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首页> 外文期刊>Proceeding of the Singapore National Academy of Science >The Black-Scholes Equation with Impulses at Random Times Via Generalized Riemann Integral
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The Black-Scholes Equation with Impulses at Random Times Via Generalized Riemann Integral

机译：布莱克-斯科尔斯随机方程的冲动通过广义黎曼积分时间

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摘要

The classical pricing theory requires that the simple sets of outcomes are extended, using the Kolmogorov Extension Theorem, to a sigma-algebra of measurable sets in an infinite-dimensional sample space whose elements are continuous paths; the process involved are represented by appropriate stochastic differential equations (using It? calculus); a suitable measure for the sample space can be found by means of the Girsanov and Radon-Nikodym Theorems; the derivative asset valuation is determined by means of an expression using Lebesgue integration. It is known that if we replace Lebesgue's by the generalized Riemann integration to obtain the expectation, the same result can be achieved by elementary methods. In this paper, we consider the Black-Scholes PDE subject to impulse action. We replace the process which follows a geometric Brownian motion by a process which has additional impulsive displacements at random times. Instead of constants, the volatility and the risk-free interest rate are considered as continuous functions which can vary in time. Using the Feynman-Ka? formulation based on generalized Riemann integration, we obtain a pricing formula for a European call option which copes with many discontinuities. This paper seeks to develop techniques of mathematical analysis in derivative pricing theory which are less constrained by the standard assumption of lognormality of prices. Accordingly, the paper is aimed primarily at analysis rather than finance. An example is given to illustrate the main results.
机译：经典的定价理论要求简单的结果集的扩展,使用柯尔莫哥洛夫扩展定理,代数可测集的无限维度样本空间的元素是连续路径;代表不同的过程适当的随机微分方程(使用它吗?样本空间可以通过找到Girsanov Radon-Nikodym定理;衍生品资产估值决定通过利用勒贝格积分的表达式。知道如果我们取代勒贝格的广义黎曼积分得到期望,可以实现相同的结果基本方法。布莱克-斯科尔斯PDE冲动的行动。我们替换的过程中遵循几何布朗运动的过程中有额外的冲动的随机位移。常量的波动率和无风险利率被认为是连续的功能,可以在不同的时间。Feynman-Ka吗?黎曼积分,我们获得一个定价公式为欧洲应对很多的看涨期权不连续。技术导数的数学分析定价理论的制约标准假设对数常态的价格。因此,本文主要针对而不是财务分析。为了说明的主要结果。

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