随机微分方程的Picard逼近及其在LIBOR模型中的应用
Picard approximation of stochastic differential equations and application to LIBOR models
摘要：
本文的目的是为LIBOR市场模型中衍生品的蒙特卡罗定价提供快速准确的近似方案。标准方法可用于求解连续LIBOR利率的随机微分方程，但这些方法通常很慢。我们的贡献是双重的。首先，我们提出了一个替代的近似计划的基础上Picard迭代。这种方法在精度上类似于欧拉离散化，但其特点是每个利率的演变独立于期限结构中的其他利率。这使得能够使用并行计算同时计算不同到期日的衍生品价格。其次，在由跳跃过程驱动的LIBOR市场模型的漂移中出现的产品项作为利率数量的函数呈指数增长，迅速使模型变得难以处理。我们减少这种增长从指数到二次使用截断扩展的产品条款。我们包括我们的方法定价caplet，掉期和远期利率协议的准确性和速度的数值说明。
英文摘要：
The aim of this work is to provide fast and accurate approximation schemes for the Monte Carlo pricing of derivatives in LIBOR market models. Standard methods can be applied to solve the stochastic differential equations of the successive LIBOR rates but the methods are generally slow. Our contribution is twofold. Firstly, we propose an alternative approximation scheme based on Picard iterations. This approach is similar in accuracy to the Euler discretization, but with the feature that each rate is evolved independently of the other rates in the term structure. This enables simultaneous calculation of derivative prices of different maturities using parallel computing. Secondly, the product terms occurring in the drift of a LIBOR market model driven by a jump process grow exponentially as a function of the number of rates, quickly rendering the model intractable. We reduce this growth from exponential to quadratic using truncated expansions of the product terms. We include numerical illustrations of the accuracy and speed of our method pricing caplets, swaptions and forward rate agreements.
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