Dynamic Hedging in a Volatile Market

In financial markets, errors in option hedging can arise from two sources. First, the option value is a nonlinear function of the underlying; therefore, hedging is instantaneous and hedging with discrete rebalancing gives rise to error. Frequent rebalancing can be impractical due to transaction costs. Second, errors in specifying the model for the underlying price movement (model specification error) can lead to poor hedge performance. In this article, we compare the effectiveness of dynamic hedging using the constant volatility method, the implied volatility method, and the recent volatility function method [3]. We provide evidence that dynamic hedging using the volatility function method [3] produces smaller hedge error. We assume that there are no transaction costs, and both the risk-free interest rate r and the dividend rate q are constant. Many studies have shown that the classical Black-Scholes constant volatility model does not adequately describe the stock price dynamics, see e.g., [13]. Implied volatility typically exhibits a dependence on both option strike and maturity, referred to as the volatility smile. Thus the constant volatility method, which assumes that the volatility is constant for all the options on the same underlying, can lead to a significant model specification error. To reduce this error in practice, the implied volatility method, which uses different constant volatilities for options with different maturities and strikes, is frequently used in pricing and hedging. Although the implied volatility method yields accurate valuation of vanilla options, it does not eliminate the model specification error. A volatility function method attempts to accurately approximate the unknown local volatility function σ(s, t) from the available market option prices in the context of a 1factor continuous model. The computed volatility function can then be used for pricing options and determining hedge factors. This type of method follows the work of Dupire [8] who shows that the local volatility function can be uniquely determined if the prices of European options of all strikes and maturities are available. Various methods have been proposed [1, 2, 3, 5, 6, 10, 11, 12, 13] to compute a local volatility function calibrating a finite set of market option data. In the volatility function method [3], the volatility function is represend as a 2-dimensional spline; this method is distinguished from the