I'm deriving the solution for European call option in the Heston Model. I follow the original paper by Heston and Fabrice Douglas Rouah's derivations in his book The Heston Model and Its Extensions in Matlab and C#. However, I'm having troubles understanding a few steps - I have 3 questions.
The hedging portfolio in the Heston Model consists of an option, $V = V(S,v,t)$, $\Delta$ stocks and $\phi$ units of the option to hedge volatility, $U(S,v,t)$, and has the value: \begin{align*} \Pi = V + \Delta S + \phi U, \end{align*} where the change in the value of the portfolio in the time interval, $dt$ is given by: \begin{align} \label{HestonPort} d\Pi = dV + d\Delta S + d\phi U. \end{align}
Next, I want to obtain the process followed by $dV$. Rouah writes that, one must apply Itô's lemma to $V$, and that one must differentiate $V$ wrt $t,S$ and $v$, and create a second-order Taylor expansion. This results in: \begin{align*} dV = \frac{\partial V}{\partial t}dt + \frac{\partial V}{\partial S}dS + \frac{\partial V}{\partial v}dv + \frac{1}{2}vS^2\frac{\partial^2V}{\partial S^2}dt + \frac{1}{2}v\sigma^2\frac{\partial^2V}{\partial v^2}dt + \sigma \rho v S \frac{\partial^2 V}{\partial S \partial v}dt. \end{align*}
- I don't understant this step. Why do I need to create a second-order Taylor expansion? And why do I differentiate $V$ wrt $t,S$ and $v$? I understand Itô's lemma as in the derivation of the Black Scholes model - is this some kind of extensions of Itô's? Or how should I know that I need the second-order Taylor expansion?
Later in the derivations, Heston writes that for at a European call option he "guesses a solution of the form": $$ C(S,v,t) = SP_1 - Ke^{-rT}P_2. $$ (page 330, equation 10). This is analogy with the Black-Scholes formula. The first term is the present value of the spot asset upon optimal exercise, and the second term is the present value of the strike-price payment. Both of these terms must satisfy the PDE given by:
\begin{align} \label{HestonPDE} \begin{split} & \frac{\partial U}{\partial t} + \frac{1}{2}vS^2\frac{\partial^2U}{\partial S^2} + \sigma \rho v S \frac{\partial^2 U}{\partial S \partial v} + \frac{1}{2}v\sigma^2\frac{\partial^2U}{\partial v^2} \\ - &rU + rS \frac{\partial U}{\partial S} + \left[ \kappa(\theta - v) - \lambda(S,v,t) \right] \frac{\partial U}{\partial v} = 0. \end{split} \end{align}
Substituting the proposed solution into the original PDE shows that P1 and P2 must satisfy:
\begin{align} \label{PPDE} \frac{\partial P_j}{\partial t} + \rho \sigma v \frac{\partial^2 P_j}{\partial v \partial x} + \frac{1}{2} v \frac{\partial^2 P_j}{\partial x^2} + \frac{1}{2} \sigma^2 v \frac{\partial^2 P_j}{\partial v^2} + (r+u_j v) \frac{\partial P_j}{\partial x} + (a-b_j v) \frac{\partial P_j}{\partial v} = 0, \end{align}
- I can see that Heston "guess" is similar to the Black Scholes equation - but how can he "guess" this solutions? Can this guess be derived from the PDE?
- Why is it that the two terms from the guessed solution also must satisfy the PDE? And why is it nessecary to derive a PDE for P1 and P2?
Thanks in advance!