Revision book 4, question 9

[NS206]

Hi,

In this question, part (iii) we need to explain whether or not to buy the option at the price offered. In the answers, the value of q is said to be less than p since the expected growth in log returns is lower, giving 1.28% vs 4.28%. Where does the 1.28% come from? This is also required later, and there is a formula given ( r - 0.5*sigma^2) but it's not clear where this is coming from to me.

Thank you!

 

[Anna Bishop]

Good morning!

In Q world (the risk-neutral world), investors expect to earn a return of r on all investments, no matter how risky.

In order for the investor to earn an expected return (expressed as a force of interest) of r on share investments, we would need the following lognormal distribution:

S1/S0 is logN(r - 0.5σ^2, σ^2)

This is because, using Page 14 of the Tables, the mean return is:

E[S1/S0] = exp(r - 0.5σ^2 + 0.5σ^2) = exp(r)

However, the question talks about the expected growth in 'log' returns.

If we have:

S1/S0 is logN(r - 0.5σ^2, σ^2)

then:

log(S1/S0) is N(r - 0.5σ^2, σ^2)

The 1.28% comes from the expected growth: r - 0.5σ^2 = 0.02 - 0.5 * 0.12^2 = 0.0128.

Does this help?

Anna