金融应用中多元跳扩散过程首达时间的蒙特卡洛模拟
Monte-Carlo Simulations of the First Passage Time for Multivariate Jump-Diffusion Processes in Financial Applications
AI总结：
针对金融中多元相关跳扩散过程的首达时间问题，提出蒙特卡洛求解方法，并实证展示其在违约率与违约相关性分析中的应用。
摘要：
金融中的许多问题需要随机过程首达时间（FPT）的信息。在数学上，此类问题通常归结为评估该过程穿越某一水平、边界或进入某一区域的时间的概率密度。虽然在其他应用领域中，FPT问题通常可以解析求解，但在金融中，我们通常不得不诉诸数值程序，特别是当我们处理跳扩散随机过程（JDP）时。在本文中，我们提出了一种基于蒙特卡洛的方法，用于求解多元（且相关）跳扩散过程背景下的首达时间问题。所开发的技术为多种应用提供了高效工具，包括信用风险和期权定价。我们通过一组经验数据，展示了该方法在分析几家不同但相关的公司的违约率和违约相关性方面的适用性。
英文摘要：
Many problems in finance require the information on the first passage time (FPT) of a stochastic process. Mathematically, such problems are often reduced to the evaluation of the probability density of the time for such a process to cross a certain level, a boundary, or to enter a certain region. While in other areas of applications the FPT problem can often be solved analytically, in finance we usually have to resort to the application of numerical procedures, in particular when we deal with jump-diffusion stochastic processes (JDP). In this paper, we propose a Monte-Carlo-based methodology for the solution of the first passage time problem in the context of multivariate (and correlated) jump-diffusion processes. The developed technique provide an efficient tool for a number of applications, including credit risk and option pricing. We demonstrate its applicability to the analysis of the default rates and default correlations of several different, but correlated firms via a set of empirical data.
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