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

Journal of Investment Strategies, 2013, Vol. 3, No. 1, pp. 19-59

37 Pages Posted: 19 Jun 2011 Last revised: 8 Dec 2015

See all articles by Artur Sepp

## Artur Sepp

Sygnum Bank Asset Management

Date Written: September 8, 2013

### Abstract

We consider the delta-hedging strategy for a vanilla option under the discrete hedging and transaction costs, assuming that an option is delta-hedged using the Black-Scholes-Merton model with the log-normal volatility implied by the market price of the option. We analyze the expected profit-and-loss (P&L) of the delta-hedging strategy assuming the four possible dynamics of asset returns under the statistical measure: the log-normal diffusion, the jump-diffusion, the stochastic volatility and the stochastic volatility with jumps. For all of the four models, we derive analytic formulas for the expected P&L, expected transaction costs, and P&L volatility assuming hedging at fixed times. Using these formulas, we formulate the problem of finding the optimal hedging frequency to maximize the Sharpe ratio of the delta-hedging strategy. Also, we show that the Sharpe ratio of the delta-hedging strategy can be improved by incorporating the price and delta bands for the rebalancing of the delta-hedge and provide analytical approximations for computing the optimal bands in our optimization approach. As illustrations, we show that our method provides a very good approximation to the actual Sharpe ratio obtained by Monte Carlo simulations under the time-based re-hedging. In contrary to Monte Carlo simulations, our analytic approach provide a fast and an accurate way to estimate the risk-reward characteristic of the delta-hedging strategy for real time computations.

Keywords: delta-hedging errors, profit & loss distribution, discrete trading, transaction costs, parameters misspecification, jump-diffusion model, stochastic volatility, Sharpe ratio

JEL Classification: C00, C0, G00

Suggested Citation

Sepp, Artur, When You Hedge Discretely: Optimization of Sharpe Ratio for Delta-Hedging Strategy under Discrete Hedging and Transaction Costs (September 8, 2013). Journal of Investment Strategies, 2013, Vol. 3, No. 1, pp. 19-59, Available at SSRN: https://ssrn.com/abstract=1865998 or http://dx.doi.org/10.2139/ssrn.1865998