationship to the stock price. The characteristics of the option contract remain essentially unchanged.)
Time to Expiration as a Model Input
Like the exercise price, the option's expiration date is fixed and will not vary. Our ABC March 55 call will not suddenly turn into an April 55 call, nor will our XYZ July 70 put turn into a June 75 put. Of course, each day that passes brings us closer to expiration, so in that sense, the time to expiration is constantly growing shorter. However, the expiration date, like the exercise price, is fixed by the exchange and will not change.
Time to expiration is entered into the BlackScholes Model as an annualized number. If we are entering raw data directly into the model, we must make the appropriate annualization. With 91 days remaining to expiration, we would enter an input of .25 (91 / 365 = .25). With 36 days remaining, we would enter .10 (36 / 365 = .10). Most option platforms already have this transformation incorporated into the software so that we need only enter the correct number of days remaining to expiration.
Underlying Price as a Model Input
Unlike the exercise price and time to expiration, the correct price of the underlying is not always obvious. At any one time, there is usually a bid price and an asked price, and it may not be clear whether we ought to use one or the other of these prices or perhaps some price in between.
We have noted that the correct use of an option's theoretical value requires us to hedge the option position with an opposing trade in the underlying contract. Therefore, the underlying price we feed into our theoretical pricing model ought to be the price at which we believe we can make the opposing trade.
If we intend to purchase calls or sell puts, both of which are long market positions, we will have to hedge by selling the underlying contract. In that case, we ought to use the bid price since that is the price at which we can sell the underlying. On the other hand, if we intend to sell calls or buy puts, both of which are short market positions, we will have to hedge by purchasing the underlying contract. Now we ought to use the asked price since that is the price at which we can buy the underlying.
In practice, the bid and offer are constantly changing, and many traders will simply use the last trade price as the basis for theoretical evaluation. However, the last trade price may not always reflect the present market. Even the settlement price quoted on the exchange may not accurately reflect the market at the close of business. The last trade price may show 75.25 for a contract, but the market at the close may have been 75.25 bid, 75.50 offered.
A trader who hoped to buy at 75.25 would have very little chance of being filled because of the difficulty of buying at the bid price. Even a purchase at some middle price, say 75.40, may be unlikely if the market is very unbalanced, with many more contracts being bid for at 75.25 than offered at 75.50. For all of these reasons, an experienced trader will rarely enter an option market without knowing the exact bid and offer in the underlying market.
Interest Rates as a Model Input
Since an option trade may result in either a cash credit or debit to a trader's account, the interest considerations resulting from this cash flow must also play a role in option evaluation. This is a function of interest rate over the life of the option.
The interest rate component plays two roles in the theoretical evaluation of options. First, it will affect the carrying costs of the underlying contract. As we raise interest rates, we raise the carrying costs, increasing the value of calls and decreasing the value of puts. Secondly, the interest rate affects the cost of carrying the option. As we raise interest rates, we decrease the value of the option. In spite of the fact that the interest rate plays two roles, in most cases, the same rate is applicable and we need only input one interest rate into the model.
If, however, different rates are applicable, such as would be the case with foreign currency options (the foreign currency interest rate plays one role, and the domestic currency interest rate plays a different role), the model will require the input of two interest rates. In some cases, as with a futures contract, there are no carrying costs associated with the underlying contract.
What interest rate should a trader use when evaluating options? Most traders cannot borrow and lend at the same rate, so the correct interest rate will, in theory, depend on whether the trade will create a debit or a credit. In the former case, the trader will be interested in the borrowing rate, while in the latter case, they will be interested in the lending rate. In practice, however, the most common solution is to use the riskfree interest rate, the rate that applies to the most creditworthy borrower.
In the U.S., the government is usually considered the most creditworthy borrower of funds, so the yield on government security with a term equivalent to the life of the option is the general benchmark. For a 60day option, one can use the yield on a 60day treasury bill; for a 180day option, one can use the yield on a 180day treasury bill.
Treasury bill
A security, often issued by a federal government, will be redeemed at a predetermined price on its maturity date. The issuance of treasury bills represents a means by which an issuer can borrow funds over some period of time, typically from 30 days to one year.
Volatility as a Model Input
Of all the inputs required for option evaluation, volatility is the most difficult for traders to understand. At the same time, volatility often plays the most important role in actual trading situations. Changes in our assumptions about volatility can have a dramatic effect on an option's value, and the manner in which the marketplace assesses volatility can have an equally dramatic effect on an option's price. For these reasons, we will devote the next section to a detailed discussion of volatility.
Dividends as a Model Input
This was not one of the original inputs into the BlackScholes model, but many stocks do pay dividends. To accurately evaluate a stock option, a trader must know both the amount of the dividend that the stock will pay and the exdividend date, the date on which a trader must own the stock to receive the dividend. The emphasis here is on ownership of the stock. A deeply inthemoney option may have many of the same characteristics as stock, but only ownership of the stock carries with it the right to collect the dividend.
Dividend
An amount paid by a company to the shareholders in the company. Dividends are usually paid from one to four times annually, with the quoted dividend representing the amount paid for each share of stock held.
Exdividend
The day on which a dividend paying stock is trading without the right to receive the dividend.
In the absence of other information, traders tend to assume that a company will continue the same dividend policy it has had in the past. If the company has been paying a 75¢ dividend each quarter, it will probably continue to do so. However, this is not always a certainty. Companies sometimes increase or decrease dividends and occasionally omit them completely.
If there is the possibility of a change in a company's dividend policy, a trader has to consider its impact on option values. Additionally, if the exdividend date is expected just prior to expiration, there is the danger that a delay of several days will cause the exdividend date to fall after expiration. For purposes of option evaluation, this is the same as eliminating the dividend completely. In such a situation, a trader ought to make a special effort to ascertain the exact exdividend date.
Summary
 The BlackScholes model is designed to evaluate European options on stock.
 The BlackScholes model assumes that when buying or selling an option, a trader will take an opposing market position in the underlying contract. The theoretically correct opposing position in the underlying is determined by the option's hedge ratio.
 The BlackScholes model for stock options requires the following inputs: exercise price, time to expiration, underlying price, interest rate, volatility, and the amount and payment date of any dividend.
 Two common variations on the BlackScholes model are the Black model, designed to evaluate European options on futures, and the GarmanKohlhagen model, designed to evaluate European options on foreign currencies.
Chapter 05
Volatility
Volatility
What is volatility and why is it so important to an option trader? The option trader, like a trader in the underlying instrument, is interested in the direction of the market. But unlike the trader in the underlying, an option trader is also extremely sensitive to the speed of the market. If the market for an underlying contract fails to move at a sufficient speed, options on that contract will have less value because of the reduced likelihood of the market going through an option's exercise price. In a sense, volatility is a measure of the speed of the market. Markets that move slowly are lowvolatility markets; markets that move quickly are highvolatility markets.
Volatility
The degree to which the price of an underlying instrument tends to fluctuate over time.
One might guess intuitively that some markets are more volatile than others are. Some commodities and high technology stocks have been known to double in short periods of time. Yet few traders would predict that a stock index like the S&P 500 might double in a similar period. If we know whether a market is likely to be relatively volatile or relatively quiet, and can convey this information to a theoretical pricing model, any evaluation of options on that market will be more accurate than if we simply ignore volatility. Since option models are based on mathematical formulas, we will need some method of quantifying this volatility component so that we can feed it into the model in numerical form.
Normal Distributions
Most option pricing models, including the BlackScholes model, assume that prices follow a random walkthrough time. That is, prices move in such a way that one cannot predict beforehand in which direction prices will move. One might say that at any moment in time, there is a 50% chance that the price will go up and a 50% chance that the price will go down.
If prices follow a random walk, the resulting distribution of prices will form a familiar normal, or bellshaped, distribution shown in the following graph:
Even if prices form a normal distribution, there are many different normal distributions that are possible for an underlying contract. Different distributions can significantly affect an option's value. The following figure shows several different normal distributions with underlying centered at 100.
If we are trying to evaluate a 120 call, we can see that if the distribution is very restricted with a high peak and narrow tails (the low volatility distribution), the option will have very little chance of finishing inthemoney.
Consequently, it will have very little value. If, however, the distribution spreads out very quickly with a low peak and very wide tails (the high volatility distribution), the option will have a much better chance of finishing inthemoney. In this case, the option may have a relatively high value. If the distribution is somewhere in between (the moderate volatility distribution), the option will have a value between these two extremes.
Normal distributions can be fully described with two numbers, the mean and the standard deviation. Graphically, we can interpret the mean as the location of the peak of the curve and the standard deviation as a measure of how fast the curve spreads out. In addition, the standard deviation can be interpreted as a probability. In particular, the standard deviation tells us the probability of getting an occurrence within a specified distance of the mean. The exact probability associated with any specific number of standard deviations can be found in a statistics book. For option traders, the following approximations will be useful:
Standard Deviations  Probability Occurrence 

1  68.3% (about 2/3) 
2  95.4% (about 19/20) 
3  99.7% (about 369/370) 
Note that each number of standard deviations is preceded by a plus or minus sign. Because normal distributions are symmetrical, the likelihood of up movement and down movement of the same magnitude is identical.
Normal distribution
A theoretical distribution resulting from an infinite number of random events. A normal distribution is symmetrical, with most of the events concentrated near the middle of the distribution and progressively fewer events falling at the tails of the distribution. A normal distribution is often referred to as a bellshaped distribution.
Standard deviation
In a normal distribution, a measure of how the events are distributed. A low standard deviation indicates that a large number of the events are concentrated near the middle of the distribution. A high standard deviation indicates that more of the events fall near the tails of the distribution.
Forward Price as a Mean
When we enter the present price of an underlying instrument, we are actually entering the mean of a normal distribution curve. An important assumption in the BlackScholes Model is that the underlying market is arbitragefree. In the long run a trade in the underlying instrument will just break even. It will neither make money nor lose money.
Given this assumption, the mean of the normal distribution curve assumed in the model must be the price at which a trade in the underlying instrument, either a purchase or a sale, will just break even. In the long run a trade in the underlying instrument will just break even.
It will neither make money nor lose money. Given this assumption, the mean of the normal distribution curve assumed in the model must be the price at which a trade in the underlying instrument, either a purchase or a sale, will just break even.
For a futures contract, the breakeven price is simply the place at which the contract is bought or sold. If a trader buys a futures contract at 100, and the contract is subsequently sold at 100, the trader will break even.
But suppose a trader purchases a stock at 100 and holds it for three months. Where does the stock price have to be at the end of the holding period for the trader to break even? Since the purchase of stock requires immediate payment, the breakeven price will have to include the cost of carrying a 100 debit for three months.
If interest rates are 8% annually the carrying cost on 100 for three months is 3/12 x 8% x 100 = 2. Therefore, the stock price must be 102 at the end of three months for the trade to break even. If the stock will pay a dividend of 1 during the holding period, then the stock price needs only be 101 for the trade to break even since the owner of the stock can expect to receive 1.
Forward price
Taking into consideration all carrying costs on a contract, the price at which a contract would have to be trading on some future date such that a trade made at today's price would just break even.
A more common name for the breakeven price of a contract is the forward price of the contract. The forward price of a stock contract can be approximated as:
Formula: Forward price
forward price = current price + carrying costs  dividends
When the carrying costs are:
Formula: Carrying Cost
carrying costs = current price x interest rate x time
Volatility as a Standard Deviation
In addition to the mean, we also need a standard deviation to describe a normal distribution curve fully. This is entered into a pricing model in the form of volatility. As an approximation, we can define the volatility number associated with an underlying instrument as a one standard deviation price change, in percent, at the end of a oneyear period.
By price change, we mean the settlement from the end of one period of time to the end of the next period of time. curve. This is entered into a pricing model in the form of volatility. As an approximation, we can define the volatility number associated with an underlying instrument as a one standard deviation price change, in percent, at the end of a oneyear period. By price change, we mean the settlement from the end of one period of time to the end of the next period of time.
Since normal distributions are centered at the mean and the mean is the forward price, the volatility represents movement away from the forward price. Suppose that interest rates are at 8.00% and that an underlying stock, which pays no dividends, is currently trading at 100. The oneyear forward price of the stock is, therefore, 108.
If the annual volatility of the stock is 20%, a one standard deviation price change is:
20% x 108 = 21.60
So, one year from now, we would expect the same stock to be trading between:
108  21.60 = 86.40
AND
108 + 21.60 = 129.60
approximately 68% of the time, between:
108  (2 x 21.60) = 108  43.20 = 64.80
AND
108 + (2 x 21.60) = 108 + 43.20 = 151.30
approximately 95% of the time, between:
108  (3 x 21.60) = 108  64.80 = 43.20
AND
108 + (3 x 21.60) = 108 + 64.80 = 172.90
approximately 99.7% of the time.
Lognormal Distributions
Volatility is defined as the standard deviation of the percent price changes in an underlying contract. However, the end result of these percent changes can vary depending on how the changes are calculated.
For example, suppose you were to invest 1,000 for one year at an interest rate of 12%. How much would you have at the end of the year? The answer depends on how the 12% interest on your investment is paid out.
Rate of Payment  Investment Value  Yield 

12% once a year  1,120.00  12.00% 
6% twice a year  1,120.00  12.00% 
3% every three months  1,125,51  12.55% 
1% every month  1,126.83  12.68% 
12% / 52 every week  1,127.34  12.73% 
12% / 365 every day  1,127.47  12.75% 
12% compounded continuously  1,127.50  12.75% 
Note that as interest is paid more often, the total yield on the investment after one year increases. The yield is greatest when interest is paid continuously. In this case, it is as if interest were paid at every moment in time.
Although less common, we can also do this type of calculation using a negative interest rate. For example, suppose you were to lose 12% on your 1,000 investment (interest rate = 12%). How much would you have at the end of one year? Again, the answer depends on the frequency at which your losses accrue.
Rate of Loss  Investment Value  Yield 

12% once a year  880.00  12.00% 
6% twice a year  883.60  11.64% 
3% every three months  885.29  11.47% 
1% every month  886.38  11.36% 
12% / 52 every week  886.80  11.32% 
12% / 365 every day  886.90  11.31% 
12% compounded continuously  886.92  11.31% 
In the case of a negative interest rate, the more often the interest is compounded, the smaller the loss in terms of yield.
An interest rate and volatility are similar in that they both represent a rate of return. The yield resulting from this rate of return will depend on the frequency of compounding. The primary difference between interest and volatility is that interest compounds in one direction only (either always positive or always negative), while volatility is a combination of both positive and negative rates of return.
Lognormal distribution
A distribution that results from the continuous compounding of a random rateofreturn.
When price changes (the rate of return) are assumed to be normally distributed, the continuous compounding of these price changes will cause the possible prices at maturity (yield) to be lognormally distributed. Such a distribution, shown below, is skewed toward the upside because upside prices resulting from a positive rate of return will be greater (in absolute terms) than downside prices resulting from a negative rate of return.
Most traditional theoretical pricing models, including the BlackScholes model, assume that the percent changes in the underlying price are normally distributed and continuously compounded. This continuous compounding results in a lognormal distribution of prices at expiration, with the mean of the distribution located at the forward price of the underlying contract.
Summary
 A normal distribution is the distribution which results from many random walks. The BlackScholes model assumes that the underlying contract follows a random walk through time.
 A normal distribution can be described by its mean and standard deviation. The mean is where the peak of the distribution is located; the standard deviation is a measure of how fast the distribution spreads out.
 The BlackScholes model assumes that the mean of the underlying price distribution is located at the forward, or breakeven, price of the underlying contract.
 Volatility is defined as the annualized standard deviation, in percent, of the underlying contract.
 When the percent price changes in an underlying contract are continuously compounded, the result is a lognormal distribution. This is the true underlying distribution assumed in the BlackScholes model.
Chapter 06
Estimating Volatility
Estimating Volatility
Since option traders very often make decisions based on an opinion about volatility, a trader will always want to know whether the volatility number that they are using is accurate. While an exact volatility calculation is difficult without the aid of a computer or calculator, it is possible to determine whether the volatility input that a trader is using represents a reasonable estimate of market conditions.
While volatility represents a yearly standard deviation, a trader can often determine whether the volatility number that they are using is accurate by calculating volatility over a shorter period of time. The trader can then translate this shortterm volatility into a point price movement in the underlying contract.
By comparing the observed price movement in the underlying contract to the estimated movement, the trader will know whether their volatility number represents a reasonable estimate.
While volatility represents a yearly standard deviation, it is possible to approximate a standard deviation over a period of time other than a year by dividing the annual volatility by the square root of the number of trading periods in a year. While not 100% accurate, this is a convenient method for a trader to estimate the accuracy of a volatility estimate.
Daily Standard Deviations
To approximate a daily standard deviation, we can divide the annual volatility by the square root of the number of trading days in a year.
Daily trading periods in a year (ignoring weekends and holidays) = 256
square root of 256 = 16
That is, we can approximate a daily standard deviation by dividing the annual volatility by 16, and for purposes of approximating a standard deviation over short periods of time, we can use the current price of the underlying contract rather than the forward price.
For example, suppose an underlying contract is trading at 48 and has an annual volatility of 40%. What is an approximate one standard deviation price change from one day to the next?
For daily volatility, we calculate:
40% / 16 x 48 = 2.5% x 48 = 1.20
If 40% is the correct volatility, a trader would expect to see a daily settlementtosettlement price change of 1.20 or less approximately 2 days out of every 3, a price change of 2.40 or less approximately 19 days out of every 20, and a price change of more than 2.40 only 1 day out of every 20 (slightly more than once a month).
If a trader were to observe price changes greater than 1.20 more often than one day in three, they would surmise that their volatility estimate of 40% is too low. If the trader were to observe price changes greater than 1.20 less often than one day in three, they would surmise that their volatility estimate of 40% is too high.
Weekly Standard Deviations
In order to approximate a weekly standard deviation, we can divide the annual volatility by the square root of the number of trading weeks in a year.
Weekly trading periods in a year = 52 square root of 52 = 7.2
That is, we can estimate a weekly standard deviation by dividing the annual volatility by 7.2. Unlike our calculations for daily volatility, where we dropped weekends and holidays, we use all 52 trading weeks for our calculations since there are no holiday "weeks" where exchanges do not open at all.
For an underlying contract trading at 48 with an annual volatility of 40%, we calculate weekly volatility as:
40% / 7.2 x 48 = 5.56% x 48 = 2.67
If 40% is the correct volatility, a trader would expect to see a weekly settlementtosettlement price change of 2.67 or less approximately 2 weeks out of every 3, a price change of 5.34 or less approximately 19 weeks out of every 20, and a price change of more than 5.34 only 1 week out of every 20 (about once every 5 months).
If a trader were to observe price changes greater than 2.67 more often than one week in three, they would surmise that their volatility estimate of 40% is too low. If the trader were to observe price changes greater than 2.67 less often than one week in three, they would surmise that their volatility estimate of 40% is too high.
Summary
 The volatility over a period shorter than one year can be approximated by dividing the annual volatility by the square root of the number of trading periods in a year.
 A trader can estimate a daily standard deviation by dividing the annual volatility by 16.
 A trader can estimate a weekly standard deviation by dividing the annual volatility by 7.2.
Chapter 07
Types of Volatility
Types of Volatility
When traders discuss volatility, even experienced traders may find that they are not always talking about the same thing. When a trader makes the comment that the volatility of XYZ stock is 25%, this statement may take on a variety of meanings. It will be useful in subsequent discussions if we define the various ways in which traders interpret volatility.
Futures Volatility
Future volatility is what every trader would like to know, the volatility that best describes the future distribution of prices for an underlying contract. In theory it is this number to which we are referring when we speak of the volatility input into a theoretical pricing model. If a trader knows the future volatility, they know the right "odds". When they feed this number into a theoretical pricing model, they can generate accurate theoretical values because they have the right probabilities. Like the casino, they may occasionally lose because of shortterm bad luck. But in the long run, with the odds on their side, a trader can be fairly certain of making a profit.
Future volatility
The degree to which the price of an instrument will fluctuate over some period in the future. In option trading, the term is often used to refer to the price fluctuation of an underlying instrument over the life of the option.
Although the BlackScholes model, as well as other pricing models, assumes that a trader will input the future volatility, traders rarely talk about the future volatility since it is impossible to know what the future holds. When traders do talk about future volatility, they are usually talking about the period in the future covering the life of the option. If a trader refers to the future volatility of a threemonth option, they are probably referring to the volatility of the underlying contract over the next three months. If a trader refers to the future volatility of a sixmonth option, they are probably referring to the volatility of the underlying contract over the next six months.
Historical Volatility
Even though one cannot know the future, if a trader intends to use a theoretical pricing model, they must try to make an intelligent guess about the future volatility. In option evaluation, as in other disciplines, a good starting point is historical data.
What typically has been the volatility of this contract over some period in the past? If, over the past ten years, the volatility of a contract has never been less than 10% or higher than 30%, a guess for the future volatility of either 5% or 40% hardly makes sense. This does not mean that either of these extremes is impossible (in option trading, the impossible always seems to happen sooner or later), but based on past performance and in the absence of any extraordinary circumstances, a guess within the historical limits of 10% and 30% is probably more realistic than a guess outside these limits. Of course, 10% to 30% is still a huge range, but at least the historical data offers a starting point. Additional information may further narrow the estimate.
Note that there are a variety of ways to calculate historical volatility, but most methods depend on choosing two parameters: the historical period over which the volatility is to be calculated and the time interval between successive price changes. The historical period may be ten days, six months, five years, or any period the trader chooses. Longer periods tend to yield an average or characteristic volatility, while shorter periods may reveal unusual extremes in volatility. To become fully familiar with the volatility characteristics of a contract, a trader may have to examine a wide variety of historical time periods.
A trader must also decide what intervals to use between price changes. Should they use daily price changes? Weekly changes? Monthly changes? Or perhaps he ought to consider some unusual interval, perhaps every other day or every week and a half. Surprisingly, the interval that is chosen does not seem to affect the result greatly. Although a contract may make large daily moves yet finish a week unchanged, this is by far the exception. A contract that is volatile from day to day is likely to be equally volatile from week to week or month to month.
As a general rule, services that supply historical volatility database their calculations on daily settlementtosettlement price changes. If this is not the case, an explanation of how the volatility was calculated will usually accompany the data. If, for example, a service gave the volatility of a contract for the month of August as 21.6%, it can be assumed that the calculations were made using the daily settlementtosettlement price changes for all the business days during that month. The historical volatility of the S&P 500 index over several different periods is shown below:
Forecast Volatility
Just as there are services that will attempt to forecast future directional moves in the price of a contract, there are also services that will attempt to forecast the future volatility of a contract. Forecasts may be for any period but most commonly cover periods identical to the remaining life of options on the underlying contract. For an underlying contract with three months between expirations, a service might forecast volatilities for the next three, six, and nine months.
For an underlying with monthly expirations, a service might forecast volatilities for the next one, two and three months. A trader's guess about the future volatility of a contract might very well take into consideration any volatility forecast to which the trader has access.
Implied Volatility
Generally speaking, future, historical, and forecast volatility are associated with an underlying contract. We can talk about the future volatility of a stock index, the historical volatility of a foreign currency, or the forecast volatility for a futures contract. In each case we are referring to the volatility of the underlying contract. There is, however, a different interpretation of volatility that is associated solely with an option rather than with the underlying contract.
Suppose a certain stock is trading at 56.50 with interest rates at 8%. Suppose also that a 60call with three months to expiration is available on this stock and that no dividend is expected over this period. If our best guess about the volatility over the next three months is 32%, and we want to know the theoretical value of the 60 call, we might feed all these inputs into a theoretical pricing model. Using the BlackScholes Model, we find that the option has a theoretical value of 2.50.
Having done this, we might compare the option's theoretical value to its price in the marketplace. To our surprise, we find that the option is trading for 3.25. How can we account for the fact that we think the option is worth 2.50 while the marketplace seems to believe it is worth 3.25? One way to answer the question is to assume that everyone in the marketplace is using the same theoretical pricing model that we are, in this case, the BlackScholes Model.
If we make this assumption, then the discrepancy between our value of 2.50 and the marketplace's value of 3.25 must be due to a difference of opinion concerning one or more of the inputs into the model.
Since all inputs, except volatility, are relatively easy to observe in the marketplace, it is common among traders to make the assumption that a disagreement about volatility causes the discrepancy. The marketplace seems to be using a volatility different than our 32%
What volatility is the marketplace using?
To find out, we can ask the following question: If we hold all other inputs constant (time to expiration, exercise price, underlying price, interest rates), what volatility must we feed into our theoretical pricing model to yield a theoretical value identical to the price of the option in the marketplace?
In our example, we want to know what volatility will yield a value of 3.25 for the 60 call. Clearly, the volatility has to be higher than 32%, so we might sit down with a computer programmed with the BlackScholes Model and start to raise the volatility. If we do, we will find that at a volatility of 38.8% percent, the 60 call has a theoretical value of 3.25.
We refer to this volatility as the implied volatility of the 60 call. It is the volatility we must feed into our theoretical pricing model to yield a theoretical value identical to the price of the option in the marketplace. We can also think of it as the volatility being implied to the underlying contract through the pricing of the option in the marketplace.
Implied volatility
Assuming all other inputs are known, the volatility would have to be input into a theoretical pricing model in order to yield a theoretical value identical to the price of the option in the marketplace.
When we solve for the implied volatility of an option we are assuming that the theoretical value (the option's price) is known, but that the volatility is unknown. In effect, we are running the theoretical pricing model backward to solve for this unknown, as shown below. In fact, this is easier said than done since most theoretical pricing models cannot be reversed. However, a number of computer programs have been written which can quickly solve for the implied volatility when all other inputs are known.
Unknowns  Unknowns 
Theoretical Value (?? = 2.50)  Implied Volatility (?? = 38.8%) 
Known  Known 
Exercise Price (60)  Exercise Price (60) 
Time to Expiration (3 months)  Time to Expiration (3 months) 
Underlying Stock Price (56.50)  Underlying Stock Price (56.50) 
Interest Rates (8%)  Interest Rates (8%) 
Volatility (32%)  Option Price (3.25) 
Since implied volatility is calculated from the price of an option in the marketplace, traders often think of the implied volatility and price as the same thing. If a trader says that implied volatility is relatively high (low), this is the same as saying that option prices are relatively high (low). Alternatively, high (low) implied volatility can be interpreted as reflecting a high (low) demand for options.
In addition to being useful as a reflection of option prices, implied volatility is also useful in helping a trader make a better estimate of future volatility. If one takes the reasonable approach that prices are determined by supply and demand, and the supply and demand are, in turn, driven by market participants who may be in a better position to estimate future volatility, it makes sense to take implied volatility into consideration when making an estimate of future volatility.
Comparing Future and Implied Volatility
Trading decisions in almost all markets, whether options or other instruments are based on a comparison of value and price. If an asset has a high value and a low price, one will prefer to buy the asset. If the asset has a low value and a high price, one will prefer to sell the asset. If the estimate of the asset's value is correct, the price should eventually move toward the value, resulting in a profit for the trader.
From an option trader's point of view, the value of an option depends on future volatility  the volatility of the underlying contract over the life of the option. Obviously, one cannot know the future, so a trader will often look to the historical volatility or forecast volatility to make an intelligent guess about the future volatility. But it is still the future volatility that they are after because the future volatility determines the value of the option.
Because the implied volatility is derived directly from the price of an option in the marketplace, most traders think of the price of an option in terms of its implied volatility. A trader might say that he bought options at 25% or that they sold options at 30%. This is simply their way of referring to the relative prices of options in the marketplace.
Since trading decisions are based on consideration of value and price, and these concepts are represented in option markets by the future volatility and implied volatility, most trading decisions begin by comparing these two volatilities. If implied volatility (the price of options) is lower than the trader's estimate of future volatility (the value of options), a trader will prefer to be a buyer of options.
If implied volatility is higher than the trader's estimate of future volatility, a trader will prefer to be a seller of options. As a trader becomes more sophisticated, other considerations, such as the risk of being wrong, also play a role in the trader's choice of strategies. But for traders relying on a theoretical pricing model to make trading decisions, the first step is almost always to compare the implied volatility to an estimate of future volatility.
Volatility Fitting
The process of automatically calculating the future volatility used in the model based on the implied volatility is called volatility fitting. In the Rival One trading platform, the system automatically calculates the implied volatility for each option based on the current market prices. The system then automatically sets the future volatility used by the model to calculate the theoretical price. The results are that the theoretical price will typically be close to the market bid's average value and the option's ask price. If there is no current bid or ask price available, Rival One calculates the implied volatility based on the previous day's option settlement price and uses a model to adjust the future volatility.
Summary
 Volatility can be interpreted in different ways.
 The future volatility is the volatility of the underlying contract that will occur over some period in the future, usually the period up to expiration of an option.
 The historical volatility is the volatility of the underlying contract over some period in the past.
 A forecast volatility is someone's attempt to estimate the future volatility.
 The implied volatility is the volatility that would yield a theoretical value identical to the price of an option in the marketplace, assuming that all other model inputs are known.
 Many option trading decisions are made by comparing the implied volatility to a trader's forecast volatility. If implied volatility is high, a trader will prefer to be a seller of options; if implied volatility is low, a trader will prefer to be a buyer of options.
Chapter 08
Using an Option's Theoretical Value
Using an Option's Theoretical Value
While the BlackScholes model or any other theoretical pricing model may look very good on paper, most traders are interested in the practical use of the model. Is it possible to use the model to profit from mispriced options, and if so, what procedure ought to be followed to ensure the best results?
Establishing a Position
In theory, a trader who uses a theoretical pricing model to make trading decisions will generally do the following:
 Calculate the theoretical value of an option using a theoretical pricing model.
 Purchase the option if its price is less than its theoretical value or sell the option if its price is more than its theoretical value.
 Hedge the option by taking an opposing market position in the underlying contract.
 As market conditions change, adjust the hedge to maintain an appropriate opposing market position in the underlying contract.
 Liquidate all contracts (options and underlying contracts) at expiration at their market prices.
Identifying Mispriced Options
For an option trader using a theoretical pricing model, an option is mispriced if its price in the marketplace differs from the theoretical value generated by the pricing model. Since an option trader usually thinks of the option's price in terms of its implied volatility and the option's theoretical value is determined primarily by the trader's estimate of future volatility, the option can be considered mispriced if the implied volatility differs from the trader's volatility estimate and the option's theoretical value is determined primarily by the trader's estimate of future volatility, the option can be considered mispriced if the implied volatility differs from the trader's volatility estimate.
One might express the amount by which an option is mispriced as the difference between its price and theoretical value. An option with a theoretical value of 4.50 and a price of 5.75 might be considered overpriced by 1.25. But just as traders express prices and values in volatility terms, they also tend to express mispricing in volatility terms.
If the theoretical value of 4.50 resulted from a volatility input of 23% and the price of 5.75 translates into an implied volatility of 27%, a trader might say that the option is overpriced by four percentage points. If one uses this approach, the option whose theoretical value differs the most in total points from its price in the marketplace may not necessarily be the option that is the most mispriced. For example, consider the following options in a market where a trader's best estimate of future volatility is 30%:
Option  Price  Theoretical Value  Implied Volatility 

A  5.50  4.75  33% 
B  2.90  2.59  35% 
C  1.80  1.55  37% 
All options are overpriced, but in point terms, option A seems to be the most overpriced (.75 vs. .40 for option B and .25 for option C). However, with a volatility estimate of 30%, most traders would consider option C to be the most overpriced since its implied volatility differs the most from the trader's volatility estimate (7 percentage points vs. 3 percentage points for option A and 5 percentage points for option B).
While a trader may later take into consideration other factors, particularly the risk of making false market assumptions, initially, a trader will consider the most mispriced option the one whose implied volatility differs most from their volatility estimate. They will tend to sell those options that are most overpriced and buy those options that are most underpriced.
In the Rival One trading platform the system will automatically highlight options where the theoretical values are outside of the market bid or ask to indicate to the user which options are cheap or expensive relative to the theoretical value.
Establishing the Hedge
To create a theoretically correct hedge, it's essential to take an opposing position in the underlying contract so that any small changes in the underlying asset's price will lead to an equal but opposite change in the value of the option position. This creates a neutral hedge that is unbiased towards the direction of the underlying contract.
The hedge ratio, also known as the delta, is the number that helps us establish a neutral hedge under the current market conditions. It is derived from the theoretical pricing model and is crucial to creating a hedge. In the subsequent chapter, we will discuss the delta in greater detail.
The Rival One trading platform has an autohedge feature that will automatically send an order to buy or sell the underlying instrument as soon as an option order is filled.
Learn More about Establishing the Hedge
Adjusting the Hedge
Because the delta can change as market conditions change, a position that is initially deltaneutral may not (indeed, almost certainly will not) remain neutral. The correct use of a theoretical pricing model requires that we not only begin with a deltaneutral position but maintain a deltaneutral position over the option's life.
For example, if we initially buy 10 calls with a delta of 50 each and sell 5 underlying contracts, we are delta neutral since:
(10 x 50) + (5 x 100) = 500  500 = 0.
But suppose at some later date the delta of the same call is 70, so that our total delta position is now:
(10 x 70) + (5 x 100) = 700  500 = 200.
This position is no longer deltaneutral. If we want to return to delta neutral, we must sell two additional underlying contracts. This trade is known as an adjustment to the position. An adjustment is a trade made with the primary intent of returning a position to delta neutral.
Adjustment
A trade made with the primary intention of maintaining certain position characteristics. The most common reason for making an adjustment is to ensure that a position remains deltaneutral.
Theoretical pricing models assume that a trader constantly adjusts a position to remain deltaneutral. This constant adjusting of a position is sometimes referred to as a dynamic hedge. Constant adjustments are not possible since one cannot trade all the time. However, disciplined theoretical traders try to remain close to deltaneutral by adjusting the position whenever the total delta gets too far away from zero.
Dynamic hedge
A position that is constantly being adjusted to maintain specific characteristics. The most common dynamic hedge is continuously adjusted to remain deltaneutral.
The Result
Suppose a trader is disciplined and does all of the following:
 Purchases (sells) options at prices less (more) than their theoretical value.
 Hedges the option position by taking an opposing delta neutral position in the underlying contract
 Adjusts the position to remain deltaneutral over the life of the option
 Liquidates all contracts (options and underlying contracts) at expiration at their market prices
If the trader does all of the above, will the trader show a profit, and if so, where does the profit come from?
Since the total profit or loss is the result of many different factors (the original hedge, the adjustment process, interest considerations, dividends), any one of which might be either positive or negative, it may seem impossible to determine beforehand whether the result will be a profit or a loss.
Indeed, one cannot say whether any one factor will result in a profit or loss. But the theory upon which the BlackScholes model is based states the following: assuming that a trader has correctly predicted the volatility over the life of the option and has therefore correctly evaluated an option, in some combination, all the market factors will come together to yield a profit approximately equal to the amount by which the option was initially mispriced.
If a trader purchases options with a theoretical value of 2.50 at 2.15 and hedges the position as previously discussed, then, on average, they will profit approximately .40 per option. If a trader sells options with a theoretical value of 4.75 at 5.50, then, on average, they will profit roughly .75 per option.
8.5 Trading Restrictions
We have assumed that the underlying contract can be freely bought and sold to hedge an option position. But this is not always true. For example, there may be price limits beyond which an underlying futures contract may not trade. And in the stock market it may not always be possible to sell stock which a trader does not own. This type of short sale, the sale of stock which is borrowed rather than owned, is prohibited in some markets. This makes it difficult to hedge certain types of option positions, even though the position may appear theoretically profitable.
The short sale of stock is not totally prohibited in U.S. markets, but it is subject to an uptick rule. This rule specifies that a short sale is always prohibited at a price lower than the price at which the previous trade took place (a downtick). A short sale is always permitted at a price higher than the price at which the previous trade took place (an uptick). Finally, a short sale may take place at the same price at which the previous trade took place if the previous trade took place on an uptick (also an uptick). Below are ten consecutive trade prices (reading from left to right) for a stock with the accompanying ticks (a positive sign for an uptick and a negative sign for a downtick).
48.50 
+48.65 
+48.65 
48.50 
48.40 
48.25 
48.25 
+48.40 
+48.40 
+48.40 
Not only might a short sale not be possible because of the uptick rule, but many brokerage firms which execute short sales of stock for their customers do not pay full interest on the proceeds from a short sale. This can further distort the interest component used in a theoretical pricing model.
Summary
 An option is mispriced if its price differs from its theoretical value.
 The correct use of a theoretical pricing model begins by purchasing underpriced options or selling overpriced options. A hedge is then established by taking a delta neutral opposing market position in the underlying stock.
 A delta neutral hedge must be adjusted periodically to remain delta neutral over the life of the option.
 It may not always be possible to hedge a position using the underlying contract.
Chapter 09
The End Result
Options hedge is a critical concept in financial derivatives and risk management. It represents the assessment of gains or losses incurred by an investor or trader due to implementing hedging strategies using options contracts.
With their unique ability to provide both downside protection and profit potential, options are widely utilized to mitigate risk and optimize returns in various financial markets. Understanding options hedge P&L is essential for anyone seeking to navigate the complex landscape of financial derivatives, as it allows for a comprehensive evaluation of the performance of hedging strategies and their impact on a portfolio's overall profitability.
In this context, we delve into the intricacies of options hedge P&L, exploring its calculation, interpretation, and significance in managing financial risk and maximizing investment gains.
The first of the P & L components is the profit or loss resulting from the original hedge. The BlackScholes model assumes that at the moment of expiration, the trader will close out the original hedge by either letting the options expire worthless if they are outofthemoney or selling them at parity (intrinsic value) if they are inthemoney.
At the same time, they will liquidate any outstanding position in the underlying, purchasing any contracts which they may be short or selling out any contracts which may be long.
Hedge PNL
A hedged options position's Profit and Loss (P&L) comprises various components. Firstly, the original hedge's P&L is based on whether options expire worthless (outofthemoney) or are sold at intrinsic value (inthemoney) at expiration, alongside liquidating any underlying positions, resulting in profit or loss.
Secondly, Adjustment P&L arises from buying and selling underlying contracts to maintain a deltaneutral position, leading to profits or losses.
Interest P&L factors in financing cash flows from the initial hedge and adjustments influenced by interest rates and time to expiration. Variations in futures contracts and stock transactions during adjustments can generate interest earnings or losses. Dividends P&L comes into play with stock options, where traders collect dividends when long on the exdividend date or pay them out when short, also with interest considerations.
Adjustment P&L
Over the option's life, the trader will be forced to buy and sell underlying contracts to remain deltaneutral. Sometimes, these adjustments will result in a profit, and sometimes they will result in a loss. Total the cash flow resulting from all the buying and selling of underlying contracts will result in an adjustment P & L.
Interest P&L
When the trader establishes the initial hedge, the purchase or sale of options and the purchase or sale of underlying contracts will result in either a positive or negative cash flow. In theory, the trader will have to finance this cash flow if it is negative or will earn interest on the cash flow if it is positive. The amount of interest gained or lost will depend on the level of interest rates as well as the amount of time remaining to expiration. This interest P & L resulting from the original hedge will be part of the total P & L to the position.
In addition to the interest considerations resulting from the initial cash flow, there are also interest considerations resulting from the adjustment process. In the futures market, there will be changes in variation as the price of the futures contract changes. This change in variation will result in interest being earned or lost. Each time the trader buys or sells stock to maintain a deltaneutral position, they create a debit or credit in the stock market. Any debit will lose interest, and any credit will earn interest so that the adjustment process will result in an additional interest P & L.
Dividends P&L
If a stock option position is being hedged, the underlying stock may pay dividends over the life of the options involved in the hedge. If the trader is long stock on the exdividend date, either due to the original hedge or the adjustment process, they will collect whatever dividend accrues. This will result in an additional profit to their position. On the other hand, if they are short stock on the exdividend date, they will be required to pay the dividend to the party from whom they borrowed the stock. This will result in a loss of their position.
In theory, whatever dividends the trader receives or pays out over the life of the hedge will also result in an interest rate consideration. They can earn interest on dividends received, while they must pay interest on dividends paid out. While this interest is usually minimal, it does exist, and the BlackScholes model considers it part of the total P & L.
Real World Considerations
The prospective option trader will have noticed that we have ignored several realworld problems using the BlackScholes model. We have already touched on one of these problems: one cannot always freely buy and sell the underlying contract.
Also, there has been no mention thus far of transaction costs, which clearly will impact the final P & L to an option position. There will be transaction costs associated with establishing the original hedge, and each adjustment will entail a transaction cost.
The BlackScholes model says nothing about the effect of such costs. Finally, the BlackScholes model assumes that a trader has access to unlimited funds and can always borrow these funds at the riskfree rate used in the pricing model. If a trader did have access to unlimited funds, which is unrealistic, the trader would probably have to pay increasingly higher interest the more funds they borrowed.
The above factors are ones with which each trader will have to deal. While they may appear to invalidate the BlackScholes model, they are, in fact, not all that severe, especially for a professional trader who will generally be able to borrow funds against their position in options or the underlying contract and who usually have relatively low transaction costs.
There are many problems associated with using the BlackScholes model.
The realworld factors of differing interest rates and transaction costs invalidate some of the basic assumptions in the model. Moreover, there are almost infinite combinations of profit and loss components, and one can't determine beforehand which components will yield a profit and which will yield a loss.
Indeed, using the BlackScholes model (or any other pricing model) requires a certain amount of blind faith. However, experienced traders have found that while the BlackScholes model is imperfect, it is still one of the most effective tools an options trader has.
Chapter 10
Option Sensitivities
How Changes in Market Conditions Affect Theoretical Values
The general effects of changing market conditions on option values can be summarized as follows:
When futures options are subject to futurestype settlement, option values are unaffected by changes in the interest rates.
Changes in the Underlying Price
When the underlying contract price rises, call values rise, and put values decline.
When the price falls, call values decline, and put values rise.
Of course, this is only true if all other market conditions (time to expiration, volatility, interest rates, and dividends) remain the same.
Note that atthemoney options tend to have the most significant time value. As an option moves away from atthemoney, moving either intothemoney or outofthemoney, its time value decreases. If an option goes deeply inthemoney or far outofthemoney, its time value goes to zero.
Changes in Time to Expiration
As the time to expiration grows shorter, and all other market conditions (underlying price, volatility, interest rates, and dividends) remain the same, both call and put values will decline. (In some unusual cases, because of interest rate considerations, it may be possible for an option to rise in value as time passes. For most practical purposes, a trader can assume that an option's value will fall as time passes.)
Note also that the atthemoney option loses the most value as time to expiration grows shorter. This results from the atthemoney option having the most time value and, therefore, having the most value to lose when the underlying market fails to move.
Changes in Volatility
As volatility rises or falls, the possibility of extreme outcomes rises or falls, and option values also rise or fall accordingly. As volatility increases and all other market conditions (underlying price, time to expiration, interest rates, dividends) remain the same, both call and put values will rise. As volatility falls, both call and put values will decline.
Note also that a change in volatility will have a more significant effect on atthemoney options than either in or outofthemoney options, which are otherwise identical.
Changes in Interest Rates
Interest rates may affect option values in two ways: changes in interest rates may affect the forward price of the underlying contract, and they may affect the cost of carrying on an option position.
For example, as interest rates rise, the forward price of a stock will increase. This will cause the value of stock option calls to rise and puts to fall. As interest rates fall, the forward price of the stock will fall. This will cause the value of stock option calls to fall and puts to rise.
While interest rates do not change the forward price of a futures contract, if an option on a futures contract is subject to a stocktype settlement, purchasing the option requires a cash outlay or debit. As interest rates rise, the cost of carrying the debit increases, reducing the value of both call and put options. As interest rates fall, the cost of carrying this debit falls, increasing the value of both call and put options.
Another way of analyzing the effect of rising or falling interest rates is to ask whether an option is a more or less desirable method of taking a position in a stock market. For example, a trader can take a long position in the stock market by purchasing stock or calls. If interest rates are high, all traders prefer to take a long position by buying calls because of the smaller cash outlay.
Calls will, therefore, be a more desirable method of taking a long position, and consequently, call values will rise. In the same way, a trader can take a short position by selling stock or purchasing puts. If interest rates are high, all traders will prefer to take a short position by selling stock to earn a high rate of interest on the cash received.
Puts will, therefore, be a less desirable method of taking a short position, and consequently, put values will fall. Note also that a change in interest rates will have a more significant effect on inthemoney options than at or outofthemoney options and a more substantial impact on longterm options than shortterm options.
Changes in Dividends
Dividends will only affect stock option values if the dividend payment is expected to occur during the option's life. If this is the case, as dividends rise, the stock's forward price will fall. This will cause the value of calls to fall and puts to rise. As dividends fall, the forward price of the stock will rise. This will cause the value of calls to grow and puts to fall.
Another way of analyzing the effect of rising or falling dividends is to ask whether an option is a more or less desirable method of taking a position in a stock market. For example, a trader can take a long position by purchasing stock or calls. If dividends are high, all traders prefer to take a long position by purchasing stock to get the dividend.
Calls will, therefore, be a less desirable method of taking a long position, and consequently, call values will fall. In the same way, a trader can take a short position by selling stock or purchasing puts. If dividends are high, all traders prefer to take a short position by buying puts to avoid losing the dividend. Puts will, therefore, be a more desirable method of taking a short position, and consequently, put values will rise.
A dividend change will affect inthemoney options more than at or outofthemoney options. Multiple dividend payouts, likely to accrue over longer periods, will cause longterm options to be more sensitive to dividend changes than shortterm options.
Changes In Market Conditions
Every trader who enters the marketplace must balance two opposing considerations: reward and risk. A trader hopes that their market condition analysis is correct and will lead to profitable trading strategies.
However, intelligent traders need to pay attention to the possibility of error. If they are wrong and market conditions change in a way that adversely affects his position, how badly might the trader get hurt?
A trader who fails to consider the risks associated with their position is certain to have a short and unhappy career. Because of the importance of risk management in option trading, learning how option values and risks change as market conditions change is one of the most important aspects of a trader's education.
For the moment, we will restrict our discussion to European stock options, where no early exercise is possible. American options, where early exercise is a possibility, will be discussed in a later chapter.
Summary
Option values are sensitive to changes in market conditions. In general, for stock options:
 If the price of the underlying contract rises, call values will rise and put values will fall. If the price of the underlying contract falls, call values will fall and put values will rise.
 As time passes, the values of both puts and calls will decline.
 If volatility rises, the values of both puts and calls will rise. If volatility falls, the values of both puts and calls will fall.
 If interest rates rise, the value of stock option calls will rise and the value of stock option puts will fall. If interest rates fall, the value of stock option calls will fall and the value of stock option puts will rise.
 If options on futures are subject to stocktype settlement, rising interest rates will cause call and put values to decline slightly. Falling interest rates will cause call and put values to rise slightly.
 If options on futures are subject to futurestype settlement, option values are unaffected by changes in interest rates.
 If the dividend is increased, the value of stock option calls will fall and the value of stock option puts will rise. If the dividend is cut, the value of stock option calls will rise and the value of stock option puts will fall.
Chapter 11
Defining the Sensitivities
Defining the Sensitivities
While we may know the general effects of changing market conditions on the value of options, we must still consider the magnitude of the changes. Will the changes be large or small, representing either a major or minor risk or something in between?
Fortunately, in addition to the theoretical value, pricing models also generate several other numbers that enable a trader to assess not only the direction of the change but also the relative magnitude of the change. While these numbers will not answer all our questions concerning changing market conditions, they will help us better assess the risks associated with individual and complex option positions.
Greeks
Greeks in options trading refer to a set of vital risk management metrics and tools that are instrumental in understanding and quantifying the dynamic nature of options contracts.
Options, which are financial derivatives that give traders the right, but not the obligation, to buy or sell an underlying asset at a specified price on or before a predetermined date, can exhibit complex price movements influenced by various factors. To navigate this complexity effectively, traders and investors rely on the Greeks, a collection of mathematical parameters that help assess and manage the risks associated with options positions.
The Greeks provide valuable insights into how an option's price will likely change in response to shifts in different market variables, such as the underlying asset's price, volatility, time until expiration, and interest rates. By using these metrics, traders can make informed decisions about their options strategies, including when to enter or exit positions, how to hedge against adverse movements, and how to optimize risk and reward ratios.
Delta, Gamma, Theta, Vega, and Rho are the five primary Greek letters used in options trading. Each of these Greeks quantifies a specific aspect of an option's behavior, helping traders assess their exposure to market movements and adjust their portfolios accordingly.
Understanding the Greeks is essential for anyone involved in options trading, from beginners looking to grasp the basics to experienced traders seeking to finetune their strategies and minimize risks in a dynamic financial landscape. We'll explore each Greek's concept and significance, shedding light on how they contribute to informed decisionmaking and risk management.
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The Delta
Delta
The sensitivity of an option's theoretical value to a change in the price of the underlying contract.
The delta is a measure of directional risk. A positive delta indicates that an option will increase in value if the underlying market rises and decline if the underlying market falls. A negative delta suggests that an option will increase in value if the underlying market falls and decline in value if the underlying market rises.
Because calls always move in the same direction as the underlying market (when the underlying price rises, call values rise), we always assign positive deltas to calls. Because puts always move in the opposite direction of the underlying market (when the underlying price rises, put values fall), we always assign negative deltas to puts.
Technically, delta values range from 0 to 1.00 for calls and from 0 to 1.00 for puts. An underlying contract is always assigned a delta of 1.00. This is the form in which the delta usually appears in textbooks. However, exchangetraded stock options in the United States typically give the holder the right to buy or sell 100 shares of the underlying stock, and it has become common to equate one delta with one share of stock.
As a result, many traders express delta values as a whole number, ranging from 0 to 100 for calls and from 0 to 100 for puts. Using this notation, an underlying contract has a delta of 100. This notation, sometimes called the percent format, will be used in the accompanying exercises. Many futures traders also follow this convention by assigning a delta of 100 to an underlying futures contract.
While the delta measures directional risk, the delta has several other interpretations, which might be helpful depending on the strategies a trader is pursuing.
 The theoretically correct ratio of underlying contracts to option contracts required to establish a deltaneutral hedge. For example, if a trader buys (sells) an option with a delta of 25, a neutral hedge will require that he sell (buy) 25% of an underlying contract. If the underlying contract is 100 shares of stock, the trader must sell (buy) 25 shares. When interpreted this way, the delta is sometimes called the hedge ratio.
 In theory, the number of underlying contracts which the purchaser (seller) of a call is long (short) or which the purchaser (seller) of a put is short (long). For example, if a trader buys (sells) a call with a delta of 50, they are, in theory, long (short) 50% of an underlying contract. If they buy (sell) a put with a delta of 25, they are, in theory, short (long) 25% of an underlying contract. When interpreted this way, the delta is sometimes called the theoretical or equivalent underlying position.
 The rate of change in the theoretical value of an option concerning a difference in the price of the underlying contract. For example, an option with a delta of 75 can be expected to change its value at 75% of the rate of change in the price of the underlying contract. If the underlying goes up (down) 1.00, the option's theoretical value can be expected to go up (down) .75 point. Note that the negative delta associated with a put indicates that its value will move in the opposite direction of the underlying market. If the underlying rises (falls), puts will fall (rise) in value.
 The probability that the option will finish in the money. This interpretation is only approximate and is rarely used by option traders to make decisions or to manage risk in option markets. It is given only in the interests of completeness and because this interpretation is occasionally referred to in option texts.
Deeply inthemoney options tend to have delta values close to 100 (100 for puts), changing their theoretical value almost point for point with the underlying contract. Far outofthemoney options have delta values close to zero. When the underlying contract moves, such options change value hardly at all. Finally, options that are atthemoney have delta values of approximately 50 (50 for puts), changing their theoretical value at roughly half the rate of the underlying contract.
The Gamma
Gamma
The sensitivity of an option's delta to a change in the price of the underlying contract.
We have already seen that when an option is far outofthemoney, its delta is close to 0. When an option is deeply inthemoney, its delta is close to 100 (100 for puts). From this, we can conclude that as the underlying price changes, the delta of an option must also be changing. As the underlying price rises, call deltas move towards 100 and put deltas move towards 0; as the underlying price falls, call deltas move towards 0 and put deltas move towards 100.
The gamma, sometimes referred to as the curvature of an option, is the rate at which an option's delta changes as the price of the underlying changes. The gamma is usually expressed in deltas gained or lost per onepoint change in the underlying, with the delta increasing by the amount of the gamma when the underlying rises and falls by the amount of the gamma when the underlying falls.
Both calls and puts have positive gammas. For both types of options, when the underlying contract rises, we add the gamma to the old delta to get the new delta, and when the underlying contract falls, we subtract the gamma from the old delta to get the new delta.
For example, if a call has a gamma of 5, for each point rise in the price of the underlying, the option will gain 5 deltas, and for each point fall in the cost of the underlying, the call will lose 5 deltas. If the call originally had a delta of 25 and the underlying moves up one full point, the new delta of the call will be
25 + 5 = 30
If the underlying moves down one full point, the new delta will be:
25  5 = 20
In the same way, if a put has a gamma of 5, for each point rise in the price of the underlying, the option will gain 5 deltas, and for each point fall in the price of the underlying, the put will lose 5 deltas. But because puts have negative deltas, when we add the gamma to a put delta, it becomes a smaller negative number; when we subtract the gamma from a put delta, it becomes a larger negative number. If the put originally had a delta of 25 and the underlying moves up one full point, the new delta of the put will be;
25 + 5 = 20
If the underlying moves down one full point, the new delta will be:
25  5 = 30
Regardless of the gamma, call deltas are still bounded by zero and 100, and put deltas are bounded by zero and 100. If a call has a delta of 75 and a gamma of 10 and the underlying rises by 4 points, we might think that the new delta of the call will be:
75 + (4 x 10) = 115
But this is not possible because call deltas cannot exceed 100. The call delta must therefore be close to 100.
The Theta
Theta
The sensitivity of an option's theoretical value to a change in the amount of time remaining to expiration.
As time passes, the time value of an option disappears until it becomes 0 at expiration. It may be useful to a trader to know how fast an option will lose value as time passes and all other market conditions remain the same. This is given by the option's theta or time decay factor.
The theta is usually expressed in points lost per day when all other conditions remain the same. An option with a theta of .05 will lose .05 in value for each day that passes with no change in other market conditions. If the option is worth 2.75 today, then tomorrow it will be worth 2.70. The day after that, it will be worth 2.65.
Time runs in only one direction, and technically, the theta is a positive number. However, as a convenient notation and to remind a trader that the theta represents a loss in the option's value as time passes, it is sometimes written as a negative number. This is the convention which will be followed here.
Therefore, the theta of an option that loses .05 per day will be written as .05. Using this notation, both calls and puts will have negative theta values since both calls and puts lose value as time passes. Note that this is just the opposite of the gamma, where all options have positive gamma values. As a general principle, an option position will have a gamma and theta of opposite signs. Moreover, the relative size of the gamma and theta will correlate. A large positive gamma goes hand in hand with a large negative theta, while a large negative gamma goes hand in hand with a large positive theta.
Regardless of the theta, an option's theoretical value can never be less than zero. If an option has a theoretical value of .75, a theta of .25 , and four days pass with no movement in the underlying contract, we might think that the new theoretical value of the option will be:
.75  (4 x .25) = .25
However, this is not possible because theoretical values cannot be negative. The option must therefore have a value close to 0.
The Vega
Vega
The sensitivity of an option's theoretical value to a change in volatility.
Just as a trader must be concerned with the effect on an option's theoretical value of movement in the underlying contract (delta) and the passage of time (theta), a trader will also be concerned with the effect of a change in volatility. While delta, gamma, and theta are found in most option texts, there is no generally accepted term for the sensitivity of an option's theoretical value to a change in volatility.
The most commonly used term among traders is vega, and this is the term that will be used here. But vega is not a Greek letter, so a common alternative in academic literature, where Greek letters are preferred, is kappa (sometimes denoted with the Greek letter K).
The vega of an option is usually expressed as the amount of change in an option's theoretical value for each one percentage point change in volatility. Since all options gain value with rising volatility, the vega for both calls and puts is positive. If an option has a vega of .15, for each percentage point increase (decrease) in volatility, the option will gain (lose) .15 in theoretical value.
Suppose the option has a theoretical value of 3.25 at a volatility of 20%. In that case, it will have a theoretical value of 3.40 at a volatility of 21% and a theoretical value of 3.10 at a volatility of 19%.
The Rho
Rho
The sensitivity of an option's theoretical value to a change in interest rates.
The sensitivity of an option's theoretical value to a change in interest rates is given by its rho. The rho is usually expressed as the amount of change in an option's theoretical value for each one percentage point change in interest rates.
A positive rho indicates that a rise in interest rates will increase the value of the option, and a decline in interest rates will decrease the value of an option. A negative rho indicates that a rise in interest rates will reduce the value of the option, and a decline in interest rates will increase the value of an option.
The effect of interest rates is not uniform and will vary depending on the underlying instrument. Stock option calls have positive rho values, and stock option puts have negative rho values. Any rise in interest rates will increase the value of calls and decrease the value of puts; any decline in interest rates will reduce the value of calls and increase the value of puts.
For options on futures, where the options are subject to stocktype settlement, both calls and puts will have negative rho values. Any rise (decline) in interest rates will reduce (increase) the values of both calls and puts. If options are subject to futurestype settlement, all rho values are zero. Option values are unaffected by changes in interest rates.
Although a change in interest rates will change the value of an option, the effect is usually minor compared to changes in other inputs. For this reason, the rho is usually of less concern to a trader than the other more critical sensitivities, the delta, gamma, theta, and vega.
Summary
Theoretical pricing models also generate numbers which reflect the sensitivity of an option's theoretical value to changes in market conditions. These sensitivities are often designated with Greek letters:
 Delta  the sensitivity of an option's theoretical value to a change in the price of the underlying contract
 Gamma  the sensitivity of an option's delta to a change in the price of an underlying contract
 Theta  the sensitivity of an option's theoretical value to a change in the amount of time remaining to expiration
 Vega or Kappa  the sensitivity of an option's theoretical value to a change in volatility
 Rho  the sensitivity of an option's theoretical value to a change in interest rates
Chapter 12
Volatility Spreads
To take advantage of a theoretically mispriced option, it is necessary to hedge the purchase or sale of the option by simultaneously taking an opposing market position. In the previous examples, the opposing market position was always taken in the underlying instrument.
It is also possible to hedge an option position with other options theoretically equivalent to the underlying instrument. For example, suppose a particular call with a delta of 50 is underpriced in the marketplace. If we buy 10 calls, giving us a total delta position of +500, we can hedge our position in any of the following ways:
 Selling five underlying contracts.
 Buy puts with a total delta of 500.
 Sell calls, different than those we purchased, with a total delta of +500.
 Combine several of these strategies to create a total delta of 500.
With many different calls and puts available and the underlying contract, there are many ways of hedging our ten calls. Regardless of which method we choose, each position will have certain features in common:
 Each position will be approximately deltaneutral.
 Each position will be sensitive to changes:
 In the underlying instrument's price.
 In implied volatility.
 To the passage of time.
Positions with the preceding characteristics fall under the general heading of volatility spreads. In this chapter, we will define the basic types of volatility spreads and look at their features, initially by examining the expiration values of the spread and then by considering the delta, gamma, theta, vega, and rho associated with each spread.
Volatility spread
Any option spread whose value is sensitive to either the underlying contract's volatility or changes in the options' implied volatility.
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Why Traders Spread
Most successful option traders engage in spread trading. Since option evaluation is based on the laws of probability, and because the laws of probability can be expected to even out only over prolonged periods, options traders must often hold positions for extended periods.
Unfortunately, over short periods, while the trader is waiting for the option price to move towards theoretical value, the trader may be subject to various changes in market conditions that threaten their potential profit. Indeed, over short periods, an option is not guaranteed to react consistently with a theoretical pricing model.
Spreading is simply a way of enabling an option trader to take advantage of theoretically mispriced options while reducing the effects of shortterm changes in market conditions to hold an option position to maturity.
What is a Spread?
A spread strategy involves taking simultaneous but opposing market positions in different instruments. A spread trader assumes that there is an identifiable price relationship between different instruments. Although he may not know in which direction the market will move, the price relationship between the instruments should remain relatively constant.
When the relationship appears to be temporarily mispriced, the spread trader will take a long position in the instrument, which seems to be underpriced, and a short position in the instrument, which appears to be overpriced. The trader hopes to profit when the prices of the instruments return to their expected relationship.
Spread
A long market position and an offsetting short market position usually, but not always, in the same underlying market.
A spread may consist of opposing positions in stocks, options, futures, or other derivative instruments. For example, a trader may buy General Motor's stock (a long market position) and General Motors puts (a short market position). Or a trader may buy a basket of stocks comprising an index (a long market position) and sell futures contracts on the index (a short market position).
Or a trader may buy General Motor's stock (a long market position) and sell Ford Motor Co. stock (a short market position). In each case, the trader assumes that there ought to be a price relationship between the various instruments but that this price relationship is temporarily violated in the marketplace.
Spreading as a Risk Management Tool
Recall our example: a casino sells a roulette bet with an expected return of 95¢ (American conditions) for 1.00. The casino knows that based on the laws of probability; it has a 5 percent theoretical edge. Suppose that one day, a bettor comes into the casino and proposes to bet 2,000 on one number at the roulette table.
The casino owner knows that the odds are on their side and that they will get to keep the 2,000 bet. But there is always a chance that the player will win, and the casino will lose 70,000 (the 72,000 payoff less the 2,000bet cost) if the player's number does come up.
Suppose two other bettors come into the casino, each proposing to place a 1,000 bet at the roulette table. They promise, however, to bet on different numbers. Whichever number one bettor chooses, the other bettor will choose some other number. As with the first bettor and their single bet of 2,000, the casino's potential reward in this new scenario is 2,000 if neither of the two numbers comes up. But the risk to the casino is now only 34,000 (the 36,000 payoff if one player wins, less the cost of two 1,000 bets). Since only one player can win, the two bets are mutually exclusive: if one wins, the other must lose.
Given our two scenarios, one bettor wagering 2,000 on one number or two players wagering 1,000 each on different numbers, what is the theoretical edge to the casino? The edge to the casino in both cases is still the same, 5 percent. Regardless of the amount wagered or the number of individual bets, the laws of probability say that overall, the casino gets to keep 5 percent of everything bet at the roulette table. In the short run, however, the risk to the casino is significantly reduced with two 1,000 bets because the bets have been spread around the table.
A casino does not like to see a bettor wager a large amount of money on one outcome, whether at roulette or any other casino game. The casino knows that, overall, the odds are on its side. But, if the bet is large enough and the bettor is lucky, shortterm bad luck can overwhelm the casino.
Indeed, if a bettor knows that the odds are against them and they want the most excellent chance of showing a profit, their best course is to wager the maximum amount on one outcome and hope that they get lucky in the short run. If they continue to make bets over an extended period, the laws of probability will eventually catch up with them, and the casino will end up with the bettor's money.
The ideal scenario from the casino's point of view is for 38 players to place 38 bets of 1,000 each on all 38 numbers at the roulette table. Now, the casino has a perfect spread position. One player will collect 36,000, but with 38,000 on the table, the casino has a sure profit of 2,000.
The option trader prefers to spread for the same reason the casino prefers the bets to be spread around the table: spreading maintains profit potential but reduces shortterm risk. There is no perfect spread position for the option trader, as there is for a casino. However, the intelligent option trader learns to spread their risk in as many ways as possible to minimize the effects of shortterm bad luck.
Chapter 13
Types of Volatility Spreads
Types of Volatility Spreads
There is an infinite variety of volatility spreads that can be done in any option market, ranging from the remarkably simple to the complex. However, certain primary spreads are very actively traded in most option markets, and once a trader understands these spreads, all other spreads become combinations of the primary spreads.
Volatility spread
Any option spread whose value is sensitive to either the volatility of the underlying contract or to changes in the implied volatility of the options.
Before defining the primary types of volatility spreads, it should be noted that spreading terminology is not uniform. Traders sometimes use different terms when referring to the same spread; they sometimes use the same term when referring to different spreads. An attempt has been made to use the most common terminology, but alternative spread definitions are also given where appropriate.
In general, a trader will say that they are long a spread if the spread trade results in a debit to their account (they pay some amount of money). A trader will say that they are short a spread if the spread trade results in a credit to their account (they receive some amount of money).
Options spread strategies are advanced trading techniques that offer traders and investors a way to manage risk, generate income, and capitalize on market movements with limited capital. While [basic options strategies] like buying calls or puts provide a foundation, intermediatelevel options spread strategies take your options trading skills to the next level.
In this intermediate level, traders begin to combine multiple options contracts, both calls and puts, to create structured positions that aim to benefit from specific market conditions. These strategies involve a deeper understanding of options pricing, volatility, and the interplay of various factors affecting option premiums.
Intermediate options spread strategies can be categorized into several types, including credit spreads, debit spreads, and ratio spreads, each serving a unique purpose and risk profile. These strategies offer a range of possibilities, from hedging existing positions to generating consistent income, all while controlling potential losses.
In exploring intermediatelevel options spread strategies, we'll delve into the key concepts, strategies, and scenarios where these techniques can be applied effectively. Whether you're an experienced options trader looking to refine your skills or a novice trader aiming to expand your knowledge, understanding these strategies will empower you to make more informed decisions and potentially enhance your trading success.
Ratio Spreads
A ratio spread is a spread that consists of unequal numbers of long (or purchased) options and short (or sold) options, where all options expire at the same time, are of the same type (either calls or puts), and the position is approximately deltaneutral.
Ratio spread
Any spread where the number of long market contracts (long underlying, long call, or short put) and short market contracts (short underlying, short call, or long put) are unequal.
For example, suppose an ABC June 95 call has a delta of 75 and an ABC June 105 call has a delta of 25. A typical ratio spread might be:
Call Backspread or Long Call Ratio Spread  Call Frontspread or Short Call Ratio Spread 

buy 3 ABC Jun 105 calls  buy 1 ABC Jun 95 call 
sell 1 ABC Jun 95 call  sell 3 ABC Jun 105 calls 
These positions are typical ratio spreads. They consist of unequal long and short contracts, expiring simultaneously, and are deltaneutral.
Ratio spreads may be made up of either calls or puts. For example, suppose an XYZ October 40 put has a delta of 20 and an XYZ October 45 put has a delta of 50. A typical put ratio spread might be:
Put Backspread or Long Put Ratio Spread  Put Frontspread or Short Put Ratio Spread 

buy 5 XYZ October 40 puts  buy 2 XYZ October 45 puts 
sell 2 XYZ October 45 puts  sell 5 XYZ October 40 puts 
These positions are also ratio spreads because they consist of unequal numbers of long and short contracts, all expiring simultaneously, and they are deltaneutral.
When a ratio spread consists of more purchased contracts than sold, it is commonly called a backspread. When the spread consists of more contracts sold than purchased, there is not one commonly accepted term. The spread may be referred to simply as a ratio spread, a front spread, a vertical spread, or a ratio vertical spread.
Backspread
A spread, usually deltaneutral, where more options are purchased than sold and where all options have the same underlying contract and expire at the same time.
Front spread
A spread, usually delta neutral, where more options are sold than are purchased and where all options have the same underlying contract and expire at the same time.
Typical expiration P&L diagrams for several types of ratio spreads are shown below.
We can determine the general characteristics of ratio spreads from the previous graphs. A backspread (more long options than short) wants the underlying market to move and will be profitable if the move is sufficiently large. Because of the unlimited profit potential, there is a preference for upward movement with a call backspread and downward movement with a put backspread.
But the most important consideration is that the market moves. A backspread will almost certainly result in a loss if the market fails to move. On the other hand, a ratio vertical spread (more short options than long) wants the underlying market to sit still. As expiration approaches, it would especially like the underlying market to move towards the short exercise price, resulting in the maximum profit.
If, however, the market makes a big move in either direction, a ratio vertical spread is likely to result in a loss. In the case of a call ratio vertical spread, the potential loss on the upside is unlimited. In the case of a put ratio vertical spread, the possible loss on the downside is unlimited.
Note that to be delta neutral, a call backspread will require the purchase of calls at a higher exercise price (calls with smaller deltas) and selling calls at a lower exercise price (calls with larger deltas). A put backspread will require the purchase of puts at a lower exercise price (puts with smaller deltas) and the sale of puts at a higher exercise price (puts with larger deltas).
A ratio vertical spread involves the opposite procedure.
A call ratio vertical spread will require the purchase of calls at a lower exercise price (calls with larger deltas) and selling calls at a higher exercise price (calls with smaller deltas). A put ratio vertical spread will require the purchase of puts at a higher exercise price (puts with larger deltas) and the sale of puts at a lower exercise price (puts with smaller deltas).
Straddles
Straddle
A long (short) call and a long (short) put, where both options have the same underlying contract, the same expiration date, and the same exercise price.
A straddle consists of either a long call and a long put or a short call and a short put, where both options have the same exercise price and expire simultaneously. If both the call and put are purchased, the trader is said to be long the straddle; if both options are sold, the trader is said to be short the straddle. Typical straddles might be:
Long Straddle  Short Straddle 

long 1 ABC June 100 call  short 1 XYZ October 45 call 
long 1 ABC June 100 put  short 1 XYZ October 45 put 
In the first example, the trader has purchased, or is long, the ABC June 100 straddle; in the second example, the trader has sold, or is short, the XYZ October 45 straddle.
If the underlying price is close to the exercise price of the straddle, the delta values of the call and put will be approximately 50 and 50, and the straddle will be delta neutral if done onetoone (one call for each put). While most straddles are executed with a onetoone ratio, this is not a requirement.
buy 1 June 95 call
buy 3 June 95 puts
Typical expiration P&L diagrams for long and short straddles are shown below.
A long straddle has many of the same characteristics as a backspread. Like a backspread, it has limited risk and unlimited profit potential. With a long straddle, however, the trader's potential profit is unlimited in either direction. If the market moves sharply up or down, the straddle will realize everincreasing profits as the market continues to move in the same direction.
A short straddle has many of the same characteristics as a ratio vertical spread. The spread has limited profit potential and will realize the maximum profit if the market stays close to the call and put exercise price. A short straddle also has unlimited risk should the market move violently in either direction.
Strangles
Strangle
A long (short) call and a long (short) put, where both options have the same underlying contract, the same expiration date but different exercise prices.
Like a straddle, a strangle consists of a long call and a long put, or a short call and a short put, where both options expire at the same time. In a strangle, however, the options have different exercise prices. If both options are purchased, the trader is said to be long the strangle; if both options are sold, the trader is said to be short the strangle. Typical strangles might be:
Long Strangle  Short Strangle 

long 1 ABC June 105 call  short 1 XYZ October 50 call 
long 1 ABC June 95 put  short 1 XYZ October 45 put 
In the first example, the trader has purchased or is long, the ABC June 95/105 strangle; in the second example, the trader has sold, or is short, the XYZ October 45/50 strangle.
As with a straddle, most strangles are executed with a onetoone ratio, one call for each put. If the delta values of the call and put are unequal, to be delta neutral, a strangle can also be ratioed so that it consists of unequal numbers of calls and puts.
For example, suppose the delta of an XYZ October 50 call is 30, and the delta of an XYZ October 40 put is 20. If a trader wants to sell the October 40/50 strangle and be deltaneutral, they can:
sell 2 October 50 calls
sell 3 October 40 puts
Typically, strangles are assumed by most traders to consist of outofthemoney options. However, it is possible, although less common, to buy or sell a strangle consisting of inthemoney options. For example, with the underlying stock at 45, a trader might:
buy 1 October 40 calls
buy 1 October 50 put
This type of inthemoney strangle is known in some markets as a guts.
Guts
A strangle where both the call
and the put are inthemoney.
Typical expiration P&L diagrams for long and short strangles are shown below.
A long strangle has many of the same characteristics as a long straddle. Like a long straddle, it has limited risk and unlimited profit potential in either direction. A long strangle will have a lower price than a straddle, but the strangle also requires more significant movement to realize a profit.
A short strangle has many of the same characteristics as a short straddle. Like a short straddle, a short strangle has limited profit potential risk and will realize the maximum profit if the underlying market remains anywhere between the two exercise prices at expiration. If the underlying contract makes an unexpectedly large move, a short strangle also has unlimited risk in either direction.
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Butterflies
Butterfly
The sale (purchase) of two options with the same exercise price, together with the purchase (sale) of one option with a lower exercise price and one option with a higher exercise price.
All options must be of the same type have the same underlying contract, and expire at the same time, and there must be an equal increment between exercise prices.
Thus far, we have looked at spreads that involve buying or selling two different option contracts. However, we need not restrict ourselves to twosided spreads. A butterfly consists of three equally spaced exercise prices, where all options are the same type (all calls or all puts) and expire simultaneously.
In a long butterfly, the outside exercise prices are purchased, the inside exercise price is sold, and vice versa for a short butterfly. Moreover, the ratio of a butterfly never varies. It is always 1 x 2 x 1, with two of each inside exercise price traded for each one of the outside exercise prices.
Wings
The extreme exercise price. The term is most often applied to the outside exercise prices in a butterfly or condor.
If the outside exercise prices (sometimes referred to as the wings of the butterfly) are purchased and the inside exercise price (sometimes referred to as the body of the butterfly) sold, a trader is said to be long the butterfly. A trader is said to be short the butterfly if the outside exercise prices are sold and the inside exercise price purchased. Typical butterflies might be:
Long Butterfly  Short Butterfly 

long 1 ABC June 95 call  short 1 XYZ October 40 put 
short 2 ABC June 100 calls  long 2 XYZ October 45 puts 
long 1 ABC June 105 call  short 1 XYZ October 50 put 
In the first example, the trader has purchased, or is long, the ABC June 95/100/105 call butterfly; in the second example, the trader has sold, or is short, the XYZ October 40/45/50 put butterfly. Typical expiration P&L diagrams for long and short butterflies are shown below.
Note that butterflies have limited profit potential and limited loss. A butterfly always has a minimum value of zero and a maximum value of the amount between exercise prices. If there are 5 points between exercise prices (for example, 95/100/105 or 40/45/50), the butterfly will have a maximum value of 5 points.
A butterfly will be worthless at expiration if the underlying contract is outside the butterfly's wings (the outside exercise prices), and it will have its maximum value if the underlying contract is right at the body (the inside exercise price) of the butterfly.
A trader who purchases a butterfly will pay some amount between zero and the amount between exercise prices and will hope that at expiration, the underlying contract is right at the inside exercise price. In such a case, the butterfly will widen to its maximum value, resulting in a profit to the trader.
A trader who sells a butterfly will receive some amount between zero and the amount between exercise prices and will hope that at expiration, the underlying contract is below the lowest exercise price or above the highest exercise price.
If that happens, the butterfly will be worthless, and the trader will realize a profit equal to the amount he received when they sold the butterfly.
Since all butterflies are worth their maximum amount when the underlying contract is right at the inside exercise price at expiration, a call and put butterfly with the same exercise prices and expiration date desire precisely the same outcome.
The June 95/100/105 call butterfly and the June 95/100/105 put butterfly will be worth a maximum of 5 points with the underlying contract right at 100 at expiration and a minimum value of zero with the underlying contract below 95 or above 105.
If all options are European (no early exercise permitted), callandput butterflies will have identical characteristics and trade at the same price. If the options are American (early exercise is a possibility), the prices of call and put butterflies will still be close but may not be identical.
Calendar Spreads
Calendar spread, time spread, or horizontal spread
The purchase of one option expiring on one date, and the sale of another option expiring on a different date. Typically, both options are of the same type, have the same exercise price, and have the same underlying stock or commodity. Calendar spreads are sometimes referred to as time spreads.
A calendar spread consists of opposing positions in two options of the same type (either both calls or both puts) and with the same exercise price but different expiration dates.
When the longterm option is purchased and the shortterm option is sold, a trader is long the calendar spread; when the shortterm option is bought and the longterm option is sold, the trader is short the calendar spread. Calendar spreads are also called time spreads or, less commonly, horizontal spreads. Typical calendar spreads might be:
Long Calendar Spread  Short Calendar Spread 

long 1 ABC June 100 call  long 1 XYZ July 40 put 
short 1 ABC March 100 call  short 1 XYZ October 40 put 
In the first example, the trader has purchased or is long the ABC March/June 100 call calendar spread; in the second example, the trader has sold or is short the XYZ July/October 40 put calendar spread.
If all options in a spread expire at the same time, the value of the spread is simply a function of the underlying price at expiration. If, however, the spread consists of options that expire at different times, the value of the spread can only be determined once both options expire.
The spread's value depends not only on where the underlying market is when the shortterm option expires but also on what will happen between that time and when the longterm option expires. Typical P&L diagrams for long and short calendar spreads at nearterm expiration are shown below.
A calendar spread has different characteristics from the other spreads we have discussed because its value depends on movement in the underlying market and other traders' expectations about future market movement as reflected in the implied volatility. Assuming that the options making up a time spread are approximately atthemoney, time spreads have two essential characteristics.
A long calendar spread wants the underlying market to sit still. A vital characteristic of an atthemoney option's theta (time decay) is the tendency to become increasingly significant as expiration approaches. As time passes, a shortterm atthemoney option, having less time to expiration, will lose its value at a greater rate than a longterm atthemoney option.
If a trader purchases a calendar spread (buy the longterm option, sell the shortterm option), they want the market to sit still because the shortterm option will decay more quickly than the longterm option. When this happens, the calendar spread will increase in value. This effect is shown in the following example:
Time to Expiration  
long term option  6 months  5 months  4 months  3 months 
short term option  3 months  2 months  1 month  None 
Option Value  
long term option  7.50  7.25  6.75  6 
short term option  6  5  3  0 
Spread Value  1.50  2.25  3.75  6 
A long calendar spread benefits from an increase in implied volatility. An essential characteristic of the vega is that longterm options have a greater vega (volatility sensitivity) than shortterm options.
This means that the theoretical value of a longterm option is always more sensitive in total points to a change in volatility than the theoretical value of a shortterm option with the same exercise price. The effects of volatility on a time spread become especially evident when there is a change in implied volatility in the options market.
When implied volatility rises, calendar spreads widen; when implied volatility falls, calendar spreads narrow. This is shown in the following example:
Volatility  15%  20%  25% 

Option Value  
long term option  6.50  7.50  8.50 
short term option  5.50  6  6.50 
Spread Value  1  1.50  2 
Thus far, we have considered only the effects of changes in the price of the underlying contract and changes in volatility on the value of volatility spreads. But calendar spreads can also be sensitive to changes in interest rates and dividends.
A longcall calendar spread wants interest rates to rise; a longput calendar spread wants interest rates to fall. If we are trading stock options and change the interest rate, we change the forward price of stock (the current stock price plus the carrying cost on the stock to expiration).
When all options expire simultaneously, as they do in ratio spreads, straddles, strangles, and butterflies, the forward stock price for all options remains the same so that the effects of the change are practically negligible. If, on the other hand, we are considering stock options with two different expiration dates, there will be two different forward prices.
As interest rates rise, the forward price of the stock increases, but the longterm forward price will increase more. The longterm call will, therefore, go up more than the shortterm call, causing the call calendar spread to rise in value. Because puts move in the opposite direction of the underlying contract, the longterm put will fall more than the shortterm put, causing the put calendar spread to fall in value.
The Effect of Changing Interest Rates on Calendar Spreads 

Stock Price = 100; Volatility = 20%; Dividend = 0 
Shortterm time to expiration (March) = 6 weeks 
Longterm time to expiration (June) = 19 weeks 
If Interest Rates Are …  0%  3%  6%  9%  12% 

June 100 call  4.81  5.35  5.92  6.52  7.15 
March 100 call  2.71  2.88  3.05  3.24  3.43 
Call Spread Value  2.10  2.47  2.87  3.28  3.72 
June 100 put  4.81  4.26  3.76  3.30  2.88 
March 100 put  2.71  2.53  2.37  2.21  2.06 
Put Spread Value  2.10  1.73  1.39  1.09  .82 
A longcall calendar spread wants dividends to fall; a longput calendar spread wants dividends to rise. Changes in dividends have the opposite effect on stock options as changes in interest rates. An increase (decrease) in dividends lowers (raises) the forward price of stock.
If all options in a volatility spread expire simultaneously, the forward stock price will be identical for all options, and the effect on the spread will be negligible. But in a calendar spread, if a dividend payment is expected between the expiration of the shortterm and longterm option, the longterm option will be affected by the lowered forward price of the stock.
Consequently, if at least one dividend payment is expected between the expiration dates, an increase in dividends will cause call calendar spreads to narrow and put calendar spreads to widen. A dividend decrease will have the opposite effect, with call calendar spreads widening and put time calendar narrowing.
The Effect of Changing Dividends on Calendar Spreads 

Stock Price = 100; Volatility = 20%; Interest Rate = 6.00% 
Shortterm time to expiration (March) = 6 weeks 
Longterm time to expiration (June) = 19 weeks 
If the Quarterly Dividend is …  0  1.00  2.00  3.00  4.00 

June 100 call  5.92  4.81  3.82  2.97  2.26 
March 100 call  3.05  2.53  2.07  1.66  1.31 
Call Spread Value  2.87  2.28  1.75  1.31  .95 
June 100 put  3.76  4.62  5.62  6.75  8.01 
March 100 put  2.37  2.84  3.37  3.96  4.62 
Put Spread Value  1.39  1.78  2.25  2.79  3.39 
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Diagonal Spreads
Diagonal Spread
A long call (put) at one exercise price and expiration date and a short call (put) at a different exercise price and expiration date. All options must have the same underlying contract. This is simply a time spread using different exercise prices.
A diagonal spread is similar to a calendar spread, except the options have different exercise prices. While many diagonal spreads are executed onetoone (one longterm option for each shortterm option), diagonal spreads can also be ratioed, with unequal numbers of long and short market contracts.
With many variations in diagonal spreads, it is almost impossible to generalize their characteristics as we can with ratio spreads, straddles and strangles, butterflies, and calendar spreads. Each diagonal spread must be analyzed separately, often using a computer, to determine the risks and rewards associated with the spread.
There is, however, one type of diagonal spread about which we can generalize. If a diagonal spread is executed one to one, and both options are of the same type and have approximately the same delta, the diagonal spread will act very much like a conventional calendar spread.
For example, suppose that ABC stock is trading at 100 and that a March 105 call and a June 110 call both have deltas of 25. If a trader buys a June 110 call and sells a March 105 call, their position will act like a long calendar spread. If a trader buys a March 105 call and sells a June 110 call, their position will act like a short calendar spread.
Other Variations (Christmas Tree, Iron Butterfly, Condor)
Christmas Tree or Ladder
A spread involving three exercise prices. One or more calls (puts) are purchased at the lowest (highest) exercise price, and one or more calls (puts) are sold at each of the higher (lower) exercise prices. All options must expire simultaneously, be of the same type, and have the same underlying contract.
A Christmas tree (also called a ladder) is a term that can be applied to various spreads. The spread usually consists of three different exercise prices where all options are the same type and expire simultaneously. In a long (short) call Christmas tree, one call is purchased (sold) at the lowest exercise price, and one call is sold (purchased) at each of the higher exercise prices.
In a long (short) put Christmas tree, one put is purchased (sold) at the highest exercise price, and one put is sold (purchased) at each of the lower exercise prices. Christmas trees are usually deltaneutral, but even with this restriction, there are many ways to execute the spread. The following are typical Christmas trees:
Long Christmas Trees  Short Christmas Trees 

buy 1 Mar 95 call  sell 1 Jun 90 call 
sell 1 Mar 100 call  buy 1 Jun 100 call 
sell 1 Mar 105 call  buy 1 Jun 105 call 
buy 1 Jun 110 put  sell 1 Mar 110 put 
sell 1 Jun 100 put  buy 1 Mar 105 put 
sell 1 Jun 95 put  buy 1 Mar 95 put 
When done deltaneutral, long Christmas trees can be considered as particular types of ratio vertical spreads. Such spreads increase in value if the underlying market sits still or moves slowly. Short Christmas trees can be regarded as specific types of backspreads and increase in value with big moves in the underlying market.
Constructing a spread with the same characteristics as a butterfly is possible by purchasing a straddle (strangle) and selling a strangle (straddle) where the straddle is executed at an exercise price midway between the strangle's exercise prices. All options must expire at the same time.
Because the position wants the same outcome as a butterfly, it is known as an iron butterfly. Like a true butterfly, at expiration, an iron butterfly has a minimum value of zero and a maximum value of the amount between exercise prices.
Iron Butterfly
Iron Butterfly
A long (short) straddle and a short (long) strangle. All options must expire simultaneously and have the same underlying contract.
If the straddle is purchased and the strangle is sold, the position will result in a debit (a long iron butterfly). Such a position will show its most significant profit at expiration if the underlying market finishes beyond the outside (strangle's) exercise prices. A long iron butterfly is, therefore, equivalent to a short butterfly.
If the straddle is sold and the strangle is purchased, the position will result in a credit (a short iron butterfly). Such a position will show its greatest profit at expiration if the underlying market finishes right at the inside (straddle's) exercise price. A short iron butterfly is, therefore, equivalent to a long butterfly. The following are typical iron butterflies:
Long Iron Butterflies  Short Iron Butterflies 

buy 1 Mar 100 call  sell 1 June 100 call 
buy 1 Mar 100 put  sell 1 June 100 put 
sell 1 Mar 105 call  buy 1 June 105 call 
sell 1 Mar 95 put  buy 1 June 95 put 
buy 1 June 100 call  sell 1 Mar 105 call 
buy 1 June 100 put  sell 1 Mar 105 put 
sell 1 June 110 call  buy 1 Mar 110 call 
sell 1 June 90 put  buy 1 Mar 100 put 
Condor
Condor
The sale (purchase) of two options with different exercise prices, together with the purchase (sale) of one option with a lower exercise price and one option with a higher exercise price. All options must be of the same type, have the same underlying contract, expire simultaneously, and there must be an equal increment between exercise prices.
Another variation on a butterfly, a condor, can be constructed by splitting the inside exercise prices. Now, the position consists of four options at consecutive exercise prices where the two outside options are purchased and the two inside options sold (a long condor), or the two inside options are purchased and the two outside options sold (a short condor). As with a butterfly, all options must be the same type (all calls or all puts) and expire simultaneously.
At expiration, a condor will have its maximum value equivalent to the amount between consecutive exercise prices when the underlying contract is at or anywhere between the two inside exercise prices. It will be worthless whenever the underlying contract is outside the extreme exercise prices at expiration.
This is like a butterfly, except that a condor has a maximum value over a broader range of underlying prices. A butterfly will achieve its maximum value at expiration at only one underlying cost when the underlying contract is right at the inside exercise price.
For this reason, a condor will usually have a higher value than a butterfly with approximately the same exercise prices. The following are typical condors:
Long Condors  Short Condors 

buy 1 Mar 90 call  sell 1 Mar 95 put 
sell 1 Mar 95 call  buy 1 Mar 100 put 
sell 1 Mar 100 call  buy 1 Mar 105 put 
buy 1 Mar 105 call  sell 1 Mar 110 put 
buy 1 June 95 put  sell 1 June 90 call 
sell 1 June 100 put  buy 1 June 95 call 
sell 1 June 105 put  buy 1 June 100 call 
buy 1 June 110 put  sell 1 June 105 call 
Quoting a Spread Order
Option spreads are typically listed on the exchange as a single contract with one bid and one ask price. Spread positions have their own sensitivities, including delta, gamma, theta, and vega. Spread risk is a tradeoff between reward and various forms of risk. Traders balance risk and reward to achieve consistent, manageable profits.
Generally, spreads with more "naked" options (uncovered) are riskier. Straddles and strangles, consisting solely of naked options, are considered riskiest. Ratio spreads combine naked and covered options, with the ratio influencing the risk. Butterflies and calendar spreads, with equally long and short options, are seen as less risky. Traders may prefer less risky positions for consistent, manageable profits, while higherrisk positions can lead to more significant gains and losses over time. Balancing risk and reward is crucial in spread trading.
With the Rival One trading platform, traders can view the market prices and submit orders on option spreads listed on the exchange. If a market is not available for an option spread, the trader can submit a Request For Quote (RFQ) to the exchange. The exchange will publish the RFQ to market makers and the market makers will reply to the RFQ by submitting their bid and ask price on the option spread.
Summary
 Option spreading terminology is not uniform. Traders sometimes use different terms when referring to the same spread; they sometimes use the same term when referring to different spreads.
 The following are the most common types of volatility spreads:
 ratio spreads (including backspreads and ratio vertical spreads)
 straddles
 strangles
 butterflies
 calendar spreads
 diagonal spreads
 Less common volatility spreads include:
 Christmas trees
 iron butterflies
 condors
 Spreads are quoted in the market place with one bid price and one ask price, regardless of the complexity of the spreads.
Chapter 14
Spread Sensitivities
Just as every individual option has a delta, gamma, theta, vega and rho, every spread position also has its own sensitivities. These numbers can help a trader determine beforehand how changing market conditions are likely to affect the spread.
Classifying Spreads by their Sensitivities
Since a volatility spread is primarily sensitive to the underlying market's volatility rather than the market's direction, all volatility spreads will be deltaneutral (the deltas will add up to approximately zero).
Some volatility spreads may prefer movement in one direction rather than the other, but the primary consideration is whether action of any type will occur. Directional considerations become more important than volatility considerations if a trader has a significant positive or negative delta. The position can no longer be considered a volatility spread.
All spreads that are helped by movement in the underlying market have a positive gamma. These include backspreads, long straddles, long strangles, short butterflies, and short calendar spreads. All spreads hurt by movement in the underlying market will have a negative gamma. These include ratio vertical spreads, short straddles, short strangles, long butterflies, and long calendar spreads.
A trader with a positive gamma is sometimes said to be a long premium and hopes for a volatile market with significant moves in the underlying contract. A trader with a negative gamma is sometimes said to be a short premium and is hoping for a quiet market with only minor moves in the underlying market.
Long Premium
A position that will theoretically increase in value should the underlying contract make a large move in either direction. The position will theoretically decrease in value should the underlying market sit still.
Short Premium
A position that will theoretically increase in value should the underlying market sit still. The position will theoretically decrease in value should the underlying contract significantly move in either direction.
Since the effect of market movement and the impact of time decay always work in opposite directions, any spread with a positive gamma will necessarily have a negative theta. Any spread with a negative gamma will necessarily have a positive theta. If market movement helps, the passage of time hurts. If a market movement hurts, the course of time helps.
Spreads helped by a rise in implied volatility have a positive vega. These include backspreads, long straddles, long strangles, short butterflies, and long calendar spreads. Spreads helped by a decline in implied volatility have a negative vega. These include ratio vertical spreads, short straddles, short strangles, long butterflies, and short calendar spreads.
Position  Delta  Gamma  Theta  Vega 

Backspread  0  +    + 
Long straddle  0  +    + 
Long strangle  0  +    + 
Short butterfly  0  +    + 
Ratio vertical spread  0    +   
Short straddle  0    +   
Short strangle  0    +   
Long butterfly  0    +   
Long calendar spread  0    +  + 
Short calendar spread  0  +     
Because they are sensitive to changes in interest rates, Calendar spreads will also have a rho associated with them. A long call or short put calendar spread will have a positive rho. An increase in interest rates will make the spread more valuable. A long put calendar spread or a short call calendar spread will have a negative rho. An increase in interest rates will make the spread less valuable.
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Relative Riskiness
Every option position is a tradeoff between reward and risk. The reward is often expressed in terms of the theoretical edge, the amount by which the spread appears to be mispriced in the marketplace. The risk can come in a variety of forms:
 The sensitivity of the spread to changes in the price of the underlying contract (the gamma)
 The passage of time (the theta)
 Changes in implied volatility (the vega)
The more sensitive a spread is to any one of these changing market conditions, the greater the risk associated with the spread. A trader who plans to take a significant volatile position should be aware of the risks associated with that spread. If the risks seem too great concerning the potential profit, the trader should look for a spread with more acceptable risk characteristics.
Suppose a trader is considering several different volatility spreads. In that case, they may want to use a trading platform with a theoretical pricing model to analyze the risk characteristics of the spreads. However, in the absence of a theoretical pricing model, the following is a good rule of thumb:
The more naked options involved, the riskier a volatility spread becomes; the fewer naked options involved, the less risky a volatility spread becomes.
From this rule, we can categorize spreads, moving from the riskiest to the least risky:
strangles (most risky)
straddles
ratio spreads
butterflies/calendar spreads (less risky)
Straddles and strangles consist of only naked options, either all long calls or puts or all short calls and puts. For this reason, they are usually considered the riskiest of all option positions. Because strangles consist of lowprice options, a trader can do several for the price of one straddle. Hence, a typical strangle will consist of more naked options than a straddle and is a riskier position.
Ratio spreads consist of some naked options and some covered options. The spread ratio will determine the number of naked options. A high ratio, for example, 5 to 1, will be considered a highrisk spread; a low ratio, 4 to 3, will be regarded as a lowrisk spread.
Because they usually consist of equal numbers of long and short options, butterflies and calendar spreads are considered the least risky volatility spreads.
Of course, riskiness is a twoway street. A highly risky spread may result in a more significant loss when market conditions turn against a trader. But such a spread will also result in a more substantial profit when market conditions turn in the trader's favor. Overall, a trader may take the approach that good luck and bad luck will even out so that whether a trader takes highrisk or lowrisk option positions, in the long run, their profit will be the same.
And indeed, this may be true. However, most experienced option traders have found it easier to live with more minor but consistent profits than large and often unpredictable profits and losses. Consistent profits result from a respect for a position's risk characteristics.
Summary
 Every volatility spread position has its own sensitivities (gamma, theta, vega).
 Not all volatility spreads are equally risky. In general, the more naked options involved, the more risky a volatility spread becomes; the fewer naked options involved, the less risky a volatility spread becomes.
Chapter 15
Adjustments
Adjustments
A volatility spread may initially be deltaneutral, but the position's delta is likely to change as the price of the underlying contract rises or falls. Moreover, changes in volatility and time to expiration can also affect the delta of a spread.
A deltaneutral spread today may not be deltaneutral tomorrow, even if all other conditions remain the same. Because of this, a trader doing volatility spreads ought to consider how they will approach the adjustment process. When will they make their adjustments, and what contracts might they use?
When to Adjust
The optimum use of a theoretical pricing model requires a trader to continuously maintain a deltaneutral position throughout the spread's life. Continuous adjustments are impossible in real life, so a trader should consider when they will adjust a position. We can consider three possibilities:

Adjust at Regular Intervals
In theory, the adjustment process is assumed to be continuous because volatility is considered a constant measure of the speed of the market. In practice, however, volatility is measured over regular intervals, so a reasonable approach is to adjust a position at similar intervals.
If a trader's volatility estimate is based on daily price changes, the trader might adjust daily. If the estimate is based on weekly price changes, they might adjust weekly. This is a trader's best attempt to emulate the assumptions built into the theoretical pricing model.

Adjust When the Position Becomes a Predetermined Number of Deltas, Long or Short
Very few traders insist on being deltaneutral all the time. Most traders realize that this is not a realistic approach because a continuous adjustment process is physically impossible and because no one can be certain that all the assumptions and inputs in a theoretical pricing model from which the delta is calculated are correct.
Even if one could be sure that all delta calculations were accurate, a trader might still be willing to take on some directional risk. But a trader ought to know how much directional risk they will accept.
Suppose the trader wants to pursue deltaneutral strategies but believes they can comfortably live with a position up to 500 deltas long or short. In that case, they can adjust the position whenever their delta position reaches this limit.
Unlike the trader who adjusts at regular intervals, a trader who adjusts based on a fixed number of deltas cannot be sure how often they will need to change their position. In some cases, they may have to adjust very frequently. In other cases, they may go for extended periods without altering. The number of deltas, long or short, a trader chooses for their adjustment points depends on the size of their positions and capitalization.
A small independent trader may be uncomfortable with a position that is even 200 deltas long or short. A large trading firm may consider a position of several thousand deltas long or short as being approximately deltaneutral.

Adjust by Feel
This suggestion is not made facetiously. Some traders have a good market feel. They can sense when the market is about to move in one direction or another. If a trader has this ability, there is no reason he should not make use of it. For example, suppose that the underlying market is 50 and a trader is deltaneutral with a gamma of 200.
If the market falls to 48, the trader can estimate that they are approximately 400 deltas long. If 400 deltas are the limit of risk, they are willing to accept; they might decide to adjust at this point. If they are also aware that 48 represents staunch support for the market, they might choose not to adjust under the assumption that the market is likely to rebound from the support level. If they are right, they will have avoided an unprofitable adjustment.
Of course, if they are wrong and the market continues downward through the support level, they will regret not having adjusted. If the trader is correct, there is no reason they should not take advantage of this skill.
An experienced trader will combine all the adjustment procedures, using whichever is appropriate, given their knowledge of market conditions and ability to deal with risk. A new trader, however, should learn to become as disciplined as possible, keeping the delta position within predetermined limits as much as possible. They should only allow the total delta to exceed their normal risk limits under extraordinary conditions.
How to Adjust
In addition to determining when to adjust a position, a trader must also consider how best to adjust. There are many ways to change the total delta position. Adjusting a trader's delta position may reduce their directional risk. Still, if they simultaneously increase their gamma, theta, or vega risk, they may inadvertently exchange one type of risk for another.
A delta adjustment made with the underlying contract is a riskneutral adjustment. By this, we mean that an adjustment made with the underlying contract will not change any of the other risks we have discussed because the gamma, theta, and vega associated with an underlying contract are zero. Therefore, if a trader wants to adjust their delta position but wants to leave the other characteristics unaffected, they can do so by purchasing or selling an appropriate number of underlying contracts.
An adjustment made with options may reduce the delta risk and change the other risk characteristics associated with the position. In addition to having a delta, every option has a gamma, theta, and vega. When an option is added to or subtracted from a position, it necessarily changes the position's total delta, gamma, theta, and vega. This is something that new traders sometimes forget.
All other considerations being equal, whenever a trader adjusts, they should do so to improve the risk/reward characteristics of the position. In addition to moving the delta position toward zero, a trader should also try to reduce one or more of the gamma, theta, or vega risks. The adjustment will be ideal and one that a trader should always be willing to make if such an adjustment can be made without reducing the theoretical edge.
Unfortunately, a trader may find that an adjustment reduces one risk but increases a different risk. If, for example, a specific adjustment reduces the gamma risk but increases the vega risk, a trader may have to ask themself which risk represents a more significant threat to their position.
If they are more worried about movement in the underlying contract than changes in implied volatility, then the adjustment is sensible. But if a change in implied volatility is their primary concern, they should look for another way to adjust the position.
Finally, a trader may find that an adjustment, while reducing one or more of the gamma, theta, or vega risks, also lowers the theoretical edge. Now, the trader will have to ask themselves what, if anything, they are willing to give up in potential profit to be able to sleep better at night. This is something that each trader must decide individually.
Summary
 Most theoretical models assume continuous adjustments to remain delta neutral. Since this is not possible in real life, most traders adjust at regular intervals or when a position becomes unbalanced by a predetermined number of deltas.
 A delta adjustment made with stock will not change the other risk characteristics (gamma, theta, vega) of a position. A delta adjustment made by either purchasing or selling calls or puts, while reducing the delta of the position, will also change the other risk characteristics of the position.
Chapter 16
Directional Strategies
While deltaneutral strategies are popular among active option traders, many prefer to trade with a bullish or bearish perspective on the underlying instrument. The trader who wishes to take a directional position in the underlying instrument has the choice of doing so in either the instrument itself, buying or selling the underlying contract, or taking the position in the options market. If they choose the option market, the trader can integrate option pricing theory into a directional strategy to take advantage of theoretically mispriced options.
Naked Positions
The simplest way of taking a directional position in any market is to buy or sell naked options. Such positions constantly have a strong directional component in the form of a positive or negative delta.
Naked Bull Positions
Bullish
An expectation that an underlying market will rise in price. The term may also be applied to any position which will profit from a rise in the underlying market.
If a trader is bullish on the market and wants to take a naked position, they can buy calls or sell puts. Both positions result in a positive delta. Since a trader will also want to take positions with a positive theoretical edge, they must decide whether option prices are generally high or low.
This, in turn, is determined by the trader's opinion of implied volatility compared to future volatility over the option's life. If implied volatility is high, a trader will prefer to sell puts; if implied volatility is low, a trader will prefer to buy calls.
Theoretically, a trader will want to create the most considerable possible theoretical edge for a given delta position. Because outofthemoney options are the most sensitive in percent terms to changes in volatility, when a trader has a strong opinion that implied volatility is either much too high or much too low, they will tend to take their naked position in outofthemoney options. When the trader has a mild opinion that implied volatility is too high or too low, they will take their naked position in atthemoney or inthemoney options.
For example, suppose a trader wants to take a bullish position of +500 deltas and believes implied volatility is too low. They might buy 10 atthemoney calls with a delta of 50 each or 20 outofthemoney calls with a delta of 25 each. If they strongly believe implied volatility is too low, they will tend to buy the outofthemoney calls since that will result in the greatest theoretical edge. If they are less convinced that implied volatility is too low, they will tend to buy the atthemoney calls to achieve the desired delta position.
In the same way, if a trader believes that implied volatility is too high, they might sell 10 atthemoney puts with a delta of 50 each, or they might sell 20 outofthemoney puts with a delta of 25 each. If they strongly believe implied volatility is too high, they will tend to sell the outofthemoney puts since that will result in the greatest theoretical edge. If they are less convinced that implied volatility is too high, they will tend to sell the atthemoney puts.
Naked Bear Positions
Bearish
An expectation that an underlying market will fall in price. The term may also be applied to any position which profits from a decline in the underlying market.
If a trader is bearish on the market and wants to take a naked position, they can buy puts or sell calls. Both positions result in a negative delta. Since a trader will also want to take positions with a positive theoretical edge, they must decide whether option prices are generally high or low. This, in turn, is determined by the trader's opinion of implied volatility compared to future volatility over the option's life.
If implied volatility is high, a trader will prefer to sell calls; if implied volatility is low, a trader will prefer to buy puts. Theoretically, a trader will want to create the most significant possible theoretical edge for a given delta position.
Because outofthemoney options are the most sensitive in percent terms to changes in volatility, when a trader has a strong opinion that implied volatility is either much too high or much too low, they will tend to take their naked position in outofthemoney options. When the trader has a mild opinion that implied volatility is too high or too low, they will take their naked position in atthemoney or inthemoney options.
For example, suppose a trader wants to take a bearish position of 500 deltas and believes implied volatility is too low. They might buy 10 atthemoney puts with a delta of 50 each or 20 outofthemoney puts with a delta of 25 each.
If the trader strongly believes implied volatility is too low, they will tend to buy the outofthemoney puts since that will result in the greatest theoretical edge. If they are less convinced that implied volatility is too low, they will tend to buy the atthemoney puts to achieve the desired delta position.
In the same way, if a trader believes that implied volatility is too high, they might sell 10 atthemoney calls with a delta of 50 each, or they might sell 20 outofthemoney calls with a delta of 25 each. If they strongly believe implied volatility is too high, they will tend to sell the outofthemoney calls since that will result in the greatest theoretical edge. If they are less convinced that implied volatility is too high, they will tend to sell the atthemoney calls.
Summary
 A trader can take a bullish position by simply purchasing calls or selling puts.
 A trader can take a bearish position by simply purchasing puts or selling calls.
 If implied volatility is high, a trader will prefer to sell options; if implied volatility is low, a trader will prefer to buy options.
 If a trader has a strong opinion that implied volatility is too high or too low, he will prefer to buy or sell outofthemoney options.
Chapter 17
Bull and Bear Volatility Spreads
While a trader can undoubtedly take a bullish or bearish position by purchasing or selling naked options, most active option traders prefer to control the risk of a position through some spreading strategy.
Volatility spreads, such as those discussed in the last chapter, can often be done so that they result in a bullish or bearish position. Suppose a trader has a volatility opinion and a directional opinion. In that case, they can choose an appropriate volatility spread but execute it in such a way that it also has either bullish or bearish characteristics.
Bull and Bear Ratio Spreads
Suppose a trader is interested in executing the following deltaneutral ratio spread (delta values are in parentheses):
buy 1 June 100 call (50)
sell 2 June 110 calls (25)
Since the spread is deltaneutral, it has no preference for upward or downward movement in the underlying market.
Now, suppose the same trader believes that this ratio spread is a sensible strategy, but at the same time, they are also bullish on the market. No law requires the trader to do this spread in a deltaneutral ratio. If they want this spread to reflect their bullish sentiment, they might change the ratio slightly:
buy 2 June 100 calls (50)
sell 3 June 110 calls (25)
The trader has the same ratio vertical spread but with a bullish bias. This is reflected in the total delta for the position of +25.
There is, however, a significant limitation in using this type of ratio strategy to create a bullish or bearish position. If the trader has underestimated volatility and the underlying market moves up too quickly, the spread can invert from a positive to a negative delta. If the market rises far enough, to 130 or 140, eventually, all options will go deep inthemoney, and the deltas of the June 100 and June 110 calls will approach 100.
Eventually, the trader will be left with a delta position of 100. Even though the trader was correct in their bullish sentiment, the position was primarily a volatility spread, so the volatility characteristics of the position eventually outweighed any considerations of market direction.
The delta can also invert with a backspread. For example, suppose a trader has decided to execute the following deltaneutral backspread:
buy 2 June 110 calls (25)
sell 1 June 100 call (50)
If, however, they are bullish on the market, they can, as in the last example, change the ratio to reflect this sentiment:
buy 3 June 110 calls (25)
sell 1 June 100 call (50)
Their delta position of +25 reflects this bullish bias.
As time passes or volatility declines, all deltas move away from 50 (see Topic 6.3.1). If time begins to pass with no movement in the underlying contract, the delta of the June 100 call will remain at 50, but the delta of the June 110 call will start to decline. If, after some time, the delta of the June 110 call declines to 10, the delta of the position will no longer be +25 but will instead be 20.
Since this spread is a volatility spread, the primary consideration, as before, is the market's volatility. Only secondarily are we concerned with the direction of movement. If the trader overestimates volatility and the market moves more slowly than expected, the spread, initially deltapositive, can instead become deltanegative.
Check out Rival One Options Functionality
Bull and Bear Butterflies
Butterflies can also be chosen to reflect a trader's bullish or bearish bias. Like ratio spreads, however, their delta characteristics can also invert as market conditions change.
With the underlying market at 100, a deltaneutral trader might buy the June 95/100/105 call butterfly (buy a 95 call, sell two 100 calls, buy a 105 call).
The trader hopes that the market will sit still at 100 so that at expiration, the butterfly will widen to its maximum value of five points. If a trader wants to buy a butterfly but is also bullish on the market, they can choose a butterfly where the inside exercise price is above the current price of the underlying contract. If the underlying market is 100, they might buy the June 105/110/115 call butterfly.
Since this position wants the underlying market at 110 at expiration and currently at 100, it is a bull butterfly. This will be reflected in the position having a positive delta. Unfortunately, if the underlying market moves too swiftly to 120, the butterfly can invert from a positive to a negative delta position.
Since, at expiration, the butterfly always has its maximum value with the underlying contract right at the inside exercise price, the trader will now want the market to fall back from 120 to 110. Whenever the underlying market is below 110, the position will be bullish; whenever the underlying market is above 110, the position will be bearish.
Conversely, if the trader is bearish, they can always buy a butterfly where the inside exercise price is below the current price of the underlying market. But again, if the market moves down too quickly and goes through the inside exercise price, the position will invert from a negative to a positive delta.
Bull and Bear Calendar Spreads
A trader can choose calendar spreads that are either bullish or bearish. A long calendar spread always wants the nearterm contract to expire precisely atthemoney. A long calendar spread will be initially bullish if the exercise price of the calendar spread is above the current price of the underlying market.
With the underlying market at 100, the June/March 110 call calendar spread (buy the June 110 call/sell the March 110 call) will be bullish since the trader will want the underlying market to rise to 110 by March expiration. The June/March 90 call calendar spread (buy the June 90 call/sell the March 90 call) will be bearish since the trader will want the underlying market to fall to 90 by March expiration.
However, the delta can invert if the market moves through the exercise price. If the market moves from 100 to 120, the June/March 110 call calendar spread, initially bullish, will become bearish. If the market moves from 100 to 80, the June/March 90 call calendar spread, which was originally bearish, will become bullish.
Chapter 18
Vertical Spreads
Vertical Spread
The purchase of one option and sale of one option, where both options are of the same type, have the same underlying contract and expire simultaneously but they have different exercise prices.
Vertical spreads are the most common and straightforward directional option spreads. A vertical spread always consists of one long (purchased) option and one short (sold) option, where both options are of the same type (either both calls, or both puts) and expire at the same time. The options are distinguished only by their different exercise prices. Typical vertical spreads might be:
Vertical Spreads 

buy 1 June 100 call 
sell 1 June 105 call or 
buy 1 March 85 put 
sell 1 March 75 put 
Bull Vertical Spreads
Regardless of whether a vertical spread consists of calls or puts, the position is bullish whenever a trader buys the lower exercise price and sells the higher exercise price.
A call with a lower exercise price will always have a greater positive delta than a higher one. The purchase of the lower exercise price call and the sale of the higher exercise price call will, therefore, result in a favorable delta position. A put with a lower exercise price will always have a smaller negative delta than a higher one.
The purchase of the lower exercise price put, and the sale of the higher exercise price put will also result in a favorable delta position.
A bull vertical spread (buy the lower exercise price, sell the higher) is not only initially bullish but will remain bullish no matter how market conditions change. Two options that have different exercise prices but which are identical in every other respect cannot have identical deltas, although the deltas can become close to 100 if the options are very deeply inthemoney, or close to zero if the options are far outofthemoney.
The maximum value of a vertical spread at expiration is the amount between exercise prices if both options are inthemoney, or zero if both options are outofthemoney. A trader who buys the June 100/105 call spread (buy the 100 call, sell the 105 call) will pay some amount between 0 and 5, say 2.25, and hope the market is above 105 at expiration.
If that happens, the spread will be worth 5 and will show a profit of 2.75. A trader selling the March 75/85 put spread (buy the 75 put, sell the 85 put) will take in some amount between 0 and 10, say 6.50, and hope the market is above 85 at expiration. If that happens, both puts will expire worthless, the spread will be worthless, and the trader will show a profit of 6.50.
Note that a bull vertical call spread will result in a debit: the trader buys the spread. A bull vertical put spread will result in a credit: the trader sells the spread.
The expiration value of a bull vertical spread is shown in the following graph:
Bear Vertical Spreads
Regardless of whether a vertical spread consists of calls or puts, the position is bearish whenever a trader buys the higher exercise price and sells the lower exercise price.
A call with a higher exercise price will always have a smaller positive delta than a lower one. The purchase of the higher exercise price call and the sale of the lower exercise price call will, therefore, result in a negative delta position.
A put with a higher exercise price will always have a greater negative delta than a lower one. The purchase of the higher exercise price put and the sale of the lower exercise price put will also result in a negative delta position.
A bear vertical spread (buy the higher exercise price, sell the lower) is not only initially bearish but will remain bearish no matter how market conditions change. Two options that have different exercise prices but which are identical in every other respect cannot have identical deltas, although the deltas can become close to 100 if the options are very deeply inthemoney, or close to zero if the options are far outofthemoney.
The maximum value of a vertical spread at expiration is the amount between exercise prices if both options are inthemoney, or zero if both options are outofthemoney. A trader selling the June 100/105 call spread (buy the 105 call, sell the 100 call) will receive some amount between 0 and 5, say 2.25, and hope the market is below 100 at expiration.
If that happens, both calls will expire worthless, the spread will be worthless, and the trader will show a profit of 2.25. A trader who buys the March 75/85 put spread (buy the 85 put, sell the 75 put) will pay some amount between 0 and 10, say 6.50, and hope the market is below 75 at expiration. If that happens, the spread will be worth 10, and the trader will show a profit of 3.50.
Note that a bear vertical call spread will result in a credit: the trader sells the spread. A bear vertical put spread will result in a debit: the trader buys the spread. The expiration value of a bear vertical spread is shown in the following graph:
Determining the Total Delta of Vertical Spreads
An option trader considering a vertical spread, or any directional position, will have to decide just how bullish or bearish they are. Are they confident and willing to take a substantial directional position? Or are they less confident and ready to take only a limited position? Two factors determine the total directional characteristics of a vertical spread:
 The delta of the specific vertical spread
 The size in which the spread is executed
The delta value of a vertical spread is determined by numerous factors: time to expiration, volatility, and distance between exercise prices. Since a trader will be required to choose an expiration date that covers the period of expected directional movement, and since a trader will always make their best estimate of volatility over a given period, in practice, the delta will be a function of the exercise prices they choose.
The greater the distance between exercise prices, the greater the delta value associated with the spread. A 95/110 bull spread will be more bullish than a 100/110 bull spread, which will, in turn, be more bullish than a 100/105 bull spread. This can be seen in the following graph, which shows the expiration value of several different vertical spreads:
Given their risk tolerance, once a trader has decided which spread they would like to execute, they will determine how many times they are willing to do the spread. For example, a trader who wants to take a position that is 1,000 deltas long (equivalent to purchasing ten underlying contracts) can either choose a vertical spread that is 50 deltas long and execute it 20 times or choose a vertical spread that is 25 deltas long and execute it 40 times. Both positions will leave him long 1,000 deltas.
The Importance of Volatility in Vertical Spreads
While the expected directional movement in the underlying market is essential in choosing a vertical spread, it is not the only consideration. Every option strategy also has a volatility component, which is true of vertical spreads.
The volatility principle determines the choice of vertical spread. If we consider three options, an inthemoney, atthemoney, and outofthemoney option, which are identical except for their exercise prices, the atthemoney option is always the most sensitive in total points to a change in volatility.
This means that when all options appear overpriced because a trader believes implied volatility is too high, the atthemoney option will be the most overpriced in total points. When all options appear underpriced because a trader believes implied volatility is too low, the atthemoney option will be the most underpriced in total points. This characteristic leads to an elementary rule for choosing bull and bear vertical spreads:
If implied volatility is too low, vertical spreads should focus on purchasing the atthemoney option. If implied volatility is too high, vertical spreads should concentrate on selling the atthemoney options.
This rule applies whether a trader considers call vertical spreads or put vertical spreads. For example, with the underlying contract at 100, a bullish trader might consider buying the 95/100 call spread (buy the 95 call, sell the 100 call), or he might consider purchasing the 100/105 call spread (buy the 100 call, sell the 105 call).
If implied volatility appears too high, the trader will prefer the 95/100 spread because they will want to sell the 100 (atthemoney) call. If implied volatility seems too low, the trader will prefer the 100/105 spread because they ill want to buy the 100 (atthemoney) call.
In practice, it is unlikely that one option will be precisely atthemoney. In such a case, it is usually best to focus on the option that is closest to atthemoney, buying that option when implied volatility appears low and selling it when implied volatility seems high.
Summary
 A vertical spread always consists of one long option and one short option, where both options are of the same type and expire at the same time.
 In a vertical spread, whenever a trader buys the lower exercise price and sells the higher exercise price, the position is bullish. Whenever a trader buys the higher exercise price and sells the lower exercise price, the position is bearish.
 The further apart the exercise prices in a vertical spread, the greater its directional characteristics.
 Vertical spreads should focus on the atthemoney option, selling the atthemoney option when implied volatility is too high and buying the atthemoney option when implied volatility is too low.
Chapter 19
Option Arbitrage
Option Arbitrage
Arbitrage
The purchase and sale of the same product in different markets to take advantage of a price disparity between the two markets.
In traditional arbitrage, a trader will try to buy and sell the same instrument in different markets to profit from a perceived price differential. If the trader can indeed buy the instrument at a lower price in one market and sell the same instrument at a higher price in the other market, they will, in effect, profit by the amount of the price difference.
And because they can simultaneously take delivery and make delivery of the same instrument, there is minimal risk associated with such a strategy. This type of arbitrage can also be accomplished with options by purchasing and selling the same contract in different markets. A trader can profit from such a strategy where there is a discrepancy in the contract prices in the two other markets.
Synthetic Equivalents
A critical characteristic of options is that they can be combined with other options or with an underlying contract to create positions with characteristics almost identical to some other contract or combination of contracts. This type of replication leads to a new category of trading strategies unique to the option market.
Synthetic Stock
Suppose a trader is long a June 100 call and short a June 100 put where all options are European (no early exercise permitted). What will happen to this position at expiration? One cannot answer the question without knowing where the underlying contract will be at expiration. Surprisingly, the price of the underlying contract does not affect the outcome.
If the underlying contract exceeds 100, the put will expire worthless, but the trader will exercise the 100 call, buying the underlying contract at 100. Conversely, if the underlying contract is below 100, the call will expire worthless, but the trader will be assigned to the 100 put. This also results in their buying the underlying contract at 100.
Ignoring for the moment the unique case when the underlying contract is right at 100 at June expiration, the above position will always result in the trader going long the underlying instrument at the exercise price of 100.
The trader will go long, either by choice (the underlying contract is above 100, and they exercise the 100 call) or by force (the underlying contract is below 100 and is assigned on the 100 put). We refer to this position as a synthetic long underlying. The position has the same characteristics as the underlying contract but won't become an underlying contract until expiration.
Synthetic Long Underlying
A long call and a short put, where both options have the same underlying contract, expiration date, and exercise price.
Synthetic Short Underlying
A short call and a long put, where both options have the same underlying contract, the same expiration date, and the same exercise price.
If the trader takes the opposite position, selling a June 100 call and purchasing a June 100 put, they have a synthetic short underlying. This position will always result in the trader selling the underlying contract at the exercise price of 100, either by choice (the underlying contract is below 100, and he exercises the 100 put) or by force (the underlying contract is above 100, and they are assigned on the 100 call).
We can express the preceding relationships as follows:
Formula: synthetic long underlying
synthetic long underlying » long call + short put
Formula: synthetic short underlying
synthetic short underlying » short call + long put
A synthetic position acts very much like its real equivalent. For each point the underlying instrument rises, a synthetic long position will gain approximately one point, and a synthetic short position will lose approximately one point. This leads us to conclude, correctly, that the delta of a synthetic underlying position is approximately 100.
If the delta of the June 100 call is 75, the delta of the June 100 put will be approximately 25. If the delta of the June 100 put is 60, the delta of the June 100 call will be around 40. Ignoring the positive sign associated with a call delta and the negative sign associated with a put delta, the deltas of calls and puts with the same underlying contract, expiration date, and exercise price will always add up to approximately 100.
As we will see, interest rates and the possibility of early exercise can cause the delta of a synthetic underlying position to be slightly more or less than 100. But traders look at a synthetic underlying position and mentally assign it a delta of 100 for most practical calculations.
The result of combining a call and put into a synthetic underlying position is shown in the following graphs:
Synthetic Calls
Synthetic call
A long (short) underlying position together with a long (short) put.
As we have just seen, the purchase of a call and sale of a put at the same exercise price and with the same expiration date is equivalent to a long underlying position:
Long Underlying Formula
long underlying = long call + short put
By rearranging the components of a synthetic underlying position, we can also create synthetic call positions:
Synthetic Long Call Formula
synthetic long call = long an underlying contract + long put
Synthetic Short Call Formula
synthetic short call = short an underlying contract + short put
Each synthetic call acts like its real equivalent. The synthetic delta is approximately equal to the delta of the actual call, so the synthetic call will gain or lose value at roughly the same rate as its real equivalent.
If the underlying instrument in our examples is stock, typical synthetic long and short call positions might be:
Long June 100 Call = Long Stock + Long June 100 Put 
Short June 100 Call = Short Stock + Short June 100 Put 
In fact, we can use any exercise price and expiration date to create a synthetic call position as long as the options expire at the same time and have the same exercise price.
The result of combining an underlying position and put into a synthetic call is shown in the following graphs:
Synthetic Puts
Each synthetic put acts almost exactly like its real equivalent. The delta of the synthetic is approximately equal to the delta of the actual put, so that the synthetic put will gain or lose value at approximately the same rate as its real equivalent. Typical synthetic long and short put positions might be:
long June 100 put » short stock + long June 100 call 
short June 100 put » long stock + short June 100 call 
Again, we can use any exercise price and expiration date to create a synthetic put position, as long as the options expire at the same time and have the same exercise price. The result of combining an underlying position and call into a synthetic put is shown in the following graphs:
Summary
 Buying a call and selling a put at the same exercise price and with the same expiration date is equivalent to buying the underlying contract.
 Selling a call and buying a put at the same exercise price and with the same expiration date is equivalent to selling the underlying contract.
 Buying the underlying contract and buying a put is equivalent to buying a call.
 Selling the underlying contract and selling a put is equivalent to selling a call.
 Selling the underlying contract and buying a call is equivalent to buying a put.
 Buying the underlying contract and selling a call is equivalent to selling a put.