1. Centre of Computational Finance and Economic Agents
Option Pricing and Implied Tail Indices under the Generalised Extreme Value (GEV) Distribution
Amadeo Alentorn
Centre of Computational Finance and Economic Agents
University of Essex, UK

2. Roadmap Objective of the paper Preliminaries on option pricing
The model
The data
Results
Conclusions
Further work

3. Objectives of the paper
To use the Generalized Extreme Value (GEV) distribution in a new option pricing model to:
Remove pricing biases associated with Black-Scholes
Capture the stylized facts of the price implied RND:
Left skewness
Excess kurtosis (fat tail)
Obtain a closed form solution for the European option price
Extract the market implied tail index for asset returns

4. The Risk Neutral Density (RND)
Using traded option prices, we can extract the Risk Neutral market expectation of the underlying price distribution at maturity T

5. Literature review
Popular methods of extracting RNDs from option prices include:
Parametric methods:
Generalized distributions (Beta, Lambda, etc)
Mixture of log normals
Non-parametric methods
Spline method (Shimko)

6. No-arbitrage European option price
Following Harrison and Pliska (1981), there exists a risk neutral density (RND) function, g(ST), such that the equilibrium call option price can be written as:
where EQ[] is the risk-neutral expectation operator, conditional on all information available at time t.

7. Graphically, this means…
Integrating the density function times the payoff from strike price K (for a Call):

8. The GEV distribution The standardized GEV distribution is given by:
where:
μ is the location parameter
σ is the scale parameter
ξ is the shape parameter

9. The GEV for different values of ξ

10. The GEV assumption
We assume that the negative returns (i.e. the losses) follow a GEV distribution, and are defined as follows:
When the shape parameter ξ > 0 the GEV distribution is of the type Fréchet and has a fat tail, being held to arise in the case of losses and hence negative returns (see Dowd 2002, p.272).

11. The model
The density function of the price (under GEV returns) is given by:
And the call option pricing equation that needs to be solved is:

12. The call option closed form solution
The closed form solution of the call option pricing equation under GEV returns is:
where:
We obtain a similar equation for put options.

13. Methodology of RND estimation
For a given day, we have a set of N traded option prices with the same maturity, but different strikes.
We use a non-linear least squares algorithm to find the set of parameters that minimize the sum of squared errors:

14. Options data from www.liffe-data.com
ATM

15. The data
We used closing prices of traded European options on the FTSE 100.
Period of study 1997 – 2003
Interest rate used: 3-months LIBOR rate, which approximating the actual market borrowing and lending rates faced by option market participants
Estimated RNDs for puts and calls separately

16. Results: Pricing performance
Pricing performance: the GEV model outperforms the Black-Scholes (BS) model at all times to maturity.
Time  90 60 30 10
Model BS
GEV
Calls
11.72
1.13
9.37
1.2
5.6
0.85
3.48
0.67
Puts
14.87
1.35
12.26
1.21
7.46
1.1
4.09
0.84

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

18. Results: Pricing bias (10 days)
For short time horizons, both models improve, but the GEV model has a smaller error.
(Price bias = Market price – Calculated price )

19. Results: Implied tail index
Shape parameter ξ for put options from GEV returns – 2003

20. RNDs before and after 9/11 events

21. Conclusions
Modelling the negative returns with the GEV yields an accurate option pricing model, which removes the pricing biases of the Black-Scholes model.
The flexibility of the GEV distribution allows us to capture the RNDs with only 3 parameters, while other methods use more, like the mixture of log normals with 5 parameters.
From the event study, we find implied RNDs reflect the market sentiment of increased fear of downward moves, but do not predict them.

22. Further work: E-VaR
Calculate Economic-Value-at-Risk, E-VaR (Ait-Sahalia and Lo, 2000) and compare it with the statistical or historical VaR.
Issues:
We need to calculate the daily 10 day VaR, but the option contracts only expiry once a month
The return distribution is NOT of daily returns, but of period returns: from t until maturity date T
Therefore, we will need to apply some time-scaling, but the popular square root of time should not be used.

23. Future work: Hedging
Performance of delta hedging of this GEV model, using the closed form solution of the delta (call price sensitivity to changes in underlying)