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.
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(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.