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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 in advance!

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    $\begingroup$ I think this will answer your question, if not let us know. Although it's for Heston the logic is the same for any SV model: frouah.com/finance%20notes/… $\endgroup$
    – Frido
    Jun 18, 2023 at 16:47
  • $\begingroup$ 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 $F_T = \phi_T S_T+ \psi G_T$? $\endgroup$ Jun 18, 2023 at 18:29
  • $\begingroup$ 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! $\endgroup$ Jun 19, 2023 at 6:25

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