– Assignment 2 –
MATH4091/7091: Financial calculus
Assignment 2
Semester I 2022
No more questions will be added.
Due Friday April 29 Weight 15%
Total marks 45 marks
Submission: Softcopy (i.e. scanned copy) of your assignment by 17:00pm Friday April 29, 2022.
Hardcopies are not required.
Notation: “Lx.y” refers to [Lecture x, Slide y]
In all questions, unless otherwise stated, assume a continuous time setting. Let (Ω,F , {Ft}t∈[0,T ],P)
be a filtered probability space over a finite time interval [0, T ]. By default, P is the physical
probability measure, Q is the risk-neutral/martingale probability measure.
For simplicity, we use the notation E[·] ≡ EP[·] and Var[·] ≡ VarP[·].
By default, {Wt}t∈[0,T ] is a standard Brownian motion under P.
The function IA is defined to take value 1 if event A occurs, and 0 otherwise.
The function N(·) is the Cumulative Distribution Function (CDF) of the standard normal distri-
bution, i.e.
N(x) =
∫ x
−∞
1√
2π
e
{
− y2
2
}
dy.
Assignment questions
1. (15 marks) Answer the questions below.
a. (3 marks) Consider a stochastic process {Xt}t∈[0,T ], where Xt = cWt , c > 0. Find
all values of c for which {Xt}t∈[0,T ] satisfies the conditional expectation property of a
martingale.
b. (3 marks) Consider a stochastic process {Xt}t∈[0,T ], where Xt = e(Wt)
4
. Is this process
adapted and square-integrable? Mathematically justify your answer.
c. (1 mark) Consider a stochastic process {Xt}t∈[0,T ], where
Xt = e
α(Wt)2+
α2
2
(T−t), α > 0.
For fixed t ∈ [0, T ], is Xt log-normally distributed? Mathematically justify your answer.
d. (3 marks) Let {Xt}t∈[0,T ] be a martingale. Assume that the process {tXt}t∈[0,T ] is also
a martingale.
Prove or disprove the assertion: Xt = 0 almost surely for all t ∈ [0, T ].
MATH 4091/7091 – 1 – Duy-Minh Dang 2022
– Assignment 2 –
e. (2 marks) Let A, B, and C be three non-empty disjoint subsets of Ω. Write out the
smallest σ-algebra containing A, B, and C.
f. (3 marks) Let T > 0 be fixed. Assume that a random variable ST satisfies
lnST = lnS0 + (r − σ2/2)T + σWT ,
where S0 > 0, r > 0 and σ > 0 are known constants.
Find E
[
e−rT (lnST )2
]
in terms of T , S0, r and σ.
2. (12 marks) The process {Xt}t∈[0,T ] is called non-random and simple if there exists a partition
Π = {tn}Nn=0 of [0, T ], where
0 = t0 ≤ t2 ≤ . . . ≤ tN = T,
such that, for every n, 0 ≤ n ≤ N , Xtn is non-random and
Xt =

Xt0 t ∈ [0, t1),
Xt1 t ∈ [t1, t2),
. . . . . .
XtN−1 t ∈ [tN−1, tN ].
For t ∈ [tk, tk+1], define the random sum
It =
k−1∑
n=0
Xtn
(
Wtn+1 −Wtn
)
+Xtk (Wt −Wtk) .
a. (2 marks) Show that whenever 0 ≤ s < t ≤ T , the increment It − Is is independent of
Fs.
b. (4 marks) Show that whenever 0 ≤ s < t ≤ T , the increment It − Is is a normally
distributed random variable. Find its mean and variance.
c. (6 marks) For each of the following process, mathematically justify whether or not it is
a martingale:
(b.1) {It}t∈[0,T ], (b.2)
{
I2t
}
t∈[0,T ], (b.3)
{
I2t −
∫ t
0 (Xu)
2du
}
t∈[0,T ]
.
Question 3. (18 marks) Let {Xt}t∈[0,T ] be a (continuous) stochastic process. We assume that
both {Xt}t∈[0,T ] and the stochastic process {X2t − t}t∈[0,T ] are martingales with respect to(
P, {Ft}t∈[0,T ]
)
.
Let 0 ≤ s < u ≤ T , where s and u are fixed. For fixed positive integer n, let Πn = {ti}ni=0,
where ti = s + i∆n, ∆n = (u− s)/n, be a partition of the interval [s, u]. For simplicity, we
also use the notation ∆Wti = Wti+1 − Wti , i = 0, . . . , n− 1.
In this question, consider a real-valued function f(x) for which f(x) and the partial derivatives
∂f
∂x ,
∂2f
∂x2
, and ∂
3f
∂x3
exist and are continuous and bounded for all x ∈ R. In the below, the
notation ∂f∂x (Xt) and
∂2f
∂x2
(Xt) means that the partial derivatives are evaluated at Xt.
a. (6 marks) Show that
E[f(Xu)|Fs] = f(Xs) +
n−1∑
i=0
E
[
E
[
∂f
∂x
(Xti)
(
Xti+1 −Xti
) ∣∣∣∣Fti] ∣∣∣∣Fs]
+
1
2
n−1∑
i=0
E
[
E
[
∂2f
∂x2
(Xti)
(
Xti+1 −Xti
)2 ∣∣∣∣Fti] ∣∣∣∣Fs]
+ RΠn , (1)
MATH 4091/7091 – 2 – Duy-Minh Dang 2022
– Assignment 2 –
where RΠn is given by
RΠn =
1
3!
n−1∑
i=0
E
[
∂3f
∂x3
(X∗ti)
(
Xti+1 −Xti
)3 ∣∣∣∣Fs] , X∗ti ∈ [Xti , Xti+1] .
Hint: Write
f (Xu)− f (Xs) =
n−1∑
i=0
(
f
(
Xti+1
)− f (Xti)) ,
and use the second-order Taylor expansion for each term f
(
Xti+1
) − f (Xti). Then,
apply conditional expectation.
b. (6 marks) Show that the terms in (1) satisfy: for i = 0, . . . , n− 1,
E
[
E
[
∂f
∂x (Xti)
(
Xti+1 −Xti
) ∣∣∣∣Fti] ∣∣∣∣Fs] = 0,
E
[
E
[
∂2f
∂x2
(Xti)
(
Xti+1 −Xti
)2 ∣∣∣∣Fti] ∣∣∣∣Fs] = E [∂2f∂x2 (Xti)
∣∣∣∣Fs] (ti+1 − ti).
Hint: Use the fact that {Xt}t∈[0,T ] and {X2t − t}t∈[0,T ] are martingales.
c. (4 marks) Let
Yn =
n−1∑
i=0
E
[
∂2f
∂x2
(Xti)
∣∣∣∣Fs] (ti+1 − ti),
Y =
∫ u
s
E
[
∂2f
∂x2
(Xt)
∣∣∣∣Fs] dt.
Show that, as n→∞, we have Yn −→L2 Y .
Hint: First, show that |Yn − Y |2 → 0 almost surely, and then apply the Dominated
Convergence Theorem.
d. (2 marks) Using the same technique as in part c, it can be shown that, as n → ∞,
RΠn −→ 0 in L2. You do not need to prove this fact. Together with previous results in
part a. - part c., we can then conclude that
E[f(Xu)|Fs] = f(Xs) + 1
2
∫ u
s
E
[
∂2f
∂x2
(Xt)
∣∣∣∣Fs] dt. (2)
Which key formula covered in the course is similar to (2)? Briefly explain the connection
between the two formulae (one sentence is enough).
MATH 4091/7091 – 3 – Duy-Minh Dang 2022