Consider a non-dividend paying stock Sthat follows the stochastic differential equation dS = m S dt + s S dz, and let r be the risk-free interest rate. The stock is worth S₀ Euros today. Let T be a fixed maturity.

a. Calculate the expected value and the variance of S_T under the risk-neutral probability using the following result: if U ~ f(0,1) is a centred Gaussian random variable, E[exp(aU)] = exp(a²/2) for any constant a.

b. Calculate the price of a European binary option whose payoff at maturity T is 100 Euros if S_T> K and 0 otherwise (Hint: calculate the risk-neutral probability that S_T> K).

c. Calculate the price of a European option whose payoff at maturity T is max(K, S_T), where K is a constant. (Hint: max(K, S_T) = K + max(0, S_T - K)).

Answer rating: 100% (QA)

a The expected value of ST is given by EST S0 emrT The variance of ST is given by VarST S02 e2mrT sigma2 T where sigma2 is the variance of U Thus the ... View the full answer