分数阶微分方程在金融中的应用
首先我们引入了分数阶微分方程的定义及其运算方法。主要利用了Liouville Riemann 分数阶积分和Caputo 公式的转化方程来求解 阶导数。我们计算到四节近似解析解，将 4 u 结果代入原方程中，并取相应的利率和波动率的值就可以进行误差分析，分别取 )。利用 a Mathematic 软件做出图像，我们可以看出误差范围在 210
。由此可以说明分数阶微分方程具有较高的准确度。通过软件的误差分析可以证明该方法的误差较小， 但是仍需要实际的期权数据来进行验证， 即该方法是否对于欧式期权及各种奇异期权都较为准确地预测出期权价格。并且本毕业论文由于时间关系未能迭代出四阶以上的结果，因此仍需要后期简化计算过程，或者找到新的算法思路进行迭代运算。因此这两点将是分数阶求解 Scholes Black 方程后期需要做的工作。20005
关键字：分数阶微分方程 Caputo 公式 Scholes - Black 方程 误差分析
The application of fractional order differential
equation on financial
Abstract: Option pricing theory is one of the most important achievements of modern
finance theory. option pricing equation can be used to set prices of various financial
derivative products and it is an effective tool for a variety of financial derivatives
valuation. it is still a research problem for solving its analytical solution. In this paper,
we put the integer order partial differential equations generalized to differential
equations and then solve it.
Firstly, the arithmetic of fractional differential equations is used to calculate B-S
equation. The main calculation formulas are Liouville Riemann fractional integral
and Caputo fractional derivative. Because this process leads to a large amount of
calculation and time-consuming, we can get the first 4 terms. Then the plots of
residual analysis is made by using fixed rate and volatility along with different .The results show that the error range is in 2
, which can prove the fractional differential equation has high accurate.
This method has less error through residual analysis, but still need be verified by real
option data. Whether this method has better results on European style options and
other exotic options. Due to time constrains, the fifth order has not been iterated out,
so we need new ways to simplify the computational process.
Key Words: fractional differential equatio, Caputo fractional derivative,
Black-Scholes equation, residual analysis
目录
一、引言1
1.1 研究背景..1
1.2 国内外研究现状与发展趋势1
二、预备知识.2
2.1 Scholes - Black 偏微分方程2
2.1.1 Scholes - Black 偏微分方程2
2.1.2 边界条件..3
2.2 分数阶微积分5
2.2.1 分数阶微积分的相关定义.5
三、变分迭代法原理及其应用6
3.1 变分迭代法.6
3.1.1 对于部分热传导方程的求解..6
3.2 变分迭代法的应用.9
3.2.1 利用变分迭代法求解整数阶 Scholes - Black 方程..9
3.2.2 利用变分迭代法求解分数阶 Scholes - Black 方程..12
四、 创新点及前景展望..16
五、致谢.17
优尔、参考文献..18
附录19
1
一、引言
1.1 研究背景
在金融市场中，一切金融产品都是以盈利为目的的，因此在这个优胜劣汰的
环境中，会产生很多赢家。然而，在金融市场中也有很多盲目投资的人，因为没
有分析好局势行情而血本无归。因此，通过合理的手段分析金融产品并做出相应
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