The following image shows a past exam question that I am attempting to answer (for which I do not have a mark scheme):


I believe that under the BMS model, the payoff of a stock at maturity $T$ is given by $$ S_T = S_0 \exp \left( \mu T + \sigma W_T \right) $$

Thus, the payoff of the stock in the question would be given by $$ S_T = \exp \left( T + W_T \right) $$

Therefore, I would expect the payoff of the option to be $$ V_0 (T) = \left| \frac{T}{2} + T + W_T \right| = \left| \frac{3T}{2} + W_T \right| $$

However, $$ \lim_{T \rightarrow \infty} \frac{\left| \frac{3T}{2} + W_T \right|}{\sqrt{T}} = \lim_{T \rightarrow \infty} \left| \frac{3\sqrt{T}}{2} + \frac{W_T}{\sqrt{T}} \right| = \infty $$

What am I doing wrong?

  • $\begingroup$ Try to avoid images and copy/paste instead. Also, if this is homework, then please use the homework tag. $\endgroup$ – byouness May 18 '18 at 12:18
  • $\begingroup$ Thanks for the advice. I can confirm that this is not homework, however. $\endgroup$ – M Smith May 18 '18 at 12:47

Two things:

  • The price of the option is the expectation under the risk-neutral measure. Under this measure, the stock price's drift is $r$ and not $\mu$: $$dS_t = S_t r dt + S_t \sigma dW_t$$

  • When you integrate to get $S_T$, you have made a mistake: $$S_T = S_0\exp\left(\left(r-\frac{\sigma^2}{2} \right)T + \sigma W_T \right)$$

Start by writing out the expression of $d\ln(S_t)$ using Itô's lemma, then deduce $\ln(S_T)$.

  • $\begingroup$ Hi, thanks for your response. However, using your $S_T$ the limit simplifies to $\lim_{T \rightarrow \infty} \left| r \sqrt{T} + \frac{W_T}{\sqrt{T}} \right|$, which still tends to infinity? $\endgroup$ – M Smith May 18 '18 at 12:46
  • $\begingroup$ $r = 0$ in your exam. $\endgroup$ – byouness May 18 '18 at 12:48
  • $\begingroup$ Of course it is, silly me. $\endgroup$ – M Smith May 18 '18 at 12:51
  • $\begingroup$ Presumably this means that the fact that the drift $\mu = 1$ is provided is just to throw us off slightly? $\endgroup$ – M Smith May 18 '18 at 12:55
  • $\begingroup$ Maybe to check whether the students understand that they shouldn't care about the real world drift when pricing the option. $\endgroup$ – byouness May 18 '18 at 12:57

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.