King’s Business School, King’s College London
This paper is part of an examination and counts towards the award of a
degree. Examinations are governed by the College Regulations under the
authority of the Academic Board. Students must not share or distribute this
examination paper.
Examination 2021/22
Module Code and Title: 7SSMM705 Asset Pricing
Examination Period: Period 2, May 2022
Time allowed: Students are recommended to spend no longer than 3 hours
on this paper. The paper will be available for 24-hours from 12:00 on 12
May 2022. Students will need to submit their answers by 12:00 on 13 May
2022.
Word count: N/A
INSTRUCTIONS TO CANDIDATES:
1.
2.
3.
4.
5.
6.
7.
8.
Answer ONLY ONE question from the TWO questions in Section A.
Answer ALL FOUR questions from Section B.
The number of marks each question is worth is written in brackets.
Paste any required diagrams and graphs for your answers directly onto
the answer sheet using software or uploaded photos.
If you answer more questions than specified, only the first answers (up
to the specified number) will be marked.
A template cover sheet has been provided on the KEATS page, you
should complete and type your answers below, or attach to the front
of your submission. Make sure you clearly indicate the questions you
are answering (e.g. Section A, Question 1).
If you have a PAA cover sheet, you should include this in addition to
your submission.
Save your work regularly, at least every 15 minutes.
ONLINE SUBMISSION INSTRUCTIONS:
1. You should submit your work via the Turnitin submission link on the
module KEATS page.
2. Ensure your document is submitted through Turnitin with the title
CANDIDATE ID – MODULE CODE- e.g. AC12345-7SSMM705
3. Once submitted please check you are satisfied with the uploaded
document via the submission link.
4. If you experience technical difficulties and are unable to upload your
assessment by the deadline, please collate evidence of the technical
issue and submit a mitigating circumstances form (MCF). Remember
that the evidence must clearly show timestamps and proof that you
attempted to upload your assessment before the deadline.
7SSMM705
Section A: Answer only ONE question from this section
Question 1 [20 marks]
For each of the statements below, state whether it is True or False, justifying your
answer:
h
i′
a. In a market in which one of the state price vectors is ψ = −1 0 1 , there
is definitely arbitrage.
b. According to the consumption CAPM, investors’ endowments play a role in
determining prices.
c. Consider a market with two assets. Given some prices, it is always the case
that we can graphically find an investor’s optimal combination of the two
assets as the point of tangency between the investor’s budget constraint
and his indifference curves.
d. Consider using the binomial model for pricing a derivative. The price we
calculate for the derivative will be exactly the same, no matter what value
we choose for the probability p of the high return in the model calibration.
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7SSMM705
Question 2 [20 marks]
A commercial bank wants to sell a new derivative security. If ST is the stock price
at expiration, this new derivative pays STγ , for some γ > 0. Suppose that stock
returns follow a geometric Brownian motion with annual mean rate of return
equal to 10% and annual volatility σ = 10% a year. The risk-free rate of return is
r = 5% per year, the contract expires in two years, and the initial stock price is
5. You have been hired as an analyst to price the power contract with γ = 0.5.
Proceed in steps:
a. Explain, in words, how a derivative is priced in continuous-time finance
using the risk-neutral method.
b. Write the SDE for the stock price using B P , the Brownian motion under the
objective probability measure.
c. Rewrite the SDE for the stock price using B Q , the Brownian motion under
the risk-neutral measure. State which theorem you are using here. Also
explain in words how the change of measure works.
d. Use Ito’s lemma to find the stochastic differential equation for ln St under
the risk-neutral measure. Integrate the SDE for ln St to find the risk-neutral
distribution of ln ST as of time 0, where T is the time to expiration.
e. Express the price of the derivative (at t = 0) as a risk-neutral expectation.
Evaluate this expectation. [Hint: If X is a lognormal distribution, i.e., ln X ∼
1 2
N m, s2 , then E eX = em+ 2 s ).]
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7SSMM705
Section B: Answer ALL questions from this section
Question 1 [20 marks]
Consider the payoff matrix

−1
0


A = −1
0

0 −1
3



2

0
and the price vector


−0.5




S = −0.5 .


1
a. Is the market complete?
b. Can you calculate the risk-free return in this market? If yes, what is it?
c. Find state prices consistent with A and S and the Law of One Price, and
then determine whether there is arbitrage.
d. If there is arbitrage, propose an arbitrage portfolio and explain what type of
arbitrage it is. If there is no arbitrage, state what the risk-neutral probabilities
are.
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7SSMM705
Question 2 [20 marks]
For each of the statements below, state whether it is True or False, justifying your
answer:
a. In a one-period model with dates today and tomorrow, a riskless asset is
one whose payoff tomorrow is known with certainty.
b. Consider the consumption CAPM model we have discussed in class. The
risk-free return does not depend on investors’ time preferences.
c. Consider the economy with two investors and two assets, as we discussed in
class, and denote with
p1
p2
the ratio of the prices for asset 1 and 2, respectively.
If, at this price ratio, there is excess supply for asset 1, then the equilibrium
price ratio is higher.
d. The annual log true return of a stock is i.i.d. normally distributed with
mean and variance 0.06 and 0.24, respectively. You want to write a 2-period
binomial model to price a derivative that expires in 4 months and whose
payoffs depend on the price of this stock. In this model, the high per-period
return for the stock (i.e., Ru in the notation used in class) is 1.324.
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7SSMM705
Question 3 [20 marks]
Consider an individual with preferences represented by u (x1 , x2 ) = x1 +
√
x2 ,
where x1 and x2 represent consumption of good 1 and good 2, respectively.
Assume that:
• consumption (of either good) cannot be negative,
• the ratio of the two goods’ prices is
p1
p2
= 2, and
• the individual is initially endowed with 0 units of good 1 and 0.5 units of good
2.
What is the individual’s optimal allocation? Solve analytically, showing steps in
detail, and also demonstrate graphically.
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7SSMM705
Question 4 [20 marks]
Consider a three-period model, with t = 0 representing the first date and t = 3
the last date. There are two assets: the risk-free bank account and the stock.
The risk-free rate of return, in each period, is constant and equal to r = 0. The
one-period stock returns are independent and identically distributed, and can
take two values, Ru = 2 or Rd = 0.5, with objective probabilities pu =
1
2
and pd = 12 .
The initial stock price is 10.
Consider a “down-and-out” barrier call option that matures at the end of the
third period (i.e., T = 3), and has strike price K = 10 and barrier B = 6. Such an
option pays
CT = max (ST − K, 0) if min St > B ,
0≤t≤T
and otherwise CT = 0.
a. Construct the stock price tree.
b. Consider an arbitrary payoff at t = 3. Is it possible to price this payoff?
c. Calculate the risk-neutral probabilities in each one-period sub-model.
d. Write the barrier call option’s payoffs at the terminal nodes of the tree at
t = 3.
e. Price the barrier call option.
f. At t = 0, find the amount deposited in (or borrowed from) the bank, and
the number of shares required for the self-financing portfolio that perfectly
replicates the payoff of the barrier call option at maturity.
END OF EXAMINATION
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