# Help With GU4265/GR5265 Midterm Exam

STAT GU4265/GR5265 Midterm Exam
Apr. 4th 00:01 AM - Apr. 6th 23:59 PM, 2020
Name and UNI:
• 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.
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.
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.
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