# 基于改进Heston模型的欧式期权定价European Option Pricing Based on Improved Heston Model

DOI: 10.12677/AAM.2020.91015, PDF, HTML, XML, 下载: 167  浏览: 309  科研立项经费支持

Abstract: Heston model is a stochastic volatility (SV) model widely used in asset management. In this paper, a dynamic mean reversion power function is set to replace the Heston model which used the mean reversion square root to describe the volatility process. An improved Heston model is proposed and the specific form of power function is determined according to the actual situation. Using the ∆-hedging principle, we construct a risk-free asset portfolio, get the PDE of option price satisfying the model, and use the inversion theorem to find the closed form solution of option price. Empirical analysis shows that the improved Heston model can effectively improve the accuracy of pricing.

1. 引言

Heston模型给出欧式期权定价封闭形式的解，有效提高参数估计方面的效力。但由于模型采用固定的均值回复平方根描述波动率过程，导致有时模型计算结果与实际有较大的误差。为了克服Heston模型的不足，本文设置动态的均值回复幂函数描述波动率过程，并根据具体的实际情况确定幂函数的具体形式，提高模型的适应性和计算精度。

2. Heston模型的改进

SV模型主要基于资产价格和波动率都遵循“有偏的随机游走”，即一个漂移项加上一个随机项 [1]，且随机项服从标准的布朗运动。以下是考虑资产及其方差波动率都是随机过程的Stein-stein模型与Heston。

2.1. Stein-Stein [11] 模型

$\text{d}{v}_{t}=k\left(\xi -{v}_{t}\right)\text{d}t+\sigma \text{d}{W}_{t}^{2}$

SV模型中含dt的项均为漂移项，含 $\text{d}{W}_{t}$ 项为随机项，例如，stein-stein模型中瞬时波动率的漂移项为，随机项为 $\sigma \text{d}{W}_{t}^{2}$，反映了波动率在该时刻先进行一个固定漂移再进行一定的随机变动。

2.2. Heston [12] 模型

Heston [12] 改进了stein-stein模型的随机项，提出如下随机波动率模型

$\frac{\text{d}{S}_{t}}{{S}_{t}}=\left(r-q\right)\text{d}t+\sqrt{{v}_{t}}\text{d}{W}_{t}^{1}$

2.3. 改进的Heston模型

$\text{d}{v}_{t}=k\left(\xi -{v}_{t}\right)\text{d}t+\sigma {v}_{t}^{\beta }\text{d}{W}_{t}^{2}$ (2-1)

3. 改进Heston模型下欧式看涨期权的PDE

(3-1)

$\text{d}G=\frac{\partial G}{\partial t}\text{d}t+\frac{\partial G}{\partial S}\text{d}S+\frac{\partial G}{\partial v}\text{d}v+\frac{1}{2}{v}^{2\alpha }{S}^{2}\frac{{\partial }^{2}G}{\partial {S}^{2}}\text{d}t+\sigma {v}^{\alpha +\beta }\rho S\frac{{\partial }^{2}G}{\partial S\partial v}\text{d}t+\frac{1}{2}{\sigma }^{2}{v}^{2\beta }\frac{{\partial }^{2}G}{\partial {v}^{2}}\text{d}t$ (3-2)

$\text{d}V=\frac{\partial V}{\partial t}\text{d}t+\frac{\partial V}{\partial S}\text{d}S+\frac{\partial V}{\partial v}\text{d}v+\frac{1}{2}{v}^{2\alpha }{S}^{2}\frac{{\partial }^{2}V}{\partial {S}^{2}}\text{d}t+\sigma {v}^{\alpha +\beta }\rho S\frac{{\partial }^{2}V}{\partial S\partial v}\text{d}t+\frac{1}{2}{\sigma }^{2}{v}^{2\beta }\frac{{\partial }^{2}V}{\partial {v}^{2}}\text{d}t$ (3-3)

$\begin{array}{c}\text{d}\Pi =\left[\frac{\partial V}{\partial t}+\frac{1}{2}{v}^{2\alpha }{S}^{2}\frac{{\partial }^{2}V}{\partial {S}^{2}}+\sigma {v}^{\alpha +\beta }\rho S\frac{{\partial }^{2}V}{\partial S\partial v}+\frac{1}{2}{\sigma }^{2}{v}^{2\beta }\frac{{\partial }^{2}V}{\partial {v}^{2}}\right]\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\phi \left[\frac{\partial G}{\partial t}+\frac{1}{2}{v}^{2\alpha }{S}^{2}\frac{{\partial }^{2}G}{\partial {S}^{2}}+\sigma {v}^{\alpha +\beta }\rho S\frac{{\partial }^{2}G}{\partial S\partial v}+\frac{1}{2}{\sigma }^{2}{v}^{2\beta }\frac{{\partial }^{2}G}{\partial {v}^{2}}\right]\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[\frac{\partial V}{\partial S}+\Delta +\phi \frac{\partial G}{\partial S}\right]\text{d}S+\left[\frac{\partial V}{\partial v}+\phi \frac{\partial G}{\partial v}\right]\text{d}v\end{array}$ (3-4)

$\left\{\begin{array}{l}\frac{\partial V}{\partial S}+\Delta +\phi \frac{\partial G}{\partial S}=0\\ \frac{\partial V}{\partial v}+\phi \frac{\partial G}{\partial v}=0\end{array}$

(3-5)

$\begin{array}{l}\left[\frac{\partial V}{\partial t}+\frac{1}{2}{v}^{2\alpha }{S}^{2}\frac{{\partial }^{2}V}{\partial {S}^{2}}+\sigma {v}^{\alpha +\beta }\rho S\frac{{\partial }^{2}V}{\partial S\partial v}+\frac{1}{2}{\sigma }^{2}{v}^{2\beta }\frac{{\partial }^{2}V}{\partial {v}^{2}}\right]\\ \text{ }+\phi \left[\frac{\partial G}{\partial t}+\frac{1}{2}{v}^{2\alpha }{S}^{2}\frac{{\partial }^{2}G}{\partial {S}^{2}}+\sigma {v}^{\alpha +\beta }\rho S\frac{{\partial }^{2}G}{\partial S\partial v}+\frac{1}{2}{\sigma }^{2}{v}^{2\beta }\frac{{\partial }^{2}G}{\partial {v}^{2}}\right]\text{d}t\\ =r\left(V+\Delta S+\phi G\right)\text{d}t\end{array}$ (3-6)

$\begin{array}{l}\frac{\left[\frac{\partial V}{\partial t}+\frac{1}{2}{v}^{2\alpha }{S}^{2}\frac{{\partial }^{2}V}{\partial {S}^{2}}+\sigma {v}^{\alpha +\beta }\rho S\frac{{\partial }^{2}V}{\partial S\partial v}+\frac{1}{2}{\sigma }^{2}{v}^{2\beta }\frac{{\partial }^{2}V}{\partial {v}^{2}}\right]-rV+rS\frac{\partial V}{\partial S}}{\frac{\partial V}{\partial v}}\\ =\frac{\left[\frac{\partial G}{\partial t}+\frac{1}{2}{v}^{2\alpha }{S}^{2}\frac{{\partial }^{2}G}{\partial {S}^{2}}+\sigma {v}^{\alpha +\beta }\rho S\frac{{\partial }^{2}G}{\partial S\partial v}+\frac{1}{2}{\sigma }^{2}{v}^{2\beta }\frac{{\partial }^{2}G}{\partial {v}^{2}}-rG+rS\frac{\partial G}{\partial S}\right]}{\frac{\partial G}{\partial v}}\end{array}$ (3-7)

$g\left(S,v,t\right)=-k\xi +\left(k+\gamma \right)v$

$g\left(S,v,t\right)$ 为关于v的线性函数，为了更加简洁，记，其中 ${\lambda }_{1}=k\xi$${\lambda }_{2}=-\left(k+\gamma \right)$ 将其代入(3-8)整理化简得到改进Heston模型下期权价格的PDE为

$\frac{1}{2}{v}^{2\alpha }{S}^{2}\frac{{\partial }^{2}G}{\partial {S}^{2}}+\sigma {v}^{\alpha +\beta }\rho S\frac{{\partial }^{2}G}{\partial S\partial v}+\frac{1}{2}{\sigma }^{2}{v}^{2\beta }\frac{{\partial }^{2}G}{\partial {v}^{2}}+rS\frac{\partial G}{\partial S}+\lambda \left(v\right)\frac{\partial G}{\partial v}-rG+\frac{\partial G}{\partial t}=0$ (3-9)

$x=\mathrm{ln}S$，代入(3-9)整理化简得

$\frac{1}{2}{v}^{2\alpha }\frac{{\partial }^{2}G}{\partial {x}^{2}}-\frac{1}{2}{v}^{2\alpha }\frac{\partial G}{\partial x}+\sigma {v}^{\alpha +\beta }\rho \frac{{\partial }^{2}G}{\partial x\partial v}+\frac{1}{2}{\sigma }^{2}{v}^{2\beta }\frac{{\partial }^{2}G}{\partial {v}^{2}}+r\frac{\partial G}{\partial x}+\lambda \left(v\right)\frac{\partial G}{\partial v}-rG+\frac{\partial G}{\partial t}=0$ (3-10)

4. 改进Heston模型下欧式看涨期权定价公式

$C\left(K\right)={S}_{t}{P}_{1}\left(S,v,t\right)-K{\text{e}}^{-r\tau }{P}_{2}\left(S,v,t\right)$ (4-1)

${P}_{1}\left(S,v,t\right)$分别表示欧式看涨期权按照股票价格行权的概率和在风险中性环境被行权的概率。

$C\left(K\right)={\text{e}}^{x}{P}_{1}\left(x,v,t\right)-K{\text{e}}^{-r\tau }{P}_{2}\left(x,v,t\right)$ (4-2)

(4-2)式分别求以下偏导数得

$\frac{{\partial }^{2}C}{\partial {x}^{2}}={\text{e}}^{x}\left[{P}_{1}+2\frac{\partial {P}_{1}}{\partial x}+\frac{{\partial }^{2}{P}_{1}}{\partial {x}^{2}}\right]-K{\text{e}}^{-r\tau }\frac{{\partial }^{2}{P}_{2}}{\partial {x}^{2}}$

$\frac{\partial C}{\partial v}={\text{e}}^{x}\frac{\partial {P}_{1}}{\partial v}-K{\text{e}}^{-r\tau }\frac{\partial {P}_{2}}{\partial v}$(4-3)

$\frac{{\partial }^{2}C}{\partial x\partial v}=\left[{\text{e}}^{x}\frac{\partial {P}_{1}}{\partial v}+\frac{{\partial }^{2}{P}_{1}}{\partial x\partial v}\right]-K{\text{e}}^{-r\tau }\frac{{\partial }^{2}{P}_{2}}{\partial x\partial v}$

${u}_{j}={\left(-\frac{1}{2}\right)}^{j-1}$ , ${b}_{j}=\rho \sigma \left(2-j\right)$ ,，则上微分方程可化为

$\begin{array}{l}\frac{1}{2}{v}^{2\alpha }\frac{{\partial }^{2}{P}_{j}}{\partial {x}^{2}}+\sigma {v}^{\alpha +\beta }\rho \frac{{\partial }^{2}{P}_{j}}{\partial x\partial v}+\frac{1}{2}{\sigma }^{2}{v}^{2\beta }\frac{{\partial }^{2}{P}_{j}}{\partial {x}^{2}}+\left(r+{u}_{j}{v}^{2\alpha }\right)\frac{\partial {P}_{j}}{\partial x}\\ +\left(\lambda \left(v\right)+{b}_{j}{v}^{\alpha +\beta }\right)\frac{\partial {P}_{j}}{\partial v}+\frac{\partial {P}_{j}}{\partial t}=0\end{array}$ (4-4)

(4-4)是满足的偏微分方程，下面通过随机变量x的特征函数给出。

(4-5)

(4-6)

$\frac{\partial {f}_{j}}{\partial x}=i\varphi {f}_{j}$

$\frac{{\partial }^{2}{f}_{j}}{\partial x\partial v}=i\varphi \theta {D}_{j}{v}^{\theta -1}{f}_{j}$

$\begin{array}{l}\left[-\frac{1}{2}{\varphi }^{2}+i{u}_{j}\varphi \right]{v}^{2\alpha }+\left[i\rho {\sigma }^{2}\varphi +{b}_{j}\right]\theta {D}_{j}{v}^{\alpha +\beta +\theta -1}+\frac{1}{2}{\sigma }^{2}\theta \left(\theta -1\right){D}_{j}{v}^{2\beta +\theta -2}\\ +\frac{1}{2}{\sigma }^{2}{\theta }^{2}{D}_{j}^{2}{v}^{2\beta +2\theta -2}+\frac{\partial {D}_{j}}{\partial t}{v}^{\theta }+\left[\frac{\partial {C}_{j}}{\partial t}+ir\varphi \right]+\theta {D}_{j}\lambda \left(v\right){v}^{\theta -1}=0\end{array}$ (4-7)

1) 因为当，所以(4-7)可化为

(4-8)

2) 将 $\theta =2\alpha ,\lambda \left(v\right)={\lambda }_{1}+{\lambda }_{2}v$ 代入(4-8)得

(4-9)

i) 若，则(4-9)化为

$\begin{array}{l}\left[-\frac{1}{2}{\varphi }^{2}+i{u}_{j}\varphi +\frac{\partial {D}_{j}}{\partial t}+{D}_{j}{\lambda }_{2}\right]v+\left[i\rho {\sigma }^{2}\varphi +{b}_{j}\right]{D}_{j}{v}^{\beta +\frac{1}{2}}\\ \text{ }+\frac{1}{2}{\sigma }^{2}{D}_{j}^{2}{v}^{2\beta }+\left[\frac{\partial {C}_{j}}{\partial t}+ir\varphi +{D}_{j}{\lambda }_{1}\right]=0\end{array}$ (4-10)

$\left[i\rho {\sigma }^{2}\varphi +{b}_{j}\right]{D}_{j}{v}^{\beta +\frac{1}{2}}+\frac{1}{2}{\sigma }^{2}{D}_{j}^{2}{v}^{2\beta }=0$

ii) 若。假设，则(4-9)化为

(4-11)

$2\beta +4\alpha -2$$2\alpha -1$ 均不等于0且不等于。由于(4-9)对任意v恒成立，故的系数都为零，(4-9)可化为

，将其代入(4-9)得

$\begin{array}{l}\left[-\frac{1}{2}{\varphi }^{2}+i{u}_{j}\varphi +\frac{\partial {D}_{j}}{\partial t}+2\alpha {D}_{j}{\lambda }_{2}\right]{v}^{2\alpha }+2\alpha \left[i\rho {\sigma }^{2}\varphi +{b}_{j}\right]{D}_{j}{v}^{\alpha }+\alpha {\sigma }^{2}\left(2\alpha -1\right){D}_{j}{v}^{-2\alpha }\\ \text{ }+2{\sigma }^{2}{\alpha }^{2}{D}_{j}^{2}+\left[\frac{\partial {C}_{j}}{\partial t}+ir\varphi \right]+2\alpha {D}_{j}{\lambda }_{1}{v}^{2\alpha -1}=0\end{array}$ (4-12)

$2\alpha \left[i\rho {\sigma }^{2}\varphi +{b}_{j}\right]{D}_{j}{v}^{\alpha }+\alpha {\sigma }^{2}\left(2\alpha -1\right){D}_{j}{v}^{-2\alpha }+2\alpha {D}_{j}{\lambda }_{1}{v}^{2\alpha -1}=0$ (4-13)

$\alpha =\frac{1}{4}$ 时，(4-13)可化为

$\frac{1}{2}\left[i\rho {\sigma }^{2}\varphi +{b}_{j}\right]{D}_{j}{v}^{\frac{1}{4}}+\frac{1}{8}{D}_{j}\left(2{\lambda }_{1}-{\sigma }^{2}\right){v}^{-\frac{1}{2}}=0$ (4-14)

$\alpha \ne \frac{1}{4}$ 时，由于(4-13)对任意v恒成立，故的系数零，(4-13)化为

$2\alpha \left[i\rho {\sigma }^{2}\varphi +{b}_{j}+{\lambda }_{1}{v}^{\alpha -1}\right]{D}_{j}=0$

，将其代入(4-9)得

$\begin{array}{l}\left[-\frac{1}{2}{\varphi }^{2}+i{u}_{j}\varphi +\frac{\partial {D}_{j}}{\partial t}+2\alpha {D}_{j}{\lambda }_{2}\right]{v}^{2\alpha }+2\alpha \left[i\rho {\sigma }^{2}\varphi +{b}_{j}\right]{D}_{j}+\alpha {\sigma }^{2}\left(2\alpha -1\right){D}_{j}{v}^{-4\alpha }\\ \text{ }+2{\sigma }^{2}{\alpha }^{2}{D}_{j}^{2}{v}^{-2\alpha }+\left[\frac{\partial {C}_{j}}{\partial t}+ir\varphi \right]+2\alpha {D}_{j}{\lambda }_{1}{v}^{2\alpha -1}=0\end{array}$ (4-15)

(4-16)

$\begin{array}{l}\left[-\frac{1}{2}{\varphi }^{2}+i{u}_{j}\varphi +\frac{\partial {D}_{j}}{\partial t}+2\alpha \left(i\rho {\sigma }^{2}\varphi +{b}_{j}\right){D}_{j}+2{\alpha }^{2}{\sigma }^{2}{D}_{j}^{2}+2\alpha {D}_{j}{\lambda }_{2}\right]{v}^{2\alpha }\\ +\left[\frac{\partial {C}_{j}}{\partial t}+ir\varphi +{\sigma }^{2}\alpha \left(2\alpha -1\right){D}_{j}\right]+2\alpha {D}_{j}{\lambda }_{1}{v}^{2\alpha -1}=0\end{array}$ (4-17)

(4-18)

(4-19)

$x=\mathrm{ln}S$

${P}_{j}=\frac{1}{2}+\frac{1}{\pi }{\int }_{0}^{\infty }Re\left[\frac{{\text{e}}^{-i\varphi \mathrm{ln}K{f}_{j}\left(x,v,t;\varphi \right)}}{i\varphi }\right]\text{d}\varphi$

${f}_{j}\left(x,v,t;\varphi \right)={\text{e}}^{{C}_{j}\left(\tau ;\varphi \right)+{D}_{j}\left(\tau ;\varphi \right){v}^{\theta }+i\varphi x}$

${C}_{j}\left(\tau ;\varphi \right)=ri\varphi \tau +\frac{2\alpha -1}{4\alpha }{h}_{j}\left(\tau ,\varphi \right)$

${d}_{j}=2\alpha \sqrt{{\left({b}_{j}+{\lambda }_{2}+i\varphi \rho \sigma \right)}^{2}-{\sigma }^{2}\left(2{u}_{j}i\varphi -{\varphi }^{2}\right)}$

${g}_{j}=\frac{2\alpha \left({b}_{j}+{\lambda }_{2}+i\varphi \rho \sigma \right)-{d}_{j}}{2\alpha \left({b}_{j}+{\lambda }_{2}+i\varphi \rho \sigma \right)+{d}_{j}}$

${u}_{j}={\left(-\frac{1}{2}\right)}^{j-1}$

${b}_{j}=\rho \sigma \left(2-j\right)$$j=1,2$

5. 数值计算实例

Table 1. Parameters of SV model

Figure 1. Implied volatility chart of Shanghai 50ETF European call options

6. 结论

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