Hedging lookback-barrier option by Malliavin calculus in a mixed fractional Brownian motion environment

Abstract

We present a Malliavin calculus approach to a mixed fractional Brownian motion option hedging model, that adequately describes, e.g., the hedging of a lookback-barrier option with the floating strike price. The Markovian setup and smooth stochastic differentials are necessary components in the payoff function for classical Δ-hedging of a contingent claim. This is in contrast to the Malliavin calculus approach, which may be used to any type of path-dependent options. Based on the fractional Clark–Ocone formula, an entirely probabilistic computation is developed to obtain the closed-form solution of the hedging portfolio for a lookback-barrier option with the floating strike price. Numerical experiments are performed to demonstrate the performance of our proposed hedging model, and we also conduct sensitivity analysis to investigate the correlation between model parameters and the hedging portfolio.