Approximating Option Prices Under Large Changes Of Underlying Asset Prices

When one invests in portfolios of derivatives (such as options), the delta-gamma approximation (DGA) is often used as a risk management strategy to reduce the risk associated with the underlying asset price. However, this approximation is locally accepted only for small changes of the underlying asset price. When these changes become large, the option prices estimated by the DGA may significantly differ from those of the market (or those that are estimated using, for instance, the Black–Scholes model), depending mainly on the time-to-maturity, implied volatility, and moneyness. Hence, in practice, before the change of the underlying asset price becomes large, rebalancing operations are demanded to minimize the losses occurred due to the error introduced by the DGA. The frequency of rebalancing may be high when the rate at which the underlying asset price significantly changes. Nonetheless, frequent rebalancing may be unattainable, as there are associated transaction costs. Hence, there is a trade-off between the losses resulting from the inaccurate performance of the DGA and the transaction costs incurring from frequent hedging operations. In the present work, we show two approaches that can outperform the DGA, in this way to increase the accuracy of estimating the option prices with the ultimate goal of reducing the losses due to the estimation error. The first method is similar to the DGA but we change the reference value that the DGA uses (that is, the initial price of the underlying asset) to the underlying asset price forecasted for the time horizon. We coin this method as the extended delta-gamma approximation (EDGA). The second method that we consider in this work is the locally weighted regression (LWR) that locally regresses the option prices from the changes of the underlying asset prices, with the same reference value that is employed in the EDGA method. Finally, we compare the performance of the two methods presented in this work to that of some existing methods.