When You Hedge Discretely: Optimization of Sharpe Ratio for Delta-Hedging Strategy under Discrete Hedging and Transaction Costs

We consider the delta-hedging strategy for a vanilla option under the discrete hedging and transaction costs. Assuming that the option is delta-hedged using the Black-Scholes-Merton model with an implied log-normal volatility, we analyze the profit-and-loss (P&L) of the delta-hedging strategy given that the actual underlying dynamics are driven by one of four alternative models: log-normal diffusion, jump-diffusion, stochastic volatility and stochastic volatility with jumps. For all of the four cases, we derive approximations for the expected P&L, expected transaction costs, and P&L volatility assuming hedging at fixed times.Using these results, we formulate the problem of finding the optimal hedging frequency that maximizes the Sharpe ratio of the delta-hedging strategy. We also show how to apply our results to spot- and delta-based hedging strategies. Finally, we provide illustrations.