辅导案例-GU4265/GR5265
STAT GU4265/GR5265 Midterm Exam Apr. 4th 00:01 AM - Apr. 6th 23:59 PM, 2020 Name and UNI: Instruction and some advices: • There are 6 problems. You have 3 days to complete the exam. Two problems (random selected) will be graded, and full solutions will be provided after the midterm. • The maximum possible score is 30 points. • Your solution should be well-explained, but keep reasoning as brief as possible. • Keep your handwriting clean and readable. Cross out things that are not part of your final solution. Do not give multiple solutions. • Please upload your solution to Courseworks before the deadline Apr. 6th 23:59 PM EST. No submission to the instructor or the TA’s email will be accepted, and all gradings will be based on the submission from Courseworks. GOOD LUCK! i 1. (15 points) Let {Bt, t ≥ 0} be a standard Brownian motion. (a) (3 points) Show that {Xt, t ≥ 0} is a martingale, when Xt = B2t − t, t ≥ 0. (b) (4 points) Compute the conditional distribution of Bs, given Bt1 = a, Bt2 = b, where 0 < t1 < t2 < s? (c) (8 points) Compute an expression for P( max t1≤s≤t2 Bs > x). 2. (15 points) Assuming Black-Scholes model for the stock price, using direct differentiation: (a) (7 points) prove that the Delta (the sensitivity to stock price s) of a Call option is equal to e−δTΦ(d1), where Φ(·) is the cumulative normal distribution. (b) (8 points) compute the sensitivity of the Put option to interest rate r. 3. (15 points) Let σ(t) be a a given deterministic function of time satisfying ∫ t 0 σ2(s)ds <∞ for all t ≥ 0. Define the process X by X(t) = ∫ t 0 σ(s)dW (s). Show that for a fixed t, the characteristic function of X(t) is given by EeiuX(t) = exp ( −u 2 2 ∫ t 0 σ2(s)ds ) , u ∈ R, thus showing that X(t) is normally distributed with mean zero and variance ∫ t 0 σ2(s)ds. (Hint : write eiuX(t) as an Itoˆ process, and derive an ODE for m(t) := EeiuX(t). You may use Domi- nated Convergence Theorem or Fubini’s Theorem for complex-valued function. Note that for x ∈ R, |eix| = 1.) 4. (15 points) The purpose of this question is to provide an example of a local martingale which is not a true martingale. (a) (3 points) Show that the function f(x, y, z) = (x2 + y2 + z2)−1/2 satisfies ∆f := fxx + fyy + fzz = 0. (b) (2 points) Let B = (B1, B2, B3) be a standard 3-dimensional Brownian motion. Use part (a) to show that for 1 ≤ t < ∞, the process defined by M(t) = f(B(t)) is a local martingale. (c) (5 points) Use direct integration (say, in spherical coordinates) to show E[M2(t)] = 1 t for all 1 ≤ t <∞. (d) (5 points) Use part (c) and Jensen’s inequality to show that M(t) is not a martingale. (Hint: prove by contradiction.) 5. (15 points) Assume the stock price follows a geometric Brownian motion St = 55 exp(0.2Bt) where B is a Brownian motion under the risk-neutral measure. Consider a down-and-in call with strike K = 60, maturity T = 5 and knock-in barrier L = 50. Page 1 of 2 (a) (3 points) Write down the (random) payoff of this option. (If your answer involves a stopping time, make sure you write down its definition.) (b) (4 points) What is the continuous compound interest rate in this model? (Hint: the discounted stock price e−rtSt is a martingale under the risk-neutral measure.) (c) (8 points) Compute the price of this option. Your final answer may contain the stan- dard normal cumulation distribution function N(·), but should be free of other integrals. (Remark: You cannot directly apply reflection principle to a GBM.) 6. (15 points) Consider the 2-period binomial model in which the stock price satisfies S0 = 16, S1(H) = 22, S1(T ) = 14, S2(HH) = 32, S2(HT ) = 14, S2(TH) = 22, S2(TT ) = 13. The interest rate is r = 1/4. (a) (10 points) Find the no-arbitrage price of an American put option expiring at time 2 and with strike price 16. (b) (5 points) Find the optimal exercise time τ ∗. You should write down τ ∗(ω) for all out- comes ω and also illustrate the exercise on the tree. Page 2 of 2