Existence and Uniqueness for Lévy Option Pricing PIDEs

Summary

The paper studies multidimensional nonlocal partial integro-differential equations (PIDEs) used in financial models with jumps. It applies abstract semilinear parabolic equation theory to establish existence and uniqueness of solutions in Bessel potential spaces, extending earlier results from one spatial dimension to multiple dimensions. The analysis covers a broad class of Lévy measures subject to growth conditions near zero and at infinity.

For option pricing, the paper considers Black–Scholes models whose underlying assets follow Lévy processes with jumps. It also examines a one-dimensional nonlinear pricing application in which a shift function can depend on a prescribed large investor’s stock-trading strategy. The results establish well-posedness for this mathematical framework; the document gives no numerical pricing comparison or empirical trading evidence, and its stated application is limited to the described model settings.

Key ideas

• The analysis extends PIDE existence and uniqueness results from one spatial dimension to multidimensional spaces.
• Solutions are studied in a scale of Bessel potential spaces.
• The framework accommodates Lévy measures meeting specified growth conditions near the origin and at infinity.
• The option-pricing applications use jump-driven Lévy processes and include a nonlinear setting tied to a large investor’s trading strategy.

Tags

# 2106.10498


# Multidimensional linear and nonlinear partial integro-differential equation in Bessel potential spaces with applications in option pricing









The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black-Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from nonlinear option pricing models taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function.

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