# How to replicate a claim in a stochastic volatility model?

Given a Markovian stochastic volatility model with an asset $$S$$ and a variance process $$V$$ given by $$dS_t = \mu_t S_tdt + \sqrt{V_t}S_tdW_t, \\ dV_t = \alpha(S_t,V_t,t)dt + \eta \beta(S_t,V_t,t)\sqrt{V_t}dZ_t, \\ dW_tdZ_t = \rho dt,$$ where $$W$$ and $$Z$$ are correlated Brownian motions. Now suppose that we have two traded assets $$F$$ and $$G$$, which can be written as $$F_t = f(t,S_t,V_t)$$ and $$G_t = g(t,S_t,V_t)$$, where $$f$$ and $$g$$ are deterministic smooth functions. Additionally I can assume that $$S,F,G$$ are martingales.

Question: How can the claim $$F_T$$ be replicated by trading in the options $$S$$ and $$G$$?

My idea was the following: Define a portfolio $$\Pi = F - \phi S - \psi G$$. Using the self financing property and Ito's formula, I can calculate $$d\Pi$$ and choose $$\phi$$ and $$\psi$$ such that the portfolio is instantaneously risk free, i.e. I choose $$\phi = F_S - G_S \frac{F_V}{G_V}, \\ \psi = \frac{F_V}{G_V}.$$ This leaves me with $$d\Pi = (...)dt$$. However I am not sure how to go on from here and how this does help me. Additionally, I am unsure on where I can/should use that the processes are martingales.

Was my initial Ansatz with the portfolio $$\Pi$$ correct? If yes, how should I proceed from here? If no, what would be the right way to start?

• Thanks, what I am not completely sure about is why this portfolio must be risk free, i.e. $d\Pi = \Pi dt$ and if so, how can I use this to argue that $(\phi,\psi)$ now indeed replicates the Option, i.e. we have $F_T = \phi_T S_T+ \psi G_T$? Jun 18 at 18:29
• Ok, I think I got it. The essential part I was missing is that when $F,G,S$ are martingales, the drift vanishes, i.e. we have $dS = (\dots)dW$ and so on. Thanks again, for the help! Jun 19 at 6:25