# Option Pricing in Illiquid Markets with Jumps

## TL;DR

This paper extends the Black--Scholes model to illiquid markets with jumps, incorporating feedback effects from large traders and Levy processes, resulting in a nonlinear PDE for option pricing.

## Contribution

It generalizes the Frey--Stremme model to include jump processes and derives a nonlinear PDE accounting for market illiquidity and large trader influence.

## Key findings

- Large trader impact significantly alters option prices.
- Jumps in asset prices affect the nonlinear PDE solution.
- Numerical experiments illustrate the effects of jumps and trader influence.

## Abstract

The classical linear Black--Scholes model for pricing derivative securities is a popular model in financial industry. It relies on several restrictive assumptions such as completeness, and frictionless of the market as well as the assumption on the underlying asset price dynamics following a geometric Brownian motion. The main purpose of this paper is to generalize the classical Black--Scholes model for pricing derivative securities by taking into account feedback effects due to an influence of a large trader on the underlying asset price dynamics exhibiting random jumps. The assumption that an investor can trade large amounts of assets without affecting the underlying asset price itself is usually not satisfied, especially in illiquid markets. We generalize the Frey--Stremme nonlinear option pricing model for the case the underlying asset follows a Levy stochastic process with jumps. We derive and analyze a fully nonlinear parabolic partial-integro differential equation for the price of the option contract. We propose a semi-implicit numerical discretization scheme and perform various numerical experiments showing influence of a large trader and intensity of jumps on the option price.