4
$\begingroup$

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.

$\endgroup$
1
4
$\begingroup$

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 $$S(t) = r(t)S(t)dt + \sigma(t) S(t) d \tilde W(t).$$

$\endgroup$
3
  • 1
    $\begingroup$ See @Frido Rolloos comment to the question. Either the question text in Shreve's book is wrong, or the link added by Frido is the way to go. But then again, there are some people who think that it's a typo : quantsummaries.com/shreve_stochcal4fin_2.pdf $\endgroup$ Nov 3 '21 at 7:24
  • 1
    $\begingroup$ 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. $\endgroup$
    – Sebastian
    Nov 3 '21 at 10:18
  • $\begingroup$ Thank you! I probably should have figured that there was a typo earlier, as stocks are assumed to be geometric brownian motions under the black-scholes model. Thanks for your help! $\endgroup$ Nov 3 '21 at 15:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.