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MATH4091/7091: Financial calculus
创建日期：2024-05-08 17:38:47
所属分类：数学mathematics
标签： MATH4091
Financial calculus

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MATH4091/7091: Financial calculus

Assignment 3
No more questions will be added.
Due Tuesday May 17 Weight 15%
Total marks xx marks
Submission: Softcopy (i.e. scanned copy) of your assignment by 17:00pm Tuesday 17 May, 2022.
Hardcopies are not required.
Notation: “Lx.y” refers to [Lecture x, Slide y]
Assignment questions
1. (10 marks) Let {Wt}t∈[0,T ] be a Brownian motion defined on (Ω,F ,P, {Ft}t∈[0,T ]), where, as
usual, we assume that {Ft}t∈[0,T ] is the natural filtration generated by {Wt}t∈[0,T ].
We consider the generalised Geometric Brownian Motion {St}t∈[0,T ] where
dSt = µtStdt + σtStdWt, S0 > 0.
Here, {µt}t∈[0,T ] and {σt}t∈[0,T ] are (positive) Itoˆ stochastic processes satisfying the usual
technical conditions (adapted and integrability on L6.37).
a. (2 marks) Find d log(St) and simplify so that you have a formula for d log(St) that does
not involve St.
b. (2 marks) Integrate the formula obtained in part (a) to obtain
St = S0 exp
{∫ t
0
σsdWs +
∫ t
0
(
µs − 1
2
σ2s
)
ds
}
.
c. (3 marks) As shown in L6, if µt = µ and σt = σ, where µ and σ are (positive) constants,
then St is log-normally distributed.
Explain whether or not this result holds when we have µt = µ(t), and σt = σ(t), with
µ : [0, T ]→ R+, and σ : [0, T ]→ R+.
d. (3 marks) Redo part (c), but with µt = µ(t), and σt = σ(St, t), where µ : [0, T ] → R+,
and σ : R+ × [0, T ]→ R+.
2. (10 marks) The Cox-Ingersoll-Ross (CIR) interest rate stochastic differential equation is given
by
drt = (α− βrt)dt+ σ√rtdWt,
where α, β, σ are positive constant. Unlike the Vasicek equation (L6, Tutorial 9), the CIR
equation does not have a closed-form solution.
a. (2 marks) Use Itoˆ rule to compute d(eβtrt) and show that
d(eβtrt) = αe
βtdt+ σeβt
√
rtdWt.
MATH 4091/7091 – 1 – Duy-Minh Dang 2022
– Assignment 3 –
b. (2 marks) Integrate from 0 to t the equation in part (a) to show that
rt = r0e
−βt +
α
β
(1− e−βt) + σe−βt
∫ t
0
eβs
√
rsdWs.
c. (6 marks) Find E [rt] and Var [rt].
3. (10 marks) Consider a filtered probability space (Ω,F , {Ft}t∈[0,T ],P), where P is the physical
probability measure. Also let {Wt}t∈[0,T ] be a Brownian motion with respect to
(
P, {Ft}t∈[0,T ]
)
.
Here, T > 0 is a fixed expiry.
Recall the Black-Scholes dynamics under P
dBt = rBtdt, B0 = 1,
dSt = µStdt+ σStdWt, S0 > 0,
where r, µ, σ are constants, r > 0 and σ > 0. Consider a European call option on S2, with
strike K > 0 with payoff CT where
CT = C(ω) : Ω→ R
ω 7→ C(ST (ω)) =
(
(ST (ω))
2 −K)+ .
Let C(St, t) be the time-t price of this call option, where 0 ≤ t < T .
Important: C(St, t) is the function which maps (St, t) to the option price; it is not the function
which maps (S2t , t) to the option price.
a. (2 marks) Write down a PDE for C = C(St, t), including the terminal condition.
b. (8 marks) Solve the PDE in part (a) by the Fourier transform to find C(St, t). You can
reuse relevant results from lectures and/or tutorials.
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