The Generalized extreme value (GEV) distribution, the implied tail index and option pricing Sheri Markose and Amadeo Alentorn Papers available at: June QQASS Conference – Brunel University

2 Contribution of the paper We develop an new option pricing model using the Generalized Extreme Value (GEV) distribution, which: –Removes pricing biases associated with Black- Scholes –Capture the stylized facts of the implied RNDs: Left skewness Excess kurtosis (fat tails) –Has a closed form solution for the European options –Delivers the market implied tail shape

3 The GEV distribution A distribution from Extreme Value Theory. It is the limiting distribution of block maxima. The standardized GEV distribution is given by: where: – μ is the location parameter – σ is the scale parameter – ξ is the tail shape parameter

4 The GEV for different values of ξ

5 Option pricing approach Our option pricing approach is based on the estimating the implied Risk Neutral Density (RND) using traded option prices. Following Harrison and Pliska (1981): there exists a risk neutral density (RND) function, g(S T ), such that the call option price can be written as: where E Q [] is the risk-neutral expectation operator, conditional on all information available at time t.

6 The Risk Neutral Density (RND) The RND is the risk neutral market expectation of the underlying price distribution at maturity T. We can extract it using a cross section of traded option prices with different strikes but same maturity.

7 Assumptions of the model We assume that the terminal distribution at maturity of negative returns (i.e. losses) follow a GEV distribution. Losses are defined as follows: Note we use simple returns, rather than log returns. This was found to be necessary in order to obtain a closed form solution (but numerical experiments showed that prices using log returns are very similar).

8 Solving the pricing equation The density function of the price (if negative returns are GEV distributed) is given by: And the call option pricing equation that needs to be solved is:

9 The GEV closed form solution Using the Gamma function, we obtain the closed form solution of the call option pricing equation under GEV returns: where: We obtain a similar equation for put options.

10 Parametric RND estimation For a given day, we have a set of N traded option prices with the same maturity, but different strikes. Using a non-linear least squares algorithm, we find the set of parameters Ө that minimize the sum of squared errors: Subject to the martingale condition E Q (S T )= F t,T

11 Results: Pricing performance We use closing prices of FTSE 100 index options from 1997 to The GEV model is found to outperform the Black-Scholes model, specially for long maturities. Time to maturity ModelBSGEVBSGEVBSGEVBSGEV Calls Puts

12 Results: Pricing bias (90 days) The GEV model removes the pricing bias (Price bias = Market price – Calculated price )

13 Results: Pricing bias (10 days) The GEV model removes the pricing bias (Price bias = Market price – Calculated price )

14 The implied tail shape parameter ξ

15 The time to maturity effect When calculating a time series of implied RNDs or RND related statistics we encounter the time to maturity effect. This happens due to the fixed maturity of index options. As we approach maturity, the time horizon of the implied RND shortens, the degree of uncertainty decreases, and thus, densities of consecutive days are not directly comparable. We need a method to remove this time to maturity effect.

16 Term structure of RNDs In most markets, there are option contracts trading for different maturities T 1, T 2, … T N. For example, for the FTSE 100, there are traded options with maturities on the closest 3 months, and also quarterly (Mar, Jun, Sep, and Dec). At any given day, we can extract a term structure of RNDs from traded option prices.

17

18 The Implied RND surface Usually, when estimating implied RNDs, a separate density is obtained for each maturity. For a given day, we obtained a different set of parameters for each of the maturities. Now, we will consider an implied RND surface for the GEV model. For a given day, we want to obtain a unique set of parameters, consistent with option prices for all strikes and maturities. This is similar to implied volatility surfaces, where the aim is to have a parametric model that fits option prices across both dimensions.

19 The implied RND surface The implied RND surface as a function of time and price. Index level

20 Extension of the GEV model We extend the GEV model to make the parameters independent of time. We modify the model by scaling two of the parameters, using the equations for the mean and variance of the GEV distribution:

21 Scaling of GEV parameters Using the GEV mean equation and the martingale condition, we can rewrite the location parameter μ to be a function of the two other parameters, and the known Futures price F t,T : Using the GEV variance equation, we introduce a new parameter b, which explicitly models the scaling of implied volatility:

22 Estimation of the RND surface The optimization problem is now across both strikes and maturities: The parameter b gives the scaling law for the implied volatility (assumed to be 0.5 in Black-Scholes). We find that in 86% of days, H 0 : b = 0.5 can be rejected. The implied tail shape parameter ξ controls the fatness of the left tail, and can be used to asses (risk neutral) market expectations of extreme outcomes.

23 Implied tail shape parameter Asian crisisLTCM9/11

24 Historical tail shape We can estimate the corresponding historical tail shape using the rolling window method for the Hill index in Quintos, Fan and Phillips (2001)

25 Event studies By interpolating/extrapolating the implied RND surface, we can obtain constant time horizon RNDs, which are useful when conducting event studies. Around 9/11, ξ went from (before) to (after)

26 Conclusions Using the GEV distribution for option pricing yields a model able to fit traded option prices accurately. It removes the well known pricing bias of the Black-Scholes model. The flexibility of the GEV distribution allows us to capture the implied RNDs with only 3 parameters, and is able to model different levels of skewness and kurtosis. Unlike other models, we don’t have to specify the type of distribution a priori (i.e. Weibull, Fréchet, Gumbel). The GEV model also delivers the implied tail shape parameter, which controls the implied skewness and the fatness of the tails, and can be used to asses (risk neutral) market expectations of extreme outcomes. It is found to increase around crisis periods. Finally, the implied RND surface is a useful method for removing maturity effects of implied RNDs and related statistics. It has useful applications such as in event studies.