Helpful hints for problem set 9

I just uploaded Problem set 9; this problem set is due at the beginning of class on Thursday, November 16. Here are a few helpful hints (specifically, for problem #3).

1. In the first part of the third problem, note that since {S_T} = S{e^{\mu T}} is lognormally distributed, this implies that ln {S_T}/S is normally distributed with mean (\mu - 0.5{\sigma ^2})T and variance {\sigma ^2}T. This is shown in equation (18) in Hull’s “Wiener Processes and Ito’s Lemma” chapter. We also derived this result in class a few class meetings ago using Ito’s Lemma.

2. In the second part of the third problem, all you have to do is use the answer from the first part of the problem; since the option is at the money, the probability that in 1 year it will be in the money is calculated by determining what the probability is for a positive return.

3. Regarding risk neutral probability, as we have shown for both the binomial and continuous cases, the “trick” we use to price the option involves calculating risk neutral probabilities. Since the expected return on the stock is not only positive but also greater than the riskless rate of return, it must be the case that the “true” probability (calculated in part 2) is greater than the risk neutral probability (calculated in part 3).

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