I'm currently working through Shreve's Volume II, and I'm having some difficulty on Exercise 5.4 of Chapter 5. The problem statement is:
Consider a stock whose price differential is $$ dS(t) = r(t) S(t) dt + \sigma(t)d \widetilde{W}(t) $$ where $r(t)$ and $\sigma(t)$ are nonrandom functions of $t$ and $\widetilde{W}$ is a Brownian motion under the risk-neutral measure $\widetilde{\mathbb{P}}$. Let $T>0$ be given, and consider a European call, whose value at time zero is $$ c(0, S(0)) = \mathbb{E}\bigg[ \exp\bigg\{-\int_{0}^{T}r(s)\,ds\bigg\}(S(T)-K)^{+}\bigg].\quad\text{(Should be risk-neutral expectation $\widetilde{\mathbb{E}}$.)} $$
(i) Show that $S(T)$ is of the form $S(0)e^{X}$, where $X$ is a normal random variable, and determine the mean and variance of $X$.
Solution Attempt:
I want to take some transformation of $S(t)$, probably the log transform, apply Ito's Lemma, and the show that the result is some normal random variable plus some non-random component for the mean.
We have \begin{align*} d(\log S(t))& = \frac{1}{S(t)}\, dS(t) - \frac{1}{2 S^{2}(t)} \, dS(t)\,dS(t)\\ &= \frac{1}{S(t)}\bigg[ r(t)S(t)\,dt + \sigma(t)\,d\widetilde{W}(t)\bigg] - \frac{1}{2S^2(t)} \sigma^{2}(t)\,dt \end{align*} where the last expression results from the equations $dt\,dt = 0$, $dt\,d\widetilde{W}(t) = 0$, and $d\widetilde{W}(t)\,d\widetilde{W}(t) = dt$. Grouping terms with the same differential gives $$ d(\log S(t)) = \bigg( r(t) - \frac{\sigma^{2}(t)}{2S^{2}(t)}\bigg) \,dt + \frac{\sigma(t)}{S(t)}\,d\widetilde{W}(t). $$ If the integrands for both differentials were non-random, this problem would be solved as Ito Integrals of non-random integrands are normally-distributed. However, the stock price $S(t)$ appears in the denominator, and that is random.
I'm stuck here, so if anyone could give me a hint, that would be very much appreciated.
If the stochastic differential equation were $dS(t) = r(t)S(t)dt + \sigma(t)S(t)d\widetilde{W}(t)$, then this problem would be easy, as the stock price would be a geometric brownian motion with non-random drift and volatility. However, I checked the errata page, and this problem doesn't appear in them.
