Pricing and Hedging Asian Basket Options with Quasi-Monte Carlo Simulations

TL;DR

This paper develops efficient Quasi-Monte Carlo methods for pricing and hedging high-dimensional Asian basket options in a Black-Scholes market with time-dependent volatilities, introducing novel algorithms for path generation and delta computation.

Contribution

It introduces a new fast Cholesky algorithm for block matrices and a Kronecker Product Approximation technique for efficient path generation in high-dimensional settings.

Findings

The new algorithms reduce computational time significantly.

The methods achieve the same accuracy as traditional techniques.

Applicable to stochastic volatility models with multi-dimensional dynamics.

Abstract

In this article we consider the problem of pricing and hedging high-dimensional Asian basket options by Quasi-Monte Carlo simulation. We assume a Black-Scholes market with time-dependent volatilities and show how to compute the deltas by the aid of the Malliavin Calculus, extending the procedure employed by Montero and Kohatsu-Higa (2003). Efficient path-generation algorithms, such as Linear Transformation and Principal Component Analysis, exhibit a high computational cost in a market with time-dependent volatilities. We present a new and fast Cholesky algorithm for block matrices that makes the Linear Transformation even more convenient. Moreover, we propose a new-path generation technique based on a Kronecker Product Approximation. This construction returns the same accuracy of the Linear Transformation used for the computation of the deltas and the prices in the case of correlated asset returns while requiring a lower computational time. All these techniques can be easily employed for stochastic volatility models based on the mixture of multi-dimensional dynamics introduced by Brigo et al. (2004).