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Stochastic Foundation of Finance (FIN 538) Master of Finance Program

Fall 2020, Mini A

FINAL EXAM (Duration: 120 min)

金融代考 Apply Ito’s lemma to write the differential form for √tB where B is a standard Brownian motion. Is √tBt a martingale? Explain.

• Pleaseclearly state your name, student ID and your section number.
• Pleaseanswer all questions and show your work!
• The exam is open-book, open notes.
• Collaborations are strictly prohibited.
• Youmust leave your camera on during the duration of the exam.
• Pleasewrite your answers on blank A4 papers or on a tablet and submit your answers under the Assignments section on  Please upload a copy of your work in a single .pdf file, no other file formats will be accepted.
• The exam ends at 8:15 pm on Monday, November 2, 2020. You must stop writing at this time.You will have 10 minutes (that is, until 8:25pm on the same day) to upload your exam to Late submissions will NOT be accepted.
• Pleaseread the following statement and sign below to acknowledge it: ”I pledge my honor that I have not violated the Honor Code during this examination.”

Question 1 (35 points in total):  金融代考

1a. (10 points) Apply Ito’s lemma to write the differential form for √tB where B is a standard Brownian motion. Is √tBt a martingale? Explain.

1b. (10 points) Suppose that the process X_t = B³ − atB_t is a martingale where B_t is a standard Brownian motion. Find a and calculate E[X_t|F_s], s < t for that value of a.

1c. (10 points) Calculate the (unconditional) expectation E[B_sB_tB_u], for s ≤ t ≤ u where B_t 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. (5 points) Apply Ito’s lemma to write the differential form for  where B is a standard Brownian motion. Calculate the expected rate of return

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    金融代考

The (net) simple interest 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?

2.b(10points) What is the current price of stock Z?

Question 3 (40 points in total): Suppose that the interest rate follows the following stochastic process:  金融代考

dr_t = (1 − r_t)dt + e⁻2 dB_t, where r₀ = 0

where B_t is a standard Brownian motion.

3a. (10 points) Denote R_t = e^tr_t. Using Ito’s lemma, find the expression for dR_t.    金融代考

3b. (10 points) Solve for R_t and then r_t. In your answer, R_t and r_t should be written as a sum of a deterministic term and an Ito integral.

3c. (10 points) Calculate E[r_t]. When t approaches infinity, what does E[r_t] approach to?Please show your work.

3d. (10 points) Calculate V ar(r_t). When t approaches infinity, what does V ar(r_t) approach to? Please show your work.

Question 4 (5 points): Calculate E[. ∫ T sdB Σ ∗ . ∫ T s²dB Σ], where T > 0 and B is a standard Brownian motion.

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