Performance Measurement for Option Portfolios in a Stochastic Volatility Framework

Abstract

Measuring the performance of stock portfolios that include options is challenging due to options' nonlinearity in the underlying, their exposure to volatility risk, and their time decay. Our contribution to the literature is twofold: First, we provide a theoretically rigorous derivation of the time-variable factor loadings in a two-factor model under stochastic volatility according to Heston (1993). Within this setting, the portfolio returns are explained by the market and an additional option factor, i.e., a portfolio of standard options exposed to volatility risk. We show that (i) any option factor is suitable to perfectly explain the portfolio behavior if simple returns are considered in instantaneous time and that (ii) the option factor's loading equals the fraction of the volatility elasticities of the portfolio and of the option factor while the option factor's underlying elasticity enters the factor loading of the underlying. Second, in applications however, time has to be discretized and factor loadings are usually estimated in a single regression over a certain time horizon, which regularly leads to a bias in performance measurement. We analytically show how the discretization error arises through nonlinearity related to both risk factors, the underlying and the volatility. We run a simulation analysis to analyze the size of this bias when different option factors from the common literature are used and propose a two-step procedure to keep the bias small.