# 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?

• See Duffie, Pan, Singleton paper : nber.org/papers/w7105.pdf Aug 5, 2015 at 20:35
• Yeah, transform methods for the win. Aug 5, 2015 at 22:40
• @q.t.f. Sorry but your paper is quite general, please add the exact required formulas for the Bates option prices. Aug 6, 2015 at 10:45

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.

## 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 $$u_1=\frac{1}{2}\,,u_2=-\frac{1}{2}\,,\beta_1=\kappa^*-\rho\sigma\,,\beta_2=\kappa^*\,$$
• Thanks for your input. I am looking for the exact option pricing formulas, since I would like to use them for calibration of the parameters to market prices. The paper you cited mixes Black-Scholes, Heston and Bates model at some points, can you please give a clear copy of the closed-form solution for Bates vanilla option prices? Aug 6, 2015 at 10:44
• you can see deriscope.com/docs/Bates_1996.pdf . page 76
– user16891
Aug 6, 2015 at 17:41
• To be accepted as answer, please state the exact formula including all parameters in your post. Aug 6, 2015 at 20:28
• I think in your change of measure in 2.) its missing the definition of $\lambda^*$ and in 4.) please also add the put formula. Please add the definition of "$\phi$". Aug 8, 2015 at 10:22
• 1.$J^*=J+J\mathbb{E^P}[\frac{\Delta J_w}{J_w}]$ where $J_w$ is marginal utility of your currency e.g dollar wealth of the word-average representative investor. 2.$P(t,S_t,v_t,J,K,T)=C(t,S_t,v_t,J,K,T)+K\,e^{-r\tau}-S_t\,e^{-q\tau}$ 3. $\phi$ is element of integration in characteristic function
– user16891
Aug 8, 2015 at 10:45