Pricing Fractal Derivatives under Sub-Mixed Fractional Brownian Motion with Jumps

TL;DR

This paper introduces a novel model for pricing derivatives using sub-mixed fractional Brownian motion with jumps, deriving a PIDE, establishing solution properties, and providing numerical methods validated by empirical tests.

Contribution

It develops a new fractional jump process model for derivatives, deriving a governing PIDE, proving solution existence, and offering a numerical scheme with empirical validation.

Findings

The model captures market memory and jump features effectively.

Closed-form solutions are derived for European options.

Numerical scheme demonstrates stability and convergence.

Abstract

We study the pricing of derivative securities in financial markets modeled by a sub-mixed fractional Brownian motion with jumps (smfBm-J), a non-Markovian process that captures both long-range dependence and jump discontinuities. Under this model, we derive a fractional integro-partial differential equation (PIDE) governing the option price dynamics. Using semigroup theory, we establish the existence and uniqueness of mild solutions to this PIDE. For European options, we obtain a closed-form pricing formula via Mellin-Laplace transform techniques. Furthermore, we propose a Grunwald-Letnikov finite-difference scheme for solving the PIDE numerically and provide a stability and convergence analysis. Empirical experiments demonstrate the accuracy and flexibility of the model in capturing market phenomena such as memory and heavy-tailed jumps, particularly for barrier options. These results underline the potential of fractional-jump models in financial engineering and derivative pricing.