Pricing Multidimensional Financial Derivatives with Stochastic Volatilities Using the Dimensional-Adaptive Combination Technique

IF 0.8 4区 经济学 Q4 BUSINESS, FINANCE Journal of Computational Finance Pub Date : 2017-11-29 DOI:10.21314/JCF.2017.335
J. Benk, D. Pflüger
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Abstract

In this paper, we present a new and general approach to price derivatives based on the Black–Scholes partial differential equation (BS-PDE) in a multidimensional setting. The first ingredient in our approach is the dimensional-adaptive sparse grid combination technique, which, in the case of underlying models with stochastic volatilities, allows for inhomogeneous discretization levels of the dimensional axes. Thus, by applying the dimensional-adaptive combination technique to such problems, one may achieve higher numerical efficiency. We combine this approach with a stretched grid discretization that is derived from the underlying’s stochastic differential equation (SDE) in a general manner. This stretching enables us to employ efficient geometrical multigrid solvers, even for the strong anisotropic convection and diffusion coefficients that frequently occur in application. Our combination of the dimensional-adaptive sparse grid combination technique with SDE-based grid stretching and an efficient multigrid solver represents a new approach designed to enable derivative pricing by directly solving PDEs in higher dimensions than were possible before. The numerical results outlined in the paper demonstrate the efficacy of this new approach and of our implementation method, which entails pricing various derivatives with up to twelve dimensions in a general and simple manner.
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基于维数自适应组合技术的随机波动多维金融衍生品定价
在本文中,我们提出了一个新的和一般的方法,基于布莱克-斯科尔斯偏微分方程(BS-PDE)在多维设置的价格衍生品。我们方法的第一个组成部分是维度自适应稀疏网格组合技术,该技术在具有随机波动的底层模型的情况下,允许维度轴的非均匀离散化水平。因此,将维数自适应组合技术应用于此类问题,可以获得更高的数值效率。我们将这种方法与从底层随机微分方程(SDE)以一般方式导出的拉伸网格离散化相结合。这种扩展使我们能够使用有效的几何多网格求解器,即使是在应用中经常出现的强各向异性对流和扩散系数。我们将维度自适应稀疏网格组合技术与基于sde的网格拉伸和高效的多网格求解器相结合,代表了一种新的方法,旨在通过在比以前可能的更高维度上直接求解pde来实现衍生品定价。文中概述的数值结果证明了这种新方法和我们的实施方法的有效性,该方法需要以一般和简单的方式为多达十二个维度的各种衍生品定价。
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来源期刊
CiteScore
0.90
自引率
0.00%
发文量
8
期刊介绍: The Journal of Computational Finance is an international peer-reviewed journal dedicated to advancing knowledge in the area of financial mathematics. The journal is focused on the measurement, management and analysis of financial risk, and provides detailed insight into numerical and computational techniques in the pricing, hedging and risk management of financial instruments. The journal welcomes papers dealing with innovative computational techniques in the following areas: Numerical solutions of pricing equations: finite differences, finite elements, and spectral techniques in one and multiple dimensions. Simulation approaches in pricing and risk management: advances in Monte Carlo and quasi-Monte Carlo methodologies; new strategies for market factors simulation. Optimization techniques in hedging and risk management. Fundamental numerical analysis relevant to finance: effect of boundary treatments on accuracy; new discretization of time-series analysis. Developments in free-boundary problems in finance: alternative ways and numerical implications in American option pricing.
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