| CHAPTER 18 |

Skewness and Kurtosis

In order to generate theoretical values that more closely approximate option prices in the marketplace, a trader may vary volatilities across exercise prices, thereby creating a volatility skew. The skew is often expressed as a mathematical function with at least two inputs: the skewness (the slope of the volatility skew) and the kurtosis (the curvature of the volatility skew).

In the following questions you are using an option pricing model that requires a volatility, skew, and kurtosis input. All options are sensitive to changes in the volatility input. The ±25 delta options are most sensitive to changes in the skewness input, and the ±5 delta options are most sensitive to changes in the kurtosis input. The at-the-money (±50 delta) options are not affected by changes in skewness or kurtosis.

1.  For each group of options below, will the given change cause the option to rise in value (+), fall in value (–), or remain unchanged (0)? If more than one option will change in the same direction, which option will change most (++ or – –)? Assume that all options expire at the same time.

a.   The volatility is increased:

b.   The skewness is reduced (becomes less positive or more negative):

c.   The skewness is increased (becomes more positive or less negative):

d.   The kurtosis is increased (becomes more positive):

e.   The kurtosis is reduced (becomes less positive):

f.   Time passes:

2.  Suppose your model is generating values that differ from the observed prices in the marketplace for options with delta values given below. The wording describes your model values compared to market prices.

What changes (increase, reduce, leave unchanged) would you need to make to the volatility, skewness, and kurtosis inputs in order to generate model prices that are more consistent with the actual market prices?

a.   

b.   

c.   

d.   

e.   

3.  You have an option position that is delta neutral and is also short skew (you want the skewness to become less positive or more negative).

a.   Which of the following positions fits the above description? (There may be more than one choice.)

______ long in-the-money puts / short in-the-money calls / short underlying contracts

______ long out-of-the-money puts / short out-of-the-money calls / long underlying contracts

______ short in-the-money puts / long in-the-money calls / long underlying contracts

______ short in-the-money puts / long in-the-money calls / short underlying contracts

______ long out-of-the-money puts / short out-of-the-money calls / short underlying contracts

______ long in-the-money puts / short in-the-money calls / long underlying contracts

Suppose your position (delta neutral / short skew) is a stock option position that consists primarily of out-of-the-money calls and out-of-the-money puts together with a stock position.

b.   If the price of the underlying stock begins to rise, what will happen to your gamma position? (get longer, get shorter, remain unchanged)

c.   If the price of the underlying stock begins to fall, what will happen to your vega position? (get longer, get shorter, remain unchanged)

d.   If two weeks pass with no change in the stock price, what will happen to your delta position? (get longer, get shorter, remain unchanged)

e.   If implied volatility rises, what will happen to your delta position? (get longer, get shorter, remain unchanged)

f.   What combination of stock direction (up or down), changes in implied volatility (rising or falling), changes in interest rates (rising or falling), and changes in dividends (increase or decrease) will most help this position if:

i.   the market moves very quickly

ii.  the market moves very slowly