多层热方程在金融中的应用
Multilayer heat equations: application to finance
摘要：
本文发展了一种求解单因子抛物型方程的多层（ML）方法。我们的方法提供了一个强大的替代著名的有限差分和蒙特卡罗方法。我们讨论了这种方法的各种优点，明智地结合了半解析和数值技术，并提供了一个快速，准确的方法找到相应的方程的解决方案。为了介绍该方法的核心，我们考虑多层热方程，在物理学中已知的时间相对较长，但从未在解决金融问题时使用。因此，我们扩展了量化金融的分析机制，通过增加ML方法。我们演示了如何使用我们的方法解决各种金融数学问题。具体来说，我们开发了有效的算法定价障碍期权的时间依赖的单因素短期利率模型，如Black-Karasinski和Verhulst。此外，我们还展示了如何快速准确地求解著名的Dupire方程。数值算例表明，该方法比传统的有限差分法更有效地求解相应的偏微分方程，而且比已知的方法更快、更精确。
英文摘要：
In this paper, we develop a Multilayer (ML) method for solving one-factor parabolic equations. Our approach provides a powerful alternative to the well-known finite difference and Monte Carlo methods. We discuss various advantages of this approach, which judiciously combines semi-analytical and numerical techniques and provides a fast and accurate way of finding solutions to the corresponding equations. To introduce the core of the method, we consider multilayer heat equations, known in physics for a relatively long time but never used when solving financial problems. Thus, we expand the analytic machinery of quantitative finance by augmenting it with the ML method. We demonstrate how one can solve various problems of mathematical finance by using our approach. Specifically, we develop efficient algorithms for pricing barrier options for time-dependent one-factor short-rate models, such as Black-Karasinski and Verhulst. Besides, we show how to solve the well-known Dupire equation quickly and accurately. Numerical examples confirm that our approach is considerably more efficient for solving the corresponding partial differential equations than the conventional finite difference method by being much faster and more accurate than the known alternatives.
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