Numerical analysis for Spread option pricing model in illiquid underlying asset market: full feedback model

TL;DR

This paper develops a numerical method for pricing Spread options in illiquid markets using a full-feedback model, addressing the impact of liquidity on option valuation and demonstrating the method's stability and applicability.

Contribution

It introduces a full-feedback model incorporating price impact into Spread option pricing and employs an asymptotic expansion with Peaceman-Rachford scheme for numerical solutions.

Findings

Liquidity increases lead to more stock being bought for replication.

The numerical scheme is stable and convergent.

Illiquidity significantly affects option pricing compared to Black-Scholes.

Abstract

This paper performs the numerical analysis and the computation of a Spread option in a market with imperfect liquidity. The number of shares traded in the stock market has a direct impact on the stock's price. Thus, we consider a full-feedback model in which price impact is fully incorporated into the model. The price of a Spread option is characterize by a nonlinear partial differential equation. This is reduced to linear equations by asymptotic expansions. The Peaceman-Rachford scheme as an alternating direction implicit method is employed to solve the linear equations numerically. We discuss the stability and the convergence of the numerical scheme. Illustrative examples are included to demonstrate the validity and applicability of the presented method. Finally we provide a numerical analysis of the illiquidity effect in replicating an European Spread option; compared to the Black-Scholes case, a trader generally buys more stock to replicate this option.