# Difficulty with stochastic calculus problem

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.

Shreve titles the question as: "Black-Scholes_Merton formula for time-varying, non-random interest rate and volatility". This model is well-known, so there is no ambiguity. The SDE should be $$S(t) = r(t)S(t)dt + \sigma(t) S(t) d \tilde W(t).$$
• Thanks @kermittfrog. I misread the question, you guys are right. Yes, there is a typo. Shreve titles the question as: "Black-Scholes_Merton formula for time-varying, non-random interest rate and volatility". So, there is no ambiguity, the SDE should be $$S(t) = r(t)S(t)dt + \sigma(t) S(t) d \tilde W(t).$$ Sorry, I answered the question with this SDE in mind without having read properly the OP's question. He even hint the possible typo at the end of his question. A proper answer would have been just to point out the typo. Nov 3, 2021 at 10:18