Cody Smith

The most challenging topic that I have encountered in Fin 622 so far is that of multi-period binomial valuation. In particular I struggled with Problem 14 in the chapter 7 exam.
For this problem I will use UU as the value after 2 up moves, UD after one up and one down, DD after 2 down moves, U after 1 up move, D after 1 down move. The goal is to find the value of the derivative at time zero given the value of the asset and derivative two periods out.
In order to find the value of the derivative at time zero you must first calculate the risk neutral probabilities. To do this, solve this equation for x:

UUx + UD(1-x) = U

where UU, UD, and U are the values of the asset at those time periods. These are given to you in the problem. x will equal the risk neutral probability of the up-state occurring. 1-x will equal the probability of the down state occurring.
Now that you have the probabilities you can solve for the value of the derivative at time U. You are given the value of the derivative at UU and UD so the equation is:

(UUx + UD(1-x))/(1+r) = U

r is equal to the risk free interest rate. This will give you the value of the derivative at time U.
In order to get the value of the derivative at time D you will have to follow the same logic. You already know the risk neutral probabilities are x and 1-x so you will solve the following equation:

(UDx + DD(1-x))/(1+r) = D

where UD, DD, and D represent the values of the derivative at those times and r is equal to the risk free asset.
In order to get the value of the derivative at time zero you will follow the same sort of logic. The equation that solves for the value of the derivative at time zero is:

(Ux + D(1-x))/(1+r) = value at time zero

where U, and D represent the values of the derivative as calculated above and r is equal to the risk free rate.
Following these steps and equations will give you the value of a derivative today that will produce given cash flows multiple time periods from now. This can be very useful to managers when evaluating different investment options. This can also be expanded to consider capacity expansions in the future.