HW: 6
Course: M339W/M389W - Financial Math for Actuaries
Page: 1 of 3
University of Texas at Austin
HW Assignment 6
Provide a complete solution to the following problem(s):
Problem 6.1. Let Z be a standard Brownian motion. Define the process Y = {Y (t), t ≥ 0} as Y (t) =
et/2 sin(Z(t)). We will show that this process is a martingale.
a. (5 points) Find the expression for d(sin(Z(t))) using Itô’s Lemma.
b. (5 points) Use your answer to part a. to find the differential representation dY (t) of the process Y .
c. (2 points) Now, it is sufficient to prove that the drift term in the representation obtained in part b. is
equal to zero. Verify this.
Solution:
a. Let f : R → R be defined as f (x) = sin(x). Then, we have that f 0 (x) = cos(x) and f 00 (x) = − sin(x).
Itô’s lemma yields
1
d(sin(Z(t))) = cos(Z(t)) dZ(t) + (− sin(Z(t))) (dZ(t))2
2
1
= cos(Z(t)) dZ(t) − sin(Z(t)) dt.
2
b. By the stochastic calculus “multiplication table”, we have that
dY (t) = d(et/2 sin(Z(t))) = et/2 d(sin(Z(t))) + sin(Z(t)) d(et/2 ) + d(et/2 ) d(sin(Z(t)))
1
1
= et/2 (cos(Z(t)) dZ(t) − sin(Z(t)) dt) + sin(Z(t)) et/2 dt
2
2
t/2
= e cos(Z(t)) dZ(t).
c. This is evident from our calculation in part b.
Problem 6.2. (14 points) Solve problem 20.3 from McDonald.
Solution: Note: In this problem, we choose to ignore the fact that in the cases of the arithmetic Brownian
motion and the mean-reverting process the processes may be equal to zero which makes taking the reciprocal
of their value not well-defined. This is not a big issue since these events are negligible.
(1 point) Let F (x) = 1/x.
(2 points) Then,
1
F 0 (x) = − 2 = −x−2 ,
x
2
F 00 (x) = 3 = 2x−3 .
x
(5 points) Using Ito’s Lemma and the “multiplication table”, we get that
2
dF (S(t)) = −(S(t))−2 α (S(t), t) + (S(t))−3 σ (S(t), t) dt − (S(t))−2 σ (S(t), t) dZt ,
for any Ito process S which satisfies the SDE
dS(t) = α(S(t), t) dt + σ(S(t), t) dZ(t)
with Z a standard Brownian motion.
(6 points) Let us define the process Y (t) = F (S(t)) (to make the formulas below more legible). For the
particular processes given in this problem, we have
2
a. dY (t) = −αY (t)2 + σ 2 Y (t)3 dt − σY (t)
dZ(t). 2
2
2
3
b. dY (t) = −λ aY (t)
− Y (t) + σ Y (t) dt − σY (t) dZ(t).
c. dY (t) = −α + σ 2 Y (t) dt − σY (t) dZ(t).
Instructor: Milica Čudina
Semester: Spring 2013
HW: 6
Course: M339W/M389W - Financial Math for Actuaries
Page: 2 of 3
Problem 6.3. (2 points) In the setting of the Black-Scholes stock-price model, let {S(t), t ≥ 0} denote the
stock price. Define the new stochastic process
X(t) = ln(S(t)), for every t ≥ 0.
Then we have that the stochastic process {X(t), t ≥ 0} is a geometric Brownian motion. True or false?
Solution: FALSE
It’s an arithmetic Brownian motion.
Problem 6.4. (2 points) Let {Z(t), t ≥ 0} denote a standard Brownian motion. Then the stochastic process
{U (t), t ≥ 0} defined as
U (t) = Z(t)2 − 2t, for every t ≥ 0
has zero drift. True or false?
Solution: FALSE
In class, we showed that Z(t)2 − t has zero drift, so the above process cannot have zero drift.
Problem 6.5. (2 points) In the setting of the Black-Scholes stock-price Model, let {S(t)} denote the stock
price with parameters α and volatility is σ. Define the new stochastic process
X(t) = ln(S(t)), for every t ≥ 0.
Then we have that
V ar[X(t + h) − X(t)] = σ 2 h, for every t ≥ 0 and h > 0.
True or false?
Solution: TRUE
Since {X(t), t ≥ 0} is a Brownian motion, its increment X(t + h) − X(t) is normally distributed with mean
αh and variance σ 2 h.
Problem 6.6. (2 points) In the setting of the Black-Scholes stock-price model, let {S(t), t ≥ 0} denote the
stock price with volatility σ and drift α. Then, we have that
V ar[S(t + h) | S(t)] ≈ S(t)2 σ 2 h, for every t ≥ 0 and infinitesimally small h > 0.
True or false?
Solution: TRUE
Since {S(t)} is a geometric Brownian motion, it satisfies the following SDE:
dS(t) = S(t)(α dt + σ dZ(t)).
and can be written as
1
S(t) = exp{(α − σ 2 )t + σZ(t)},
2
for Z a standard Brownian motion. From the last display, we get that
1
S(t + h) = S(t) · exp{(α − σ 2 )h + σ(Z(t + h) − Z(t))}.
2
Then, we have that
V ar[S(t + h) | S(t)] = E[S(t + h)2 | S(t)] − (E[S(t + h)|S(t)])2 .
We get
V ar[S(t + h) | S(t)] = S(t)2 · e2αh (eσ
2
h
− 1).
For “very small” positive h, we have that
e2αh ≈ 1, eσ
2
h
− 1 ≈ σ 2 h.
Instructor: Milica Čudina
Semester: Spring 2013
HW: 6
Course: M339W/M389W - Financial Math for Actuaries
Page: 3 of 3
Problem 6.7. (10 points) Use Ito’s Lemma to express dF (S(t)) for F : R+ → R+ given as F (x) =
where the stochastic process {S(t), t ≥ 0} satisfies the stochastic differential equation
p
dS(t) = a(b − S(t)) dt + σ S(t) dZ(t)
√
x,
with a, b and σ positive constants and {Z(t), t ≥ 0} a standard Brownian motion.
Solution: Let us start by finding the relevant derivatives of F .
1 1
F 0 (x) = x− 2 ,
2
1 3
F 00 (x) = − x− 2 .
4
By Ito’s Lemma for Ito processes:
1
3
1
1
1
df (S(t)) = S(t)− 2 dS(t) + · − S(t)− 2 (dS(t))2 .
2
2
4
From the “multiplication table” from class, we conclude that (dS(t))2 = S(t)σ 2 dt. So,
1 1
1 1
p
− 12
− 32
2
· S(t)σ 2 dt
d S (t) = S(t) · a(b − S(t)) dt + σ S(t) dZ(t) + · − S(t)
2
2
4
1
1 2
1
−1/2
= S(t)
· a(b − S(t)) − σ
dt + σ dZ(t).
2
4
2
Problem 6.8. (2 points) Let {Z(t), t ≥ 0} be a standard Brownian motion. Then the process
Z t
V (t) = t2 Z(t) − 2
sZ(s) ds
0
has zero drift. True or false?
Solution: TRUE
By Itô’s lemma,
d(t2 Z(t)) = t2 dZ(t) + Z(t) d(t2 ) + d(Z(t)) d(t2 ) = t2 dZ(t) + 2tZ(t) dt + 2d(Z(t)) dt = t2 dZ(t) + 2tZ(t) dt.
Thus,
dV (t) = t2 dZ(t) + 2Z(t) dt − 2tZ(t) dt = t2 dZ(t).
Instructor: Milica Čudina
Semester: Spring 2013