欧式看涨期权定价微分方程的有限差分求解方法

发布时间：2017-12-30 23:38

本文关键词：欧式看涨期权定价微分方程的有限差分求解方法　出处：《哈尔滨工业大学》2012年硕士论文　论文类型：学位论文

【摘要】：本文主要研究讨论Black-Scholes欧式期权定价微分方程在股票价格波动率为常数和非常数两种情况下的有限差分方法求解过程和差分格式稳定性及收敛性分析。本文的研究属于计算数学中微分方程数值解法理论和金融数学的期权定价理论相交叉的新型领域。Black和Scholes于上世纪70年代初开创性的给出了一种能够针对金融市场中的期权定价的经典公式，这一重大发现推动了金融学科和数学学科的融合，促进了金融行业的科学、快速发展。本文对抛物型微分方程的有限差分方法进行了系统总结，并对各类格式的相容性、稳定性和收敛性进行了分析，为运用差分方法求解期权定价微分方程奠定了基础。本文从期权这一金融衍生工具的起源及其相关理论进行了深入分析，并设定了贴近于金融市场实际情况的假设条件，运用随机过程、投资组合和微分方程等相关理论推导了Black-Scholes期权定价微分方程。通过对微分方程定解区域进行网格剖分，对微分方程中各微分项及初始、边界条件进行差分化处理后，给出了相应的差分方程组和求解方法。本文首先对无交易成本欧式看涨期权定价微分方程运用有限差分方法进行了求解，并对差分格式进行了截断误差分析，并且通过Fourier分析方法对差分格式的稳定性、收敛性进行了讨论，给出了相应的定理。进一步，，本文对更贴近于现实情况的股票价格波动率为非常数的欧式看涨期权定价微分方程进行离散化处理，亦通过差分方法对其进行了求解，并对差分格式进行了稳定性等分析。最后，通过数值模拟试验对两种情况下的差分格式稳定性等性质加以验证。本文的研究对完善期权定价理论，及如何更加稳定、准确的获得期权定价进行了深入研究，对数学理论更加广泛的应用于金融行业起到了促进作用。
[Abstract]:This article mainly discusses the Black-Scholes European option pricing equation in stock price volatility is constant and the number of very finite difference case method solving process and analysis of differential format stability and convergence.
Option pricing theory this study belongs to the calculation of differential equations in mathematics theory and numerical solution of mathematical finance crossing the field of.Black and Scholes model in the last century, the early 70s invasively provides a classic formula for option pricing in financial markets, a major discovery that promotes the integration of financial discipline and mathematics the financial industry, promote the rapid development of science.
In this paper, the finite difference method for parabolic differential equations is systematically summarized, and the compatibility, stability and convergence of various schemes are analyzed. It lays the foundation for solving differential equations of options with difference method.
This paper makes a deep analysis from the origin of the option of financial derivatives and related theories, and the assumed conditions are close to the actual situation of the financial market, the use of stochastic process, portfolio theory and differential equation of Black-Scholes option pricing differential equation is derived.
Based on the differential equations of regional mesh, the differential equation of the differential term and initial boundary conditions, poor differentiation treatment, given the corresponding differential equations and solving method. Firstly, no transaction costs of European option pricing differential equation is solved by finite difference method. And the difference format of the truncation error analysis method, and the stability of difference scheme by Fourier analysis, convergence is discussed, given the corresponding theorem. Further, the more close to the reality of the volatility of the stock price for European call option pricing equation is the number of discretization also through the difference method for solving them, and the difference of stability analysis. Finally, through the numerical simulation test of two cases of the stability of difference scheme Quality is verified.
This research has done in-depth research on the option pricing theory and how to get the option pricing more accurately and steadily, which has promoted the wider application of mathematical theory to the financial industry.

【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：F830.91;F224

【共引文献】