布萊克-休斯之分數微分方程之探討

Study to The Fractional Black-Scholes Equation

洪子恩(Tzu-En Hung)

指導教授 ： 林賜德

中原大學/理學院/應用數學系/碩士(2008年)

https://doi.org/10.6840/cycu200800467

摘要

選擇權在現今社會中，是十分普遍的投資工具，它吸引人的地方在於，付少部分的權利金即可交易，槓桿效益很高，許多企業也會以選擇權做為避險工具。選擇權如何訂出合理價格，一直是學者所努力的目標。西元一九七三年費雪•布萊克及麥倫•休斯所提出的布萊克-休斯選擇權評價模型 (以下簡稱布萊克-休斯模型)，直到今日仍是選擇權定價模式的基礎。本文主要探討布萊克-休斯模型，從一開始的模型假設直到解出封閉解的過程，更以不同的角度去了解布萊克休斯偏微分方程，如：無風險規避法、複製法、風險貼水的市場均衡價格法。無風險規避法是說假設一投資組合包含某單位的衍生性金融商品及某單位股票，則此投資組合無風險；複製法是以無風險債券及股票做一個投資組合而此投資組合為自我融資的交易策略，藉由此投資組合來複製當作歐式買權的收益；風險貼水的市場均衡價格法是假設衍生性金融商品和股票每單位風險補償相同。以上三種方法皆可導出布萊克休斯偏微分方程。解方程式則用了熱擴散方程法以及二項式模型；二項式模型用了Cox、Ross、Rubinstein所提出的CRR二元樹的概念推導選擇權期初價格，所得到的二項式模型極限值將逼近於布萊克-休斯模型。本文以分數微積分的方法，去解釋布萊克休斯偏微分方程，而得到布萊克休斯分數微分方程式，發現經過兩次變數變換後，當參數α=1，β=0時仍可以用熱擴散方程式的方法去解出布萊克休斯偏微分方程式的封閉解。

關鍵字

布萊克-休斯偏微分方程 ； 分數微積分

並列摘要

In nowadays, options are very common in investment. They are so charming because we just pay little money, then we can trade. Options have high lever benefit. Many enterprises also take it to be a hedge instrument. How to decide reasonable price of options are experts trying hard. In 1973, Fischer Black and Myron Scholes advanced Black-Scholes Options Pricing Model(for short B-S model), until nowadays, also the base of the pricing of option. In this research, studying B-S model is central; from how to assume the B-S model to how to solve the close form of Black-Scholes partial differential equation, and we use others method to solve it. For example: riskless hedge, replication and risk premium of balance market. All of three methods can introduce Black-Scholes partial differential equation. We use heat equation and binomial model to solve the function. In this research, we use the fractional calculus to explain Black-Scholes partial differential equation. We can solve the close form of Black-Scholes partial differential equation by using twice changing variables. So, in this research, we find that the close form of Black-Scholes partial differential equation is a special case of the fractional Black-Scholes equation.

並列關鍵字

fractional calculus ； Black-Scholes partial differential equation

參考文獻

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