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Journal of Algorithms & Computational Technology

Parallel Randomized Quasi-Monte Carlo Simulation for Asian Basket Option Pricing

Yonghong Hu, Da-Qian Chen

Published Mar 1, 2012• 6 citations
Publisher

Abstract

High-dimensional derivatives pricing, such as Asian basket options, poses great computational challenges in practice. In this paper, parallel Randomized Quasi-Monte Carlo (RQMC) simulation method is investigated to tackle this kind of intractable problems. By using the intrinsic nature of “embarrassingly parallel”, parallel RQMC algorithm for Asian basket option pricing is effectively implemented using MPI. Numerical experiments are performed on supercomputer DeepComp6800 and the parallel performance is then analyzed. Our parallel algorithm has exerted perfect performance and good scalability. Parallel computing has greatly improved the computational efficiency of derivatives pricing.

DOI
10.1260/1748-3018.6.1.101

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High-dimensional derivatives pricing, such as Asian basket options, poses great computational challenges in practice. In this paper, parallel Randomized Quasi-Monte Carlo (RQMC) simulation method is investigated to tackle this kind of intractable problems. By using the intrinsic nature of “embarrassingly parallel”, parallel RQMC algorithm for Asian basket option pricing is effectively implemented using MPI. Numerical experiments are performed on supercomputer DeepComp6800 and the parallel performance is then analyzed. Our parallel algorithm has exerted perfect performance and good scalability. Parallel computing has greatly improved the computational efficiency of derivatives pricing.
Abstract Financial derivative pricing is a significant challenge in finance, involving the valuation of instruments like options based on underlying assets. While some cases have simple solutions, many require complex classical computational methods like Monte Carlo simulations and numerical techniques. However, as derivative complexities increase, these methods face limitations in computational power. Cases involving Non-Vanilla Basket pricing, American Options, and derivative portfolio risk analysis need extensive computations in higher-dimensional spaces, posing challenges for classical computers. Quantum computing presents a promising avenue by harnessing quantum superposition and entanglement, allowing the handling of high-dimensional spaces effectively. In this paper, we introduce a self-contained and all-encompassing quantum algorithm that operates without reliance on oracles or presumptions. More specifically, we develop an effective stochastic method for simulating exponentially many potential asset paths in quantum parallel, leading to a highly accurate final distribution of stock prices. Furthermore, we demonstrate how this algorithm can be extended to price more complex options and analyze risk within derivative portfolios.
The pricing of American options is one of the most challenging problems in financial engineering due to the involved optimal stopping time problem, which can be solved by using dynamic programming (DP). But applying DP is not always practical, especially when the state space is high dimensional. However, the curse of dimensionality can be overcome by Monte Carlo (MC) simulation. We can get lower and upper bounds by MC to ensure that the true price falls into a valid confidence interval. During the recent decades, progress has been made in using MC simulation to obtain both the lower bound by least-squares Monte Carlo method (LSM) and the upper bound by duality approach. However, there are few works on pricing American options using quasi-Monte Carlo (QMC) methods, especially to compute the upper bound. For comparing the sample variances and standard errors in the numerical experiments, randomized QMC (RQMC) methods are usually used. In this paper, we propose to use RQMC to replace MC simulation to compute both the lower bound (by the LSM) and the upper bound (by the duality approach). Moreover, we propose to use dimension reduction techniques, such as the Brownian bridge, principal component analysis, linear transformation and the gradients based principle component analysis. We perform numerical experiments on American–Asian options and American max-call options under the Black–Scholes model and the variance gamma model, in which the options have the path-dependent feature or are written on multiple underlying assets. We find that RQMC in combination with dimension reduction techniques can significantly increase the efficiency in computing both the lower and upper bounds, resulting in better estimates and tighter confidence intervals of the true price than pure MC simulation.
We present a theoretical framework where any randomized quasi-Monte Carlo method can be viewed and analyzed as a parameterization method for parallel quasi-Monte Carlo. We present deterministic and stochastic error bounds when different processors of the computing environment run at different speeds. We implement two parameterization methods, both based on randomized quasi-Monte Carlo, and apply them to pricing digital options and collateralized mortgage obligations. Numerical results are used to compare the parameterization methods by their parallel performance as well as their Monte Carlo efficiency.
We study the pricing of American options using least-squares Monte Carlo combined with randomized quasi-Monte Carlo (RQMC), viewed as a variance reduction method. We find that RQMC reduces both the variance and the bias of the option price obtained in an out-of-sample evaluation of the retained policy, and improves the quality of the returned policy on average. Various sampling methods of the underlying stochastic processes are compared and the variance reduction is analyzed in terms of a functional ANOVA decomposition.
ABSTRACT For a trajectory simulated from s standardized independent normal variates ε = (ε1, . . . , εs )1, the payoff of a European option can be represented as max[g(ε), 0], where the function g(ε) is assumed to be differentiable and it relates to the nature of the derivative securities. In this paper, we develop a new simulation technique by introducing an orthogonal class of transformation to ε so that the function g is instead generated from g(Aε), where A is an s-dimensional orthogonal matrix. The matrix A is optimally determined so that the effective dimension of the underlying function is minimized, thereby enhancing the quasi-Monte Carlo (QMC) method. The proposed simulation approach has the advantage of greater generality and has a wide range of applications as long as the problem of interest can be represented by g(ε). The flexibility of our proposed technique is illustrated by applying it to two high-dimensional applications: Asian basket options and European call options with a stochastic volatility model. We benchmark our proposed method against well-known efficient simulation algorithms that have been advocated in these applications. The numerical results demonstrate that our proposed technique can be an extremely powerful simulation method when combined with QMC.
As one of the major methods for the simulation of option pricing, Monte Carlo method assumes random fluctuations in the distribution of asset prices. Under certain uncertainties process, different evolution paths could be simulated so as to finally yield the expectation value of the asset price, which requires a lot of simulations to ensure the accuracy based on huge and expensive calculations. In order to solve the above computational problem, quantum Monte Carlo (QMC) has been established and applied in the relevant systems such as European call options. In this work, both MC and QM methods are adopted to simulate European call options. Based on the preparation of quantum states in QMC algorithm and the construction of quantum circuits by simulating a quantum hardware environment on a traditional computer, the amplitude estimation (AE) algorithm is found to play a secondary role in accelerating the pricing of European options. More importantly, the payoff function and the time required for the simulation in QMC method show some improvements than those in MC method.