Fast Estimation of True Bounds on Bermudan Option Prices under Jump-diffusion Processes

TL;DR

This paper introduces a fast, efficient algorithm for estimating tight upper bounds on Bermudan option prices under jump-diffusion models, avoiding nested simulations and improving computational speed.

Contribution

The paper presents a novel non-nested Monte Carlo method leveraging martingale representation for jump processes to estimate Bermudan option bounds efficiently.

Findings

Significantly reduces computational time compared to traditional methods.

Provides theoretical guarantees for martingale approximation quality.

Numerical experiments confirm the efficiency and accuracy of the proposed algorithm.

Abstract

Fast pricing of American-style options has been a difficult problem since it was first introduced to financial markets in 1970s, especially when the underlying stocks' prices follow some jump-diffusion processes. In this paper, we propose a new algorithm to generate tight upper bounds on the Bermudan option price without nested simulation, under the jump-diffusion setting. By exploiting the martingale representation theorem for jump processes on the dual martingale, we are able to explore the unique structure of the optimal dual martingale and construct an approximation that preserves the martingale property. The resulting upper bound estimator avoids the nested Monte Carlo simulation suffered by the original primal-dual algorithm, therefore significantly improves the computational efficiency. Theoretical analysis is provided to guarantee the quality of the martingale approximation. Numerical experiments are conducted to verify the efficiency of our proposed algorithm.