浮动利率下基于不确定分数阶微分方程的期权定价研究  

Option Pricing Based on Uncertain Fractional Differential Equation with Floating Interest Rate

作　　者：雷子琦 周清[1] LEI ZIQI;ZHOU QING(School of Science,Beijing University of Posts and Telecommunications,Beijinp 100876,China)

机构地区：[1]北京邮电大学理学院,北京100876

出　　处：《应用数学学报》2022年第3期401-420,共20页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(11871010,11971040);中央高校基本科研业务费专项资金(2019XD-A11)资助项目。

摘　　要：期权定价是金融数学领域中最复杂的问题之一.随着不确定理论公理化的建立,利用不确定理论进行期权定价的研究逐步展开,而分数阶微分方程的分数阶导数项可以很好地刻画金融市场的记忆特性.本文在机会空间中提出了一种新的不确定市场模型,假设股票价格满足Caputo型的不确定分数阶微分方程,且随机利率满足随机微分方程.基于该模型,利用Mittag-Leffler函数和微分方程的α-轨道我们给出了蝶式期权和欧式价差期权的定价公式及数值例子.Option pricing is one of the most complex problems in all financial mathematics.With the establishment of axiomatization of uncertainty theory,the research of option pricing based on uncertainty theory is gradually expanded and the fractional derivative term of fractional differential equation can well describe the memory characteristics of the market.Uncertainty and randomness are two basic types of indeterminacy.Chance space is initialized for modelling complex systems with not only uncertainty but also randomness.In order to model the financial market with uncertainty,randomness and memory characteristics,this paper presents a new version of uncertain market model on a chance space by assuming that stock price satisfies an uncertain fractional differential equation for Caputo type and stochastic interest rate satisfies a stochastic differential equation.By using the Mittag-Leffler function andα-path of the differential equation,the pricing formulas of butterfly option and European spread option based on the proposed model are investigated as well as some numerical examples.

关 键 词：机会理论 不确定分数阶微分方程 蝶式期权 欧式价差期权 

分 类 号：O211.6[理学—概率论与数理统计]