Exercise 3

In the previous lecture the following simple model for a financial market was discussed.

Recall that a trading strategy is said to be a free lunch without risk in case V0 = α S0+βM0=0, V1,u = α S1,u+βM1>0, and V1,d = α S1,d+βM1>0.

(a) Suppose S0=M0=1, M1=1.05, S1,u=1.10, and S1,d=1.07. Consider the trading strategy α=1 and β= -1 (M0 β= 1 x -1= -1 means that you borrow 1 euro from the bank).

IIIa: Is there a free lunch without risk in the financial market described above?

(answer IIIa)The price at t=0 is equal to 0: V0 =α S0+βM0=1-1=0. At t=1 we have:

So there is a free lunch without risk. The reason for this, of course, is that even in the down scenario the stock earns more than you need to pay on your debt with the bank (10% > 7% > r = 5%).

IIIb: Show that there is no free lunch without risk if 0<S_(1,d)<S_0 (1+r)<S_(1,u)

(answer IIIb) This question is answered in Tutorial 3