Sharif Mozumder

Revisiting Variance Gamma pricing : An application to S&P500 index options

This paper reformulates the Levy–Kintchine formula to make it suitable for modeling the stochastic time-changing effects of Levy processes. Using the variance gamma (VG) process as an example, it illustrates the dynamic properties of a Levy process and revisits the earlier work of Geman (2002). It also shows how the model can be calibrated to price options under a Levy VG process, and calibrates the model on recent S&P500 index options data. It then compares the pricing performance of fast Fourier transform (FFT) and fractional Fourier transform (FRFT) approaches to model calibration and investigates the trade-off between calibration performance and required calculation time.

Pricing and hedging options with GARCH-stable proxy volatilities

ABSTRACT This article considers modelling nonnormality in return with stable Paretian (SP) innovations in generalized autoregressive conditional heteroskedasticity (GARCH), exponential generalized autoregressive conditional heteroskedasticity (EGARCH) and Glosten-Jagannathan-Runkle generalized autoregressive conditional heteroskedasticity (GJR-GARCH) volatility dynamics. The forecasted volatilities from these dynamics have been used as a proxy to the volatility parameter of the Black–Scholes (BS) model. The performance of these proxy-BS models has been compared with the performance of the BS model of constant volatility. Using a cross section of S&P500 options data, we find that EGARCH volatility forecast with SP innovations is an excellent proxy to BS constant volatility in terms of pricing. We find improved performance of hedging for an illustrative option portfolio. We also find better performance of spectral risk measure (SRM) than value-at-risk (VaR) and expected shortfall (ES) in estimating option portfolio risk in case of the proxy-BS models under SP innovations. Abbreviation: generalized autoregressive conditional heteroskedasticity (GARCH), exponential generalized autoregressive conditional heteroskedasticity (EGARCH) and Glosten-Jagannathan-Runkle generalized autoregressive conditional heteroskedasticity (GJR-GARCH)

Back-testing extreme value and Lévy value-at-risk models: Evidence from international futures markets

Purpose - The purpose of the paper is to back-test value-at-risk (VaR) models for conditional distributions belonging to a Generalized Hyperbolic (GH) family of Levy processes – Variance Gamma, Normal Inverse Gaussian, Hyperbolic distribution and GH – and compare their risk-management features with a traditional unconditional extreme value (EV) approach using data from future contracts return data of S&P500, FTSE100, DAX, HangSeng and Nikkei 225 indices. Design/methodology/approach - The authors apply tail-based and Levy-based calibration to estimate the parameters of the models as part of the initial data analysis. While the authors utilize the peaks-over-threshold approach for generalized Pareto distribution, the conditional maximum likelihood method is followed in case of Levy models. As the Levy models do not have closed form expressions for VaR, the authors follow a bootstrap method to determine the VaR and the confidence intervals. Finally, for back-testing, they use both static calibration (on the entire data) and dynamic calibration (on a four-year rolling window) to test the unconditional, independence and conditional coverage hypotheses implemented with 95 and 99 per cent VaRs. Findings - Both EV and Levy models provide the authors with a conservative proportion of violation for VaR forecasts. A model targeting tail or fitting the entire distribution has little effect on either VaR calculation or a VaR model’s back-testing performance. Originality/value - To the best of the authors’ knowledge, this is the first study to explore the back-testing performance of Levy-based VaR models. The authors conduct various calibration and bootstrap techniques to test the unconditional, independence and conditional coverage hypotheses for the VaRs.

MARKET RISK OF INVESTMENT IN US SUBPRIME CRISIS: COMPARISON OF A PURE DIFFUSION AND A PURE JUMP MODEL

We consider the oldest financial model to estimate the market risk of investment underlying the world indexes and compare its risk management features with those of a newer model. Our concern is the risk underlying the world indexes in the recent US subprime crisis period. We illustrate how the recent variance gamma (VG) pure jump model is comparable with one of the earliest pure diffusion (Bachelier (BC)) model in estimating investment risk in financial markets using the tail risk measure value-at-risk (VaR) and its coherent version expected shortfall (ES). We observe that for pure jump VG model single quantile VaR is consistently a better performer across indexes; however for tail average risk measure ES, VG is not a consistently better performer; pure diffusion Bachelier model gives ES estimates which are often — not always — better than VG. This provides one more empirical indication that the combination of diffusion and jump is likely to be more effective in turbulent times, e.g., in US subprime crisis period.