# Derivation of Stochastic Vol PDE

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 $f = -(\alpha - \varphi \beta)$?

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.