FIN 538 Stochastic Foundation of Finance Fall 2022
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Stochastic Foundation of Finance (FIN 538)
Fall 2022
SAMPLE FINAL EXAM
Question 1
1a. Apply Ito’s lemma to write the differential form for ^tBt where Bt is a standard Brownian motion. Is ^tBt a martingale? Explain.
1b. Suppose that the process Xt = Bt(3) - atBt is a martingale where Bt is a standard Brownian motion. Find a and calculate E[Xt |Fs], s < t for that value of a.
1c. Calculate the (unconditional) expectation E[Bs Bt Bu], for s < t < u where Bt is a standard Brownian motion. Hint. The following formula might be useful: If X has the distribution N(µ, σ2 ) then E[X3] = µ3 + 3µσ2 .
1d. Apply Ito’s lemma to write the differential form for Xt = Bt e t(t)Bsds where Bt is a standard Brownian motion. Calculate the expected rate of return Et [Xt(dX)t ].
Question 2
Let’s consider a world with only two dates: Today and Tomorrow. There are three possible states tomorrow: Burst, Normal, and Boom. We have three risky stocks X , Y , Z traded in the market. The current prices and future possible payoffs of these risky stocks, if they are known, are reported in the following table
Stock |
Today’s price |
Tomorrow payoff |
||
Burst |
Normal |
Boom |
||
X |
$ 2 |
$ 1 |
$ 2 |
$ 3 |
Y |
$ 2 |
$ 4 |
$ 0 |
$ 0 |
Z |
??? |
$ 0 |
$ 1 |
$ 2 |
The (net) simple interest rate in the market is given to be r = 0%. Assume that there are no arbitrage opportunities in the market.
2a. What are the risk neutral probabilities of the states Burst, Normal, Boom? 2b. What is the current price of stock Z?
Question 3
Suppose that the interest rate follows the following stochastic process:
drt = (1 - rt )dt + e − 2 dBt , where r0 = 0
where Bt is a standard Brownian motion.
3a. Denote Rt = etrt . Using Ito’s lemma, find the expression for dRt .
3b. Solve for Rt and then rt . In your answer, Rt and rt should be written as a sum of a determin- istic term and an Ito integral.
3c. Calculate E[rt]. When t approaches infinity, what does E[rt] approach to? Please show your work.
3d. Calculate Var(rt ). When t approaches infinity, what does Var(rt ) approach to? Please show your work.
2022-10-12