I have this question. Consider the Heston stochastic volatility model:
\begin{array}{rl} dS_t = uS_t dt + \sqrt{v_t} S_t dB_{1,t} \\ dV_t = \kappa(\theta - V_t)dt + \sigma\sqrt v_t dB_{2,t} \end{array}
where $B^1, B^2$ are related Brownian motions. Find the differential equation of an evaluation of a derivative $f = f (t, S_t, v_t)$
As for the differential equation of evaluation of a generic derivative, from a payoff at maturity $ F (S_t, v_t) $, we define the price through the following equation:
$$f(t, s, v) = E^Q [e^{-r(T-t)}F(S_T,v_T)]$$
dividing by per $e^{rt}$ we get the discounted price:
$$\tilde{f}(t,s,v) = e^{-rt} f(t,s,v) = E^Q [e^{-r(T-t)}F(S_T,v_T)]$$
with relative dynamics: $d \tilde{f} = e^{-rt} df - re^{-rt} f dt \ $.
We now derive through the equations of the Heston model (in compact form according to the Q dynamics) using the Ito formula for a correlated two-dimensional process:
$ df = (\partial_{t}f + rS_t \partial_{S}f + \tilde{k} (\tilde{\theta} - v) \partial_{v} f+ \frac{1}{2} v_t S_t^2 \partial_{ss} f + \rho \eta S_t v_t \partial_{sv} f + \frac{1}{2} \eta^2 v_t \partial_{vv} f) dt + S_t \sqrt{v_t} \partial_s f dB_{1,t}^{\lambda} + \eta \sqrt{v_t} \partial_v f dB_{2,t}^{\lambda}$
Finally we get the condition of martingality:
$e^{-rt} (\partial_{t} f + rS_t\partial1_{s} f + \tilde{k} (\tilde{\theta} - v) \partial_{v} f + \frac{1}{2} v_t S_t^2 \partial_{ss} f + \delta \nu S_t v_t \partial_{sv} f + \frac{1}{2} \nu^2 v_t \partial_{vv} f)) - re^{-rt} f = 0$
The pricing of an generic derivate is correct? Thanks for support.