Lévy betas: Static hedging with index futures

Abstract

This study considers calibration to forward-looking betas by extracting information on equity and index options from prices using Lévy models. The resulting calibrated betas are called Lévy betas. The objective of the proposed approach is to capture market expectations for future betas through option prices, as betas estimated from historical data may fail to reflect structural change in the market. By assuming a continuous-time capital asset pricing model (CAPM) with Lévy processes, we derive an analytical solution to index and stock options, thus permitting the betas to be implied from observed option prices. One application of Lévy betas is to construct a static hedging strategy using index futures. Employing Hong Kong equity and index option data from September 16, 2008 to October 15, 2009, we show empirically that the Lévy betas during the sub-prime mortgage crisis period were much more volatile than those during the recovery period. We also find evidence to suggest that the Lévy betas improve static hedging performance 2 Wong, Cheung, and WongJournal of Futures Markets relative to historical betas and the forward-looking betas implied by a stochastic volatility model.

Equity Option Pricing with Systematic and Idiosyncratic Volatility and Jump Risks

“…Based on Theorems 1 and 2, we employed a two step calibration procedure (see, for example, Wong et al 2012;Christoffersen et al 2018) to estimate the model parameters. First, we calibrated the market index dynamic Θ I based on the S&P 500 index option price alone.…”

“…Supposing that stock return is linearly related to market index return in terms of the beta parameter, Carr and Madan (2012) developed a factor model for individual equity option pricing under a purely discontinuous Lévy process via fast Fourier transform, in which the variance gamma process for the dynamics of both the market index and stock was taken as an example for illustration. By supposing a continuous-time CAPM with Lévy processes, Wong et al (2012) also derived analytical solutions to the index and equity options and explored the corresponding static hedging with index futures. Christoffersen et al (2018) empirically studied the equity volatility levels, skews, and term structures by using equity option prices and principal component analysis.…”

Mathematical Finance with Applications

“…Supposing that stock return is linearly related to market index return in terms of the beta parameter, Carr and Madan ( 2012) developed a factor model for individual equity option pricing under a purely discontinuous Lévy process via fast Fourier transform, in which the variance gamma process for the dynamics of both the market index and stock was taken as an example for illustration. By supposing a continuous-time CAPM with Lévy processes, Wong et al (2012) also derived analytical solutions to the index and equity options and explored the corresponding static hedging with index futures. Christoffersen et al (2018) empirically studied the equity volatility levels, skews, and term structures by using equity option prices and principal component analysis.…”