When you hedge discretely: optimization of the Sharpe ratio for the Delta-hedging strategy under discrete hedging and transaction costs

ABSTRACT We consider the Delta-hedging strategy for a vanilla option under discrete hedging and transaction costs. Assuming that the option is Delta-hedged using the Black-Scholes-Merton model with an implied lognormal 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: lognormal 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 formaulate 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 price- and Delta-based hedging strategies. Finally, we provide illustrations.