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MATH4091/7091: Financial calculus

创建日期：2024-04-30 15:29:04

所属分类：数学mathematics

标签： MATH4091

Financial calculus

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MATH4091/7091: Financial calculus

Assignment 1
Semester I
Final version
Due Friday April 1 Weight 15%
Total marks 35 marks
Submission: Please upload scanned copy of your solutions onto Blackboard by 17:00 Friday April
Notation: “Lx.y” refers to [Lecture x, Slide y]
Assignment questions
1. (10 marks) Although in L1, we discussed the perfect replication technique in a one-period
binomial model, it is more realistic to consider a multi-period binomial model as we do in
this question.
Our model consists of a risky asset S and a bond B. Let St, where t = 0, 1, 2, . . . , T , denote
the time-t price of a tradeable (non-dividend-paying) asset. Let S0 = 100 and let each random
increment St+1 − St take value +1 with physical probability 40%, and value -1 with physical
probability 60%, independently of all other increments. The bond has a constant price 1 at
all times (i.e. r = 0%).
We assume that there exists a call option on S, with strike K = 105 and expiry at time
T = 10 (years).
(a.) (5 marks) A student came up with the following argument to prove that the no-
arbitrage time-0 value of the call is zero.
Consider the trading strategy Θrept , t = 0, 1, 2, . . . , T , specified as follows.
Θrept =
{
(0 share of asset , 0 bonds ) if St < K
(1 share of asset ,−K bonds ) if St ≥ K
This trading strategy replicates the call payoff, because either ST ≥ K, in which
case the call payoff matches the time-T portfolio value ST −K, or else ST < K,
in which case the call payoff matches the time-T portfolio value 0. The time-0
value of the replicating strategy Θrept is zero, because S0 < K. So if the time-0
call price is not zero, then arbitrage exists. Specifically, if the time-0 call price is
strictly positive, then shorting the call and going long the replicating strategy is
an arbitrage; in other words,
Θt = Θ
rep
t −Θcallt
is an arbitrage, where we let Θcallt be the portfolio consisting of 1 call at all times.
(And, likewise, if the time-0 price of the call is strictly negative, then −Θcallt is
an arbitrage.) Therefore the no-arbitrage time-0 call price must be zero.
Identify and explain the specific flaw in this “proof”.
MATH 4091/7091 – 1 – Duy-Minh Dang 2022
– Assignment 1 –
(b.) (5 marks) Find the true time-0 value of the call.
Do not induct backwards step-by-step in a tree, and do not use a computer (unless you
want to check your answer).
Although your answer should be explicit, you may leave it un-simplified. For example,
you may leave binomial coefficients (numbers of the form: n choose k) un-simplified.
2. (10 marks) Suppose that the value of a certain stock at time T is a random variable with
distribution P. Note we are not assuming a binary model. An option written on this stock
has payoff CT at time T . Consider a portfolio consisting of α units of the underlying and
β units of bond, held until time T . Let V0 be the portfolio’s value at time-0. Assume that
interest rate is zero.
(a.) (7 marks) Show that the extra cash required by the holder of this portfolio to replicate
the claim CT is
Ψ = CT − V0 − α (ST − S0) .
Find expressions for the values of V0 and α (in terms of E [ST ] ,E[CT ], var [ST ] and
cov (ST , CT )) that minimise
E
[
Ψ2
]
.
Verify that for these values, we have E[Ψ] = 0.
(b.) (3 marks) Prove that for the one-period binomial model in L1, any CT depends linearly
on ST − S0. Deduce that in this case, we can find V0 and α such that Ψ = 0.
3. (7 marks) Let X be a random variable on a probability space (Ω,F ,P). As discussed in L2.36,
if g : R→ R is a Borel-measurable function, then
EP|g(X)| =
∫
R
|g(x)|dPX(x), (1)
and if this quantity is finite, then
EPg(X) =
∫
R
g(x)dPX(x), (2)
where PX is the probability distribution of X.
a. (3 marks) Let IB : Ω→ R be the indicator function (random variable) of the form
IB(ω) =
{
1 ω ∈ B
0 otherwise,
where B is a Borel subset of R1. Show (1)-(2) when g(x) = IB(x).
b. (4 marks) Show (1)-(2) when g(x) is non-negative simple random variable, i.e.
g(x) =
n∑
k=1
ckIBk(x),
where ck, k = 1, . . . , n, are non-negative constants and Bk, k = 1, . . . , n, are Borel subset
of R.
1That is, B ∈ B(R), the Borel σ-algebra of R.
MATH 4091/7091 – 2 – Duy-Minh Dang 2022
– Assignment 1 –
4. (8 marks) Suppose X is a random variable on some probability space (Ω,F ,P), and A is a set
in F , and for every Borel subset B of R, we have∫
A
IB(X(ω))dP(ω) = P(A) · P{X ∈ B}
Then we say that X is independent of the event A.
Show that if X is independent of an event A, then∫
A
g(X(ω))dP(ω) = P(A) · Eg(X)
for every nonnegative, Borel-measurable function g.
Hint: you can consider the two cases as in Question 3
indicator function : g(x) = IB(x)
non-negative simple function: g(x) =
∑n
k=1 ckIBk(x), where c1, . . . cn are non-negative
constants and B1, . . . , Bn, are Borel subsets of R.
MATH 4091/7091 – 3 – Duy-Minh Dang 2022

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