September 2020 Q.2

[George Philip]

Question:
The chief executive officer (CEO) of a company benefits from an executive reward plan that includes company shares currently worth €100,000. The shares currently trade at €1 each. The CEO wishes to retire in 4 years’ time and hopes the share fund value at that time will be at least a target value of €150,000.

The share price St at time t (measured in years) follows the stochastic differential equation:
St = exp (0.06875 t + 0.25 Wt )

where Wt is a Standard Brownian Motion. The ‘surplus amount’ is defined as the difference between the share fund value in 4 years’ time and the CEO’s target value.
(i) Calculate the standard deviation of the surplus amount.

The answer given in the Examiner's Report uses the following formula:

Var(St) = E[St]^2 . {exp(0.252t)–1}
where E[St] = exp(0.06875t+0.252t/2)
So E = 1.49182

Where is this formula/similar question given in the course notes?

 

[Steve Hales]

Hi
We know that standard Brownian motion is normally distributed so:

0.06875 * t + 0.25 Wt ~ N(0.06875 * t , 0.25^2 * Wt)

The exponential of a normally distributed random variable has a lognormal distribution with the same parameters. Then we can use the formulae on page 14 of the Tables to find the mean and the variance.