1) The price of a stock follows the following binomial tree with 3 time steps.

binomial tree

(15 points) Please price an American put written on this stock with the strike price K=$75 and maturing in 3 months. Suppose the risk-free rate is r=1.6% and one tick-time t= One month. (5 points) After pricing, please also provide hedge ratio on each node before maturity.

2)

Let P be a price index, such as the Consumer Price Index (CPI). Let M be the nominal supply of money in the economy. For example, M might be designated as the amount of bank deposits and currency in circulation. Assume P and M each follows geometric Brownian motion processes

dPP=pdt+pdBp

dMM=mdt+mdBm

with dBpdBm=dt. Monetary economists define real money balances, m to be m=MP. Derive the stochastic process for real money balance m.

3)

Consider the following SDE:

dSt=rStdt+StdBt

where Bt is a standard Brownian motion under risk-neutral measure .

(8 points) Compute the probability of ST being more than a constant K under measure , that is Pr(ST>K) , with clear derivation steps. (4 points) Name the economic meaning of Pr(ST>K) if we consider the European put option of the underlying asset St with strike price K.

4)

If Bt is a standard Brownian motion under measure , then evaluate the following expectations:

(3 points) [(Bt)2]=? (No need to give derivation steps) (5 points) [(eBt)n]=? with clear derivation steps.

5)

Suppose Bt is a standard Brownian motion under measure , answer the following questions:

(8 points) Compute t0BsdBs=? with clear derivation steps. (12 points) Is the result in part 1 a martingale under measure ? Justify your answer with clear derivation steps.

6)

Let

yT=T0etdBt

where is a constant and Bt is a standard Brownian motion under measure .

(5 points) Find out the distribution followed by yT under measure . (5 points) Find [yT]=? (10 points) Find Var[yT]=? Show clear derivation steps.