You are given the price of a nondividend paying stock St and a European call option Ct in a world where there are only two possible states
The true probabilities of the two states are given by {Pu = .5, Pd = .5}. The current price is St = 280. The annual interest rate is constant at r = 5%. The time is discrete, with Δ = 3 months. The option has a strike price of K = 280 and expires at time t + Δ.
(a) Find the risk-neutral martingale measure P* using the normalization by risk-free borrowing and lending.
(b) Calculate the value of the option under the risk-neutral martingale measure using
Ct = 1 / 1 +rΔ EP*[Ct+Δ]
(c) Now use the normalization by St and find a new measure P under which the normalized variable is a martingale
(d) What is the martingale equality that corresponds to normalization by St?
(e) Calculate the option's fair market value using the P.
(f) Can we state that the option's fair market value is independent of the choice of martingale measure?
(g) How can it be that we obtain the same arbitrage-free price although we are using two different probability measures?
(h) Finally, what is the risk premium incorporated in the option's price? Can we calculate this value in the real world? Why not?