Approximate hedging with proportional transaction costs in stochastic volatility models with jumps

TL;DR

This paper extends the approximate hedging framework with proportional transaction costs to models combining stochastic volatility and jumps, showing that jump risk can be mitigated and classical results still hold.

Contribution

It demonstrates that the Leland adjusting volatility principle effectively applies to jump-diffusion models, generalizing prior diffusion-only results.

Findings

Transaction costs can be approximately offset using Leland's method.

Jump risk can be effectively eliminated under mild jump size conditions.

Classical asymptotic results are valid in jump-diffusion models with deterministic volatility.

Abstract

We study the problem of option replication under constant proportional transaction costs in models where stochastic volatility and jumps are combined to capture the market's important features. Assuming some mild condition on the jump size distribution we show that transaction costs can be approximately compensated by applying the Leland adjusting volatility principle and the asymptotic property of the hedging error due to discrete readjustments is characterized. In particular, the jump risk can be approximately eliminated and the results established in continuous diffusion models are recovered. The study also confirms that for the case of constant trading cost rate, the approximate results established by Kabanov and Safarian (1997)and by Pergamenschikov (2003) are still valid in jump-diffusion models with deterministic volatility using the classical Leland parameter in Leland (1986).