Exercise 4

Consider the simple financial market with M0=1 and M1=1 so we actually assume that the interest rate r=0. This is only for simplicity. For the stock we have S0=100, S1,u=120, and S1,d=80.

IVa: Determine the price, at t=0, of a put option with strike price 100 euro (recall (Exercise 1) that its payoff, at t=1, is given by max{ 100 - S1,0} ).

(answer IVa)To determine the replicating portfolio we have to solve the following system of linear equations:

Subtracting the second equation from the first one yields -20= α (S1,u -S1,u). We thus obtain α= - 20/(120-80) = -1/2. Now we can solve for β (from each of the equations): 0= -1/2 x 120 + β x1 → β=60. The price of the option, at t=0, is thus given by 100 x (-1/2) + 60 x 1=10 euro.