## Option Pricing Method and Parameter Calibration for Jump-Diffusion Model

Xu Congcong1,2, Xu Zuoliang,1

 基金资助: 国家自然科学基金.  11571365国家自然科学基金.  11401162河北省高等学校科学技术研究重点项目.  ZD2019080

 Fund supported: the NSFC.  11571365the NSFC.  11401162the Key Projects of Science and Technology Research in Colleges and Universities in Hebei Province.  ZD2019080

Abstract

In this paper, the pricing method and parameter calibration of jump-diffusion model are investigated. First, the risk-neutral characteristic function of jump-diffusion model is derived under the mean correctiong equivalent martingale measure. The option under jump-diffusion model is priced by using the COS pricing method. Then, the pricing error of the COS algorithm is analyzed and the effectiveness of the COS pricing method is verified through numerical experiment. Subsequently, the parameters of the jump-diffusion model are calibrated by the relative entropy regularization method. Numerical experiments demonstrate the accuracy and reliability of the proposed method. Finally, the calibration method is tested by analyzing the S&P500 market data. The results show that the values of calibrated parameter are qualitatively for each maturity. Moreover, the results indicate a better fitting to the market data for the Merton jump-diffusion model in comparison to the Black-Scholes model.

Keywords： Jump-diffusion models ; Option pricing ; Parameter calibration ; COS method ; Regularization

Xu Congcong, Xu Zuoliang. Option Pricing Method and Parameter Calibration for Jump-Diffusion Model. Acta Mathematica Scientia[J], 2019, 39(3): 649-663 doi:

## 1 引言

1973年Black和Scholes推导出著名的Black-Scholes期权定价公式[1],是金融数学领域的一个里程碑.然而, Black-Scholes模型的一系列假设(如波动率为常数,价格过程服从对数正态分布、市场是完备的)使得期权价格与市场观察价格存在较大差异.尤其1987年金融危机的爆发, Black-Scholes模型中期权价格与市场报价的偏差更加显著.从那时起,人们不断的探寻能够精确复制股票价格动态的定价模型.指数Lévy模型通常被当作Black-Scholes模型的替代模型,因为Lévy过程允许股票价格跳跃,能够有效的描述股票价格具有"尖峰厚尾"及对数收益非对称等特点.

### 2.1 指数Lévy模型

$$$S_{t}=S_{0}\exp(X_{t}), \, 0\leq t\leq T,$$$

由于这种变换只改变了$X_t$特征三元组的漂移项,扩散项$\sigma$和Lévy测度$\nu$没变,因此$X_t$在概率测度${\mathbb{Q}}$下是特征三元组为$(\sigma^2, \nu, \gamma+m)$的Lévy过程.要想使$\{{\rm e}^{-rt}S_t\}$${\mathbb{Q}}鞅,则对于0\leq \tau\leq t\leq T,应有{\rm e}^{-r\tau}S_{\tau}={\bf E}^Q[{\rm e}^{-rt}S_t|{\cal F}_{\tau}],其中{\bf E}^Q[\cdot]为概率测度{\mathbb{Q}}下的期望算子.事实上, \begin{eqnarray}{\bf E}^Q[{\rm e}^{-rt}S_t|{\cal F}_{\tau}]&=&{\rm e}^{-rt}S_{\tau}{\bf E}^Q[{\rm e}^{X_t-X_{\tau}}]\\&=&{\rm e}^{-rt}S_{\tau}{\rm e}^{(t-\tau)[m+\phi(1)]}\\&=&{\rm e}^{-r\tau}S_{\tau}{\rm e}^{(t-\tau)[m+\phi(1)-r]} \end{eqnarray} 有唯一解m=r-\phi(1),此时\{{\rm e}^{-rt}S_t\}$${\mathbb{Q}}$鞅.

$$$\gamma^Q=\gamma+\beta\sigma^2.$$$

$$$\frac{{\rm d}{\mathbb{Q}}}{{\rm d}{\mathbb{P}}}\bigg|{\cal F}_t=\frac{\exp(\beta W_t)}{{\bf E}^P[\exp(\beta W_t)]}.$$$

在均值修正鞅测度${\mathbb{Q}}$下, $\gamma^Q=\gamma+m$, ${\sigma^Q}^2=\sigma^2$, $\nu^Q=\nu$.由文献[5]中引理2.1可知(2.8)和(2.9)式成立,且参数$\beta$满足方程

$\sigma>0$,则$\beta$有唯一解$\beta=\frac{r-\phi(1)}{\sigma^2}$.

$$$\int_{{\bf R}\setminus[a, b]}f(y|x){\rm d}y<{\rm TOL}.$$$

$$$v_1(x, t)={\rm e}^{-r(T-t)}\int_a^bv(y, T)f(y|x){\rm d}y.$$$

(2)由$v_2(x, t)$代替$v_1(x, t)$时,由前$N$项和代替无穷级数的截断误差

$$$\renewcommand\arraystretch{1.5} \begin{array}{ll}\displaystyle\epsilon_2(x, t;N, [a, b])&:=v_1(x, t;[a, b])-v_2(x, t;N, [a, b])\\&\displaystyle=\frac{1}{2}(b-a){\rm e}^{-r(T-t)}\sum\nolimits_{k=N}^{'\infty}A_k(x)\cdot V_k.\end{array}$$$

### 4.2 相对熵正则化方法

(1)相对熵函数是凸函数;

(2)相对熵函数保证了校准测度${\Bbb Q}$关于先验测度${\mathbb{P}}$的绝对连续性;

(3)如果先验测度是风险中性测度,相对熵函数保证了校准测度${\Bbb Q}$${\mathbb{P}}的等价鞅测度; (4)相对熵函数易于计算. 定义4.1[10] 令{\mathbb{P}}$${\mathbb{Q}}$$(\Omega, {\cal F})上的两个等价概率测度. {\mathbb{Q}}关于{\mathbb{P}}的相对熵定义为 $$\varepsilon({\mathbb{Q}}|{\mathbb{P}})={\bf E}^Q\bigg[\ln\frac{{\rm d}{\mathbb{Q}}}{{\rm d}{\mathbb{P}}}\bigg]\={\bf E}^P\bigg[\frac{{\rm d}{\mathbb{Q}}}{{\rm d}{\mathbb{P}}}\ln\frac{{\rm d}{\mathbb{Q}}}{{\rm d}{\mathbb{P}}}\bigg].$$ f(x)=x\ln x,则 f(x)=x\ln x为严格凸函数,可知相对熵\varepsilon({\mathbb{Q}}|{\mathbb{P}})是关于{\mathbb{Q}}的凸函数,并且\varepsilon({\mathbb{Q}}|{\mathbb{P}})\geq 0,当且仅当{\mathbb{Q}}={\mathbb{P}}时等号成立.下面给出Merton跳-扩散模型的相对熵. 引理4.1[10] 假设{\mathbb{P}}$${\mathbb{Q}}$是指数Lévy模型下的等价概率测度,特征三元组分别为$(\sigma^{2}, \nu^P, \gamma^P)$$(\sigma^{2}, \nu^Q, \gamma^Q),且\sigma>0.那么,在风险中性条件下, {\mathbb{P}}$${\mathbb{Q}}$相对熵函数可表示为

$$$\varepsilon({\mathbb{Q}}|{\mathbb{P}})=\frac{T}{2\sigma^{2}}\left[\int_{-\infty}^{+\infty}({\rm e}^{x}-1)(\nu^Q-\nu^P)({\rm d}x)\right]^2+ T\int_{-\infty}^{+\infty}(\frac{{\rm d}\nu^Q}{{\rm d}\nu^P}\ln(\frac{{\rm d}\nu^Q}{{\rm d}\nu^P}) +1-\frac{{\rm d}\nu^Q}{{\rm d}\nu^P})\nu^P({\rm d}x).$$$

$\begin{eqnarray}\varepsilon({\mathbb{Q}}|{\mathbb{P}})&=&\displaystyle\frac{T}{2\sigma^{2}}\left[\lambda^Q\left({\rm e}^{\mu^Q+\frac{1}{2}{\delta^Q}^2}-1\right)-\lambda^P\left({\rm e}^{\mu^P+\frac{1}{2}{\delta^P}^2}-1\right)\right]^2 +\displaystyle T\lambda^Q\ln\left(\frac{\lambda^Q\delta^P}{\lambda^P\delta^Q}\right)\\ &&+T\lambda^P+T\lambda^Q\left(-\displaystyle\frac{3}{2}+\frac{{\delta^Q}^2+\left(\mu^Q-\mu^P\right)^2}{2{\delta^P}^2}\right). \end{eqnarray}$

由引理4.1可知

$\begin{eqnarray}\displaystyle\frac{1}{T}\varepsilon({\mathbb{Q}}|{\mathbb{P}})&=&\displaystyle\frac{1}{2\sigma^2}\left[\int_{-\infty}^{+\infty}({\rm e}^x-1)(v^Q-v^P)\right]^2 +\displaystyle\int_{-\infty}^{+\infty}v^P{\rm d}x-\displaystyle\int_{-\infty}^{+\infty}v^Q{\rm d}x+\int_{-\infty}^{+\infty}\ln(\frac{{\rm d}v^Q}{{\rm d}v^P}){\rm d}v^Q\\ &=&\displaystyle\frac{1}{2\sigma^2}[\lambda^Q{\bf E}^{Q}({\rm e}^x-1)-\displaystyle\lambda^P{\bf E}^{P}({\rm e}^x-1)]^2+\lambda^P-\lambda^Q\\&&+\displaystyle\int_{-\infty}^{+\infty}\left(\displaystyle\ln(\frac{\lambda^Q\delta^P}{\lambda^P\delta^Q})-\frac{(x-\mu^Q)^2}{2{\delta^Q}^2} +\frac{(x-\mu^P)^2}{2{\delta^P}^2}\right){\rm d}v^Q\\ &=&\displaystyle\frac{T}{2\sigma^{2}}\left[\lambda^Q\left({\rm e}^{\mu^Q+\frac{1}{2}{\delta^Q}^2}-1\right)-\lambda^P\left({\rm e}^{\mu^P+\frac{1}{2}{\delta^P}^2}-1\right)\right]^2 +\lambda^P-\lambda^Q\\&&+\displaystyle\lambda^Q\ln\left(\frac{\lambda^Q\delta^P}{\lambda^P\delta^Q}\right)-\displaystyle\int_{-\infty}^{+\infty}\frac{(x-\mu^Q)^2}{2{\delta^Q}^2}{\rm d}v^Q +\displaystyle\int_{-\infty}^{+\infty}\frac{(x-\mu^P)^2}{2{\delta^P}^2}{\rm d}v^Q.\end{eqnarray}$

$\begin{eqnarray}\displaystyle\int_{-\infty}^{+\infty}(x-\mu^Q)^2{\rm d}v^Q&=&\displaystyle\int_{-\infty}^{+\infty}(x^2-2\mu^Qx+{\mu^Q}^2)^2{\rm d}v^Q\\&=&\displaystyle\lambda\left({\bf E}^{Q}(x^2)-2\mu^Q{\bf E}^{Q}(x)+{\mu^Q}^2\right)\\&=&\displaystyle\lambda\left({\delta^Q}^2+{\mu^Q}^2-2{\mu^Q}^2+{\mu^Q}^2\right)\\&=&\displaystyle\lambda^Q{\delta^Q}^2, \end{eqnarray}$

$\begin{eqnarray}\displaystyle\int_{-\infty}^{+\infty}(x-\mu^P)^2{\rm d}v^Q&=&\displaystyle\int_{-\infty}^{+\infty}(x^2-2\mu^Px+{\mu^P}^2)^2{\rm d}v^Q\\&=&\displaystyle\lambda\left({\bf E}^{Q}(x^2)-2\mu^P{\bf E}^{Q}(x)+{\mu^P}^2\right)\\&=&\displaystyle\lambda\left({\delta^Q}^2+{\mu^Q}^2-2\mu^Q\mu^P+{\mu^P}^2\right)\\&=&\displaystyle\lambda^Q\left({\delta^Q}^2+(\mu^Q-\mu^P)^2\right).\end{eqnarray}$

$$$J(\sigma^Q, \lambda^Q, \mu^Q, \delta^Q):=\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{M_i}\omega_{ij}\left|C^{\theta^Q}(T_{i}, K_{j})-C_{ij}\right|^{2}+\alpha \varepsilon({\mathbb{Q}}|{\mathbb{P}}),$$$

### 图 2

$$$\varepsilon^2(\alpha^*)=c\hat{\varepsilon}_0^{2} , (c>1).$$$

### 5.2 实证分析

(1)选择权重:选用市场买卖差价利用(4.2)计算;

(2)先验参数选取:利用非正则化的NLS方法对(4.1)最小化的解作为先验参数;

(3)选择正则化参数$\alpha$:令$c=1.2$,利用市场数据通过求解(4.11)得到;

(4)在给定的$\alpha$${\mathbb{P}}$条件下,对函数$J$进行最小化求解.

$$${\rm RMSE}=\sqrt{\sum\limits_{i=1}^n\sum\limits_{j=1}^{m}\frac{(C^{\theta}(T_i, K_j)-C_{ij})^2}{mn}}.$$$

 到期日 模型 $\sigma^Q$ $\lambda^Q$ $\mu^Q$ $\delta^Q$ RMSE $T_1=0.038$ Merton模型 0.0656 8.2454 -0.0341 0.0440 0.8416 Black-Scholes模型 0.1247 2.8718 $T_2=0.115$ Merton模型 0.0520 5.9989 -0.0388 0.0396 0.3094 Black-Scholes模型 0.1136 4.9638 $T_3=0.211$ Merton模型 0.0648 1.6550 -0.0998 0.0176 0.2825 Black-Scholes模型 0.1164 6.2999 $T_4=0.460$ Merton模型 0.0662 0.8905 -0.1412 0.0467 0.5395 Black-Scholes模型 0.1252 11.9545

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