### Rival Systems Options Knowledge Base

# Advanced Guide to Options

- Select Your Level
- Beginner
- Intermediate
- Advanced

###### Chapters

- Advanced Options
- Options Sensitivities
- Position Analysis
- Option Arbitrage
- Arbitrage Strategies
- Arbitrage Risk
- Other Synthetic Applications
- Early Exercise
- The Effect of Early Exercise on Arbitrage Strategies
- Volatility Skews
- Models and the Real World
- Using a Volatility Skew
- Changing Skew Shapes
- How a Volatility Skew Affects Option Values
- Skew Trading Strategies

## Advanced Options

Advanced Options is the final installment in our option's training series. Before beginning, you should know the concepts covered in our Beginning Options and Intermediate Options courses. This course is designed to prepare a new options trader for the real world of options trading by presenting those concepts and developing skills necessary for successful trading.

While an in-depth knowledge of option theory is not a prerequisite for successful option trading, experience has shown that most successful option traders have at least a basic understanding of option theory. Most chapters, therefore, include some aspects of option theory, together with a discussion of how this theory relates to the real world.

However, we have attempted to keep theoretical discussions to a minimum, and no advanced mathematics is required. This is not meant to discourage traders interested in option theory and comfortable with higher mathematics from seeking a more detailed discussion of option theory from any of the excellent texts available. The emphasis in this program, however, is on the practical aspects of option trading.

The material is presented as a series of chapters, with each chapter covering a significant aspect of option evaluation or trading. Chapters are broken down into sections and topics.

Since each chapter builds on the material in previous chapters, we strongly urge new traders to resist the temptation to skip to a new chapter without fully mastering previous chapters' material. By covering the material in an orderly fashion, new traders will find that each chapter becomes the foundation for each subsequent chapter and, consequently, a solid foundation to build a successful trading career.

While a new trader may initially find some of the material difficult, no one needs to become discouraged. Like any new undertaking, options require a willingness to learn and practice new or unfamiliar concepts. By moving through the material at a comfortable pace, doing the exercises, and then reviewing those areas that have presented the most significant difficulty, we feel confident that anyone can master the material in this course.

##### 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

## Option Sensitivities

### Option Sensitivities

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. But no intelligent trader can afford to ignore the possibility of error.

If the trader is wrong and market conditions change in a way that adversely affects their position, how badly might the trader get hurt? A trader who fails to consider the risks of their work will 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, can be found here.

### How Changes in Market Conditions Affect the Risk Sensitivities

Not only do changing market conditions affect the theoretical value of options, but changing conditions also affect risk sensitivities. This means that the risk to a position under one set of market conditions may differ from the risk to the same position under a different set of market conditions. Because market conditions are constantly changing, the risk to a position is continually changing.

An essential part of an options trader's education is learning how risks change as market conditions change. In this section, we will look at the effect of changing market conditions on the risk sensitivities: the delta, gamma, theta, vega, and rho.

### How Changes in Market Conditions Affect the Delta

The following are some of the more essential characteristics of delta:

- Call delta values range from 0 for far out-of-the-money calls to 100 for deeply in-the-money calls. At-the-money calls have deltas of approximately 50.
- Put delta values range from 0 for far out-of-the-money puts to -100 for deeply in-the-money puts. At-the-money puts have deltas of approximately -50.
- As volatility rises, call deltas move towards 50, and put deltas move towards -50. At-the-money calls, with deltas of 50, remain unchanged.
- As time passes or volatility falls, call deltas move away from 50 and put deltas move away from -50. At-the-money puts, with deltas of -50, remain unchanged.

Note the effect of volatility and time. As volatility rises, all options act more and more as if they are at-the-money. As volatility falls or time passes, in-the-money options go more deeply into-the-money, and out-of-the-money options go further out-of-the-money.

This means a trader's delta position will change over time, even if all other market conditions remain the same. Also, while a trader may do their best to stay delta-neutral, they must realize that they can never be sure that they are, in fact, delta-neutral.

The correct delta depends on knowing the accurate volatility, but the correct volatility cannot be known with certainty since it will occur in the future.

### How Changes in Market Conditions Affect the Gamma

The following are some of the more critical characteristics of gamma:

- When all other contract specifications are the same, an at-the-money option will always have a greater gamma than an in- or out-of-the-money option. The option's gamma rises as the underlying market moves toward an option's exercise price (the option becomes more at-the-money).
- As volatility increases, the gamma of an at-the-money option falls, while the gamma of an in- or out-of-the-money option rises.
- As time passes, or as volatility declines, the gamma of an at-the-money option rises, while the gamma of an in- or out-of-the-money option falls. In a very low volatility market, or with very little time remaining to expiration, the gamma of an at-the-money option can increase dramatically.
- When all other contract specifications are the same, calls and puts with the same exercise price, have the same gamma.

### How Changes in Market Conditions Affect the Theta

While almost all options have negative thetas (they lose value as time passes), note that a very deeply in-the-money stock option put, which is European (no early exercise permitted), can sometimes take on a positive theta.

This means it will become more valuable over time, even if there is no movement in the underlying contract.

The following are some of the more essential characteristics of theta:

- Since at-the-money options have the most significant time value when all other contract specifications are the same, an at-the-money option will have a greater theta than an in- or out-of-the-money option. As the underlying market moves toward an option's exercise price (the option becomes more at-the-money), the option's theta rises.
- As volatility rises, the theta of an option will increase. A higher volatility means an option has a greater time value, so the option will decay more quickly when no movement occurs. As volatility falls, the theta of an option will fall. A lower volatility means an option has less time value, so the option will decay less quickly when no movement occurs.
- When all other contract specifications are the same, the theta of a short-term at-the-money option will always be greater than that of a long-term at-the-money option. Short-term at-the-money options Hi Hi Ki HI decay more quickly than long-term at-the-money options. As expiration approaches, the theta of an at-the-money option can become infinitely large.

This last point is not necessarily valid for in-the-money or out-of-the-money options. A short-term option that is deeply in-the-money or far out-of-the-money has almost no time value, so it has very little value to lose.

Hence, its theta will be very small. Because of this, under some conditions, a long-term in-the-money (out-of-the-money) option can have a greater theta than a short-term in-the-money (out-of-the-money) option, even if all other contract specifications are the same.

### How Changes in Market Conditions Affect the Vega

The following are some of the more important characteristics of vega:

- When all other contract specifications are the same, an at-the-money option will always have a greater vega than an in- or out-of-the-money option. As the underlying market moves toward an option's exercise price (the option becomes more at-the-money), the option's vega rises.
- As a percent of theoretical value, out-of-the-money options have a greater vega than in- or at-the-money options. In percent terms, out-of-the-money options are more sensitive to a change in volatility.
- When all other contract specifications are the same, a long-term option will always have a greater vega than a short-term option. An option's sensitivity to a change in volatility declines as time passes.
- As volatility rises, the vega of in- and out-of-the-money options will generally increase. As volatility falls, the vega of in- and out-of-the-money options will generally fall. The vega of at-the-money options is relatively constant concerning changes in volatility.
- When all other contract specifications are the same, calls and puts with the same exercise price, have the same vega.

### How Changes in Market Conditions Affect the Rho

The following are some of the more important characteristics of rho:

- When all other contract specifications are the same, calls with lower exercise prices have greater rhos than calls with higher exercise prices, and puts with higher exercise prices have greater rhos than puts with lower exercise prices.
- When all other contract specifications are the same, long-term options have greater rhos than short-term options.

Check out Rival One Options Functionality

### Implied Sensitivities

In theory, a trader who uses a theoretical pricing model must feed into the model their best estimate of the future volatility over the option's life. From this number, they will derive a theoretical value that can be used to identify mispriced options in the marketplace.

At the same time, they will also derive the theoretical sensitivities of the option - the delta, gamma, theta, vega, and rho - which can be used to assess the risk of taking a position in the option.

Of course, the trader's estimate of future volatility is only an educated guess, and they will often want to consider the marketplace's opinion about volatility. Suppose they use the prices of options to determine the implied volatility and replace their volatility input with the implied volatility. In that case, this will change the theoretical value of each option (making it equal to the option's price) and the various risk sensitivities.

The risk sensitivities that result from using the implied volatility are called the implied sensitivities: the implied delta, implied gamma, implied theta, implied vega, and implied rho.

##### Implied sensitivities

The delta, gamma, theta, vega, and rho are generated when the implied volatility is used as the volatility input in a theoretical pricing model.

Of the implied sensitivities, the most commonly used are the implied delta and the implied theta. The implied delta is vital because a trader attempting to remain delta-neutral will often consider the implied volatility when deciding how to hedge a position.

If volatility is increased, the delta values of options move towards 50 (-50 for puts); if volatility is decreased, delta values move away from 50 (-50 for puts). Suppose the implied volatility in the marketplace is higher than the trader's volatility estimate. In that case, the implied deltas of both in-the-money and out-of-the-money options will be closer to 50 than their theoretical delta values.

Suppose the implied volatility in the marketplace is lower than the trader's volatility estimate. In that case, the implied deltas of in-the-money options will be closer to 100, and those of out-of-the-money options will be closer to zero than their theoretical delta values. A trader must decide which values to hedge the option position. Some traders use the theoretical delta, some use the implied delta, and some use a combination.

Also of interest to many traders is the implied theta. A trader will know from the theoretical theta how fast an option's theoretical value will decay. However, the theoretical value may differ significantly from the option's price in the marketplace. Because the value of a trader's account depends on the prices of options in the market, traders are usually more interested in the rate of decay in an option's price than in its theoretical value.

Since the price of an option determines the implied volatility, a trader can replace their volatility estimate with the implied volatility to determine the implied theta. This theta value will enable the trader to determine the decay rate in an option's price as time passes with no movement in the price of the underlying contract.

### Summary

Option sensitivities, like theoretical values, also change as market conditions change. The most important of these changes are:

- As the price of the underlying contract rises, call deltas move towards 100 and put deltas move toward zero. As the price of the underlying contract falls, call deltas move towards zero and put deltas move toward -100.
- As volatility rises, call deltas move toward 50 and put deltas move toward - 50. As volatility falls, or as time passes, call deltas move away from 50 and put deltas move away from -50.
- At-the-money options have greater gamma, theta and vega values than either in-the-money or out-of-the-money options.
- The gamma and theta of an at-the-money option rises as the option approaches expiration.
- Long-term options have greater vega values than short-term options.
- When the implied volatility is used to calculate an option's sensitivities, the result is the option's implied sensitivity.

##### Chapter 03

## Position Analysis

### Position Analysis

With even limited trading experience, a trader will quickly become familiar with the risks associated with holding a small option position. But when a trader begins to build up more complex options positions, with many options at different exercise prices and expiration dates, the risk to the position may not be readily apparent.

If a trader expects to survive and prosper in an option market, they must be able to analyze the risks of a position, no matter how complex it may be. Otherwise, they will not be prepared to take protective action when market conditions turn against them. Nor will they be able to maximize their profits when market conditions work in their favor.

##### Position

The sum total of a trader's open contracts in a particular underlying market.

### Determining the Total Risk

While the delta, gamma, theta, vega, and rho help analyze the sensitivity of a single option to changes in market conditions, these numbers are particularly valuable in analyzing more complex positions. Knowing the total delta, gamma, theta, and vega of an option position can help a trader determine how the position will likely react to changing market conditions beforehand.

Since all the option sensitivities are additive, the total sensitivity of a position can be calculated by multiplying each option's sensitivity by the number of contracts traded (using a plus sign for a purchase and a minus sign for a sale) and then adding up the resulting numbers. The following examples demonstrate this procedure.

A trader who has purchased ten calls with a delta of 35 each and sold seven puts with a delta of -65 each has a total delta position of:

(10 x 35) + (-7 x -65) = 350 + 455 = 805

A trader who has purchased five options with a gamma of 2.5 each and sold two options with a gamma of 4.0 each has a total gamma position of:

(5 x 2.5) + (-2 x 4.0) = 12.5 - 8.0 = 4.5

A trader who has purchased nine options with a theta of -.05 each and sold four options with a theta of -.08 each their total theta position is:

(9 x -.05) + (-4 x -.08) = -.45 +.32 = -.13

A trader who has sold three options with a vega of .15 each and sold five options with a vega of .25 each has a total vega position of:

(-3 x .15) + (-5 x .25) = -.45 -1.25 = -1.70

It is sometimes complicated for new traders to remember which sign (positive or negative) is associated with which type of position in options or the underlying contract. The signs for the delta, gamma, theta, and vega can be summarized as follows:

Position | Delta | Gamma | Theta | Vega |
---|---|---|---|---|

long underlying | positive | zero | zero | zero |

short underlying | negative | zero | zero | zero |

long calls | positive | positive | negative | positive |

short calls | negative | negative | positive | negative |

long puts | negative | positive | negative | positive |

short puts | positive | negative | positive | negative |

While the rho for an underlying position is zero regardless of the market, interest rates affect option positions differently, depending on the underlying contract and settlement procedure. We must, therefore, summarize the rho separately:

Market | Position | Rho |
---|---|---|

Stock Market | long calls | positive |

short calls | negative | |

long puts | negative | |

short puts | positive | |

Futures market (futures-type settlement) | long or short calls | zero |

long or short puts | zero | |

Futures market (stock-type settlement) | long calls | negative |

short calls | positive | |

long puts | negative | |

short puts | positive |

Even though a trader can analyze the effect of changing market conditions on their position through its delta, gamma, theta, and vega characteristics, their first and primary concern is that the position will be profitable if their assumptions about market conditions are correct. This means that the position ought to have a positive theoretical edge.

The theoretical edge can be calculated similarly to the total delta, gamma, theta, and vega. One need only multiply the theoretical edge of each option (the difference between its trade or settlement price and its theoretical value) by the number of contracts traded and add up all the contracts involved. The amount of positive or negative theoretical edge reflects the position's potential profit or loss.

For example, a trader who pays 1.75 for three options with a theoretical value of 2.25 each and receives 3.50 for four options with a theoretical value of 3.75 each has a total theoretical edge of:

[3 x (2.25 - 1.75)] + [4 x (3.50 - 3.75)] = (3 x .50) + (4 x - .25) = 1.50 -1.00 = .50

Check out Rival One Options Functionality

### Interpreting the Total Risk

Each option sensitivity has a specific interpretation reflecting the changes in market conditions, which will either help or hurt a trader's position. The variations can be summarized as follows:

Since graphs often evaluate option positions, we can interpret several option sensitivities graphically. Using a typical grid where the underlying price movement is measured along the X-axis and the profit or loss is measured along the Y-axis, the interpretation of the sensitivities can be defined as follows:

Delta - A positive delta indicates that the graph is sloping from the lower left to the upper right at the current underlying price. As the underlying contract moves up, the position becomes more valuable. A negative delta indicates that the graph is sloping from the upper left to the lower right at the current underlying price. As the underlying contract moves up, the position becomes less valuable. This is shown below.

Gamma: A positive gamma indicates that at the current underlying price, the general shape of the graph is concave (a smile). As the underlying moves up or down, the position becomes more valuable. A negative gamma indicates that at the current underlying price, the general shape of the graph is convex (a frown). As the underlying moves up or down, the position becomes less valuable. This is shown below.

Theta: A positive theta indicates that the graph will shift upward. As time passes, the position will become more valuable. A negative theta suggests that the chart will move downward over time. As time passes, the position will become less valuable. Since theta and gamma are always of opposite signs, a positive theta position will take on the shape of a frown (a negative gamma), and a negative theta position will take on the shape of a smile (a positive gamma). This is shown below.

Vega: A positive vega indicates that a higher volatility will shift the graph upward, and a lower volatility will shift the graph downward. A negative vega suggests that a lower volatility will shift the graph upward, and a higher volatility will shift the graph downward. This is shown below.

### The Total Contract Position

While the option sensitivities can be useful to a trader in determining the risk to an option position, these numbers are only reliable under relatively narrow market conditions. When conditions change dramatically, the delta, gamma, theta, vega, and rho are unlikely reliable risk determinants. Experience has shown that market conditions occasionally change so drastically that the changes are inconsistent with the assumptions built into most pricing models.

For example, in theory, a move of five or more standard deviations will occur less than one time in a million. Yet most traders know that such movements occur more often in the real world, as witnessed in the stock market crash of 1987. When such moves occur, the option sensitivities are no longer practical in analyzing a position.

Suppose a trader is concerned with the effect on their position of an unusually large move. In that case, they may attempt to analyze their position by asking the following questions: If the market makes such a significant upward move that all the calls trade at intrinsic value and all the puts collapse to zero, what am I left with? What am I left with if the market makes such a significant downward move that all the puts trade at intrinsic value and all the calls collapse to zero?

If the market makes such a significant upward move that all the calls trade at intrinsic value and all the puts collapse to zero, then the puts essentially disappear while the calls act like long underlying contracts. The trader can, therefore, add their call position to any position they may have in the underlying contract to get their upside contract position.

Suppose the market makes such a significant downward move that all the puts trade at intrinsic value, and all the calls collapse to zero. In that case, the calls disappear while the puts act like short underlying contracts. The trader can, therefore, add their put position to any position in the underlying contract to get their downside contract position.

For example, suppose a trader's net position consists of:

- Long 30 calls
- Short 25 puts
- Short 20 underlying contracts

Regardless of their current delta, gamma, theta, vega, and rho, if the market makes a violent upward move, such that all the calls act like underlying contracts and all the puts collapse to zero, the trader is left with a position which is equivalent to being long 10 underlying contracts: long 30 underlying contracts in the form of long 30 calls, no put position, and short 20 actual underlying contracts. Therefore, the trader's upside contract position is long 10, and their potential reward on the upside is unlimited.

In the same way, regardless of their current delta, gamma, theta, vega, and rho, if the market makes a violent downward move, such that all the puts act like underlying contracts and all the calls collapse to zero, the trader is left with a position which is equivalent to being long 5 underlying contracts:

- No call position
- Long 25 underlying contracts in the form of short 25 puts
- Short 20 actual underlying contracts

Therefore, the trader's downside contract position is long 5, and their potential risk on the downside is unlimited.

### Summary

- All option sensitivities are additive. In a complex position, the total sensitivity (delta, gamma, theta, vega or rho) of the position is the sum of the individual sensitivities.
- The total sensitivities of a position determine what changes in market conditions will help or hurt the position.
- The total contract position is the theoretical number of underlying contracts resulting on the upside if all calls go deeply into-the-money and all puts collapse to zero, or on the downside if all puts go deeply into-the-money and all calls collapse to zero.

##### Chapter 04

## 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 different markets.

### Synthetic Pricing

Traders use synthetic pricing to compare the advantages of a position in an actual contract or its synthetic counterpart. The Put-Call Parity concept helps them assess the best strategy by comparing call and put option prices.

Based on the difference between call and put option prices with the same exercise price and expiration date, the synthetic market plays a pivotal role in their evaluation. Interest and dividend considerations further influence the choice between real and synthetic positions. Put-Call Parity provides a framework for understanding these relationships, making it an essential tool for financial decision-making.

### Put-Call Parity

Suppose an underlying contract is trading at 102, and we want to take a long position in the market. We can go into the underlying market and buy the contract at 102. But we now have an additional choice. We can take a long synthetic position by purchasing a June call and selling a June put at the same exercise price.

Which of these strategies is best? As with any option strategy, the decision depends on the prices of the options in the marketplace. Suppose the June 100 call is trading at 5 and the June 100 put is trading at 3. If we buy the June 100 call for 5 and sell the June 100 put for 3, we will show a debit of 2. If, at expiration, the underlying contract is at 110, we will show a credit of 10 when we exercise our June 100 call for a total profit of 8. If we ignore interest considerations, this is identical to the profit we would have realized had we bought the underlying contract at 102 instead.

Suppose the underlying contract is still trading at 102, but now the June 100 call is trading at 4.75, and the June 100 put is trading at 3.25. If we take a synthetic long underlying position by purchasing the June 100 call and selling the June 100 put, we will show a debit of 1.50. If the underlying contract is 110 at expiration, we will show a profit of 8.50 (the debit of 1.50 from the options trades, plus the 10 credits when we exercise the 100 call). This is .50 better than we would do by taking a long position in the underlying contract at 102.

##### Synthetic market

In an option market, the bid-ask spread on a synthetic underlying contract. Since a synthetic underlying contract is either a long call and short put (a long synthetic underlying contract) or a short call and long put (a short synthetic underlying), the synthetic market is the difference between a call and put price where both options have the same exercise price and expiration date.

As long as the price of the June 100 call is exactly two points greater than that of the June 100 put, the profit or loss resulting from a synthetic position will be identical to an actual position taken in the underlying contract at 102. The difference between the call and the put price is often called the synthetic market. In the absence of any interest or dividend considerations, the value of the synthetic market can be expressed as:

Formula: Synthetic market value

call price - put price = underlying price - exercise price

If this equality holds, there is no difference between taking a position in the underlying market or an equivalent synthetic position in the options market. With the June 100 call at 5 and the June 100 put at 3, we can write:

5 - 3 = 102 - 100 |

2 = 2 |

There is no difference between the synthetic and its real equivalent. With the June 100 call at 4.75 and the June 100 put at 3.25, we can see that:

4.75 - 3.25 ¹ 102 - 100 |

1.50 ¹ 2 |

Here, the synthetic side is cheaper, and we, therefore, prefer to take a long underlying position synthetically by purchasing the call and selling the put.

Suppose the price of the June 100 call is still 5, but the price of the June 100 put is 2.75, with the underlying contract still trading at 102. We have:

5 - 2.75 ¹ 102 - 100 |

2.25 ¹ 2 |

The real side is cheaper, so we prefer to take a long position by purchasing the underlying contract. On the other hand, if we want to take a short underlying position, we will likely sell the underlying contract synthetically by selling the call and purchasing the put. If we do this, we are selling a contract, which is only worth 2, for 2.

The three-sided relationship between a call, a put, and its underlying contract means that we can always express the value of any one of these contracts in terms of the other two:

##### Formula: Underlying Price

underlying price = call price - put price + exercise price

##### Formula: Call Price

call price = underlying price + put price - exercise price

##### Formula: Put Price

put price = call price - underlying price + exercise price

This three-sided relationship is sometimes referred to as put-call parity. Traders often refer to a trade that involves purchasing and selling a put or selling a call and buying a put as a combination or combo.

##### Put-call parity

In an arbitrage-free market, the relationship between the price of the underlying contract and a call and a put with the same exercise price and expiration date.

##### Combination or Combo

A two-sided option spread which does not fall into any well-defined category of spreads. Most commonly it is used to refer to a long call and short put, or short call and long put, which together make up a synthetic position in the underlying contract.

Check out Rival One Options Functionality

### Interest Considerations

Consider an underlying contract trading at 103, with a June 100 call trading at 5 and a June 100 put trading at 2. If there are no interest or dividend considerations, such as in a futures market where options are subject to futures-type settlement, a trader who wants to buy the underlying contract will have no preference between actually buying the underlying contract for 103 or buying the underlying contract synthetically by purchasing the June 100 call and selling the June 100 put, since:

call price - put price = underlying price - exercise price |

5 - 2 = 103 - 100 |

3 = 3 |

But suppose interest rates are not zero. The value of the synthetic relationship will then depend on any cash flows that result from the synthetic trade. To determine the true value, we must consider a stock option market and a futures option market where the options are subject to stock-type settlement.

### Futures options markets

Suppose, in our example, the trades are made in a futures option market where the options are subject to stock-type settlement. If the trader buys the futures contract synthetically (buy the call, sell the put), they will incur a debit of 3.00.

They cannot offset this debit until the June expiration when they can buy the futures contract at 100. If interest rates are currently 8 percent per annum, and three months are remaining to June expiration, the cost of carrying this debit of 3.00, in terms of interest considerations, will be:

8% x 3/12 x 3.00 = 2% x 3.00 = .06

Since the purchase of a futures contract requires no outlay of cash and, therefore, entails no carrying costs (in theory, there is no loss of interest associated with a margin deposit), the trader will prefer to take a long position in the futures market by purchasing the actual futures contract rather than by buying a synthetic futures contract.

For the trader to have no preference, the cost of the synthetic would have to be .06 cheaper. Because interest rates in the real world are not zero, we must adjust the put-call parity relationship slightly when futures options are subject to stock-type settlement. Initially, we wrote:

##### Formula: Original put-call parity

call price - put price = underlying - exercise price

Taking into consideration carrying costs, we can now express the relationship as follows:

##### Formula: Put-call parity w/ carry costs

call price - put price = underlying price - exercise price + carrying costs

where carrying costs are the costs of financing the difference between the call price and the put price to expiration.

In our example, the difference between the call and put price was 3.00. Note that this was equal to the difference between the futures and exercise prices. Since the exercise price is usually a whole number, when calculating synthetic equivalents, most traders approximate the carrying costs on a synthetic position by calculating the carrying costs on the difference between the futures price and the exercise price. In our example, the arithmetic for the synthetic must be:

call price - put price = futures price - exercise price - carrying costs = 103 - 100 - .06 = 2.94

With the underlying futures contract trading at 103.00, for a trader to have no preference between a position in the futures contract and a synthetic position, the difference between the price of the June 100 call and the June 100 put must be 2.94. If the call is trading at 5.00, the put must be trading at 2.06; if the put is trading at 2.00, the call must be trading at 4.94. In a futures market where options are subject to stock-type settlement, the difference between the call and put price must be reduced by the carrying cost on this difference.

### Stock options markets

Suppose that our original trades were made in a stock option market. If the trader buys the stock, they will incur a debit of 103.00. If the trader buys the stock synthetically (buy the call, sell the put), they will incur a debit of only 3.00. In either case, they will have to finance this debit. However, the debit incurred by purchasing the stock is 100 points greater than the debit incurred from the synthetic position. If interest rates are 8 percent per annum, and three months are remaining to expiration, the cost of buying the stock will exceed by 2 points the cost of purchasing the synthetic since:

8% x 3/12 x 100.00 = 2% x 100.00 = 2.00

In such a case, the trader will strongly prefer taking a long position in the stock by purchasing the synthetic rather than the stock. For the trader to have no preference, the cost of the synthetic would have to be 2 points greater.

With the underlying stock trading at 103.00, for a trader to have no preference between a position in the stock and a synthetic position, the difference between the price of the 100 call and the 100 put must be 5.00. If the call is trading at 7.00, the put must be trading at 2.00; if the put is trading at 4.00, the call must be trading at 9.00.

Taking into consideration carrying costs, we can express the relationship in the stock market as follows:

call price - put price = stock price - exercise price + carrying costs

The carrying costs are the costs of financing the difference incurred by purchasing the stock and the debit incurred by buying the synthetic equivalent. This difference is equal to the exercise price. In our example, the carrying costs were the amount required to finance a debit of 100, which was the same as the exercise price of the synthetic equivalent. The arithmetic for the synthetic must be:

call price - put price = stock price - exercise price + carrying costs = 103 - 100 + 2.00 = 5.00

If the synthetic equivalent (the difference between the call price and put price) equals 5, a trader will have no preference between buying the stock and the

### Dividend Considerations

Taking into consideration interest rates, we saw that a trader would have no preference for an outright position in the underlying stock or a synthetic position if:

call price - put price = stock price - exercise price + carrying costs

If the underlying stock is trading for 103, the June 100 call is trading for 8, the June 100 put is trading for 3, and carrying costs are 2, an outright long position in the stock and a synthetic long position are equivalent since:

8 - 3 = 103 - 100 + 2 |

5 = 5 |

Suppose that before expiration, the stock will pay a dividend of 1.50. If a trader takes a long synthetic position in the stock (buy the call, sell the put), they may have a position equivalent to a long stock position but won't own the stock. So they won't receive the dividend. Because of this, a trader will do 1.50 better by purchasing the stock rather than the synthetic.

If the stock is expected to pay a dividend over the life of the options, we must adjust the put-call parity relationship slightly. Taking into consideration dividends, we can now express the relationship as:

##### Formula: Put-call parity w/ dividends

call price - put price = stock price - exercise price + carrying costs - dividends

The dividend amount is the expected dividend payout over the life of the options. If a dividend of 1.50 is expected, the arithmetic for the synthetic must be:

stock price - exercise price + carrying costs - dividends

103 - 100 + 2 - 1.50 = 3.50

Suppose the synthetic equivalent (the difference between the call and put prices) equals 3.50. In that case, a trader will have no preference for buying the stock for 103 or buying the stock synthetically by purchasing the June 100 call and selling the June 100 put.

### Summary

- If all options are European (no early exercise), a synthetic position is fairly priced if:
- call price - put price = underlying price - exercise price + carrying costs on the exercise price - expected dividends

- If the call price minus the put price is less than the given relationship, a trader will prefer to take any long position in the stock by purchasing the stock synthetically (purchase the call, sell the put).
- If the call price minus the put price is greater than the given relationship, a trader will prefer to take any short position in the stock by selling the stock synthetically (sell the call, purchase the put).

##### Chapter 05

## Arbitrage Strategies

Since a synthetic position acts essentially the same as its real equivalent, we could create an arbitrage by purchasing a contract in one market and selling it synthetically in a different market. This will be profitable if the prices of the real underlying contract and the synthetic equivalent are different.

### Conversions and Reversals

If we take a synthetic long or short underlying position, our primary concern, as with an actual underlying position, is the market's direction. If the market moves in our favor, we expect to show a profit; if it moves against us, we expect a loss. If we execute the synthetic at favorable prices, we may gain more or lose less, but it is still primarily the direction of the market in which we are interested.

Suppose an underlying contract trades at 103.00 without interest rates or dividend considerations. The June 100 synthetic relationship (the difference between the price of the June 100 call and the June 100 put) should be:

underlying price - exercise price = 103.00 - 100 = 3.00

Suppose the June 100 call is trading for 5.25, and the June 100 put is trading for 2.00. The synthetic trades at 5.25 – 2.00 = 3.25, or .25 more than its expected price. The two instruments are essentially the same but trading for different prices. In such circumstances, a trader will try to execute an arbitrage by purchasing the instrument in the cheaper market and selling the instrument in the more expensive market.

In our example, the trader will attempt to buy the underlying contract at 103.00 and sell the synthetic at 3.25. At expiration, this strategy will yield a profit equal to the amount the synthetic was overpriced, in this case, .25.

##### Conversion

A long underlying position together with a short call and long put, where both options have the same exercise prices and expire at the same time.

This type of arbitrage, where a trader buys the underlying contract and synthetically sells the underlying contract (sell the call, buy the put), is known as a conversion:

##### Formula: Conversion

= long underlying + synthetic short underlying

= long underlying + short call + long put

Suppose we have the same market conditions as in the last example, but now the June 100 call is trading for 5.35, and the June 100 put is trading for 3.70. The synthetic is trading at 5.35 – 3.70 = 2.65, or .35 less than its expected price of 3.00.

Now, a trader will try to execute an arbitrage by purchasing the instrument in the synthetic market and selling the instrument in the underlying market. In our example, the trader will attempt to sell a futures contract at 103.00 and buy the synthetic for 2.65. At expiration, this strategy will yield a profit equal to the amount the synthetic was underpriced, in this case, .35.

##### Reversal

A short underlying position together with a long call and short put, where both options have the same exercise price and expire at the same time.

This type of arbitrage, where a trader sells the underlying contract and synthetically buys the underlying contract (buy the call, sell the put), is known as a reverse conversion, or more commonly, a reversal:

##### Formula: Reversal

= short underlying + synthetic long underlying

= short underlying + long call + short put

Because interest rates are not zero in the real world, and stocks also pay dividends, the exact value of the conversion or reversal will be determined by put-call-parity. It will depend on both the underlying market and settlement procedure.

##### Box

A long call option and short put option at one exercise price, together with a short call and long put at a different exercise price. All options must have the same underlying contract and expire at the same time.

A trader can combine a conversion and reversal at different exercise prices in the same expiration month. This results in a box.

##### Formula: Box

conversion (long stock, short call, long put) +

reversal (short stock, long call, short put)

For example, a trader might:

Buy a June 100 Call | Sell a June 110 Call |

Sell a June 100 Put | Buy a June 110 Put |

Sell Stock | Buy Stock |

The trader has executed a June 100 reversal and a June 110 conversion. However, the long and short stock cancel out, leaving:

Buy a June 100 Call | Sell a June 110 Put |

Sell a June 110 Call | Buy a June 110 Put |

The remaining position is a long synthetic stock position at one exercise price and a short one at a different exercise price, where all options expire simultaneously. In our example, at expiration, the trader will simultaneously buy stock at 100 (they exercise their 100 call or are assigned on the 100 put) and sell stock at 110 (they are assigned on the 110 call or exercise their 110 put).

The result will be a credit at expiration equal to the amount between exercise prices, in this case, 10. Since the position will be worth 10 at expiration, its value today should be 10 less the carrying cost to expiration on this amount. If options are subject to stock-type settlement, interest rates are 8%, and 3 months are remaining to June expiration, the current value of the box is:

10.00 - (3/12 x 8.00% x 10.00) = 10.00 - .20 = 9.80

Note that a box combines a call vertical spread and a put vertical spread. Our June 100/110 box is:

Buy Call Spread | Long June 100 Call / Short June 110 Call |

Bear Put Spread | Short June 100 Put / Long June 110 Put |

Since the call and put vertical spreads together make up a box, their values should add up to the value of a box. If the June 100/110 box is worth 9.80, and the June 100/110 call vertical spread is worth 3.50, then the June 100/110 put vertical spread must be worth 6.30 since 9.80 - 3.50 = 6.30.

Check out Rival One Options Functionality

### Jelly Rolls

##### Jelly roll or Roll

A long call and short put with one expiration date and a short call and long put with a different expiration date. All four options must have the same exercise price and underlying contract.

Instead of combining a conversion and reversal in the same expiration month but at different exercise prices to create a box, a trader might combine a conversion and reversal in different months at the same exercise price. This type of strategy is common in the stock options market and results in a jelly roll or roll:

##### Formula: Jelly roll

Conversion in One Month (long stock, short call, long put) + reversal in a different month (short stock, long call, short put)

For example, a trader might:

Buy a March 100 Call | Sell a June 100 Call |

Sell a March 100 Put | Buy a June 100 Put |

Sell Stock | Buy Stock |

The trader has executed a March 100 reversal and a June 100 conversion. However, the long and short stock cancel out, leaving:

Buy a March 100 Call | Sell a June 100 Call |

Sell a March 100 Put | Buy a June 100 Put |

The remaining position is a long synthetic stock in March and a short synthetic stock in June, where all options have the same exercise price. At the March expiration, the trader will buy stock at 100 (they exercise their March 100 call or are assigned on the March 100 put), and at the June expiration, the trader will sell stock at 100 (they are assigned on the June 100 call or exercises their June 100 put). The cost of this strategy should be the cost of carrying stock from March expiration to June expiration, less any dividends paid over this period. That is:

##### Formula: Jelly roll

Carrying Costs on the Exercise Price - Expected Dividends

In our example, the trader must carry a debit of 100 from March expiration to June expiration, a period of approximately three months. If interest rates are 8%, the carrying cost is:

100 x 3/12 x 8.00% = 2

If no dividends are expected between March and June, the value of the jelly roll is 2. If a dividend of .75 is expected between March and June, the trader who buys stock in March will receive the dividend. The value of the jelly roll then becomes the carrying cost of 2 less the expected dividend of .75, or 1.25.

Note that a jelly roll is a combination of long and short calendar spreads, one a call spread and one a put spread:

##### Formula: Jelly roll

= Long-Term Synthetic - Short-Term Synthetic

= (long-term call - long-term put) - (short-term call - short-term put)

= (long-term call - short-term call) - (long-term put - short-term put)

= carrying costs = expected dividends

The difference between the call and put time spread should equal the cost of carrying on the exercise price less the expected dividends. Our March/June 100 jelly roll is:

Short Call Calendar Spread = Long March 100 Call / Short June 100 Call |

Long Put Calendar = Short March 100 Put / Long June 100 Put |

Since the long and short calendar spreads make up a jelly roll, their values should differ by the value of a jelly roll. If the March/June 100 jelly roll is worth 1.25 and the March/June 100 call calendar spread is worth 3.50, then the March/June 100 put calendar spread must be worth 2.25 since 3.50 - 1.25 = 2.25.

### Summary

- A conversion consists of buying the underlying contract in the underlying market and selling the underlying synthetically in the option market (sell a call, buy a put with the same exercise price and expiration date).
- A reversal consists of selling the underlying contract in the underlying market and buying the underlying synthetically in the option market (buy a call, sell a put with the same exercise price and expiration date).
- A box consists of long and short synthetic positions in the same expiration month but at different exercise prices. A box can also be thought of as the combination of a bull (bear) vertical call spread and a bear (bull) vertical put spread.
- A jelly roll consists of long and short synthetic stock positions at the same exercise price but in different months. A jelly roll can be thought of as the combination of a long (short) call calendar spread and a short (long) put calendar spread.

##### Chapter 06

## Arbitrage Risk

### Arbitrage Strategies

Arbitrage strategies, including conversions, reversals, boxes, and jelly rolls, are considered the least risky of all option strategies. This does not mean that such strategies are "no risk." If a trader is unaware of the risks, they may find that even an arbitrage strategy represents a greater risk than they are willing to accept.

### Interest Rate Risk

Whenever there is a cash flow associated with a trade, the value of the trade depends, among other factors, on the interest that can be earned on a cash credit or which must be paid on a cash debit. The cash flow resulting from a conversion or reversal will depend on the type of options and the settlement procedure.

The cash flow is approximately equal to the exercise price for stock options. In contrast, for futures options subject to stock-type settlement, the cash flow is approximately equal to the difference between the underlying futures and exercise prices. The value of the conversion or reversal will depend on the cash flow, the level of interest rates, and the amount of time remaining to expiration.

For example, suppose that with two months remaining to expiration and an interest rate of 6.00%, a June 50 stock option call is trading at 4, and a June 50 stock option put is trading at 2.25. If interest rates suddenly rise to 9.00%, there will be an additional carrying cost on the exercise price of 2/12 x 3.00% x 50, or .25.

If the stock price does not change, the difference between the call and put must narrow by .25. If the June 50 call continues to trade at 4.00, the June 50 put will have to decline to 2.00. Alternatively, if the June 50 put continues to trade at 2.25, the June 50 call must rise to 4.25.

A jelly roll in the stock option market will also be sensitive to changes in interest rates. Suppose that interest rates are currently 6.00% and that there are three months between September and December expiration. Suppose no dividends are expected over this period. In that case, the value of the September/December 100 jelly roll (the difference between the September 100 synthetic and the December 100 synthetic) should be the cost of carrying the exercise price of 100 from September to December:

3/12 x 6.00% x 100 = 1.50

If, however, interest rates rise to 8.00%, the jelly roll will widen to:

3/12 x 8.00% x 100 = 2.00

The above examples will not apply to futures options markets because the credit or debit resulting from a conversion or reversal will be much less than in the stock market. Moreover, while a conversion in the stock market (sell call, buy put, buy stock) will always result in a credit, a conversion in the futures option market may result in either a credit or debit, depending on whether the exercise price is less than or greater than the underlying futures price.

Regardless of the instruments involved, significant changes in interest rates over the relatively short life of most exchange-traded options. For this reason, the interest rate risk associated with conversions and reversals, especially in the futures option market, is relatively small.

### Dividend Risk

Consider a stock trading at 102.50, with three months to June expiration, interest rates at 8%, and a dividend of 1.50 expected before expiration. The value of the June 100 combination (the difference between the 100 call and the June 100 put) is:

Stock Price | - Exercise Price | + Carry on 100 to Expiration | - Expected Dividends | = | |

102.50 | - 100 | + (100 x 3/12 x 8%) | - 1.50 | = | 3 |

Suppose a trader can sell a June 100 call for 7.75, buy a June 100 put for 4.50, and purchase stock for 102.50. If interest rates do not change at expiration, the trader should realize a profit of .25 since they have done the June 100 conversion at .25, better than its value.

Since the trader owns the stock, part of their profit comes from the dividend of 1.50, which they expect to receive when the stock goes ex-dividend. Unfortunately, if the company is doing poorly and decides to cut its dividend in half to .75, the trader's conversion will be worth .75 less, and their profit of .25 will turn into a loss of .50.

Of course, suppose the company is doing well and decides to increase its dividend to two points. In that case, the conversion will be worth .50 more, and the trader's profit will increase from .25 to .75. Clearly, the possibility of a change in the expected dividend represents a risk to a conversion or reversal.

Moreover, if multiple dividends are expected over the life of the strategy, the impact of a change in the company's dividend policy can be significantly magnified. For example, suppose the expected dividend is 1.50, with one dividend payout expected before March expiration and a second dividend payout expected between March and June expiration.

Suppose the dividend is unexpectedly increased to 2. In that case, the March combination will fall by 2 - 1.50, or .50. But the June combination, because it includes two dividend payments, will fall in value by 2 x (2 - 1.50), or 1. That is, the jelly roll will decline in value by the amount of the dividend increase, or .50.

Rising dividends make a combination (the difference between the call price and put price) less valuable; falling dividends make a combination more valuable.

Boxes tend to be immune to dividend changes because a box does not involve a position in the underlying stock. Boxes in the stock options market are also much less sensitive to changes in interest rates than conversions or reversals because the credit or debit is equal to the difference between exercise prices rather than the amount of the exercise price.

A June 100 conversion results in a cash flow of 100, while a June 100/110 box results in a cash flow of only 10. For these reasons, a trader with many conversions or reversals in their account will often try to turn them into boxes to reduce the risk of changes in interest rates or dividends.

### Pin Risk

Suppose a trader has executed a June 100 conversion: they are short a June 100 call, long a June 100 put, and long the underlying. If the underlying market is above or below 100 at expiration, there is no problem. They will either be assigned on the call or will exercise the put. In either case, they will offset their long underlying position to have no market position on the day following expiration.

But suppose that the underlying market is right at 100 at the moment of expiration. The trader would like to be rid of their underlying position. If they don't get assigned on the call, they plan to exercise their put; if they do get assigned on the call, they will let the put expire worthless. To decide, they must know whether the call will be exercised. Unfortunately, they won't know this until the day after expiration, when they either do or do not receive an exercise notice. But then it will be too late because the call will have expired.

The trader who is short an at-the-money option at expiration has a problem. What can they do? While there can be no certain solution to this problem, sometimes referred to as pin risk (the price of the underlying is pinned to the exercise price), the practical solution is to avoid carrying conversions and reversals to expiration when there is a real possibility of expiration right at the exercise price.

Suppose a trader has many conversions or reversals, and expiration is approaching with the underlying market close to the exercise price. In that case, the sensible course is to reduce the pin risk by reducing the position size. If the trader doesn't reduce the size, they may be under considerable pressure to get out of many risky positions at the last moment.

##### Pin Risk

The risk to the seller of an option is that at expiration, the option will be exactly at-the-money. The seller will not know whether the option will be exercised.

Sometimes, even a careful trader will find that they have some at-the-money conversions or reversals outstanding as expiration approaches. One way to eliminate the pin risk that still exists is to liquidate the position at the prevailing market prices. Unfortunately, this is likely a losing proposition since the trader will be forced to trade each contract at an unfavorable price, either buying at the offer or selling at the bid. Fortunately, trading out of such a position at a fair price is often possible.

Since conversions and reversals are common strategies in all options markets, a trader who has an at-the-money conversion and is worried about pin risk can be reasonably confident that there are also traders who have at-the-money reversals and are concerned with pin risk. If the trader with the conversion could find a trader with a reversal and cross positions with him, both would eliminate the pin risk associated with their positions.

This is why, on options exchanges, one often finds traders looking for other traders who want to trade conversions or reversals at even money. This means that a trader wants to trade out of their position at a fair price to everyone involved so that everyone can avoid the problem of pin risk. Whatever profit a trader expected to make from the conversion or reversal presumably resulted from the initial trade, not the closing trade.

Specific options, such as those on stock indices, are settled at expiration in cash rather than with the delivery of an actual underlying contract. When such an option expires, the amount of cash that flows into or out of a trader's account is simply the amount by which the option is in-the-money, i.e., the difference between the underlying price and the option's exercise price. No pin risk is associated with options that settle in this manner because no underlying position results from exercise or assignment.

### Summary

- Changes in interest rates and, in the case of stock options, changes in the expected dividend payout will change the value of arbitrage strategies.
- In the stock option market, if interest rates rise conversions become less valuable and reversals become more valuable. If interest rates fall, conversions become more valuable and reversals become less valuable. If interest rates rise, jelly rolls also become more valuable. Changes in interest rates can also affect the value of conversions and reversals in the futures option market, but to a lesser degree than the stock option market.
- If the dividend is increased, conversions become more valuable and reversals become less valuable. If the dividend is cut, conversions become less valuable and reversals become more valuable. If the expected dividend between expiration months is increased, jelly rolls become less valuable.
- Pin risk results when a trader is short an option which expires exactly at-the-money. Since the trader will not know whether he will be assigned on the option until the day after expiration, he will not know whether to hedge the position.

##### Chapter 07

## Other Synthetic Applications

While most strategies involving synthetics focus on arbitrage, synthetics can also be used in any strategy for which a synthetic equivalent is available. To determine whether the synthetic strategy or the real equivalent is preferable, a trader needs to calculate the value of the synthetic and compare it to its price in the marketplace.

### Using Synthetic Relationships in Volatility Spreads

Consider this situation:

Bid Price | Ask Price | |
---|---|---|

Underlying Stock | 51.25 | 51.40 |

Mar 50 Call | 4.40 | 4.75 |

Mar 50 Put | 2.50 | 2.75 |

Suppose a trader wants to buy the March 50 straddle (buy a 50 call, buy a 50 put). What is the best way to do this?

Of course, the trader can buy the call and put for a total of 4.75 + 2.75 = 7.50. But instead of purchasing the straddle outright, the trader might also consider buying it synthetically. The trader might, for example, buy two March 50 calls and sell one stock contract. Or they might buy two March 50 puts and buy one stock contract. Both alternatives are synthetic straddles since:

In the first case, the March 50 put has been purchased synthetically (long call, short stock), while in the second case, the March 50 call has been purchased synthetically (long put, long stock). In either case, the result is a long straddle.

Is either synthetic priced better than the outright straddle?

Suppose the carrying cost to expiration on the exercise price of 50 is .50 and that no dividends are expected before March expiration. From the synthetic relationship, we know that:

call price - put price = stock price - exercise price + carrying costs

In terms of the call price and the stock price, the put price should be:

##### Formula: Put Price

Put Price = Call Price - Stock Price + Exercise Price - Carrying Costs

If the trader chose to buy the straddle synthetically by purchasing two March 50 calls and selling one stock contract at the listed prices, they are effectively paying 3 for the put since:

3 = 4.75 - 51.25 + 50 - .50

But the trader can buy the March 50 put for its listed price of 2.75, so there is no advantage to purchasing the put synthetically. What about buying the call synthetically? In terms of the put price and stock price, the call price should be:

##### Formula: Call Price

Call Price = Stock Price - Exercise Price + Carrying Costs + Put Price

If the trader chose to buy the straddle synthetically by purchasing two March 50 puts and purchasing one stock contract at the listed price, they would be paying 4.65 for the call since:

4.65 = 51.40 - 50 + .50 + 2.75

Since the trader would have to pay 4.75 for the March 50 call if they were to purchase it outright, the synthetic call is cheaper. By purchasing two puts and one stock contract, the trader has effectively paid 7.40 for the straddle, or .10 better than simply purchasing a call and a put. .10 may seem a small amount. But if done in large size over many trades such small amounts can add up. Whenever a trader can do any strategy at a better price using a synthetic relationship, they certainly ought to do so.

Check out Rival One Options Functionality

### Buy / Writes

One common hedging strategy involving options involves buying the underlying stock and selling a call. The position is known as a buy/write when both trades are made simultaneously. When the stock is purchased first or already owned, and the call is later sold, the position is known as a covered write.

##### Buy / Write

The purchase of an underlying contract together with the sale of a call option on that contract.

Buy/writes are actively traded in most stock option markets, and, like other spreading strategies, they are usually quoted with one bid price and one ask price. The price of the buy/write is quoted as the stock price less the option premium. For example, suppose the following are current prices in the marketplace:

Bid Price | Ask Price | |
---|---|---|

Underlying Stock | 61.25 | 61.50 |

June 60 call | 4.40 | 4.75 |

June 65 call | 1.50 | 1.75 |

The current market for the June 60 buy/write is therefore:

56.50 bid (buy the stock at 61.25, sell the June 60 call at 4.75),

57.10 offer (sell the stock at 61.50, buy the June 60 call at 4.40)

The current market for the June 65 buy/write is:

59.50 bid (buy the stock at 61.25, sell the June 65 call at 1.75),

60 offer (sell the stock at 61.50, buy the June 65 call at 1.50)

Of course, a buy/write or covered write is simply a synthetic put, so the value of the combination can be calculated using the synthetic relationship:

##### Formula: Buy/Write

call price - put price = stock price - exercise price + carrying costs - dividends

Rearranging, we can express the value of the buy/write as:

##### Formula: Buy/Write

stock price - call price = exercise price - put price - carrying costs + dividends

For example, suppose an underlying stock is trading at 61.25, a June 60 put is trading at 1.10, and the carrying costs on 60 to expiration is .65. A dividend of .25 is expected before expiration. The value of the buy/write is therefore:

60 - 1.10 - .65 + .25 = 58.50

### Fences

Another common hedging strategy using options involves simultaneously selling a covered option and buying a protective option to hedge a position in the underlying contract. This is most often done using out-of-the-money options. For example, suppose a trader has purchased stock at 62. They might hedge the position by selling a June 65 call (a covered call) and buying a June 60 put (a protective put).

This type of strategy goes by many different names. Hedgers often refer to the strategy as a fence, collar, or range forward. When the hedge includes a long underlying position, it is known as a long fence or collar. When the hedge consists of a short underlying position, it is known as a short fence or collar.

Among traders, this type of position is sometimes referred to as a risk conversion or a split price conversion when the position is long the underlying, or a risk reversal or split price reversal when the position is short the underlying.

##### Fence or Range Forward

A long (short) underlying position, together with a long (short) out-of-the-money put and a short (long) out-of-the-money call. All options must expire at the same time.

Hedgers often use fences to reduce the cost of purchasing a protective option. By selling a covered option, a hedger will receive some premium, which will at least partially offset the premium paid for the purchased option. If the premiums for the two options are identical, the position is said to be a zero-cost fence or zero-cost collar.

Traders sometimes construct fences because of the perceived differences in implied volatilities across different exercise prices. This subject will be discussed further in the chapter on volatility skews.

Note that a fence is just a bull or bear vertical spread. For example, we can rewrite our long fence as follows:

+1 Stock | |

+1 June 60 Put | +1 June 60 Call |

-1 June 65 Call | -1 June 65 Call |

The long stock position and the long June 60 put is a synthetic long June 60 call. The position is, therefore, equivalent to being long the June 60/65 bull vertical call spread. The position wants the underlying to rise, has limited upside profit potential because of the covered call, and limited downside risk because of the protective put.

### Summary

- While synthetic equivalents are most often used for arbitrage, any option strategy can be done using a synthetic equivalent. The synthetic equivalent should be used if it offers a pricing advantage over the real counterpart.
- A common hedging strategy consists of buying the underlying contract and selling a call option against the position. The combination is known as a buy/write or covered/write. In theory, the position is equivalent to a short put.
- Another common hedging strategy consists of simultaneously buying a protective option and selling a covered option against a position in the underlying contract. The total position goes by several different terms, including fence, collar, risk conversion (reversal) and split price conversion (reversal). In theory, the position is equivalent to a bull or bear vertical spread.

##### Chapter 08

## Early Exercise

## Early Exercise

The ownership of an option confers on the owner the right to exercise that option, thereby taking a long position in the underlying contract in the case of a call or a short position in the underlying contract in the case of a put. The terms specifying when the owner may exercise this right can be essential in deciding whether to purchase an option and what price to pay for it.

## American and European Options

Most options fall into two exercise categories: European and American. A European option can be exercised only at expiration, while an American option can be exercised any time prior to expiration.

##### American Option

An option which can be exercised at any time prior to expiration.

##### European Option

An option which can only be exercised at expiration.

### Early Exercise of Stock Option Calls for Dividends

In simple terms, we can express the value of a stock option call in terms of its components:

##### Formula: Value of a Stock Option Call

call value = intrinsic value + interest rate value + volatility value - dividend value

Since the intrinsic value, interest rate, and volatility components can never be less than zero, these factors always enhance the call's value. As any one of them rises, the call value rises. Only the dividend component might affect the option's value negatively. As the dividend rises, the call's value falls.

If the underlying stock pays no dividend or no dividend is expected prior to expiration of the option, a call option can never have a value less than parity (intrinsic value). If, however, the negative effects of the dividend are greater than the positive effects of interest rates and volatility, it might be possible for a call, if it is European, to be worth less than parity. For example, suppose a certain stock is trading at 100, and the stock will go ex-dividend 2.00 tomorrow.

Suppose also that there is a 90 call available which will expire in two weeks. When we evaluate the option, we find that it has a theoretical value of 10 and a delta of 100. This means that the option has essentially the same characteristics as the stock. If the option is American, and we want to maintain the same delta position, we have three possible choices:

- Hold the option
- Exercise the option
- Sell the option and buy stock

Which of these choices is best?

Suppose we simply hold the option. Certainly, we will maintain our delta position. But what will happen tomorrow when the stock gives up its dividend? If the stock opens unchanged, it will open ex-dividend at 98 since the two-point dividend will be deducted from its price. Since the option has a value of parity, it will open not at 10, the previous day's parity price, but at 8, today's parity price. In other words, if we hold the option we can be certain of losing two points on our position.

Can we do any better with the second choice, exercising the option? If we exercise the option, we will pay the exercise price of 90 for the stock, and we will discard the 10-point value of the option, effectively purchasing the stock at 100. When the stock goes ex-dividend we will lose two points when it opens two points lower the next day, but we will also receive the dividend since we now own the stock. Hence, we will break even.

The two-point loss on the stock price will be offset by the two-point dividend we receive. Clearly, we are better off exercising the option than holding it, not because we will show some additional profit, but because we will avoid a two-point loss. We must exercise the option to ensure that we break even.

What about the third choice, selling the option and buying stock ourselves? This seems to be very similar to early exercise. In both cases we are replacing the option with the stock. If the option is trading at parity, in this case, 10, there is no difference between exercising the option or selling the option and buying the stock.

But suppose the 90 call is trading for more than parity, say 10.50? Now, if we sell the option and purchase the stock, we will still receive the dividend since we will own the stock. But we will end up with an additional .50, which we would not have collected through exercise of the call. Hence, our third choice, selling the call and buying the stock, is the optimum choice.

Since the only reason a trader would ever consider exercising a stock option call early is to receive the dividend, if a stock pays no dividend, there is no reason to exercise a call early. If the stock does pay a dividend, the only time a trader ought to consider early exercise is the day before the stock goes ex-dividend. At no other time in its life is a stock option call an early exercise candidate.

### Early Exercise of Stock Option Puts for Interest

As we did with a stock option call, we can express a stock option puts value in terms of its components:

##### Formula: Value of a Stock Option Put

put value = intrinsic value - interest rate value + volatility value + dividend value

In the case of a put, the only component that affects its value negatively is the interest rate component. If the negative effects of interest rates are greater than the positive effects of volatility and dividends, it might be possible for a put, if it is European, to be worth less than parity. Consider this situation:

Stock price = 100, Time to expiration = 8 weeks, Volatility = 20%, Interest rate = 8%, Dividend = 0

With these assumptions, the value of a 110 call is approximately .70. Using the put/call parity relationship, we can approximate the value of the 110 put:

##### Formula: Value of a Put Option

put value = call value + exercise price - stock price - carrying costs on the exercise price

The carrying costs on the exercise price of 110 are:

110 x 56/365 x 8% = 1.35

The European put value is therefore:

.70 + 110 - 100 - 1.35 = 9.35

Since the 110 put is only worth 9.35 if we hold it to expiration but is worth 10 if we exercise it today, the option is apparently worth more dead (exercised) than alive (unexercised). When we exercise the 100 put, we get to sell stock at 110, and we can earn interest on this 110-point credit to expiration.

Whereas a stock option call can only be an early exercise candidate on the day prior to the stock's ex-dividend date, a stock option put can become an early exercise candidate anytime the interest that can be earned through the sale of the stock at the exercise price is sufficiently large.

Determining exactly when this happens is a difficult problem, but if the stock pays a dividend, it is most likely to occur on the day after the stock goes ex-dividend. Since a put is a substitute for a short stock position, one of the advantages of holding a put is to avoid paying the dividend. Hence, a trader will almost always want to hold the put through the ex-dividend date. Then, if the interest considerations are sufficient, the trader will exercise their put.

Check out Rival One Options Functionality

### The Cox-Ross-Rubenstein and Whaley Models

The problem of early exercise, when to exercise an option early, and how much to pay for the right of early exercise was never addressed by Black and Scholes. The Black-Scholes model was designed to evaluate European options only, where no early exercise is permitted.

But because the great majority of exchange traded options carry with them the right of early exercise, models were eventually developed to accurately evaluate American options. The most widely used of these are the Cox-Ross-Rubenstein model, developed by John Cox, Stephen Ross, and Mark Rubenstein; and the Whaley model, developed by Giovanni Barone-Adesi and Robert Whaley.

Unlike the Black-Scholes model, which results in an exact theoretical value, the Cox-Ross-Rubenstein and Whaley models approximate the value of an option by means of an algorithm or loop. Each time the user makes a pass through the loop, the closer they come to the true American value of the option. In theory, if one goes through the loop enough times, both models eventually will converge to the true value of an American option.

##### Cox-Ross-Rubenstein Model

A widely used option theoretical pricing model developed by John Cox, Stephen Ross, and Mark Rubenstein is sometimes referred to as the binomial model. The model approximates the price distribution of the underlying contract by constructing a binomial tree where the price of the underlying is assumed to move up or down a given amount over each time interval. The Cox-Ross-Rubenstein model is often used to evaluate American options on different underlying instruments.

##### Whaley Model

An option pricing model developed by Robert Whaley and Giovanni Barone-Adesi. The Whaley model can be used to evaluate American options on different underlying instruments.

Most stock option traders prefer the Cox-Ross-Rubenstein model because it can more accurately evaluate an American call option when a dividend is expected. If no dividend is expected, the Whaley model has the advantage of converging more quickly to the true American value. In addition to generating theoretical values for American options, both the Cox-Ross-Rubenstein and Whaley models also determine when an option should be exercised early. When using either of these models, an option is optimally exercised early when its theoretical value is exactly parity and its delta is exactly 100.

### Summary

- American options carry with them the right of early exercise, and this additional right usually makes an American option more valuable than a similar European option.
- Early exercise is a possibility for a stock option call only if the stock is expected to pay a dividend prior to expiration. In such a case the option may be exercised optimally only on the day prior to the stock's ex-dividend date.
- Early exercise for a stock option put may be optimal any time, providing the interest that can be earned on the sale of the stock at the exercise price becomes sufficiently great.
- The Cox-Ross-Rubenstein and Whaley models are the two most widely used models for the evaluation of American options.

##### Chapter 09

## The Effect of Early Exercise on Arbitrage Strategies

The possibility of early exercise affects not only the value of individual options but also the value of various options strategies. The effects of early exercise are most crucial concerning arbitrage strategies. Strategies that may be almost risk-free if carried to expiration can sometimes have unforeseen risks if the possibility of early exercise exists.

### The Effect of Early Exercise on Conversions and Reversals

The possibility of early exercise can have an impact on arbitrage strategies. For example, suppose a stock option trader executes a reverse conversion:

- Buy a call
- Sell a put
- Sell stock

If they execute this strategy at what they believe to be profitable prices, part of their profit will come from the interest they expect to earn on the stock sale. But what will happen if the stock price begins to fall and falls so far that the trader is assigned to the put? This will eliminate the interest earnings since they ust buy back the stock.

Of course, they can still sell the call and take in some cash. But if the value of the call is insufficient to offset the interest loss, the trader may find that their profitable reversal has become unprofitable.

In our example, the trader worries about being assigned on their put. If the market drops, there will be a greater likelihood of assignment; if the market rises, there will be a lesser likelihood of assignment. Since the trader prefers the market to rise, they ust be delta-long. The following delta values confirm this:

Stock price = 100; Time to expiration = 3 months; Volatility = 25%; Interest rate = 8%; Dividend = 0

European | American | |||
---|---|---|---|---|

Option | Value | Delta | Value | Delta |

100 Call | 5.97 | 58.8 | 5.97 | 58.8 |

100 Put | 4.00 | -41.2 | 4.20 | -44.0 |

If these are European options, the total delta of the reversal is:

58.8 + 41.2 -100 = 0

If, however, the options are American, the total delta is:

58.8 + 44.0 -100 = 2.8

The positive delta of 2.8 reflects a slight preference for the market to rise, so the trader will avoid being assigned on the put.

Similarly, if the trader executes a conversion (sell call, buy put, buy stock), they are 2.8 deltas short. They want the stock to fall to exercise their put early and avoid the interest costs of carrying a long stock position.

Because the desirability of early exercise and the likelihood of early exercise can increase or decrease as the underlying market rises or falls, conversions and reversals using American options are not delta-neutral. While these strategies may be unbalanced by only two or three deltas, the fact that they are often done in large size can result in an additional risk that the trader must pay attention to.

### The Effect of Early Exercise on Boxes and Jelly Rolls

The possibility of early exercise also affects the value of boxes and jelly rolls. Consider these option values:

Stock price = 100; Volatility = 25%; Interest rate = 8%; Dividend = 0; Time to expiration: March = 3 months, June = 6 months

European | American | |||
---|---|---|---|---|

Option | Value | Delta | Value | Delta |

Mar 95 call | 9.00 | 73.7 | 9.00 | 73.7 |

Mar 95 put | 2.13 | -26.3 | 2.24 | -27.8 |

Mar 100 call | 5.97 | 58.8 | 5.97 | 58.8 |

Mar 100 put | 4.00 | -41.2 | 4.20 | -44.0 |

Jun 95 call | 11.96 | 72.7 | 11.96 | 72.7 |

Jun 95 put | 3.24 | -27.3 | 3.49 | -29.9 |

Jun 100 call | 9.03 | 62.3 | 9.03 | 62.3 |

Jun 100 put | 5.11 | -37.7 | 5.55 | -42.1 |

Strategy | Value | Delta | Value | Delta |

Mar 95/100 box | 4.90 | 0 | 4.99 | -1.3 |

Jun 95/100 box | 4.80 | 0 | 4.99 | -1.8 |

Mar/Jun 95 jelly roll | 1.85 | 0 | 1.71 | +1.1 |

Mar/Jun 100 jelly roll | 1.95 | 0 | 1.71 | +1.6 |

We can see how the possibility of early exercise affects the values and deltas of various arbitrages: Boxes become more valuable using American options because a trader who is long the box owns a put with a higher exercise price.

If a trader buys the 95/100 box, they are long a 95 call and a 100 put and short a 95 put and 100 call. The 100 put will become an early exercise candidate before the 95 put, making the American box more valuable than the same European box. The delta of the box is negative with American options because a trader who owns the box would like the market to decline so that they can exercise the 100 put as quickly as possible.

Jelly rolls become less valuable using American options because a trader who owns the jelly roll has sold a long-term put. If a trader buys the March/June 100 jelly roll, they are long a March 100 put and June 100 call and are short a March 100 call and June 100 put. The June 100 put, being a longer-term option, has more potential for early exercise than the March 100 put.

The delta of the jelly roll is, therefore, positive because there is a greater danger of being assigned on the June 100 put. Hence, the trader who owns the jelly roll would like the market to rise so that they are less likely to be assigned on the June 100 put.

Check out Rival One Options Functionality

### Dividend and Interest Plays

Because early exercise of an option is not automatic, there are strategies that depend on someone making an error and not exercising an option early when they ought to do so. For example, a stock options trader might try to execute a dividend play.

This strategy consists of buying stock and selling deep in-the-money calls as the ex-dividend date for the stock approaches. If the trader is not assigned to the calls, they will break even on the stock (the stock price will fall, but they will collect the dividend). At the same time, they will profit when the deep-in-the-money calls that they have sold fall by the amount of the dividend.

Of course, if they are assigned on the calls, as they ought to be, they will only break even. But for each call that goes unexercised the trader will profit by the amount of the dividend. Dividend plays were much more common in the early days of option trading when the market was less sophisticated, and many options that should have been exercised were not.

As markets have become more efficient, only a professional trader, with very low transaction costs, can afford to take advantage of such a possibility. Even then, they may find that they are assigned on the great majority of the calls they have sold.

##### Dividend Play

In a market where early exercise is a possibility, a strategy in which deep in-the-money calls are sold and stock is purchased just before the stock's ex-dividend date. The strategy will profit from the dividend amount if the calls remain unexercised through the ex-dividend date.

A trader can execute a similar type of interest play by selling stock and simultaneously selling deep in-the-money puts. Now, instead of profiting by the amount of the dividend, the trader will profit by the amount of the interest they can earn on the exercise price (the proceeds of the stock sale and the put sale combined).

This profit will continue to accrue as long as the puts remain unexercised. If the puts are exercised, the trader does no worse than break even. Again, with their own transaction costs, only a professional trader is likely to employ such a strategy.

##### Interest Play

In a market where early exercise is possible, a strategy in which deep in-the-money puts are sold against a short stock position. The strategy will profit from the amount of interest that can be earned on the resulting cash credit as long as the puts remain unexercised.

A variation on dividend and interest plays can also be executed using deep in-the-money vertical spreads. For example, suppose a stock is trading at 100 with the ex-dividend date approaching. If both the 80 and 85 calls are deep enough in-the-money to be early exercise candidates, a trader might try to purchase the 80/85 call vertical spread for 5.

If they do so, they will exercise their 0 call to collect the dividend and, at the same time, hope not to be assigned on the 85 call. Oddly, if both the 80 and 85 calls are early exercise candidates, the trader should also be willing to sell the 80/85 call vertical for 5.

If they do so, they will exercise their 5 call to collect the dividend and, at the same time, hope not to be assigned on the 80 call. In other words, a professional trader could make a market under these circumstances of 5 bid/ 5 ask. They are willing to buy or sell the 80/85 call vertical at 5. They intend to immediately exercise whichever call they buy, hoping to avoid being assigned on the call they sell.

### Summary

- If early exercise is a possibility, conversions and reversals are not delta neutral. A conversion (sell calls, buy puts, buy stock) is slightly delta negative; a reversal (buy calls, sell puts, sell stock) is slightly delta positive.
- A box consisting of American options is slightly delta negative and has a greater value than a similar box consisting of European options.
- A jelly roll consisting of American options is slightly delta positive and has a value less than a similar jelly roll consisting of European options.
- A trader can attempt a dividend or interest play by selling options which ought to be exercised early for their dividend or interest value, and hedging the position with stock. If the trader is not assigned, they will profit by the amount of the dividend or the amount of interest they can earn.

##### Chapter 10

## Volatility Skews

## Volatility Skews

In most option markets, options with different exercise prices tend to trade at different implied volatilities, even when the options expire simultaneously. The distribution of implied volatilities across different exercise prices is often referred to as a volatility skew.

Every active option trader needs at least a basic understanding of volatility skewing:

- Why the skew occurs.
- What the skew characteristics of a particular market are.
- How to incorporate skew information into the decision-making process.

##### Volatility Skewing

The assignment of different volatility to different exercise prices, even though all options have the same underlying contract and expire simultaneously.

### Skew Basics

In this section, we will define the various types of volatility skews and look at some market forces that tend to cause a skew.

### What is a Volatility Skew?

Consider the following table of option prices and implied volatilities calculated using the Black-Scholes model:

Underlying contract price = 99.75

Time to expiration = 8 weeks; Interest rate = 0.00%

Strike | Call Price | Call Volatility | Put Price | Put Volatility |
---|---|---|---|---|

80 | 19.77 | 23.16% | .02 | 23.16% |

85 | 14.86 | 22.45% | .10 | 22.08% |

90 | 10.81 | 21.53% | .41 | 21.38% |

95 | 6.06 | 20.57% | 1.30 | 20.49% |

100 | 3.03 | 20.21% | 3.27 | 20.08% |

105 | 1.15 | 19.33% | 6.43 | 19.57% |

110 | .35 | 19.09% | 10.61 | 19.23% |

115 | .07 | 18.39% | 15.33 | 18.80% |

120 | .01 | 17.90% | 20.26 | 17.90% |

Except for minor discrepancies, calls and puts at the same exercise price have approximately the same implied volatility, indicating that put-call parity is maintained. However, the fact that implied volatilities differ across exercise prices poses a problem since this is inconsistent with the assumptions on which the Black-Scholes model is based.

The underlying contract's volatility will determine the options' value, and the underlying contract can have only one volatility over the life of the options.

Since the same trader will usually have positions at many different exercise prices, it doesn't make sense to assume that everyone trading the 85 calls believes that the volatility over the 8-week life of the options will be 22.45% while everyone trading the 115 calls believes that the volatility will be 18.39%.

In a perfect theoretical world, if we were to graph the implied volatilities of options with the same expiration date across different exercise prices, the graph should be a straight horizontal line. If, however, we graph the above-implied volatilities across exercise prices, the chart might look something like this:

The fact that the graph is not a straight horizontal line indicates that the market exhibits volatility skewing, with the severity of the skew determined by the degree to which the graph deviates from a straight horizontal line.

Volatility skewing results from two forces at work in the marketplace: how market participants use options to achieve their goals and how the real world differs from the world assumed by a theoretical pricing model. Together, these forces cause prices to be distributed in a way a trader might not expect, primarily if the trader relied strongly on model-generated theoretical values.

Check out Rival One Options Functionality

### Types of Volatility Skews

One of the most common uses of options is a hedge against a position in the underlying instrument. Someone with a long position will tend to hedge by purchasing puts and/or selling calls against their long position. Someone with a short position will tend to hedge by purchasing calls and/or selling puts.

Most often, these hedging strategies involve out-of-the-money options. A hedger will prefer to buy out-of-the-money options because such options are cheap and, therefore, reduce the hedging costs. A hedger will prefer to sell out-of-money options because this will allow for additional profit should the underlying market move in the hedger’s favor.

Underlying Position | Hedging Strategy |
---|---|

Long | buy puts with a lower exercise price |

sell calls with a higher exercise price | |

Short | buy calls with higher exercise price |

sell puts with a lower exercise price |

In some markets, those desiring to hedge long underlying positions will be approximately equal to those wishing to hedge short positions. Neither the longs nor the shorts will predominate. In other markets, however, one side may dominate, either because of natural forces or the adverse consequences of a move in one direction rather than another.

For example, the longs far outnumber the shorts in the stock market. As a result, stock and stock index options with lower exercise prices will tend to be inflated because hedgers will want to purchase out-of-the-money puts. At the same time, options with higher exercise prices tend to be deflated because hedgers wish to sell out-of-the-money calls. If we were to graph the resulting volatility skew, it might look something like this:

This type of skew, where the lower exercise prices carry higher implied volatilities and the higher exercise prices have lower implied volatilities, is sometimes referred to as an investment skew because it typically occurs in markets dominated by investment activity.

##### Investment Skewing

A volatility skew where lower exercise prices tend to carry higher implied volatilities and higher exercise prices tend to carry lower implied volatilities.

A different type of skew might occur in markets where the shorts outnumber the longs, or upside movement represents a greater danger to participants than downside movement. For example, in some commodity markets, prices may be artificially supported, so a severe drop in prices is unlikely.

On the other hand, there is nothing to prevent a swift rise in prices. Anyone who must regularly purchase the commodity as part of their business activity has a natural short position in the commodity since any decline in commodity prices will reduce their costs. Because of this natural short position, a possible rise in commodity prices represents a real risk.

As a result, in markets with a low risk of collapsing prices but a high risk of rising prices, higher exercise price options will tend to be inflated because hedgers will want to purchase out-of-the-money calls. At the same time, lower exercise price options will tend to be deflated because hedgers will want to sell out-of-the-money puts. If we were to graph the resulting volatility skew, it might look something like this:

This type of skew, where the higher exercise prices carry higher implied volatilities and the lower exercise prices have lower implied volatilities, is sometimes referred to as a demand skew because it occurs in markets with strong demand for a particular commodity.

##### Demand Skew

A volatility skew where higher exercise prices tend to carry higher implied volatilities and lower exercise prices tend to have lower implied volatilities.

Finally, some markets will tend to have equal numbers of longs worried about a decline in the market and shorts concerned about a rise in the market. This might occur in a foreign currency market with equal commerce in both directions. Some people will worry about the value of the domestic currency in terms of the foreign currency, and others will worry about the foreign currency in terms of the domestic currency.

If each side’s desire to hedge is approximately equal, the buying and selling pressure on options, whether higher or lower exercise prices, will be roughly equal. The resulting skew might look something like this:

Even though this type of balanced skew is symmetrical, it is still not a straight line. In terms of implied volatility, there appears to be greater demand for higher and lower exercise price options than at-the-money options. One might reasonably conclude that forces other than those which result from hedging activity must also be at work.

##### Balanced Skew

A volatility skew where both higher and lower exercise prices tend to carry higher implied volatilities than the at-the-money exercise prices.

### Summary

- In theory, under the assumptions of the Black-Scholes model, every option on the same underlying and with the same time to expiration ought to have the same implied volatility.
- Volatility skewing results when options on the same underlying contract with the same expiration date, but with different exercise prices, carry different implied volatilities.
- The severity of a volatility skew is determined by the degree to which the implied volatilities differ across exercise prices.
- Volatility skewing is often the result of options being used to hedge a position in the underlying contract.
- When those desiring to hedge a long position in the underlying contract outnumber those desiring to hedge a short position, the result is an investment skew. In an investment skew lower exercise prices carry higher implied volatilities than higher exercise prices.
- When those desiring to hedge a short position in the underlying contract outnumber those desiring to hedge a long position, the result is a demand skew. In a demand skew higher exercise prices carry higher implied volatilities than lower exercise prices.
- When those desiring to hedge a long position in the underlying contract are approximately equal to those desiring to hedge a short position, the result is a balanced skew. In a balanced skew both lower and higher exercise prices will tend to carry higher implied volatilities than at-the-money options.

##### Chapter 11

## Models and the Real World

All theoretical pricing models are predicated on a set of assumptions. The accuracy of model-generated values will, therefore, depend not only on the accuracy of the inputs into the model but also on the accuracy of the assumptions on which the model is based. If these assumptions are inaccurate, then the values will not be reliable.

The Black-Scholes model makes several assumptions about the characteristics of the underlying contract and how trading is conducted. The following are the most important of these assumptions:

- Any trade made in an underlying contract will break even in the long run. Given all considerations of interest, dividends, or other factors, no profits can be made trading the underlying contract.
- The percentage price changes in the underlying contract are normally distributed and continuously compounded. The result of this is that prices at expiration will be lognormally distributed.
- Trading is continuous, with no gaps in the price of the underlying contract.
- Whatever the correct volatility is over the life of an option, it is assumed to be evenly distributed. If one were to draw a volatility graph over time, it would be a straight line.
- The volatility of an underlying contract is independent of the direction in which the contract moves.

Are these assumptions realistic, and if not, what changes might a trader make to correct the inaccuracy of the assumptions?

Check out Rival One Options Functionality

### Price Distribution in the Real World

Traditional pricing models assume that the percent price changes in an underlying contract are normally distributed. When these percent price changes are continuously compounded, the result is that the prices at the end of some period will be lognormally distributed. Is this a realistic assumption?

One way to answer the question is to construct a histogram, or bar graph, showing the frequency of price changes of various magnitudes. Compare the shape of the histogram to a normal distribution.

For example, the graph below is a histogram of the daily percent changes in the S&P 500 index over the ten years from 1988 through 1997. Each bar represents the number of times a price change of a given magnitude occurred, with the price changes broken down into increments of .2%.

If these price changes formed a perfectly normal distribution, which is what most pricing models assume, then there ought to be a standard distribution curve that exactly fits the data in the histogram. But in fact, one can see from the normal distribution curve, which has been overlaid on the data, that the real world (the bars in the histogram) differs significantly from the theoretical world (the normal distribution curve).

For example, one can easily see that there are more days with more minor price changes than is predicted by a normal distribution. There are also fewer days with intermediate moves than is denoted by a normal distribution. And finally, there are more days with substantial moves than is predicted by a normal distribution.

This last point that significant moves occur more often than a normal distribution predicts is especially dramatic. Although it may be difficult to see on the graph, there was one day over the ten years when prices rose by 5.05%.

Since the standard deviation for the distribution is .83%, there was one day with an upward move of greater than 6 standard deviations (5.05/.83 » 6.1 standard deviations). In fact, in a true normal distribution, we would not expect a price change of that magnitude more than once in 500 million. In addition, one-day prices fell by 6.87%, or more than 8 standard deviations (6.87/.83 » 8.2).

We would not expect to see a move of this size more than one time in 300 billion. Yet both these moves occurred over a period of 2529 days.

The S&P 500 does not seem to correspond to a normal distribution. What about other markets? Below is a histogram of the daily percent changes in the Treasury Bond futures market over the same ten-year period. Note that this distribution has similar characteristics to the S&P 500:

- More days with small price changes
- Fewer days with intermediate price changes
- More days with significant price changes

Studies that have been done confirm that almost all markets differ from a normal distribution in ways similar to those in our sample histograms. There are more days with small price changes, fewer days with intermediate price changes, and more days with significant price changes.

This leads to significant problems for anyone using a traditional option pricing model. What should a trader do if the model-generated values are based on a normal distribution, but the world is not a normal distribution?

### Price Movement in the Real World

The Black-Scholes model assumes that prices follow a continuous diffusion process. This simply means that no price gaps can occur. If one were to graph the prices of an underlying contract, one would never lift the pencil from the paper. The graph of a typical diffusion process is shown below.

##### Diffusion Process

A process whereby changes in the price of a contract do not contain any gaps. In a diffusion process, if the contract rises or falls from one price to another, the contract must also have traded at all intermediate prices.

A diffusion process is not the only way in which prices might move. In a jump process, prices are fixed for some time at a certain level until they instantaneously jump to a new level, where they again remain fixed for some time. This combination of fixed prices and instantaneous jumps repeats itself over the contract's life. The graph below represents a jump process.

##### Jump Process

A process whereby changes in the price of a contract consist only of gaps. The contract's price is fixed in a jump process until a gap occurs. It then remains fixed at the new price until another gap occurs.

The real world is a combination of these processes or a jump-diffusion process. Prices usually follow an orderly diffusion process but occasionally gap to a new level. At this point, they revert to a diffusion process. The graph below represents a jump-diffusion process.

##### Jump-diffusion Process

A process whereby changes in the price of a contract follow a diffusion process but with occasional gaps.

The fact that there may be price gaps in the real world is an essential consideration for any trader. A disciplined trader usually tries to remain delta-neutral to capture the difference between an option’s price and value. If prices follow a diffusion process, as assumed in most pricing models, a trader can adjust their position at any price level to remain delta-neutral.

But if there is a price gap, the trader cannot adjust at any price within the gap. The fact that the delta position may become instantaneously unbalanced represents a risk the trader must consider.

### Volatility in the Real World

When we feed a volatility into a theoretical pricing model, the model assumes that the volatility is the correct volatility over the option's life. Moreover, the model assumes the volatility is constant over the option's life.

If we were to graph the volatility over the option's life, the graph would be a perfectly horizontal line equal to the volatility input. But every trader knows that volatility constantly changes like everything else in the trading world. Sometimes, volatility rises, and sometimes it falls. Rarely does it stay at precisely the same level for long periods.

Even though we may have the correct volatility input into the model, how the volatility occurs may impact an option's value. For example, the underlying contract's volatility over an option's life is 15%.

Regarding model-generated values, it won't make any difference whether the volatility is a constant 15% or is 17% early in the option's life and 13% later, or 13% early and 17% after that. The model assumes that 15% is the correct volatility. In other words, the model cannot differentiate between any of these three scenarios:

Because a model assumes constant volatility when, in fact, volatility is not constant, a trader may ask whether the order in which volatility occurs, whether it is rising or falling over the life of the option, will affect model-generated values.

### Volatility and Price Level

When we feed a volatility into a pricing model, the model assumes that the volatility will be the same over the option's life, regardless of the price of the underlying contract.

For example, suppose the price of the underlying contract is currently 100.00, and we input a volatility of 15% into the model. In that case, the model will continue to make all calculations using the 15% volatility, regardless of whether the price of the underlying contract moves up to 150 or down to 50. The volatility the model uses is independent of the price of the underlying contract.

Unfortunately, this goes against most traders' experience. Some contracts, such as stock indexes and interest rates, seem more volatile at lower prices and less volatile at higher prices. Other contracts, such as physical commodities, seem to have the opposite characteristics, becoming more volatile at higher prices and less volatile at lower prices.

This might logically lead a trader to conclude that the model is wrong since the volatility depends on the underlying contract's price level. If the model is wrong, what can a trader do about it?

### Summary

- The Black-Scholes model makes several assumptions which may not be accurate when applied to the real world.
- The price changes in most markets do not appear to be normally distributed, so that prices at expiration are unlikely to be lognormally distributed.
- Because trading in the real world is not continuous, gaps can occur. Prices in the real world seem to follow a jump-diffusion process.
- Most models assume a constant volatility over the life of the option. In the real world, however, volatility sometimes rises over the life of the option and sometimes falls over the life of the option. Volatility is rarely constant over any given period of time.
- Volatility in many markets is dependent on the price of the underlying contract. Some markets become more volatile at higher prices; some markets become more volatile at lower prices.

##### Chapter 12

## Using a Volatility Skew

Because the assumptions on which theoretical pricing models are based do not seem consistent with the real world, one might wonder why anyone would use a theoretical pricing model. Almost all traders use theoretical pricing models, but they do so in a way that is likely consistent with real-world conditions.

One method of doing this is to assume that the distribution of option prices in the marketplace, as reflected in a volatility skew, contains helpful information that can be incorporated into a pricing model. This is similar to assuming that the implied volatility contains valuable information about future volatility.

If one assumes that the marketplace is reasonably efficient, then any information garnered from option prices, whether implied volatility data or skew data, will be helpful to a trader.

### Creating a Volatility Skew

The simplest way to create a volatility skew is to plot the various implied volatilities for a particular expiration month and then draw a graph that best fits this plotted data. The implied volatility data for a volatility skew is typically taken from at- and out-of-the-money options since these usually are more liquid and, therefore, a better representation of market conditions than in-the-money options.

The at-the-money implied volatilities usually average the implied volatility for the call and put. Very often, the implied volatility for an option series is calculated using the weighted average of the two closest-to-the-money calls and puts.

The chart below shows a series of implied volatilities (the red dots) for an underlying contract trading at 207.00 with 29 days remaining to expiration. A graph has been overlaid on these implied volatilities to create a representative volatility skew, which fits these points best. Note that not all points will fall precisely on the graph.

Even in an efficient market, it is unlikely that all implied volatilities will fall precisely along the graph. The bid/ask spread alone will cause option prices to oscillate through the skew. But implied volatilities that fall significantly above or below the chart represent a mispricing, either because the prices were mismarked on the close or because the option prices and underlying prices do not represent contemporaneous information. In our example, skew the 190 put and the 225 call seem slightly inflated, while the 195 put appears somewhat depressed. As trading continues, one would expect these discrepancies to disappear.

We can also use the skew to see the volatility for a theoretically at-the-money option. With the underlying price at 207.00, an option with an exercise price of 207 would have an implied volatility of 22.1%. This number is helpful since the implied volatility of a theoretically at-the-money option is often the basis for other volatility calculations.

If a trader believes that the implied volatility and the skew are accurate indicators of option values, they will use this skew as part of their evaluation process. If market conditions remain the same, the trader will assign volatilities to the various exercise prices as follows:

Exercise Price | Implied Volatility |
---|---|

180 | 32.3% |

185 | 30.3% |

190 | 28.3% |

195 | 26.5% |

200 | 24.5% |

205 | 22.7% |

210 | 21.6% |

215 | 21.5% |

220 | 22.2% |

225 | 23.0% |

230 | 23.8% |

235 | 24.5% |

### Shifting the Volatility Skew

What will happen to this skew if market conditions change? The price of the underlying contract may vary. Or implied volatility will change.

The graph below shows the original skew in the last section and the skew one day later, with the underlying contract trading at 214.00. Notice that two things have occurred. The skew has shifted to the right and downward. The shift to the right is approximately equal to the change in price of the underlying contract, or 7 points.

The shift downward is approximately 1.5 percentage points, equal to the change in implied volatility of a theoretically at-the-money option, which is now 20.5%. From this, one can conclude that the volatility skew will shift from right to left or left to right as the price of the underlying contract moves up or down, and the skew will shift upward or downward as implied volatility rises or falls.

The magnitude of the shifts will be equal to the volumes of the price change or volatility change. Option evaluation software traders will typically use these characteristics to reevaluate options under different underlying prices and volatility assumptions.

We have now focused on the implied volatility and the corresponding volatility skew to evaluate options. Suppose, however, that a trader believes that the implied volatility is either too low or too high but still believes the skew is a good representation of the relative value of options. How should the trader proceed?

The solution is to raise or lower the skew by the difference between the implied volatility and the trader’s volatility opinion. If, in the original skew, the trader believes that 24% represents more realistic volatility than 22.1% (the implied volatility), the trader can shift the whole skew upwards by 1.9 percentage points. They can then use the new volatilities to evaluate the options.

A trader can also shift their new theoretical evaluation skew to the right or left as the price of the underlying contract rises or falls. In our example, if the underlying falls to 197.00 but the trader still wants to use a volatility of 24%, they can shift the whole skew 10 points to the left.

The shifted skews in relation to the original skew are shown below.

### Summary

- A volatility skew can be created by generating a graph which best fits the implied volatilities of option across exercise prices. The data points are most often taken from at- and out-of-the-money options.
- The implied volatility for a series of options usually consists of the weighted average of the two closest-to-the-money calls and puts.
- As a trader changes their opinion about volatility, or as the price of the underlying contract moves up or down, the trader can shift the skew up or down, or from right to left, so that it is consistent with current market conditions.

##### Chapter 13

## Changing Skew Shapes

Up to now, we have assumed that the shape of the volatility skew will remain the same. Unfortunately, the shape of the skew, like everything else in the marketplace, is constantly changing. Sometimes, these changes occur because the market changes its opinion about the likelihood of different outcomes. Changes of this type can often seem random and unpredictable. At other times, however, changes in market conditions can have a more predictable effect on the shape of a volatility skew.

### How Changes in the Underlying Price Affect a Volatility Skew

In our previous examples, we assumed that if the underlying price moved up or down, the shape of the skew would remain the same. With the underlying contract at 207.00, a time to expiration of 29 days, and an at-the-money implied volatility of 22.1%, we had the following skew volatilities:

Exercise Price | Implied Volatility |
---|---|

180 | 32.3% |

185 | 30.3% |

190 | 28.3% |

195 | 26.5% |

200 | 24.5% |

205 | 22.7% |

210 | 21.6% |

215 | 21.5% |

220 | 22.2% |

225 | 23.0% |

230 | 23.8% |

235 | 24.5% |

Suppose the underlying contract were to double in price to 414.00. How would the skew look? Obviously, implied volatility would change dramatically with a move of that magnitude. But let’s assume that implied volatility remains at 22.1%.

Recall that all price relationships in theoretical pricing models are percentage relationships. With an underlying market at 100, a 105 call is 5% out-of-the-money. With an underlying market at 200, a 210 call is 5% out-of-the-money.

Therefore, if all relationships remained constant, we expect the 210 call with the market at 200 to have the same implied volatility as the 105 call at 100. Using the transformation, with the underlying contract now at 414.00, we would end up with the following implied volatilities:

Original Exercise Price | New Exercise Price | Implied Volatility |
---|---|---|

180 | 360 | 32.3% |

185 | 370 | 30.3% |

190 | 380 | 28.3% |

195 | 390 | 26.5% |

200 | 400 | 24.5% |

205 | 410 | 22.7% |

210 | 420 | 21.6% |

215 | 430 | 21.5% |

220 | 440 | 22.2% |

225 | 450 | 23.0% |

230 | 460 | 23.8% |

235 | 470 | 24.5% |

If we were to graph out the new skew with the underlying contract at 414.00, it might appear that the shape of the skew has changed. However, the skew hasn't changed regarding the percent relationship of the exercise price to the underlying price. Since the underlying price doubled, the exercise prices, when doubled, have precisely the same implied volatilities.

### How Changes in Implied Volatility Affect a Volatility Skew

What would happen to the skew if implied volatility changed dramatically but all other conditions remained the same? Suppose, for example, implied volatility were to double to 44.2%. How would the skew look?

If the implied volatility of a theoretically at-the-money option doubles, one would expect the implied volatility of each exercise price to double. So, the new skew might consist of the following implied volatilities:

Exercise Price | Original Implied Volatility | New Implied Volatility |
---|---|---|

180 | 32.3% | 64.6% |

185 | 30.3% | 60.6% |

190 | 28.3% | 56.6% |

195 | 26.5% | 53.0% |

200 | 24.5% | 49.0% |

205 | 22.7% | 45.4% |

210 | 21.6% | 53.2% |

215 | 21.5% | 43.0% |

220 | 22.2% | 44.4% |

225 | 23.0% | 46.0% |

230 | 23.8% | 47.6% |

235 | 24.5% | 49.0% |

With a new at-the-money implied volatility of 44.2%, the shape of the skew might have changed. But if we consider the implied volatility of the various exercise prices concerning the at-the-money implied volatility, the skew has not changed.

### How Changes in Time to Expiration Affect a Volatility Skew

Finally, what will happen to a volatility skew as time passes? The following graph shows the volatility skew for our underlying contract under three different times to expiration: 28 days, 14 days, and 7 days.

In each case, the underlying price is 214.00, and the implied volatility is 20.5%, so neither has changed. But the skew shape has changed, so the passage of time must be affecting the skew. The shape of the skew becomes more severe or sloped as time passes. Is this predictable?

In our skew examples, higher and lower exercise prices carry higher implied volatilities than the at-the-money exercise prices. This indicates that the marketplace thinks there is a greater likelihood of a significant move than the theoretical pricing model predicted.

But how does one define a significant move? Among other things, it depends on the amount of time to expiration. A price change of 10.00 is a much smaller move over a period of six months than the exact price change over a period of one day. Another way of expressing this relationship is to express the magnitude of a move in terms of standard deviations.

With the underlying contract at 214.00, a price change of 10.00 may represent a move of only a fraction of a standard deviation over six months, while the exact price change may represent a move of multiple standard deviations over one day. A logical consequence is that exercise prices, which are an equal number of standard deviations out-of-the-money, are likely to carry similar implied volatilities.

For example, with the underlying contract at 214.00, an implied volatility of 20.5%, and a time to expiration of 28 days, the 195 put is approximately 1.6 standard deviations out-of-the-money. However, using the same volatility and underlying price but a time to expiration of 14 days, it is now the 200 put, which is 1.6 standard deviations out-of-the-money.

And if there are only 7 days to expiration, it is the 205 put, which is 1.6 standard deviations out-of-the-money. If each exercise price (the 195 put with 28 days, the 200 put with 14 days, and the 205 put with seven days) is the same standard deviations out-of-the-money, one would expect them to have approximately the same implied volatility.

And this is indeed born out by the graph. Each exercise price has an implied volatility of approximately 27%.

Check out Rival One Options Functionality

### The Volatility Skew as a Model Input

A trader who uses a volatility skew as part of the option evaluation process can think of the skew as an additional input into the theoretical pricing model:

Most professional options software platforms will allow a trader to feed in not only the traditional inputs into the model but will also enable the trader to feed in a volatility skew. The user may be required to draw what they feel is an appropriate skew, from which the software will generate a mathematical formula to replicate the skew.

Rival One and Rival Risk automatically generate a skew that best fits the implied volatilities.

### Summary

- Changes in market conditions can cause the shape of a volatility skew to change.
- As the price of the underlying rises (falls), a volatility skew will appear to become flatter (steeper).
- As the implied volatility rises (falls), a volatility skew will appear to become steeper (flatter).
- As time passes, a volatility skew will appear to become steeper.
- One method of normalizing a volatility skew is to express the exercise price (the x-axis) in standard deviations.
- The volatility skew can be thought of as an additional input into a theoretical pricing model.

##### Chapter 14

## How Volatility Skew Affects Option Values

Any trader who expects to succeed in options needs to know the effect of changing market conditions on option values and how changes in inputs to the model will affect the risk/reward of an option position.

To achieve this, traders continually monitor the sensitivities of their position: the delta, gamma, theta, vega, and rho. Unfortunately, with the addition of a volatility skew as an input into the model, option values and the option sensitivities may not change as they do without a skew.

While including skew can affect all option sensitivities, the two most affected are the delta and the vega. For this reason, it will be worthwhile to consider how a volatility skew affects delta and vega values.

### How Changes in the Underlying Price Affect Skewed Option Values

The delta is the standard measure of how an option’s value will change if the underlying price changes while all other model inputs remain constant. If a trader uses a volatility skew as an input into the model and wants to calculate the delta, the trader must also hold the volatility skew constant.

As we saw, the skew will shift horizontally as the underlying price changes. If we assume a constant skew, we are assuming that the skew retains the same shape with respect to the price of the underlying contract.

##### Skewed Delta

The delta generated by a theoretical pricing model when the effects of a volatility skew are also included in the calculation.

For example, look at the following graph. With the underlying contract at 207, the implied volatility of the 195 put is 26.4%. But if the price of the underlying rises to 214, the skew will shift to the right. As the skew shifts, the implied volatility of the 195 put will rise to 28.9% because the implied volatility of lower exercise prices along the skew are inflated.

As the 195 put moves further out-of-the-money, its implied volatility is rising since it is moving up the skew. If we were to ignore the skew, the delta of 195 put at an underlying price of 207, and a volatility of 26.4% would be approximately -20. If the underlying were to rise (fall) 1 point, we would expect the put to fall (rise) by about .20.

But if we take into consideration the skew, where a rise in the price of the underlying contract will also cause the implied volatility of the 195 put to rise, the skewed delta of the 195 put is approximately -14. If the underlying were to rise (fall) 1 point, the put would only fall (rise) .14. Inclusion of the skew has changed the option’s delta, in this case reducing the delta. The option’s value will fall less than expected when the underlying market rises; its value will also rise less than expected when the underlying market falls.

Now consider the 230 call. With the underlying contract at 207, the implied volatility of the 230 call is 23.8%. But if the price of the underlying rises to 214, the skew will shift, causing the implied volatility of the 230 call to fall to 22.6%. As the 230 call becomes less far out-of-the-money its implied volatility is falling since it is moving down the skew. Without the skew, the delta of the 230 call at a volatility of 23.8% would be 6.

If the underlying were to rise (fall) 1 point, we would expect the call to rise (fall) by about .06. But if we include the effects of the skew, a rise in the price of the underlying contract will also cause the implied volatility of the 230 call to fall. As a result, the skewed delta of the 230 call is approximately 5. If the underlying were to rise (fall) 1 point, the call would rise (fall) only .05. Inclusion of the skew has caused the option’s value to rise less than expected when the underlying market rises. Its value will also fall less than expected when the underlying market falls.

As a general rule, when using an investment skew (lower exercise prices carry higher implied volatilities), put deltas will be less than expected, and call deltas will be greater than expected. When using a demand skew (higher exercise prices carry higher implied volatilities), put deltas will be greater than expected, and call deltas will be less than expected.

This, of course, assumes that all other market conditions, including the at-the-money volatility, remain the same. The following table compares delta values in our example with and without skew considerations. Put deltas are in parentheses.

Exercise Price | Implied Volatility | Implied Delta | Skewed Delta |
---|---|---|---|

180 | 32.3% | 94 (-6) | 96 (-3) |

185 | 30.3% | 91 (-9) | 94 (-5) |

190 | 28.3% | 86 (-13) | 91 (-9) |

195 | 26.5% | 80 (-20) | 85 (-14) |

200 | 24.5% | 70 (-30) | 77 (-22) |

205 | 22.7% | 57 (-43) | 64 (-35) |

210 | 21.6% | 42 (-58) | 44 (-55) |

215 | 21.5% | 27 (-72) | 26 (-74) |

220 | 22.2% | 17 (-82) | 15 (-85) |

225 | 23.0% | 10 (-89) | 9 (-91) |

230 | 23.8% | 6 (-93) | 5 (-95) |

235 | 24.5% | 4 (-96) | 3 (-97) |

Check out Rival One Options Functionality

### How Changes in Volatility Affect Skewed Option Values

The vega is the standard measure of how an option’s value will change if volatility changes while all other model inputs remain constant. The vega is especially useful when a trader is trying to analyze the effect of a change in implied volatility on the value of their position.

If a trader uses a volatility skew as an input into the model and wants to calculate the vega, the trader must also hold the volatility skew constant. If we assume a constant skew, we are assuming that the skew retains the same shape with respect to the volatility of a theoretically at-the-money option.

##### Skewed Vega

The vega generated by a theoretical pricing model when the effects of a volatility skew are also included in the calculation.

For example, consider what will happen to the implied volatility of the 195 put if the at-the-money implied volatility rises 4 percentage points from 22.1% to 26.1%. We can see from the graph below that the implied volatility of the 195 put will rise from 26.4% to 31.2%, a change of 4.8 percentage points. This means that for each percent change in the implied volatility of an at-the-money option, the implied volatility of the 195 put will change by

4.8 / 4.0 = 1.20.

The change in the implied volatility will be 20% greater than the change in the implied volatility of an at-the-money option. This means the vega value for the 195 put will be 20% greater than it would be if one simply ignored the volatility skew. If a trader wanted to analyze their vega risk and wanted to also include the effects of the skew on their position, they would find that the skewed vega of the 195 put is 20% greater than the vega value without taking into consideration the volatility skew.

The effect is just the opposite for the 215 call. As the implied volatility of a theoretically at-the-money option rises from 22.1% to 26.1% (4 percentage points), the implied volatility of the 215 call will rise from 21.5% to 25.3% (only 3.8 percentage points). This means that for each percent change in the implied volatility of an at-the-money option, the implied volatility of the 215 call will change by

3.8 / 4.0 = .95.

The change in the implied volatility will be 5% less than the change in the implied volatility of an at-the-money option. The vega value for the 215 call will therefore be 5% less than it would be if one simply ignored the volatility skew. If a trader wanted to analyze their vega risk and wanted to also include the effects of the skew on their position, they would find that the skewed vega of the 215 call is 5% less than the vega value without taking into consideration the volatility skew.

As a general rule, when using an investment skew (lower exercise prices carry higher implied volatilities), lower exercise prices will have vega values that are greater than expected, and higher exercise prices will have vega values that are less than expected. When using a demand skew (higher exercise prices carry higher implied volatilities), lower exercise prices will have vega values that are lower than expected, and higher exercise prices will have vega values that are greater than expected.

### Summary

- The use of a volatility skew to evaluate options will also affect the option sensitivities.
- When using an investment volatility skew put deltas will be less than expected and call deltas will greater than expected. When using a demand skew put deltas will be greater than expected and call deltas will be less than expected.
- When using an investment volatility skew lower exercise prices will have vega values which are greater than expected and higher exercise prices will have vega values which are less than expected. When using a demand volatility skew lower exercise prices will have vega values which are lower than expected and higher exercise prices will have vega values which are greater than expected.

##### Chapter 15

## Skew Trading Strategies

Just as traders make trading decisions based on directional and volatility considerations, it is also possible to make trading decisions based on volatility skew considerations. In this sense, the skew is analogous to volatility. If the marketplace is exhibiting a specific skew, what can a trader do if they feel that the skew will change or that the characteristics of the market do not justify the skew?

### Strategies Dependent on Changes in the Shape of the Skew

Suppose a trader faces an investment skew of the type shown below. Suppose the trader also has no opinion on implied volatility, but they expect the volatility skew to become flatter.

What strategies might the trader pursue?

Note that as the skew becomes flatter, the implied volatilities of the lower exercise prices fall while the implied volatilities of the higher exercise prices rise. The implied volatility of the at-the-money options remains unchanged.

Suppose a trader believes that the volatility skew is about to become flatter. In that case, the most straightforward course is to sell options at lower exercise prices and buy options at higher exercise prices. For example, one might sell twenty 195 puts and buy twenty 220 calls. If the skew becomes flatter, while all other market conditions remain the same, the price of the 195 puts will fall while the price of the 220 calls will rise.

Of course, all other market conditions are unlikely to remain unchanged. If the trader sells twenty 195 puts, each with a delta of -14, and buys twenty 220 calls, each with a delta of 15, their total delta position will be:

(-20 x -14) + (20 x +15) = +580

If they want to neutralize their delta risk, they might sell the underlying contract so that they are approximately delta neutral, in this case, by selling six underlying contracts.

Even if the trader establishes a delta-neutral spread, they still have gamma risk. If the underlying contract moves significantly, the trader will get long deltas because they own twenty calls but are only short six underlying contracts.

If the market makes a significant downward move, the trader will also get long deltas because they will be short twenty puts but are only short six underlying contracts. In other words, if the market becomes more volatile, the trader would like the volatility to result in upward rather than downward movement.

Finally, the trader will have vega risk because an increase in implied volatility will cause the price of the 195 puts to go up more quickly than the price of the 220 calls. Of course, any decline in implied volatility will help the position because the price of the 195 puts will decline more quickly than that of the 220 calls.

Instead of believing that the skew will become flatter, a trader might have the opposite opinion that the skew will become more severely sloped.

As the skew becomes steeper, the implied volatilities of the lower exercise prices rise while the implied volatilities of the higher exercise prices fall. The implied volatility of the at-the-money options remains unchanged.

Suppose a trader believes that the volatility skew is about to become steeper. In that case, the most straightforward course is to buy options at lower exercise prices and sell options at higher exercise prices. If the skew does become steeper, and there are no changes in other market conditions, the trader will profit as the price of the purchased options rises and the price of the sold options falls.

But as in the previous example, even if the trader decides to hedge the position with the underlying contract to be delta-neutral, the position still has gamma risk from a significant move in the underlying contract (in this case, a large upward movement). The position also has vega risk because a decline in implied volatility might accompany the steepening skew. If this happens, the lower exercise price options, because of their greater skewed vega values, will lose value more quickly than the higher exercise prices.

A skew strategy will always be profitable if the volatility skew changes in a way the trader can predict and all other market conditions remain unchanged. But as with any strategy, a trader must also consider what will happen if other market conditions change. If the risk of a change in other conditions is too significant, the skew strategy may not be sensible.

Check out Rival One Options Functionality

### Combining Skew and Volatility Strategies

A trader might have an opinion about the severity of the skew (whether it is too flat or too steep) and implied volatility (whether it is too high or too low). How can they combine these opinions?

In many cases, a skew strategy is simply a vertical spread. One either buys the lower exercise price and sells the higher (a bull spread) or sells the lower exercise price and buys the higher (a bear spread).

The fundamental rule for vertical spreads is that one buys an at-the-money option when implied volatility is too low and sells an at-the-money option when implied volatility is too high.

Even if the trader wants to stay delta-neutral by hedging their position against the underlying contract, this same principle can be applied to skew strategies.

For example, suppose an investment volatility skew (lower exercise prices carry higher implied volatilities) is expected to become flatter. In that case, a trader will sell options with lower exercise prices and buy options with higher exercise prices. But if the trader believes implied volatility is low, they should purchase at-the-money options as part of their skew strategy. With the underlying contract at 207, the trader might buy 205 calls and sell 195 calls.

Not only will the trader be helped by a flattening skew, but they will also be helped by an increase in implied volatility since the at-the-money options have the most significant vega. If the skew is expected to become flatter, but the trader believes implied volatility is too high, the trader should sell at-the-money options as part of their skew strategy. With the underlying at 207, they might sell 210 calls and buy 220 calls. Not only will the trader be helped by a flattening skew, but they will also be helped by a decline in implied volatility since the at-the-money options have the greatest vega.

### Summary

- In addition to volatility and directional strategies, a trader who has an opinion about the volatility skew can execute a skew strategy.
- A trader who believes a skew is too steep will try to buy options at the low end of the skew and sell options at the high end of the skew. The trader can also hedge the position with the underlying contract if they have no opinion on market direction.
- A trader who also has an opinion about volatility can include this in their skew strategy. As part of the skew strategy, when implied volatility is high a trader will try to sell options which are close to at-the-money; when implied volatility is low a trader will try to buy options which are close to at-the-money.