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

question

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?

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  • $\begingroup$ 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
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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 $\ln(S_T)$.

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  • $\begingroup$ 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

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