(Binomial) Probability of up-movement when risky security is numeraire

operationsres · November 23rd, 2012, 8:31 am

Hi, I have a tutorial question that I need help with (completely stuck). It's as follows. Consider a binomial tree with one time step, from 0 to T. Up factor is u and down factor is d.At t=0, the stock is valued S, and the riskless asset is valued 1.At t=T, the stock is u*S in the up state and d*S in the down state. The riskless asset is R:=(1+r) in both states.I'm asked "What is the probability of an up-movement in the tree under the measure associated with the risky security as the numeraire?".Thanks.

mj · November 24th, 2012, 9:55 pm

find the the value of p for the up move so thatE( B_1/S_1) = B_0/S_0given that E( B_1/S_1) = (1+r)(p/d + (1-p)/d)/S