– Assignment 4 –
MATH4091/7091: Financial calculus
Assignment 4
Semester I 2022
No more questions will be added.
Due Friday June 3 Weight 15%
Total marks 35 marks
Submission: Softcopy (i.e. scanned copy) of your assignment by 23:59pm Friday June 3, 2022.
Hardcopies are not required.
Notation:“Lx.y”refers to [Lecture x, Slide y]
Assignment questions
- (15 marks) In L8, you learned how to perform a probability measure change using Girsanov’s
Theorem (statement on L8.16). In particular, the Radon-Nikody′m derivative process {Zt}t∈[0,T]
is constructed from the process {αt}t∈[0,T], and, as emphasized in class, the choice for this
process depends on the objective of the measure change, which in turn, depends on the pricing
problem you are dealing with (e.g. the payoff function).
In constructing the P→ Q measure change for the (standard) Black-Scholes model (in L8), we
were motivated by the objective that the discounted price process
{
St
Bt
}
t∈[0,T]
is a martingale
under Q.1 For this objective, αt was chosen to be the constant market price of risk αt = μ?rσ .
Suppose that now you want to perform a measure change from P to another probability measure,
denoted by QS , under which
{
Bt
St
}
t∈[0,T]
is a martingale. (Note this process is
{
Bt
St
}
t∈[0,T]
, not{
St
Bt
}
t∈[0,T]
.) We denote by
{
W?St
}
a Brownian motion under QS . The probability measure QS
is often referred to as the“share measure”.
a. (4 marks) Mathematically derive a formula for αt used in the Radon-Nikody′m derivative
process {Zt}t∈[0,T]. In your derivation, clearly express W?St (or dW?St) in terms of Wt (or
dWt) and αt. (Note that αt should be a constant in this case.)
Find dSt and dBt under QS .
b. (3 marks) LetΘt = (at, bt) be a self-finance portfolio ofXt = (St, Bt) under the real-world
measure P. Let its value process be {Vt}t∈[0,T], where Vt = Θt ·Xt. Here, as usual, we
assume {at}t∈[0,T] and {bt}t∈[0,T] are Ft-predictable processes.
What can you conclude about“self-financing”under the P→ QS measure change?
1As such, with this choice of αt, the price process of any (admissible) portfolio of (S,B) is a martingale under Q,
and hence, we arrive at the general expectation pricing formula on L8.28.
MATH 4091/7091 – 1 – Duy-Minh Dang 2022
– Assignment 4 –
c. (4 marks) Consider a financial contract with payoff CT . Let Ct be the price at time-t of
this contract. Prove that
Ct = StEQS
[
CT
ST
∣∣∣∣Ft] . (1)
d. (4 marks) Suppose CT = ST ln (ST). In this case, while it is possible to use the risk-neutral
pricing formula on L8.28 (via EQ [·]), it is more convenient to use EQS , since CTST = ln (ST)
which is much simpler than CT /BT .
Use the pricing formula (1) to find C0. - (12 marks) Let {Wt}t∈[0,T] be a Brownian motion on a probability space (?,F ,P), and let
{Ft}t∈[0,T] be the filtration generated by this Brownian motion.
Let {ht}t∈[0,T] be an {Ft}t∈[0,T]-predictable process satisfying EP
(∫ T - (ht)
- dt
)
<∞. Define an
a. (7 marks) Show that Y can be decomposed into
b. (3 marks) Is the representation (3) unique? Justify your answer.
c. (2 marks) Conclude that any FT -measurable random variable X, where X =
∑
i aiYi,
ai ∈ R, and Yi has form (2), can be represented in form (3).
- (8 marks) In this question, we consider the Vasicek interest rate model given under the risk-
neutral measure Q
drt = a(b? rt)dt+ σdW?t, r0 > 0,
where a, b and σ all are positive constants, and W?t is a Brownian motion (BM) with respect
to Q. Here, recall that the risk-neutral probability measure Q has the bank account account
as the numeraire. The time-t value of the bank account is Bt = e
∫ t - rsds.
For a fixed time T , and 0 ≤ t ≤ T , we denote by Zt(T) the time-t price of a zero-coupon bond
with maturity T . It can be shown that
dZt(T) = rtZt(T)dt? σC(t, T)Zt(T)dW?t. (4)