**Financial Engineering With Extreme Events and Computational Finance**

**Forecasting Extreme Volatility of FTSE-100 With Model Free VFTSE, Carr-Wu and Generalized Extreme Value (GEV) Option Implied Volatility Indices (2012)**

**Forecasting Extreme Volatility of FTSE-100 With Model Free VFTSE, Carr-Wu and Generalized Extreme Value (GEV) Option Implied Volatility Indices (2012)**

###### Abstract:

Since its introduction in 2003, volatility indices such as the VIX based on the model-free implied volatility (MFIV) have become the industry standard for assessing equity market volatility. MFIV suffers from estimation bias which typically underestimates volatility during extreme market conditions due to sparse data for options traded at very high or very low strike prices, Jiang and Tian (2007). To address this problem, we propose modifications to the CBOE MFIV using Carr and Wu (2009) moneyness based interpolations and extrapolations of implied volatilities and so called GEV-IV derived from the Generalised Extreme Value (GEV) option pricing model of Markose and Alentorn (2011). GEV-IV gives the best forecasting performance when compared to the model-free VFTSE, Black-Scholes IV and the Carr-Wu case, for realised volatility of the FTSE-100, both during normal and extreme market conditions in 2008 when realised volatility peaked at 80%. The success of GEV-IV comes from the explicit modelling of the implied tail shape parameter and the time scaling of volatility in the risk neutral density which can rapidly and flexibly reflect extreme market sentiments present in traded option prices.

Markose, Sheri M and Peng, Yue and Alentorn, Amadeo (2012) ‘Forecasting Extreme Volatility of FTSE-100 With Model Free VFTSE, Carr-Wu and Generalized Extreme Value (GEV) Option Implied Volatility Indices.’ Economics Department Discussion Paper No. 713, University of Essex.

**T****he Generalized Extreme Value (GEV)Distribution, Implied Tail Index and Option Pricing (2005)**

**T**

**he Generalized Extreme Value (GEV)Distribution, Implied Tail Index and Option Pricing (2005)**

###### Abstract:

Crisis events such as the 1987 stock market crash, the Asian Crisis and the bursting of the Dot-Com bubble have radically changed the view that extreme events in financial markets have negligible probability. This paper argues that the use of the Generalized Extreme Value (GEV) distribution to model the Risk Neutral Density (RND) function provides a flexible framework that captures the negative skewness and excess kurtosis of returns, and also delivers the market implied tail index of asset returns. We obtain an original analytical closed form solution for the Harrison and Pliska (1981) no arbitrage equilibrium price for the European option in the case of GEV asset returns. The GEV based option prices successfully remove the well known pricing bias of the Black-Scholes model. We explain how the implied tail index is efficacious at identifying the fat tailed behaviour of losses and hence the left skewness of the price RND functions, particularly around crisis events.

Markose, Sheri M and Alentorn, Amadeo (2005) The Generalized Extreme Value (GEV) Distribution, Implied Tail Index and Option Pricing. Working Paper. Economics Department, University of Essex, Discussion Paper 594.

**Papers from the EDDIE Project**

**Papers from the EDDIE Project**

**Generalized Extreme Value Distribution and Extreme Economic Value at Risk (EE-VaR) (2008)**

**Generalized Extreme Value Distribution and Extreme Economic Value at Risk (EE-VaR) (2008)**

###### Abstract:

In 2000, Ait-Sahalia and Lo have argued that Economic VaR (E-VaR) calculated under option market implied risk neutral density (RND) is a more relevant measure of risk than historically based VaR. As industry practice requires VaR at high confidence level of 99%, Extreme Economic Value at Risk (EE-VaR) based on the Generalized Extreme Value (GEV) distribution has been proposed as a new risk measure. This follows from a GEV option pricing model developed by Markose and Alentorn in 2005 which shows that the GEV implied RND can accurately capture negative skewness and fat tails, with the latter explicitly determined by the market implied tail index. Here, the term structure of the GEV based RNDs is estimated which permits the calibration of an empirical scaling law for EE-VaR, and thus, obtain daily EE-VaR for any time horizon. Backtesting results for the FTSE 100 index from 1997 to 2003, show that EE-VaR has fewer violations than historical VaR. Further, there are substantial savings in risk capital with EE-VaR at 99% as compared to historical VaR corrected by a factor of 3 to satisfy the violation bound. The efficiency of EE-VaR arises because an implied VaR estimate responds quickly to market events and in some cases even anticipates them. In contrast, historical VaR reflects extreme losses in the past for longer.

Markose, S.M., and A. Alentorn (2008) Generalized Extreme Value Distribution and Extreme Economic

Value at Risk (E-EVaR). *Chapter in Computational Methods in Financial Engineering edited by EJ Kontoghiorghes, B. Rustem and P. Winker in honour of Manfred Gilli, Springer Verlag. *

**Removing Maturity Effects of Implied Risk Neutral Densities and Related Statistics (2006)**

**Removing Maturity Effects of Implied Risk Neutral Densities and Related Statistics (2006)**

###### Abstract:

When studying a time series of implied Risk Neutral Densities (RNDs) or other implied statistics, one is faced with the problem of maturity dependence, given that option contracts have a fixed expiry date. Therefore, estimates from consecutive days are not directly comparable. Further, we can only obtain implied RNDs for a limited set of expiration dates. In this paper we introduce two new methods to overcome the time to maturity problem. First, we propose an alternative method for calculating constant time horizon Economic Value at Risk (EVaR), which is much simpler than the method currently being used at the Bank of England. Our method is based on an empirical scaling law for the quantiles in a log-log plot, and thus, we are able to interpolate and extrapolate the EVaR for any time horizon. The second method is based on an RND surface constructed across strikes and maturities, which enables us to obtain RNDs for any time horizon. Removing the maturity dependence of implied RNDs and related statistics is useful in many applications, such as in (i) the construction of implied volatility indices like the VIX, (ii) the assessment of market uncertainty by central banks (iii) time series analysis of EVaR, or (iv) event studies.

Alentorn, Amadeo and Markose, Sheri M (2006) Removing Maturity Effects of Implied Risk Neutral Densities and Related Statistics. *WP002-06: Centre for Computational Finance and Economic Agents*

**Evolutionary Decision Trees For Stock Index Options And Futures Arbitrage (2002)**

**Evolutionary Decision Trees For Stock Index Options And Futures Arbitrage (2002)**

###### Abstract:

EDDIE-ARB (EDDIE stands for Evolutionary Dynamic Data Investment Evaluator) is a genetic program (GP) that implements a cross market arbitrage strategy in a manner that is suitable for online trading. Our benchmark for EDDIE-ARB is the Tucker (1991) put-call-futures (P-C-F) parity condition for detecting arbitrage profits in the index options and futures markets. The latter presents two main problems, (i) The windows for profitable arbitrage opportunities exist for short periods of one to ten minutes, (ii) Prom a large domain of search, annually, fewer than 3% of these were found to be in the lucrative range of £500-£800 profits per arbitrage. Standard ex ante analysis of arbitrage suffers from the drawback that the trader awaits a contemporaneous signal for a profitable price misalignment to implement an arbitrage in the same direction. Execution delays imply that this naive strategy may fail. A methodology of random sampling is used to train EDDIE-ARB to pick up the fundamental arbitrage patterns. The further novel aspect of EDDIE-ARB is a constraint satisfaction feature supplementing the fitness function that enables the user to train the GP how not to miss opportunities by learning to satisfy a minimum and maximum set on the number of arbitrage opportunities being sought. Good GP rules generated by EDDIE-ARB are found to make a 3-fold improvement in profitability over the naive ex ante rule.

Markose, S., Tsang, E. and Er, H. (2002). Evolutionary Decision Trees for Stock Index Options and Futures Arbitrage. *Genetic Algorithms and Genetic Programming in Computational Finance*, pp.281-308, Edited by Shu-Heng Chen, *Kluwer Academic Publishers*,(ISBN 0-7923-7601-3).

**The Black (1976) Effect And Cross Market Arbitrage In FTSE-100 Index Futures And Options (2000)**

**The Black (1976) Effect And Cross Market Arbitrage In FTSE-100 Index Futures And Options (2000)**

Markose, Sheri M and Er, Hakan (2000) The Black (1976) effect and cross market arbitrage in FTSE-100 index futures and options. Economics Department Discussion Paper,No. 522, University of Essex.