# How might I answer this past exam question relating to the limiting price of an option?

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?

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

• 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? – M Smith May 18 '18 at 12:46
• $r = 0$ in your exam. – byouness May 18 '18 at 12:48
• Of course it is, silly me. – M Smith May 18 '18 at 12:51
• Presumably this means that the fact that the drift $\mu = 1$ is provided is just to throw us off slightly? – M Smith May 18 '18 at 12:55
• Maybe to check whether the students understand that they shouldn't care about the real world drift when pricing the option. – byouness May 18 '18 at 12:57