Efficient European and American option pricing under a jump-diffusion process

TL;DR

This paper introduces efficient numerical methods for pricing European and American options under a jump-diffusion model, significantly reducing computational complexity through lattice truncation techniques.

Contribution

It develops a theoretical framework to reduce the complexity of lattice-based option pricing methods for jump-diffusion models, from polynomial to near-linear time.

Findings

Reduced complexity to O(n log n) for European options

Reduced complexity to O(n^2 log n) for American puts

Provides closed-form formulas for lattice truncation limits

Abstract

When the underlying asset displays oscillations, spikes or heavy-tailed distributions, the lognormal diffusion process (for which Black and Scholes developed their momentous option pricing formula) is inadequate: in order to overcome these real world difficulties many models have been developed. Merton proposed a jump-diffusion model, where the dynamics of the price of the underlying are subject to variations due to a Brownian process and also to possible jumps, driven by a compound Poisson process. Merton's model admits a series solution for the European option price, and there have been a lot of attempts to obtain a discretisation of the Merton model with tree methods in order to price American or more complex options, e. g. Amin, the $O(n^3)$ procedure by Hilliard and Schwartz and the $O(n^{2.5})$ procedure by Dai et al. Here, starting from the implementation of the seven-nodes procedure by Hilliard and Schwartz, we prove theoretically that it is possible to reduce the complexity to $O(n \ln n)$ in the European case and $O(n^2 \ln n)$ in the American put case. These theoretical results can be obtained through suitable truncation of the lattice structure and the proofs provide closed formulas for the truncation limitations.