Rival Systems Options Knowledge Base
Intermediate Guide to Options
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Chapter 02
Theoretical Pricing Models
Chapter 03
Applying Probability to Option Pricing
Chapter 04
The Black-Scholes Model
The Black-Scholes Model
In its most basic form, the Black-Scholes 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 Black-Scholes 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 risk-free 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 model-generated 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 Black-Scholes 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 risk-free interest rate, the rate that applies to the most credit-worthy borrower.
In the U.S., the government is usually considered the most credit-worthy 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 60-day option, one can use the yield on a 60-day treasury bill; for a 180-day option, one can use the yield on a 180-day 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 Black-Scholes 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 ex-dividend 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 in-the-money 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.
Ex-dividend
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 ex-dividend date is expected just prior to expiration, there is the danger that a delay of several days will cause the ex-dividend 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 ex-dividend date.
Summary
- The Black-Scholes model is designed to evaluate European options on stock.
- The Black-Scholes 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 Black-Scholes 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 Black-Scholes model are the Black model, designed to evaluate European options on futures, and the Garman-Kohlhagen 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 low-volatility markets; markets that move quickly are high-volatility 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 Black-Scholes model, assume that prices follow a random walk-through 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 bell-shaped, 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 in-the-money.
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 in-the-money. 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 bell-shaped 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 Black-Scholes Model is that the underlying market is arbitrage-free. 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 break-even 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 break-even 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 break-even 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 one-year 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 one-year 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 one-year 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 rate-of-return.
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 Black-Scholes 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 Black-Scholes 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 Black-Scholes model assumes that the mean of the underlying price distribution is located at the forward, or break-even, 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 Black-Scholes model.
Chapter 06
Estimating Volatility
Chapter 07
Types of Volatility
Chapter 08
Using an Option's Theoretical Value
Chapter 09
The End Result
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 futures-type 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 at-the-money options tend to have the most significant time value. As an option moves away from at-the-money, moving either into-the-money or out-of-the-money, its time value decreases. If an option goes deeply in-the-money or far out-of-the-money, 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 at-the-money option loses the most value as time to expiration grows shorter. This results from the at-the-money 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 at-the-money options than either in- or out-of-the-money 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 stock-type 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 in-the-money options than at- or out-of-the-money options and a more substantial impact on long-term options than short-term 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 in-the-money options more than at- or out-of-the-money options. Multiple dividend payouts, likely to accrue over longer periods, will cause long-term options to be more sensitive to dividend changes than short-term 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 stock-type 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 futures-type 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
Chapter 12
Volatility Spreads
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, intermediate-level 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 intermediate-level 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 delta-neutral.
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 delta-neutral.
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 delta-neutral.
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 delta-neutral, 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 one-to-one (one call for each put). While most straddles are executed with a one-to-one 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 ever-increasing 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 one-to-one 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 delta-neutral, they can:
sell 2 October 50 calls
sell 3 October 40 puts
Typically, strangles are assumed by most traders to consist of out-of-the-money options. However, it is possible, although less common, to buy or sell a strangle consisting of in-the-money 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 in-the-money strangle is known in some markets as a guts.
Guts
A strangle where both the call
and the put are in-the-money.
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 two-sided 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), call-and-put 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 long-term option is purchased and the short-term option is sold, a trader is long the calendar spread; when the short-term option is bought and the long-term 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 short-term option expires but also on what will happen between that time and when the long-term option expires. Typical P&L diagrams for long and short calendar spreads at near-term 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 at-the-money, time spreads have two essential characteristics.
A long calendar spread wants the underlying market to sit still. A vital characteristic of an at-the-money option's theta (time decay) is the tendency to become increasingly significant as expiration approaches. As time passes, a short-term at-the-money option, having less time to expiration, will lose its value at a greater rate than a long-term at-the-money option.
If a trader purchases a calendar spread (buy the long-term option, sell the short-term option), they want the market to sit still because the short-term option will decay more quickly than the long-term 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 long-term options have a greater vega (volatility sensitivity) than short-term options.
This means that the theoretical value of a long-term option is always more sensitive in total points to a change in volatility than the theoretical value of a short-term 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 long-call calendar spread wants interest rates to rise; a long-put 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 long-term forward price will increase more. The long-term call will, therefore, go up more than the short-term call, causing the call calendar spread to rise in value. Because puts move in the opposite direction of the underlying contract, the long-term put will fall more than the short-term 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 |
| Short-term time to expiration (March) = 6 weeks |
| Long-term 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 long-call calendar spread wants dividends to fall; a long-put 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 short-term and long-term option, the long-term 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% |
| Short-term time to expiration (March) = 6 weeks |
| Long-term 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 one-to-one (one long-term option for each short-term 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 delta-neutral, 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 delta-neutral, 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 trade-off 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 higher-risk 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
Chapter 15
Adjustments
Chapter 16
Directional Strategies
Chapter 17
Bull and Bear Volatility Spreads
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 in-the-money, or close to zero if the options are far out-of-the-money.
The maximum value of a vertical spread at expiration is the amount between exercise prices if both options are in-the-money, or zero if both options are out-of-the-money. 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 in-the-money, or close to zero if the options are far out-of-the-money.
The maximum value of a vertical spread at expiration is the amount between exercise prices if both options are in-the-money, or zero if both options are out-of-the-money. 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 in-the-money, at-the-money, and out-of-the-money option, which are identical except for their exercise prices, the at-the-money 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 at-the-money option will be the most overpriced in total points. When all options appear underpriced because a trader believes implied volatility is too low, the at-the-money 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 at-the-money option. If implied volatility is too high, vertical spreads should concentrate on selling the at-the-money 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 (at-the-money) call. If implied volatility seems too low, the trader will prefer the 100/105 spread because they ill want to buy the 100 (at-the-money) call.
In practice, it is unlikely that one option will be precisely at-the-money. In such a case, it is usually best to focus on the option that is closest to at-the-money, 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 at-the-money option, selling the at-the-money option when implied volatility is too high and buying the at-the-money 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.