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MTH6155: Financial Mathematics

创建日期：2024-05-31 15:44:14

所属分类：数学mathematics

标签： MTH6155

Financial Mathematics

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Main Examination period 2023 – May/June – Semester B

MTH6155: Financial Mathematics II
Duration: 2 hours
The exam is intended to be completed within 2 hours. However, you will have a period
of 4 hours to complete the exam and submit your solutions.
You should attempt ALL questions. Marks available are shown next to the
questions.
All work should be handwritten and should include your student number. Only
one attempt is allowed – once you have submitted your work, it is final.
In completing this assessment:
• You may use books and notes.
• You may use calculators and computers, but you must show your working for any
calculations you do.
• You may use the Internet as a resource, but not to ask for the solution to an exam
question or to copy any solution you find.
• You must not seek or obtain help from anyone else.
When you have finished:
• scan your work, convert it to a single PDF file, and submit this file using the
tool below the link to the exam;

Examiners: I. Goldsheid, E. Katirtzoglou

1. The following convention is used in this paper. If Y(t) is a random process then Yt
may be used to describe the same process; a similar convention applies to any other
random process. In particular, throughout this paper both W(t) and Wt denote the
standard Wiener process.
2. E˜ denotes the expectation over a risk-neutral probability.
3. Time involved in calculations should be expressed in years. E. g., 3 months should
be converted into 0.25 years.
4. The precision of calculations should be to 3 decimal places.
5. You may use without proof the following equalities. If X ∼ N (0, σ2) then E(eX) = eσ22 .
In particular, E(ebWt) = eb
2
2
t, where b is any real number.
Question 1 [18 marks]. This question is about the Wiener process and the
geometric Brownian motion.
Remark. You are reminded that two random variables X1, X2 with joint normal
distribution are independent if and only if Cov(X1, X2) = 0.
(a) The random process Y(t), t > 0, is defined by
Y(t) = tW(t−1),
where W(t), t > 0, is the standard Wiener process.
(i) Prove that Cov(Yt, Ys) = min(t, s), where t > 0, s > 0. [5]
(ii) For time moments 0 < s < t < u < v, compute the covariance of the
increments Yt − Ys and Yv − Yu. Are the increments of the process
Y(t), t > 0, independent? [5]
(b) Consider the geometric Brownian motion of the form S(t) = S0e
µt+σW(t). For
t > 0, compute the expectation of the product S(t/3)S(t). [8]
Question 2 [12 marks]. This question is about the Arbitrage Theorem.
(a) State the Arbitrage Theorem. [3]
(b) Three players A, B, and C are competing in a game.
If you bet £1 on A and A wins, then you are paid £3 (your pure gain is £2). If A
doesn’t win then you loose your £1.
If you bet £1 on B, and B wins you are paid £4. If B doesn’t win then you loose
your £1.
If you bet £1 on C, and C wins then you are paid £x. If C doesn’t win then you
loose your £1.
(i) Write down the return functions for the three possible bets. [3]
(ii) For what value of x there would be no arbitrage in this game? [6]

Question 3 [19 marks]. The price of a share is described by a geometric Brownian
motion St, 0 6 t 6 T, with parameters s = S(0),, µ, and σ. The continuously
compounded interest rate is r.
(a) A derivative on this share has expiration time T and a payoff function R(T) = S
1
3
T .
Find the formula for the price C of this derivative. [5]
(b) The trader of the derivative described in (a) has to devise a hedging strategy so
that to meet his financial obligation at time T . The hedging portfolio should
consist of underlying shares and of money deposited in the bank.
(i) In the case of the derivative described above, derive the formulae allowing
one to compute the total capital of the hedging portfolio, the number of
shares in the portfolio, and the amount of money deposited in the bank at
time t, 0 6 t 6 1.5. [6]
(ii) Suppose now that s = S(0) = £28, µ = 0.2, σ = 0.25, the expiration time is
15 months, and the continuously compounded interest rate is r = 5%.
Suppose that after 9 months the price of the share fell down to £25.
What should be the total value of the hedging portfolio in 9 months from
now?
How many shares should be in the portfolio and how much money should be
deposited in the bank? [8]

Question 4 [22 marks].
(a) Yt is a random process defined for t > 0 by
Yt =
∫ t
0
s3/2dWs.
(i) Compute the variance of this process and state the distribution of Yt. [4]
(ii) Compute the expectation E
(
e2Y(t)
)
. [5]
(b) A random variable V is defined by
V =
∫ 2
1
s
3
2WsdWs.
Compute the expectation and the variance of V . [4]
(c) Compute the stochastic integral
∫3
1
W3s dWs in terms of a function of Wt and the
ordinary integral of a function of Ws. [4]
(d) The random process Xt, t > 0, satisfies the following stochastic differential
equation:
dXt = X
2
tdt+ tXtdWt.
A new process Yt is defined by Yt = ln(X
2
t + t
2). Compute the stochastic
differential dYt. [5]
Question 5 [17 marks]. The following stochastic differential equation describes
the behaviour of the price St, t > 0 of a share.
dSt = (a+ b sin(t))Stdt+ σStdWt, (1)
where a, b, σ are constants such that a > 0, |b| < a, and σ > 0.
(a) Compute the differential of the function F(t) defined by F(t) = ln(St). [5]
(b) Solve the equation for St with the initial value S(0) = S0 [6]
(c) Compute E(ln(St)) and Var(ln(St)) and state the distribution of ln(St). [6]

Question 6 [12 marks]. This question is about the Merton model.
The total capital F(t) of a company follows the geometric Brownian motion with
parameters µ = 0.2 and σ = 0.3. At present, the total capital of the company is £4
million.
The company has just sold zero-coupon bonds with the total nominal value of £2.5
million which it is supposed to repay in 15 months’ time.
The continuously compounded annual interest rate r = 5%.
Within the framework of the Merton model, establish the following.
(a) How much money did the company raise by selling the bonds? [7]
(b) What is the probability that the company would not default on its promise to
bond holders? [5]
End of Paper – An appendix of 1 page follows.

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