finance mathematics - Problem on Characteristic function in Heston model

I know the Heston model .In this model, we have

$$f(\Phi,x_t,v_t)=\exp(C_j(\tau,\Phi)+D_j(\tau,\Phi)+i * \Phi * x_t)$$

How can we extract the Characteristic function as follows

$$f(\Phi_1,\Phi_2,x_t,v_t)=\mathbb{E}[\exp(i * \Phi_1 * x_T+i*\Phi_2*v_T)]$$

Thanks.

Answer

The Heston model is represented by the bivariate system of stochastic differential equations

$$\begin{align} & dS_t=rS_tdt+{\sqrt\upsilon_t}S_t dW_1(t) \\ & dv_t=\kappa(\theta-v_t) dt+\sigma{\sqrt v_t}dW_2(t) \tag 1\\ &\mathbb{E}[dW_1(t),dW_2(t)]=\rho dt \end{align}$$ set $x_t=\ln S_t$, By application of Ito's lemma, we have $$\begin{align} & dx_t=\left(r-\frac12 v_t\right)dt+{\sqrt\upsilon_t} dW_1(t) \\ & dv_t=\kappa(\theta-v_t) dt+\sigma{\sqrt v_t}dW_2(t) \tag 2\\ \end{align}$$ Let $B_1(t)$ and $B_2(t)$ be two independent Wiener processes, we have $$\begin{align} & dx_t=\left(r-\frac12 v_t\right)dt+{\sqrt\upsilon_t} dB_1(t) \\ & dv_t=\kappa(\theta-v_t) dt+\sigma{\sqrt v_t}\left(\rho\,dB_1(t)+\sqrt{1-\rho^2}dB_2(t)\right) \tag 3\\ \end{align}$$ Now we can write the Heston model as follow $$dy_t=\mu(t,y_t)dt+\Sigma(t,y_t)dB_t\tag 4$$ where $$y_t=\left( \begin{matrix} {x_t} \\ {v_t} \\ \end{matrix} \right)$$ $$\mu(t,y_t)=\left( \begin{matrix} r-\frac{1}{2}{{v}_{t}} \\ \kappa (\theta -{v_t}) \\ \end{matrix} \right) \\ \Sigma (t,y_t)=\left( \begin{matrix} \sqrt{{{v}_{t}}} & 0 \\ \sigma \rho \sqrt{v_t} & \sigma \sqrt{1-{{\rho }^{2}}}\sqrt{{{v}_{t}}} \\ \end{matrix} \right)\tag 5$$ and $$B(t)=\left( \begin{matrix} {{B}_{1}}(t) \\ {{B}_{2}}(t) \\ \end{matrix} \right)$$ The drift $\mu$ and the matrix $\Sigma\Sigma^{\text{T}}$ can both be written in the affine form $$\begin{align} &\quad\,\, \mu (t,{{y}_{t}})={{\alpha}_{0}}+{{\alpha}_{1}}{{x}_{t}}+{{\alpha}_{2}}{{v}_{t}} \\ & \Sigma {{\Sigma }^{\text{T}}}(t,{{y}_{t}})={{\beta}_{0}}+{{\beta}_{1}}{{x}_{t}}+{{\beta}_{2}}{{v}_{t}} \\ \end{align}\tag 6$$ where $${{\alpha }_{0}}=\left( \begin{matrix} r \\ k\theta \\ \end{matrix} \right),\,{{\alpha }_{1}}=\left( \begin{matrix} 0 \\ 0 \\ \end{matrix} \right),{{\alpha }_{2}}=\left( \begin{matrix} -0.5 \\ -\kappa \\ \end{matrix} \right)\tag 7$$ and $${{\beta }_{0}}={{\beta }_{1}}=\left( \begin{matrix} 0 & 0 \\ 0 & 0 \\ \end{matrix} \right),{{\beta }_{2}}=\left( \begin{matrix} 1 & \rho \sigma \\ \rho \sigma & {{\sigma }^{2}} \\ \end{matrix} \right)\tag 8$$ The result of Duffie, Pan, and Singleton (2000) is that the characteristic function has the log-linear form $$f(\phi_1,\phi_2,x_t,v_t)=\exp\left(A(\tau,\phi_1,\phi_2)+B(\tau,\phi_1,\phi_2)x_t+C(\tau,\phi_1,\phi_2)v_t\right)$$

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Note

Duffie, Pan, and Singleton (2000) show that the characteristic function of a wide class of multivariate affine models (of which the Heston model is a special case) has a log linear form .

For more details, see it: