Exercise 6.7 (Heston stochastic volatility model). Suppose that under a risk-neutral measure $\widetilde{\mathbb{P}}$ a stock price is governed by
$$
d S(t)=r S(t) d t+\sqrt{V(t)} S(t) d \widetilde{W}_1(t),
$$
where the interest rate $r$ is constant and the volatility $\sqrt{V(t)}$ is itself a stochastic process governed by the equation
$$
d V(t)=(a-b V(t)) d t+\sigma \sqrt{V(t)} d \widetilde{W}_2(t) .
$$
6.9 Exercises
289
The parameters $a, b$, and $\sigma$ are positive constants, and $\widetilde{W}_1(t)$ and $\widetilde{W}_2(t)$ are correlated Brownian motions under $\widetilde{\mathbb{P}}$ with
$$
d \widetilde{W}_1(t) d \widetilde{W}_2(t)=\rho d t
$$
for some $\rho \in(-1,1)$. Because the two-dimensional process $(S(t), V(t))$ is governed by the pair of stochastic differential equations (6.9.23) and (6.9.24), it is a two-dimensional Markov process.
So long as trading takes place only in the stock and money market account, this model is incomplete. One can create a one-parameter family of risk-neutral measures by changing the $d t$ term in (6.9.24) without affecting (6.9.23).
At time $t$, the risk-neutral price of a call expiring at time $T \geq t$ in this stochastic volatility model is $\mathbb{E}\left[e^{-r(T-t)}(S(T)-K)^{+} \mid \mathcal{F}(t)\right]$. Because of the Markov property, there is a function $c(t, s, v)$ such that
$$
c(t, S(t), V(t))=\tilde{\mathbb{E}}\left[e^{-r(T-t)}(S(T)-K)^{+} \mid \mathcal{F}(t)\right], \quad 0 \leq t \leq T .
$$
This problem shows that the function $c(t, s, v)$ satisfies the partial differential equation
$$
c_t+r s c_s+(a-b v) c_v+\frac{1}{2} s^2 v c_{s s}+\rho \sigma s v c_{s v}+\frac{1}{2} \sigma^2 v c_{v v}=r c
$$
in the region $0 \leq t<T, s \geq 0$, and $v \geq 0$. The function $c(t, s, v)$ also satisfies the boundary conditions
$$
\begin{aligned}
c(T, s, v) & =(s-K)^{+} \text {for all } s \geq 0, v \geq 0, \\
c(t, 0, v) & =0 \text { for all } 0 \leq t \leq T, v \geq 0, \\
c(t, s, 0) & =\left(s-e^{-r(T-t)} K\right)^{+} \text {for all } 0 \leq t \leq T, s \geq 0, \\
\lim _{s \rightarrow \infty} \frac{c(t, s, v)}{s-K} & =1 \text { for all } 0 \leq t \leq T, v \geq 0, \\
\lim _{v \rightarrow \infty} c(t, s, v) & =s \text { for all } 0 \leq t \leq T, s \geq 0 .
\end{aligned}
$$
In this problem, we shall be concerned only with (6.9.27).
(i) Show that $e^{-r t} c(t, S(t), V(t))$ is a martingale under $\widetilde{\mathbb{P}}$, and use this fact to obtain (6.9.26).
(ii) Suppose there are functions $f(t, x, v)$ and $g(t, x, v)$ satisfying