MGT 821/ECON 873 Volatility Smiles & Extension of Models

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

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

The Volatility Smile for Foreign Currency Options Implied Volatility Strike Price

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

The Volatility Smile for Equity Options Implied Volatility Strike Price

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

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

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)

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

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

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

Example of a Volatility Surface

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

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

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

CEV Models Implied Volatilities K

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

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

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

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

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

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

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)

The IVF Model

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

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