基于FFT的Regime-Switching指数Lévy模型的期权定价

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浙江财经学院硕士学位论文

摘要①

期权作为一种金融衍生品投资工具，是市场经济发展到高级阶段的产物。随

着商品交易风险和市场不确定因素的增加，期权作为一种有效的套期保值和防范

风险的手段，其应用已日趋广泛。期权价格反映的是期权的买卖双方对某一权利

作出的价值判断，但我们很难从市场中直接得到期权价格，因此期权定价一直是

金融工程中的一个重要课题。在过去的30多年里，学术工作者和实际操作者对期

权定价做出了许多的尝试和贡献，具有划时代的意义的突破性进展是Black和

Scholes（1973）年提出的BS期权定价模型，尽管BS期权定价模型在期权定价方面

取得了很大的成功，但这个纯对数正态模型却不能反映以下三种经济现象：（1）

特大的随机波动，例如股灾。（2）股票收益分布的非正态特征：负的偏态和尖峰

性。（3）隐含波动率微笑，也就是布莱克—斯科尔斯模型中的隐含波动率不是常

数。

针对上述缺陷，金融界展开了对 BS 期权定价模型修正的研究，传统的修正模

型主要有随机利率模型、随机波动率模型、跳扩散模型、纯跳模型等。但是上述

传统修正模型都是时齐的，Konikov 和Madan（2002）指出了那些模型矩的期限结

构的理论值和实证结果不符。例如，理论上讲，方差以速率 t（持有股票的时间）

增加，偏态以速率 t1/2 递减，峰度以速率 t增加，但时齐模型的这些矩并不能反映

上述模式增加或减少。鉴于此，考虑到非时齐性，用 Regime-Switching 过程来描

述股票收益更为合适。

Regime-Switching 模型大都被应用于传统的随机利率模型、随机波动率模型

等，鲜有将 Regime-Switching 模型应用到带跳的 Lévy 模型中。所以，本文在传统

的Lévy 模型的基础上提出了带跳的 Regime-Switching 指数 Lévy 模型，并具体提

出并研究了在 Regime-Switching 市场中的三种跳扩散模型和两种纯跳模型。本文

在对原生资产过程的 Regime-Switching 指数 Lévy 模型进行期权定价时采用了快速

傅里叶变换法，期权价格的傅里叶变换可由马尔科夫链逗留时间的联合特征函数

得到。

首先，本文给出了一般意义上的带跳的 Regime-Switching 指数 Lévy 模型，得

出了风险中性条件下欧式看涨期权价格，推导出了 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，以及 m个状态

下的 m维微分方程。然后通过快速傅里叶反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。但此表达式的具体结果

①该论文为国家自然科学基金项目“期权组合非线性 VaR 度量模型及数值方法研究”的一个子课题，基金编号：

70771099（G0115）。

浙江财经学院硕士学位论文

还要依赖于 Regime-Switching 指数 Lévy 模型的 Lévy 测度的具体形式。

其次，将 FFT 算法分别应用到具体 Regime-Switching 有限活动 Lévy 模型、

Regime-Switching 无限活动 Lévy 模型上和 Regime-Switching 布朗运动模型（即

Regime-Switching BS 模型），其中 Regime-Switching 有限活动 Lévy 模型我们选取

了Regime-Switching Merton 跳扩散模型、Regime-Switching Kou 跳扩散模型、

Regime-Switching 对数均匀跳扩散模型；扩散部分由马尔可夫 Regime-Switching 的

几何布朗运动来刻画，跳跃部分由马尔科夫 Regime-Switching 指数 Lévy 测度下的

复合泊松过程来描述；Regime-Switching 无限活动 Lévy 模型我们选取了有代表性

的Regime-Switching VG 纯跳模型和 Regime-Switching CGMY 纯跳模型；而

Regime-Switching BS 模型是 Regime-Switching 跳扩散模型的特例（令其跳跃次数

为零即可）。根据五种具体 Regime-Switching 指数 Lévy 模型的 Lévy 测度和特征函

数的具体形式，我们分别推导出 Regime-Switching 市场中两个状态（m=2）的马尔

科夫链逗留时间联合特征函数的显示形式。进而计算出五种具体带跳的

Regime-Switching 指数 Lévy 模型的欧式看涨期权价格的 FFT 算法的最终表达式。

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、期权到期日的两个方面的关系，对本文提出五种具体形式的

Regime-Switching 指数 Lévy 模型与 Regime-Switching BS 模型进行比较分析，得出

这五种 Regime-Switching 指数 Lévy 模型都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，原因是跳跃和 Regime-Switching 因素增加了

风险补偿。

关键词：指数 Lévy 模型；Regime-Switching；跳扩散模型；VG 模型；CGMY 模

型；快速傅里叶变换法；期权定价

ABSTRACT

浙江财经学院硕士学位论文

III

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

increasing development of the market economy. As the increase of the risk of

commodity trading and the market uncertainty, options have been widely applied as

instruments for effective hedging and risk management. The research of option pricing

problem has significant theoretical and realistic meanings. The price of option means

the value judgment made by both of counterparts, but it is rather difficult to watch it.

Hereby, the option valuation is always an important subject in financial engineering.

Despite the great success of Black-Scholes model (1973) in option pricing, this

pure lognormal diffusion model fails to reflect the three empirical phenomena: (1) the

large random fluctuations such as crashes and rallies; (2) the non-normal features, that is,

negative skewness and leptokurtic (peakedness) behavior in the stock log-return

distribution; (3) the implied volatility smile, that is, the implied volatility is not a

constant as in the Black-Scholes model.

Therefore, many different models are proposed to modify the Black-Scholes model

so as to represent the above three empirical phenomena, such as stochastic interest

models, stochastic volatility models, diffusion jump models, pure jump models and so

on. However, most models considered are time homogeneous and as Konikov and

Madan (2002) have shown, the theoretical behavior of the term structure of their

moments does not match empirical observations. For example, the variance theoretically

increases with a factor t (the length of the holding period), skewness decreases with a

factor t1/2, and kurtosis decreases with a factor t, while empirically, these moments do

not show patterns of growth or decay that are even close to these factors. Given all this,

in order to allow for time-inhomogeneity, there has developed an interest in modeling

asset returns using switching processes. Regime-Switching model is mostly used to the

traditional stochastic interest models, stochastic volatility models, but rarely applied to

the Lévy model with jumps.

In this paper, we present a new model of Lévy process in a Regime-Switching

market based on the traditional Lévy models. Three specific Regime-Switching

diffusion models and two pure jump models. By adopted the methodology of Liu,

Zhang and Yin (2006), we give a fast Fourier transform approach to option pricing for

regime switching models of the underlying asset process. The Fourier transform of the

option price is obtained in terms of the joint characteristic function of the sojourn times

of the Markov chain. We present the joint characteristic function in explicit form for

two-state (m=2) Markov chains, and in terms of solutions of systems of m-dimensional

浙江财经学院硕士学位论文

differential equations for m-state case.

Firstly, we begin with risk-neutral valuation for European option, where the asset

price follows a general Lévy process in a regime-switching market. The Fourier

transform of the option price is obtained in terms of the joint characteristic function of

the sojourn times of the Markov chain. We present the joint characteristic function in

general form for two-state Markov chains, and in terms of solutions of systems of

m-dimensional differential equations for m-state case. However, the explicit form option

price depends on the specific Lévy measure of each regime-switching models.

Furthermore, Fast Fourier transform is adopted for calculating option prices, h

where the asset price follows four specific regime-switching Lévy process respectively.

The four models are regime-switching jump diffusion model with log-double

exponential distribution’s jump-amplitudes ，Merton jump diffusion model with

regime-switching, Regime-Switching jump diffusion model with log-uniform

jump-amplitudes, Regime-Switching jump diffusion model with log-uniform jump-

amplitudes ，Regime-Switching pure jump model with VG process and

Regime-Switching pure jump model with CGMY process. The diffusion component is

given by Markov-switching geometric Brownian motion and the jump component is

modeled by a compound Poisson process with Markov-switching Lévy measure. The

Fourier transform of the option price is obtained in terms of the joint characteristic

function of the sojourn times of the Markov chain. We present the joint characteristic

function in explicit form for two-state (m= 2) Markov chains based on the explicit form

of the Lévy measure of the five regime-switching models respectively.

At last, the option prices from the regime-switching jump diffusion models and the

regime-switching pure jump model are compared with those of regime switching

Black-Scholes (RS-BS) model. As expected, call option prices of Lévy model are

higher than those of SBS model with respect to the strike price. The reason is the jump

and the regime switching factors increase the risk premium.

Keywords: Exponential Lévy Model; Jump Diffusion Model; Regime-Switching; VG

Model; CGMY Model; 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······································································· 10

第二章 Regime-Switching 指数 Lévy 模型································································· 12

第一节指数 Lévy 模型························································································ 12

第二节 Regime-Switching 指数 Lévy 模型·························································· 13

第三章 Regime-Switching 指数 Lévy 定价模型中的 FFT 方法································ 18

第一节快速傅里叶变换法（FFT）···································································· 18

第二节期权价格的傅里叶变换的推导······························································· 18

第三节马尔科夫链逗留时间联合特征函数的推导··········································· 21

第四节期权定价的 FFT 算法·············································································· 23

第四章 Regime-Switching 指数 Lévy 模型中 FFT 方法的应用································ 25

第一节 Regime-Switching Merton 跳扩散模型··················································· 27

第二节 Regime-Switching Kou 跳扩散模型························································ 30

第三节 Regime-Switching 对数均匀跳扩散模型················································ 34

第四节 Regime-Switching VG 纯跳模型····························································· 38

第五节 Regime-Switching CGMY 纯跳模型······················································· 43

第五章数值结果及分析····························································································· 49

第一节参数的选择······························································································ 49

第二节期权价格与执行价格的关系··································································· 50

第三节期权价格和到期日的关系······································································· 54

第六章结论与展望····································································································· 60

第一节结论·········································································································· 60

第二节展望·········································································································· 60

参考文献······················································································································· 62

附录······························································································································· 68

致谢 ……………………………………………………………………………………………………………… 79

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摘要：

浙江财经学院硕士学位论文I摘要①期权作为一种金融衍生品投资工具，是市场经济发展到高级阶段的产物。随着商品交易风险和市场不确定因素的增加，期权作为一种有效的套期保值和防范风险的手段，其应用已日趋广泛。期权价格反映的是期权的买卖双方对某一权利作出的价值判断，但我们很难从市场中直接得到期权价格，因此期权定价一直是金融工程中的一个重要课题。在过去的30多年里，学术工作者和实际操作者对期权定价做出了许多的尝试和贡献，具有划时代的意义的突破性进展是Black和Scholes（1973）年提出的BS期权定价模型，尽管BS期权定价模型在期权定价方面取得了很大的成功，但这个纯对数正态模型却不能反映以下三种经济现...

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