Calibration of Heston model with stochastic short rate

Question

I have following Heston model with stochastic short rate: \begin{eqnarray*}dS\left(t\right)&=&r\left(t\right)S\left(t\right)dt+\nu\left(t\right)S\left(t\right)dW^{S}\left(t\right)\\dr\left(t\right)&=&\beta\left(\alpha-r\left(t\right)\right)dt+\sigma\sqrt{r\left(t\right)}dW^{r}\left(t\right)\\d\nu\left(t\right)&=&\kappa\left(\theta-\nu\left(t\right)\right)dt+\xi\sqrt{\nu\left(t\right)}dW^{\nu}\left(t\right)\end{eqnarray*}

where all Wiener processes can be correlated. How can I calibrate this model? I heard that if short rate is not correlated with stock price process, then we can firstly calibrate short rate (how to do this? which instruments should we use?) and then calibrate stock process to for example call option prices. Why this is possible only when these two components are not correlated? What is the correlation between stock and interest rate in practice? If we have correlated stock and interest rate, how to calibrate this model?

Answer 1

If we take your model literally (with the correction that I suggested as a comment), then there exists no (semi-)closed form, IMHO, that you can use for asset pricing. What you could do is then to make the model a bit simpler or to simulate.

Simulation

This is the nasty part. Based on your model, you simulate a very large number of the discount factor(s) and asset prices and price your reference options as average discounted payoffs. Keeping the random seed fixed, of course. Then you calibrate the model parameters trying to minimize some function of the pricing errors.

Decrease complexity and gain tractability

If you could work with a slightly simpler version, you could use the standard machinery of European type derivatives valuation under affine diffusion processes.

You might assume zero correlations between the risk free rate and asset / variance stochastics, i.e. $\mathrm{E}(dW_S(t)dW_r(t))=\mathrm{E}(dW_v(t)dW_r(t))=0dt$ and recover the models that you have already quoted in your question. Or you could reformulate your model as:

$$ \begin{eqnarray*}dS\left(t\right)&=&r\left(t\right)S\left(t\right)dt+\color{red}{\sqrt{\nu\left(t\right)}}S\left(t\right)dW^{S}\left(t\right)\\ dr\left(t\right)&=&\beta\left(\alpha-r\left(t\right)\right)dt+\sigma\color{red}{\sqrt{\nu\left(t\right)}}dW^{r}\left(t\right)\\d\nu\left(t\right)&=&\kappa\left(\theta-\nu\left(t\right)\right)dt+\xi\sqrt{\nu\left(t\right)}dW^{\nu}\left(t\right)\end{eqnarray*} $$

In this formulation, the three processes share the same variable driving their variance, $\nu(t)$. Hence the process' covariance structure, after transforming $S_t\to ln(S_t)\equiv y_t$,

$$ \mathrm{E}(dXdX^T)=\mathrm{E}\begin{pmatrix}dy_tdy_t&dy_tdr_t&dy_td\nu_t\\ dy_tdr_t & dr_tdr_t & dr_tdy_t\\ dy_td\nu_t & dr_td\nu_t & d\nu_td\nu_t\end{pmatrix}=\nu_t\begin{pmatrix}1&\sigma\rho_{S,r}&\xi\rho_{S,\nu}\\ \sigma\rho_{S,r}&\sigma^2&\sigma\xi\rho_{r,\nu}\\\xi\rho_{S,\nu}&\sigma\xi\rho_{r,\nu}&\xi^2\end{pmatrix}dt $$

This is linear in $X_t$, i.e. in $\begin{pmatrix}y_t\equiv \ln(S_t)&r_t&\nu_t\end{pmatrix}^T$.

From here, you could, quite traceably, derive the discount bond pricing equations and the characteristic equations of the option price formula, plug everything into some Fourier transform method. As a result, you obtain tractable (quasi) closed form bond and option pricing formulas that can be calibrated to observed prices.

HTH a bit?

NB: One question will remain for the practitioner: Which financial instrument out there will yield information to pinpoint the relationship between the innovation in the short rate and the innovation in the stock prices?