算子分裂方法在金融中的应用
Application of Operator Splitting Methods in Finance
摘要：
金融衍生工具定价的目的是找到一个金融合同的基础资产的公允价值。在这里，我们考虑期权定价的偏微分方程框架。当代模型导致一维或多维对流扩散型抛物问题及其推广。概述了各种算子分裂方法，这些问题的有效数值解。讨论了多维问题的交替方向隐式（ADI）型分裂格式，例如由随机波动率（SV）模型给出的多维问题。对于跳跃模型，隐式显式（IMEX）方法被认为是有效地处理非局部跳跃算子。对于美式期权的线性互补问题，给出了一种易于实现的算子分裂方法。数值实验证明了分裂格式的稳定性和收敛性。本文在经典的布莱克-斯科尔斯模型、Merton跳跃扩散模型、Heston SV模型和带跳跃的Bates SV模型四种资产价格模型下研究了欧式和美式看跌期权。
英文摘要：
Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or multidimensional parabolic problems of the convection-diffusion type and generalizations thereof. An overview of various operator splitting methods is presented for the efficient numerical solution of these problems. Splitting schemes of the Alternating Direction Implicit (ADI) type are discussed for multidimensional problems, e.g. given by stochastic volatility (SV) models. For jump models Implicit-Explicit (IMEX) methods are considered which efficiently treat the nonlocal jump operator. For American options an easy-to-implement operator splitting method is described for the resulting linear complementarity problems. Numerical experiments are presented to illustrate the actual stability and convergence of the splitting schemes. Here European and American put options are considered under four asset price models: the classical Black-Scholes model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV model with jumps.
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