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Given the Heston model $$dS_t=\mu S_tdt+\sqrt{\nu_t}S_tdB_{1,t}\\ d\nu_t=k(\theta-\nu_t)dt+\eta\sqrt\nu_tB_{2,t}$$ how should the replicating portfolio $V_t$ for the derivative $F_t$ be composed?

I could not clearly understand whether the replicating portfolio $$V_t=\alpha S_t+\beta B_t$$ with $dB_t=rB_tdt$ would be sufficient to hedge the price of the derivative $F_t$, or whether the replicating portfolio should also contain another asset, whose dynamics depend on $\nu_t$, such as $$V_t=\alpha S_t+\beta B_t+\gamma X_t$$

In this second case, which dynamics should the asset $X_t$ follow?

I am trying to determine the pricing PDE in the framework of Heston using the self-financing approach, according to which, the values of the derivative $F_t$ and that of the replicating portfolio $V_t$ coincide ($F_t=V_t$).

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    $\begingroup$ Because you have two stochastic variables (two sources of uncertainty), your hedge portfolio needs to contain a third assets: a bond, the stock and a third asset whose value depends on volatility. Because the portfolio is self-financing, $\text{d}V=\alpha\text{d}S+\beta\text{d}B+\gamma\text{d}X$. Then, you use Itô's Lemma on $\text{d}S$ and $\text{d}X$ and require that $\text{d}V=rV\text{d}t$. Eliminating the random shocks shows you how to choose $\alpha$, $\beta$ and $\gamma$. It's very similar to the derivation of the Black-Scholes PDE. $\endgroup$
    – Kevin
    Apr 5 at 12:47
  • $\begingroup$ Yes, but what should it be dynamics of $X_t$? $\endgroup$
    – Mr Frog
    Apr 5 at 13:01
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    $\begingroup$ You don't need them to specify them. Suppose $X(t,S_t,v_t)$ is the value of a second traded asset. Just write $\text{d}X=(...)\text{d}t+(...)\text{d}S_t+(...)\text{d}v_t$. Do the same with $F$, the option you want to hedge. A lot of stuff will cancel out. $\endgroup$
    – Kevin
    Apr 5 at 13:08
  • $\begingroup$ I do not still see why you suggested applying Ito's lemma on $dS$ and $dX$ if their representation is known. In addition, I am trying not to require $dV=rVdt$ because that does not lead to a rigorous derivation. I am trying to set equal the martingale and drift components of the Ito representation of the price of the derivative $dF_t$ and its replicating strategy $dV_t=\alpha dS_t+\beta dB_t + \gamma dX_t$ $\endgroup$
    – Mr Frog
    Apr 5 at 14:31

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