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?
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.
