Final Exam Solutions FIN 538 Fall 2019 Mini A
Question 1 (40 points in total): 1a. (10 points) Apply Ito’s lemma to calculate
martingale? Explain. Answer. By Ito’s lemma,
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d(t Bt)=2√tBtdt+ tdBt
t Bt is not a martingale because the drift term of its differential form is not equal to zero.
1b. (10 points) Suppose that the process Xt = Bt3 − atBt is a martingale. Find a and calculate E[Xt|Fs], s < t for that value of a.
Answer. By Ito’s lemma
dXt = (−aBt + 3Bt)dt + (3Bt2 − at)d is a martingale if the drift term is zero: −aBt +3Bt = Bt(3−a) = 0. Thus a = 3. Since Xt = Bt3 − 3tBt is a martingale, E[Xt|Fs] = Xs = Bs3 − 3sBs.
1c. (10 points) Calculate the (unconditional) expectation E[BsBtBu], for s ≤ t ≤ u.
Hint. The following formulas might be useful: If X has the distribution N(μ,σ2) then E[X3] = μ3 + 3μσ2.
E[BsBtBu] = E[BsBt(Bu − Bt) + BsBt2]
= E[BsBt(Bu − Bt)] + E[BsBt2]
= E[BsBt]E[Bu − Bt] + E[Bs(Bt − Bs)2 + 2Bs2Bt − Bs3] = E[Bs(Bt − Bs)2] + E[2Bs2Bt − Bs3]
= E[2Bs2(Bt − Bs) + Bs3]
1d. (10 points) Apply Ito’s lemma to write the differential form for X = B e t Bsds. Calculate the
Answer. LetY =tBds,andZ =eYt =etBsds sothatX =BZ. WehavedY =Bdtand
expected rate of return 1 Et[dXt ]. dt Xt
t0st0ttttt so dZt = BtZtdt. Therefore
This implies d
dXt = d(BtZt) = BtdZt + ZtdBt + dZtdBt = Bt2Ztdt + ZtdBt + BtZtdtdBt
= Zt(Bt2dt + dBt)
= Btdt + 1 dBt, so that Bt
1 dXt dtEt[X ]=Bt
Question 2 (20 points in total): 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
Today’s price Tomorrow payoff Burst Normal Boom
X $2 $1$2$3 Y $2 $4$0$0 Z ??? $0 $1 $2
The (net) risk-free rate in the market is given to be r = 0%. Assume that there are no arbitrage opportunities in the market.
2.a (10 points) What are the risk neutral probabilities of the states Burst, Normal, Boom? Answer. Let p1,p2,p3 be the risk neutral probabilities of the states Burst, Normal, Boom corre- spondingly. Since these are the only possible states tomorrow, p1 + p2 + p3 = 1. The risk neutral pricing formula says
2=p1 +2p2 +3p3 2 = 4p1
Solving these equations yield p1 = 12 , p2 = 0, p3 = 21 .
2.b (10 points) What is the current price of stock Z?
Answer. The price of the stock Z is equal to
0∗p1 +1∗p2 +2∗p3 = $1
Question 3 (40 points in total): Suppose that the risk-free rate follows the following stochastic
3b. (10 points) Solve for Rt and then rt. In your answer, Rt and rt should be written as a sum of a deterministic term and an Ito integral.
3a. (10 points) Denote Rt = etrt. Using Ito’s lemma, find the expression for dRt.
drt =(1−rt)dt+e−2tdBt
dRt =(etrt+et(1−rt))dt+ete−2tdBt =etdt+e2tdBt
rt =r0e +1−e +e
Rt = r0 + e − 1 +
−t −t−tts
3c. (10 points) Calculate E[rt]. When t approaches infinity, what does E[rt] approach to? Please show your work.
e2 dBs e2dBs
E[rt] = r0e−t + 1 − e−t, lim E[rt] = 1 t→∞
3d. (10 points) Calculate V ar(rt). When t approaches infinity, what does V ar(rt) approach to? Please show your work.
−2t ts
V ar(rt) = e V ar
e2 dBs esds = et − 1
tst e2 dBs =
lim V ar[rt] = lim e−2t(et − 1) = 0
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