Option pricing in constant elasticity of variance model with liquidity costs

TL;DR

This paper extends the analysis of illiquidity costs on option hedging to the constant elasticity of variance model, deriving PDE systems for optimal strategies and highlighting the significance of small transaction costs.

Contribution

It generalizes previous models to the CEV framework, providing new PDE-based solutions for optimal hedging under nonlinear liquidity costs.

Findings

Optimal hedging strategies are characterized by a system of three PDEs.

Small transaction costs have a non-negligible impact on option valuation.

The model reduces to Black-Scholes as a special case when elasticity parameter is set accordingly.

Abstract

Paper is based on "The cost of illiquidity and its effects on hedging", L. C. G. Rogers and Surbjeet Singh, 2010. We generalize its thesis to constant elasticity model, which own previously used Black-Schoels model as a special case. The Goal of this article is to find optimal hedging strategy of European call/put option in illiquid environment. We understand illiquidity as a non linear transaction cost function depending only on rate of change of our portfolio. In case this function is quadratic, optimal policy is given by system of 3 PDE. In addition we show, that for small $\epsilon$ costs of selling portfolio in time $T$ be important ($O(\epsilon)$) and shouldn't be neglected in Value function ($o(\epsilon^k)$- our result).