Question

Consider a market with Ω = {ω1,ω2,ω3), r = 0 and one asset S. Suppose that S(0) = 2 and S has claim S̄ = (1,3,3) at time 1. Find all the risk-neutral probability measures on Ω.

I have worked out the risk neutral probability measure for w1, which is 1/2, by using the definition of probability measure EQ(Sn∗(1)) = Sn∗(0) (i.e. p1+3*p2+3*p3=2) and the fact that p1+p2+p3=1. So I'm left with p2+p3=1/2, not sure what to do next.

Answer 1

Consider a market with Ω = {ω1,ω2,ω3), r = 0 and one asset S. Suppose that S(0) = 2 and S has claim S ̄ = (1,3,3) at

time 1.

A probability Q on Ω is said to be a risk neutral probability measure if

(a) Q(ω) > 0 for all ω ∈ Ω.

(b) EQ(∆Sn∗ ) = 0, n = 1, 2, ..., N

Hence for the risk neutral probability we have EQ(Sn∗(1)) = Sn∗(0)

The risk neutral probability measure on ω1 is 1/(2+1)= 1/3 = 0.333

The risk neutral probability measure on ω2 is 1/(2+3) = 1/5 = 0.2

The risk neutral probability measure on ω3 is 1/(2+3) = 1/5 = 0.2

The risk neutral probability measure on Ω is (0.33, 0.2, 0.2)