Approximate hedging problem with transaction costs in stochastic volatility markets

TL;DR

This paper analyzes option hedging in stochastic volatility markets with transaction costs, introducing new asymptotic results that improve understanding of hedging errors and suggest ways to enhance convergence and reduce costs.

Contribution

It develops new limit theorems for hedging errors in stochastic volatility models with transaction costs, extending previous results and addressing under-hedging issues.

Findings

Generalized existing asymptotic results for hedging errors.

Fixed under-hedging property in Leland's algorithm.

Suggested methods to improve convergence and reduce costs.

Abstract

This paper studies the problem of option replication in general stochastic volatility markets with transaction costs, using a new specification for the volatility adjustment in Leland's algorithm \cite{Leland}. We prove several limit theorems for the normalized replication error of Leland's strategy, as well as that of the strategy suggested by L\'epinette. The asymptotic results obtained not only generalize the existing results, but also enable us to fix the under-hedging property pointed out by Kabanov and Safarian. We also discuss possible methods to improve the convergence rate and to reduce the option price inclusive of transaction costs.