Jump Models with delay -- option pricing and logarithmic Euler-Maruyama scheme

TL;DR

This paper develops a mathematical framework for delayed stochastic differential equations with jumps, applies it to option pricing, and introduces a new numerical scheme with proven convergence for simulation.

Contribution

It establishes existence, uniqueness, and positivity of solutions, derives analytical option prices, and proposes a logarithmic Euler-Maruyama scheme with convergence proof.

Findings

Analytical European option prices obtained using Fourier transform.

Proposed scheme maintains positivity of solutions.

Convergence rate of the scheme is 0.5.

Abstract

In this paper, we obtain the existence, uniqueness and positivity of the solution to delayed stochastic differential equations with jumps. This equation is then applied to model the price movement of the risky asset in a financial market and the Black-Scholes formula for the price of European options is obtained together with the hedging portfolios. The option price is evaluated analytically at the last delayed period by using the Fourier transformation technique. But in general, there is no analytical expression for the option price. To evaluate the price numerically we then use the Monte-Carlo method. To this end, we need to simulate the delayed stochastic differential equations with jumps. We propose a logarithmic Euler-Maruyama scheme to approximate the equation and prove that all the approximations remain positive and the rate of convergence of the scheme is proved to be $0.5$.