Rival Systems Options Knowledge Base
Advanced Guide to Options
- Select Your Level
- Beginner
- Intermediate
- Advanced
Chapter 02
Option Sensitivities
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
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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
Chapter 05
Arbitrage Strategies
Chapter 06
Arbitrage Risk
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.
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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
Chapter 09
The Effect of Early Exercise on Arbitrage Strategies
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.
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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?
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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%.
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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) |
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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.
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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.