基于FFT的regime-switching的相关期权定价

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浙江财经学院硕士学位论文

摘要

期权作为一种金融衍生工具，是市场经济发展到高级阶段的产物。近年来，

期权在金融领域变得越来越重要，许多交易所正在进行大量的期货和期权交易，

而其中的期权定价是金融数学的核心问题之一。

1973 年，Black 和Scholes 提出的 B-S 期权定价模型对期权定价做出突破性进

展，具有划时代的意义。1997 年，美国经济学家 Merton 和Scholes 被授予 Nobel

经济学奖，以表彰他们在期权定价上的开创性的贡献。他们创立和发展了 B-S 期

权定价模型，为包括股票、债券、货币、商品在内的新兴衍生金融市场的各种以

市价变动定价的期权定价奠定了理论基础。随着期权的不断发展，国际金融衍生

市场上出现了大量欧式、美式期权之外的由标准期权变化组合、派生出的新品种，

即新型期权。近年发展起的多标的资产期权就是其中的一种。本文研究这种标的

资产之间的股价运动具有相关关系的新型期权的定价模型——相关资产的期权定

价模型。

而在期权定价领域和套期保值中，传统的 B-S 期权定价公式虽然被广泛的应

用，但这个模型并不能反映某些经济现象，如：特大的随机波动，股票收益分布

的非正态特征及在布莱克斯-斯科尔斯模型中的隐含波动率微笑不是常数等。大量

的实证也表明资产价格满足几何布朗运动并不符合实际情况。过去三十年，已经

有大量学者提出了许多不同的期权定价模型，这些模型包括跳扩散模型、Lévy 过

程、随机波动率模型和 GARCH 模型等。近年来，马尔可夫调制的区域转换模型

在期权定价中的应用已经引起了大量研究学者的兴趣。模型中连续时间马尔可夫

链的状态被看作市场经济状态，经济状态的转移是由于经济和商业周期的结构变

化而引起的。因此，本文在前人研究成果的基础上，研究了区域转换模型下的期

权定价问题，并提出了一个两状态体制转换下的相关期权定价模型。

首先，本文给出了一般意义上的相关期权定价模型，得出了风险中性条件下

的欧式看涨期权价格，然后引入 Regime-Switching 因素后，通过 Regime-Switching

Esscher 变换得到风险中性条件下 Regime-Switching 模型下的欧式看涨相关期权价

格。

其次，对期权价格进行傅里叶变换，将期权价格的决定问题转化为计算随机

向量马尔科夫链逗留时间问题。采用 Carr 和 Madan 提出的方法，即将欧式期权价

格公式看成一个卷积，然后利用卷积的拉普拉斯变换等于两因子拉普拉斯变换的

乘积的性质，求得卷积的拉普拉斯变换，再进行一次逆拉普拉斯变换从而推导得

浙江财经学院硕士学位论文

出Regime-Switching 市场中两个状态（m=2）下的马尔科夫链逗留时间联合特征函

数的显示形式，得到期权的价值。然后通过快速傅里叶逆变换求出欧式看涨期权

的价格，在经过离散化处理最终得出欧式看涨期权的 FFT 算法表达式。

最后，在数值结果与分析方面，利用 MATLAB 软件根据快速傅里叶变换算

法得出了马尔科夫链为两个状态情况下的欧式看涨期权价格，着重从欧式期权价

格和执行价格、期权到期日、波动率等方面的关系来进行比较分析。并将其与不

含Regime-Switching 因素下的期权价格进行比较分析。得出 Regime – Switching 的相

关期权模型拥有较高的期权价格，并且两者之间的差距随着期权到期日的增加而

增加，原因是 Regime-Switching 因素增加了风险补偿。

关键词：相关期权；区域转换模型；Esscher 变换；快速傅里叶变换法；期权定价

浙江财经学院硕士学位论文

III

ABSTRACT

As an investment tool of derivative securities, the option emerges from the

increasing development of the market economy. In recent years, options become more

and more important in the financial field, many exchange have a lot of futures and

options trading. Among them, option pricing is one of the core issues in mathematical

finance.

In 1973, Black and Scholes proposed BS model, they make a breakthrough of

option pricing, which has an epoch-making significance. In 1997, American economists

Robert C.Merton and Myron S.Scholes were awarded the Nobel prizes in economics, to

commend them the tremendous contribution in option pricing. They established and

developed the Black-Scholes model, ,under market pricing changes, they laid the

theoretical basis for the various kinds of options pricing in the emerging derivative

financial market, Which has include stocks, bonds, currencies, commodities. With the

continuous development of options, despite the European and American options, the

international financial derivatives markets emerges a lot of new options ,which was

combination, derived out from the standard options. the multi underlying assets option

is a kind among them. In this paper, we studied the underlying assets which have a

correlation between the Share price movement, namely the related assets option pricing

model.

The traditional Black-Scholes Option Pricing formula has been widely used for

Pricing Option and hedging in finance industry, but this model fails to reflect some

empirical phenomena, such as the larger random fluctuations, the non-normal features

and the implied volatility is not a constant as in Black-Scholes model. And there are a

large number of empirical results indicate that the asset price follows the geometric

Brownian motion is not realistic. Over the past three decades，many different option

valuation models have been proposed. Some of these include jump-diffusion models，

Lévy processes, stochastic volatility models, GARCH model and others. Recently, there

has been considerable interest in applications of a regime switching model which is

modulated by a continuous time Markov chain to option pricing problem. The states of

the continuous time Markov chain can be interpreted as the states of the economy. The

transitions of the states of the economy may be attributed to structural changes of the

浙江财经学院硕士学位论文

economy and business cycles. Therefore, based on the previous research result，this

thesis discuss the Option Pricing under Regime-Switching models and a two-state

Regime-Switching model of the related assets option pricing model is provided.

Firstly, we present a model of the related assets option pricing model, and

concluded the European call option price under the risk neutral conditions, then

introduce the Regime-Switching factors, through the Regime-Switching Esscher

transformation, we can get the European call option price under the risk neutral

conditions.

Furthermore, through Fourier transform, the option price decision problem is

transformed into computing random vector markov chain sojourn time problem. Using

the method proposed by Carr and Madan, taking the European option price formula as a

convolution, then using the properties that the Laplace transform about the convolution

is equal to the product of the two factors Laplace transform, get the Laplace transform

about the convolution, then using an inverse Laplace transform again, we can get the

display form of the markov chain sojourn time joint characteristic function under the

Regime-Switching market with two state(m = 2), and get the value of the option. Then

through fast Fourier inverse transform find out the European call option price, after the

processing about discretization, we can find out the FFT algorithm expression of the

European call option.

Finally, in the numerical results and analysis, based on fast Fourier transform

algorithm, we using the MATLAB software concluded the European call option price

where the markov chain with two state cases, emphatically comparative and analysis the

relationship about the European option price with executive price, options maturity date,

volatility. And comparative the option price which without Regime - Switching factors.

Getting that Regime - Switching related option model has higher option price, and the

gap is increasing with the increase of the due date, the reason is that with Regime -

Switching factor the pricing getting the risk compensation.

Keywords: Related options；Regime-Switching model；Esscher transformation；Fast

Fourier Transform；Option Pricing

浙江财经学院硕士学位论文

第一章导论.................................................................................................................. 1

第一节研究背景及意义 ......................................................................................... 1

第二节国内外相关的文献综述 ............................................................................. 2

第三节本文的研究框架及创新点 ......................................................................... 8

第二章 Regime-Switching的相关期权定价 ................................................................ 10

第一节相关期权 ................................................................................................... 10

第二节 Regime-Switching的相关期权的定价 ..................................................... 12

第三章 Regime-Switching相关期权定价模型中的FFT方法 ..................................... 22

第一节快速傅里叶变换法（FFT）.................................................................... 22

第二节期权价格中傅里叶变换的推导 ............................................................... 22

第三节马尔科夫链逗留时间联合特征函数的推导 ........................................... 28

第四节期权定价中的FFT算法 ............................................................................ 30

第四章数值结果及分析 .............................................................................................. 32

第一节参数的选择 ............................................................................................... 32

第二节期权价格与执行价格的关系 ................................................................... 32

第三节期权价格与到期期限的关系 ................................................................... 34

第四节期权价格与波动率的关系 ....................................................................... 35

第五节期权价格与Regime-Switching因素的关系 ............................................. 36

第五章结论与展望 ...................................................................................................... 37

第一节结论 ........................................................................................................... 37

第二节展望 ........................................................................................................... 37

参考文献 ........................................................................................................................ 39

附录.............................................................................................................................. 42

致谢.............................................................................................................................. 46

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摘要：

浙江财经学院硕士学位论文I摘要期权作为一种金融衍生工具，是市场经济发展到高级阶段的产物。近年来，期权在金融领域变得越来越重要，许多交易所正在进行大量的期货和期权交易，而其中的期权定价是金融数学的核心问题之一。1973年，Black和Scholes提出的B-S期权定价模型对期权定价做出突破性进展，具有划时代的意义。1997年，美国经济学家Merton和Scholes被授予Nobel经济学奖，以表彰他们在期权定价上的开创性的贡献。他们创立和发展了B-S期权定价模型，为包括股票、债券、货币、商品在内的新兴衍生金融市场的各种以市价变动定价的期权定价奠定了理论基础。随着期权的不断发展，国际金融衍生市场上...

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