1. MGT 821/ECON 873 Volatility Smiles & Extension of Models

2. What is a Volatility Smile?
It is the relationship between implied volatility and strike price for options with a certain maturity
The volatility smile for European call options should be exactly the same as that for European put options
The same is at least approximately true for American options

3. Why the Volatility Smile is the Same for Calls and Put
Put-call parity p +S0e-qT = c +Ke–r T holds for market prices (pmkt and cmkt) and for Black-Scholes prices (pbs and cbs)
It follows that pmkt−pbs=cmkt−cbs
When pbs=pmkt, it must be true that cbs=cmkt
It follows that the implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity

4. The Volatility Smile for Foreign Currency Options
Implied
Volatility
Strike
Price

5. Implied Distribution for Foreign Currency Options
Both tails are heavier than the lognormal distribution
It is also “more peaked” than the lognormal distribution

6. The Volatility Smile for Equity Options
Implied
Volatility
Strike
Price

7. Implied Distribution for Equity Options
The left tail is heavier and the right tail is less heavy than the lognormal distribution

8. Other Volatility Smiles?
What is the volatility smile if
True distribution has a less heavy left tail and heavier right tail
True distribution has both a less heavy left tail and a less heavy right tail

9. Ways of Characterizing the Volatility Smiles
Plot implied volatility against K/S0 (The volatility smile is then more stable)
Plot implied volatility against K/F0 (Traders usually define an option as at-the-money when K equals the forward price, F0, not when it equals the spot price S0)
Plot implied volatility against delta of the option (This approach allows the volatility smile to be applied to some non-standard options. At-the money is defined as a call with a delta of 0.5 or a put with a delta of −0.5. These are referred to as 50-delta options)

10. Possible Causes of Volatility Smile
Asset price exhibits jumps rather than continuous changes
Volatility for asset price is stochastic
In the case of an exchange rate volatility is not heavily correlated with the exchange rate. The effect of a stochastic volatility is to create a symmetrical smile
In the case of equities volatility is negatively related to stock prices because of the impact of leverage. This is consistent with the skew that is observed in practice

11. Volatility Term Structure
In addition to calculating a volatility smile, traders also calculate a volatility term structure
This shows the variation of implied volatility with the time to maturity of the option

12. Volatility Term Structure
The volatility term structure tends to be downward sloping when volatility is high and upward sloping when it is low

13. Example of a Volatility Surface

14. Greek Letters
If the Black-Scholes price, cBS is expressed as a function of the stock price, S, and the implied volatility, simp, the delta of a call is
Is the delta higher or lower than

15. Three Alternatives to Geometric Brownian Motion
Constant elasticity of variance (CEV)
Mixed Jump diffusion
Variance Gamma

16. CEV Model When a = 1 the model is Black-Scholes
When a > 1 volatility rises as stock price rises
When a < 1 volatility falls as stock price rises
European option can be value analytically in terms of the cumulative non-central chi square distribution

17. CEV Models Implied VolatilitiesK

18. Mixed Jump Diffusion k is the expected size of the jump
Merton produced a pricing formula when the asset price follows a diffusion process overlaid with random jumps
dp is the random jump
k is the expected size of the jump
l dt is the probability that a jump occurs in the next interval of length dt

19. Jumps and the Smile
Jumps have a big effect on the implied volatility of short term options
They have a much smaller effect on the implied volatility of long term options

20. The Variance-Gamma Model
Define g as change over time T in a variable that follows a gamma process. This is a process where small jumps occur frequently and there are occasional large jumps
Conditional on g, ln ST is normal. Its variance proportional to g
There are 3 parameters
v, the variance rate of the gamma process
s2, the average variance rate of ln S per unit time
q, a parameter defining skewness

21. Understanding the Variance-Gamma Model
g defines the rate at which information arrives during time T (g is sometimes referred to as measuring economic time)
If g is large the change in ln S has a relatively large mean and variance
If g is small relatively little information arrives and the change in ln S has a relatively small mean and variance

22. Time Varying Volatility
Suppose the volatility is s1 for the first year and s2 for the second and third
Total accumulated variance at the end of three years is s12 + 2s22
The 3-year average volatility is

23. Stochastic Volatility Models
When V and S are uncorrelated a European option price is the Black-Scholes price integrated over the distribution of the average variance

24. Stochastic Volatility Models continued
When V and S are negatively correlated we obtain a downward sloping volatility skew similar to that observed in the market for equities
When V and S are positively correlated the skew is upward sloping. (This pattern is sometimes observed for commodities)

25. The IVF Model

26. The Volatility Function
The volatility function that leads to the model matching all European option prices is

27. Strengths and Weaknesses of the IVF Model
The model matches the probability distribution of asset prices assumed by the market at each future time
The models does not necessarily get the joint probability distribution of asset prices at two or more times correct