Dynamic Hedging With a Deterministic Local Volatility Function Model

We compare the dynamic hedging performance of the deterministic local volatility function approach with the implied/constant volatility method. Using an example in which the underlying price follows an absolute diffusion process, we illustrate that hedge parameters computed from the implied/constant volatility method can have significant error even though the implied volatility method is able to calibrate the current option prices of different strikes and maturities. In particular the delta hedge parameter produced by the implied/constant volatility method is consistently significantly larger than that of the exact delta when the underlying price follows an absolute diffusion. In order to compute a better hedge parameter, accurate estimation of the local volatility function in a region surrounding the current asset price is crucial. We illustrate that a suitably implemented volatility function method can estimate this local volatility function sufficiently accurately to generate more accurate hedge pa-rameters. Hedging using this volatility function for the absolute diffusion example leads to a smaller average absolute hedging error when compared with using the implied/constant volatility rate. When comparing the hedging performance in the S&P 500 index option market as well as the S&P 500 futures option market, we similarly observe that the delta hedge parameter from the implied/constant volatility method is typically greater than that using the volatility function approach. Examination of the hedging error reveals that using a larger delta factor greater than that of the true volatility yields more positive 1 average hedging error, assuming the underlying follows a deterministic volatility model. We observe that, in both the S&P 500 index option market and futures option market, the average absolute hedging error using the volatility function approach is smaller than that of the implied/constant volatility method for a sufficiently long hedging horizon, approximately 17 days for the S&P 500 index options and 6 days for the S&P 500 futures options. In addition, the average hedging error using the volatility function approach is always smaller than that of the implied/constant volatility method. 1