An Approximate Distribution of Delta-Hedging Errors in a Jump-Diffusion Model with Discrete Trading and Transaction Costs

We introduce a jump-diffusion model for asset returns with jumps drawn from a mixture of normal distributions and show that this model adequately fits the historical data of the Samp;P500 index. We consider delta-hedging strategy for vanilla options under the diffusion model (DM) and the proposed jump-diffusion model (JDM) assuming discrete trading intervals and transaction costs, and derive an approximation for the probability density function (PDF) of the profit-and-loss (Pamp;L) of the delta-hedging strategy under the both models. We find that, under the log-normal model by Black-Scholes-Merton, the actual PDF of the Pamp;L can be well approximated by the chi-squared distribution with specific parameters. We derive an approximation for the Pamp;L volatility in the DM and JDM. We show that, under the both DM and JDM, the expected loss due to transaction costs is inversely proportional to the square root of the hedging frequency. We apply the mean-variance analysis to find the optimal hedging frequency given the hedger's risk tolerance. Since under the JDM it is impossible to reduce the Pamp;L volatility by increasing the hedging frequency, we consider an alternative hedging strategy, following which the Pamp;L volatility can be reduced by increasing the hedging frequency