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Exam information
Course code and title
MATH4091
Financial calculus
Semester Semester 1, 2021
Exam type Online, non-invigilated, final examination
Exam technology File upload to Blackboard Assignment
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Semester One Final Examinations, 2021 MATH4091 Financial Calculus
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, and QS is the share 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 distribution, i.e.
N(x) =
∫ x
−∞
1√
2pi
e
{
−y22
}
dy.
Question 1. (20 marks) Answer the following questions.
a. (4 marks) Give an example of a stochastic process covered in MATH4091
which is a Markov process, but not a martingale. Mathematically justify
your answer.
b. (4 marks) Consider a stochastic process {Xt}t∈[0,T ], where
Xt =
∫ t
0
sdWs +
∫ t
0
Wsds.
Prove or disprove the assertion: {Xt}t∈[0,T ] is a martingale.
c. (4 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.
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Semester One Final Examinations, 2021 MATH4091 Financial Calculus
d. (4 marks) Consider a stochastic process {Xt}t∈[0,T ], where Xt = e(Wt)
4
.
Is this process in H2T , i.e. adapted and square-integrable? Mathematically
justify your answer.
e. (4 marks) Recall the Black-Scholes model. We denote by {St}t∈[0,T ] the
price process of a risky-asset. Assume that this process follows the risk-
neutral dynamics
dSt = rStdt+ σStdW˜t, S0 > 0, r > 0, σ > 0,
where
{
W˜t
}
t∈[0,T ]
is a Brownian motion under Q. Consider a financial
contract with the time-T payoff given by CT = (ST )
2.
Prove or disprove the following assertion: The time-0 no-arbitrage price
of this contract is (S0)
2.
Question 2. (20 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.
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Semester One Final Examinations, 2021 MATH4091 Financial Calculus
a. (7 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)
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. (7 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.
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Semester One Final Examinations, 2021 MATH4091 Financial Calculus
c. (5 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. (1 mark) 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) +
∫ 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).
Question 3. (20 marks) Consider a stochastic process {St}t∈[0,T ] whose dynam-
ics are given by
dSt = µtStdt+ σtStdWt, S0 > 0. (3)
Here, {µt}t∈[0,T ] and {σt}t∈[0,T ] are both deterministic functions of t, satisfying
the usual technical conditions. We assume that σt > 0 for all t ∈ [0, T ]. The
specific forms of µt and σt are not important for this question.
Let β ∈ {1, 2, 3, . . .} be fixed. Define
At(β) =
∫ T
t
β
(
µu +
β − 1
2
σ2u
)
du,
(Dt(β))
2 =
∫ T
t
(βσu)
2 du.
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Semester One Final Examinations, 2021 MATH4091 Financial Calculus
a. (7 marks) Show that, conditional on Ft, the random variable ln
(
(ST )
β
)
is normally distributed with the conditional mean and variance given by
E
[
ln
(
(ST )
β
) ∣∣Ft] = ln((St)β)+ At(β)− (Dt(β))2
2
,
Var
[
ln
(
(ST )
β
) ∣∣Ft] = (Dt(β))2 .
b. (6 marks) Let K be a nonnegative constant. Show that
E
[
(ST )
β I{(ST )β>K}
∣∣∣∣Ft] = (St)β eAt(β)N (β lnSt − lnK + At(β)Dt(β) + Dt(β)2
)
.
c. (7 marks) Assume that the interest rate is a positive constant r. Let
{Bt}t∈[0,T ] be the bank-account process, where Bt = ert with B0 = 1.
Consider a T -maturity, K-strike asset-or-nothing β-power option with
payoff CT given by
CT =
{
0 if (ST )
β ≤ K,
(ST )
β if (ST )
β > K.
Here, the dynamics of underlying asset price under the physical proba-
bility measure P are given by (3). Denote by Ct the time-t price of this
option.
Find a closed-form formula for Ct. Any correct risk-neutral approach is
acceptable.
Question 4. (20 marks) Assume that we have a stochastic interest rate process,
denoted by {rt}t∈[0,T ]. Let {Bt}t∈[0,T ] be the bank-account process, where
Bt = exp
(∫ t
0
rudu
)
, B0 = 1.
We denote by {Zt}t∈[0,T ] the price process of a T -maturity financial contract
that pays $1 at maturity, i.e. ZT = 1.
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Semester One Final Examinations, 2021 MATH4091 Financial Calculus
a. (5 marks) Use the First Fundamental of Asset Pricing to show that
Zt = EQ
[
exp
(
−
∫ T
t
rsds
) ∣∣∣∣Ft] . (4)
For the rest of Question 4, we assume that the dynamics of rt under the
risk-neutral measure Q are given by
drt = (θt − art)dt+ σdW˜t, r0 > 0,
where {W˜t}t∈[0,T ] is a Brownian motion under the risk-neutral probability
measure Q, θt is a positive deterministic function, and a and σ are positive
constants.
b. (5 marks) Show that, given rt at fixed t, for t ≤ s, we have
rs = e
−a(s−t)rt +
∫ s
t
e−a(s−u)θudu+ σ
∫ s
t
e−a(s−u)dW˜u.
What is the distribution of rs given rt?
c. (7 marks) Show that, for fixed t, the random variable
∫ T
t rsds is nor-
mally distributed. Find its mean and variance, namely EQ
[∫ T
t rsds
]
and
VarQ
[∫ T
t rsds
]
d. (3 marks) Evaluate the conditional expectation (4) to obtain a mathe-
matical expression for Zt in terms of model parameters, namely θ, a, and
σ, as well as t, T .
You may use EQ
[∫ T
t rsds
]
and/or VarQ
[∫ T
t rsds
]
in your answer even if
you were not able to find them in part c.
END OF EXAMINATION
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