Option prices in Bates SVJ model?

In this [post] discussed the European put and call price formulas under the Heston Stochastic Volatility model.

There exists an important extension of Heston model to include diffusion jumps, known as Bates Stochastic Volatility Jump (SVJ) model, as referred to in this paper.

What are the option price formulas in SVJ model?

Answer

The Bates model is represented by the bivariate system of stochastic differential equations \begin{align} &dS_t=(r-q)S_tdt+\sqrt{v_t}S_tdW_1(t)+S_tdN_t\\ &dv_t=\kappa(\theta-v_t)dt+\sigma\sqrt{v_t}dW_2(t)\\ \end{align} where $$\mathbb{E^Q}[dW_1(t)dW_2(t)]=\rho dt$$ and $N_t$ is a compound Poisson process with intensity $\lambda$ and independent jumps $J$ with $$\ln{(1+J)}\tilde{~} N\left(\ln(1+\beta)-\frac{1}{2}\alpha^2\,\, ,\,\,\alpha^2\right)$$ The parameters $\beta$ and ${\alpha}$ determine the distribution of the jumps and the Poisson process is assumed to be independent of the Wiener processes.By application of delta-hedging argument,we have $$\frac{\partial U}{\partial t}+\frac{1}{2}{{v}_{t}}{{S}_{t}}^{2}\frac{{{\partial }^{2}}U}{\partial {{S}^{2}}}+\frac{1}{2}{{\sigma }^{2}}{{v}_{t}}\frac{{{\partial }^{2}}U}{\partial {{v}^{2}}}+\rho \sigma \,{{v}_{t}}{{S}_{t}}\frac{{{\partial }^{2}}U}{\partial S \partial v}+\,(r-q-\lambda\,\bar{k}){{S}_{t}}\frac{\partial U}{\partial S}+\kappa (\theta -{{v}_{t}})\,\frac{\partial U}{\partial v}-rU+I_U=0\,\,\,(1)$$ where $$I_U=\lambda\int_{0}^{\infty}[U(S\xi,v,t)-U(S,v,t)]\,g(\xi)\,d\xi$$ $$g(\xi)=\frac{1}{\sqrt{2\pi}\alpha\xi}e^{-\frac{1}{2\alpha^2}(\ln{\xi-m})^2}$$ $$m=\ln(1+\beta)-\frac{1}{2\alpha^2}$$ $$\bar{k}=e^{\frac{1}{2}\alpha^2+m}-1$$ It should be noted that closed-form solutions for vanilla-option payoff do exist but PIDE (1) is easily approximated by Numerical Methods.

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close form solution

I edited my Answer for emcor.

1. Dynamic of $S_t$ under historical measure \begin{align} &\frac{dS_t}{S_t}=(\mu-\lambda\,\bar{J})dt+\sqrt{v_t}dW_1(t)+J\,dN(t)\\ &\hspace{0.3cm}dv_t=\kappa(\theta-v_t)dt+\sigma\sqrt{v_t}dW_2(t),\\ \end{align} where $$\mathbb{E^P}[dW_1(t)dW_2(t)]=\rho dt,$$ $N_t$ is a compound Poisson process with intensity $\lambda$ and independent jumps $J$ with $$\ln{(1+J)}\tilde{~} N\left(\ln(1+\bar{J})-\frac{1}{2}\alpha^2\,\, ,\,\,\alpha^2\right).$$ The parameters $\bar{J}$ and ${\alpha}$ determine the distribution of the jumps and the Poisson process is assumed to be independent of the Wiener processes.


2. Change measure: $\mathbb{P\rightarrow Q}$ \begin{align} &\frac{dS_t}{S_t}=(r-q-\lambda^*\,\bar{J^*})dt+\sqrt{v_t}dW_1^{\mathbb{Q}}(t)+J^*\,dN^*(t)\\ &\hspace{0.3cm}dv_t=\kappa^*(\theta^*-v_t)dt+\sigma\sqrt{v_t}dW_2^{\mathbb{Q}}(t),\\ \end{align} where $$\mathbb{E^Q}[dW_1^{\mathbb{Q}}(t)dW_2^{\mathbb{Q}}(t)]=\rho dt$$ $$\kappa^*=\kappa +\xi$$ $$\hspace{0.3cm}\theta^*=\frac{\kappa\theta}{\kappa+\xi},$$ such that $\xi$ is volatility market price and $$J^*=J+J\,\mathbb{E^P}\left[\frac{\Delta J_w}{J_w}\right]$$ $$\bar{J}^*=\bar{J}+\frac{\mathbb{Cov}\left(J,\frac{\Delta J_w}{J_w}\right)}{1+\mathbb{E^P}\left[\frac{\Delta J_w}{J_w}\right]}.$$ Where $\frac{\Delta J_w}{J_w}$ is random percentage jump conditional on a jump occurring and $\frac{dJ_w}{J_w}$ is percentage shock in the absence of jump.


3. Note that,when $\xi=0$ we have $\kappa^*=\kappa$ and $\theta^*=\theta$. We set $\xi=0$, because when we estimate the risk-neutral parameters to price options we do not need to estimate $\xi$. Also, when $\Delta J_w/J_w\rightarrow0$ thus we have $J^*=J$ and $\bar{J}^*=\bar{J}.$


4. let $x_t=\ln S_t$ then \begin{align} C(t\,,{{S}_{t}},{{v}_{t}},J,K,T)={{S}_{t}}{{P}_{1}}-K\,{{e}^{-r\tau }}{{P}_{2}} \end{align} where,for $j=1,2$ \begin{align} & {{P}_{j}}({{x}_{t}}\,,\,{{v}_{t}}\,;\,\,{{x}_{T}},\ln K)=\frac{1}{2}+\frac{1}{\pi }\int\limits_{0}^{\infty }{\operatorname{Re}\left( \frac{{{e}^{-i\phi \ln K}}{{f}_{j}}(\phi ;t,x,v)}{i\phi } \right)}\,d\phi \\ &\\ &\hspace{1.9cm}{{f}_{j}}(\phi ;{{v}_{t}},{{x}_{t}})=\exp\left[{{C}_{j}}(\tau ,\phi) +{{D}_{j}}(\tau ,\phi ){{v}_{t}}+i\phi{{x}_{t}}+{{\Xi }_{j}}\right]\\ &\\ &\hspace{3.8cm}\Xi_j={{\lambda }^{*}}\tau\,{{(1+{{\kappa }^{*}})}^{{{u}_{j}}+\frac{1}{2}}}\left[ {{(1+{{\kappa }^{*}})}^{\phi }}{{e}^{{{\alpha }^{2}}({{u}_{j}}\phi +0.5{{\phi }^{2}})}}-1 \right],\\ \end{align} such that \begin{align} &C_j(\tau ,\phi)=(r-q-\lambda^*\bar{\kappa}^*)\phi\,\tau-\frac{\kappa^*\theta^*\,\tau}{\sigma^2}(\rho\sigma\phi-\beta_j-\gamma_j)-\frac{2\kappa^*\theta^*\,\tau}{\sigma^2}\ln\left(1+\frac{1}{2}(\rho\sigma\phi-\beta_j-\gamma_j)\frac{1-e^{\gamma_j\tau}}{\gamma_j}\right)\\ &\\ &D_j(\tau ,\phi)=\frac{-2(u_j\phi+\frac{1}{2}\phi^2)}{\rho\sigma\phi-\beta_j+\gamma_j\frac{1+e^{1-\gamma_j\tau}}{1-e^{1-\gamma_j\tau}}}\\ &\\ &\hspace{1.1cm}\gamma_j=\sqrt{(\rho\sigma\phi-\beta_j)^2-2\sigma^2(u_j\phi+0.5\phi^2)}\\ \end{align} and $$u_1=\frac{1}{2}\,,u_2=-\frac{1}{2}\,,\beta_1=\kappa^*-\rho\sigma\,,\beta_2=\kappa^*\,$$