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

ArXiv ID: 2308.08760 “View on arXiv”

Authors: Unknown

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.

Keywords: American Options, Jump-Diffusion Models, Fredholm-Volterra Equations, Option Pricing, Greeks, Options (Derivatives)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper presents a highly mathematical framework involving partial integro-differential equations, integral transforms, and nonlinear algebraic systems, but offers no backtests or numerical implementations, focusing instead on theoretical derivations.

  flowchart TD
    A["Research Goal"] --> B["Methodology: Semi-Analytic Generalization"]
    B --> C["Formulate Problem: American Option PDE with Jumps"]
    C --> D["Key Steps: Reduce to Algebraic & Fredholm-Volterra Equations"]
    D --> E["Input Parameters: Time-Dependent Models & Exponential Jumps"]
    E --> F["Computation: Solve Exercise Boundary & Option Price/Greeks"]
    F --> G["Outcome: Closed-Form Solution for Price & Greeks"]