Semi-analytic pricing of American options in time-dependent jump-diffusion models with exponential jumps

TL;DR

This paper introduces a semi-analytic method for pricing American options in complex time-dependent jump-diffusion models with exponential jumps, generalizing previous approaches and providing explicit formulas.

Contribution

The paper develops a novel semi-analytic approach that handles arbitrary time dependencies and simplifies the American option pricing problem to solving algebraic and integral equations.

Findings

Provides a closed-form solution for American options in jump-diffusion models.

Reduces complex pricing to solving algebraic and Fredholm-Volterra equations.

Extends previous methods to more general time-dependent models.

Abstract

In this paper we propose a semi-analytic approach to pricing American options for time-dependent jump-diffusions models with exponential jumps The idea of the method is to further generalize our approach developed for pricing barrier, [Itkin et al., 2021], and American, [Carr and Itkin, 2021; Itkin and Muravey, 2023], options in various time-dependent one factor and even stochastic volatility models. Our approach i) allows arbitrary dependencies of the model parameters on time; ii) reduces solution of the pricing problem for American options to a simpler problem of solving a system of an algebraic nonlinear equation for the exercise boundary and a linear Fredholm-Volterra equation for the the option price; iii) the options Greeks solve a similar Fredholm-Volterra linear equation obtained by just differentiating Eq. (25) by the required parameter. Once done, the American option price is presented in close form.