– 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 ( ,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 expect
ation 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 . 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 =