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A couple questions regarding stochastic vol PDE derivation. Following Gatheral, a general stochastic vol model is given by \begin{align*} dS(t) & = \mu(t) S(t) dt + \sqrt{v(t)}S(t) dW_1, \\ dv(t) & = \alpha(S,v,t) dt + \eta \beta(S,v,t)\sqrt{v(t)} dZ_2, \\ dZ_1dZ_2 = \rho dt \end{align*} To price an option on a stock whose price process follows $S$, we construct a portfolio consisting of the option whose price is $V(S,v,t)$, short $\Delta$ shares of the stock and short $\Delta_1$ units of some other asset whose value $V_1$ depends on volatility.

First Question: Is this "other asset" absolutely anything such that $V_1 = V_1(v)$? E.g., another option on $S$, or some other option on another stock, or another stock, or...?

The value $\Pi$ of this portfolio is $$ \Pi = V - \Delta S - \Delta_1 V_1. $$

We then derive the SDE satisfied by $\Pi$, select $\Delta$ and $\Delta_1$ to make the portfolio riskless, argue that $d\Pi = r\Pi dt$ else arbitrage, and finally get $$ \frac{\frac{\partial V}{\partial t} + \frac{1}{2}vS^2\frac{\partial^2 V}{\partial S^2} + \rho \eta v \beta S \frac{\partial^2 V}{\partial v \partial S} + \frac{1}{2}\eta^2v\beta^2\frac{\partial^2 V}{\partial v^2} + rS\frac{\partial V}{\partial S} - rV}{\frac{\partial V}{\partial v}} \\ = \frac{\frac{\partial V_1}{\partial t} + \frac{1}{2}vS^2\frac{\partial^2 V_1}{\partial S^2} + \rho \eta v \beta S \frac{\partial^2 V_1}{\partial v \partial S} + \frac{1}{2}\eta^2v\beta^2\frac{\partial^2 V_1}{\partial v^2} + rS\frac{\partial V_1}{\partial S} - rV_1}{\frac{\partial V_1}{\partial v}} $$ Since the LHS only depends explicitly on $t,v,S,V$ and the RHS only on $t,v,S,V_1$, they must each be a function of only $t,v,S$, say $f(t,v,S)$. In particular, the price $V$ of the option must satisfy $$ \frac{\partial V}{\partial t} + \frac{1}{2}vS^2\frac{\partial^2 V}{\partial S^2} + \rho \eta v \beta S \frac{\partial^2 V}{\partial v \partial S} + \frac{1}{2}\eta^2v\beta^2\frac{\partial^2 V}{\partial v^2} + rS\frac{\partial V}{\partial S} - rV = \frac{\partial V}{\partial v}f(t,v,S). $$

Then, for whatever reason, we choose $f = -(\alpha - \varphi \beta)$, and as Gatheral states following eqn (3), "...$\varphi(S,v,t)$ is called the market price of volatility risk because it tells us how much of the expected return of $V$ is explained by the risk (i.e. standard deviation) of $b$ in the CAPM framework."

Second question: How does this market price of risk ($\varphi$) relate to the market price of risk I'm familiar with in the Black-Scholes model, $\frac{\mu - r}{\sigma}$? More importantly, how did they (Heston?) settle on $f = -(\alpha - \varphi \beta)$?

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  1. This other asset can be anything that can be priced with the very same equation. So in case you considering pricing the original option that expires on a single underlying, then any other option on that underlying would work. In case you use an option on a different underlying, you would need to have a model for the joint evolution of the two underlyings, and even if you do, things may not cancel nicely, so there's no big help in that. Think of that from the practical perspective: you'd like to hedge your position (make it indifferent) to changes in stock. For that you'd need a security for which you know for sure how does it depend on the stock price. Stock itself is a good choice. Another stock maybe highly correlater with the original one, and you may want to use it for hedging both in pricing and in theory - however in both domain you'd have some risk left in case the correlation is not perfect.

  2. According to "Willmot on QF", the motivation is as follows. If you only consider a $\Delta$-hedge portfolio $\Pi = V - \Delta S$ you'd get $$ \mathrm d\Pi = \eta\beta\frac{\partial V}{\partial \sigma}(\varphi \;\mathrm dt + \mathrm dZ_2) $$ so that with focus on the term in brackets, for every unit of risk you receive $\varphi$ as a reward.

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