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2015 Examinations

This document outlines an examination for a course in financial stochastic processes. It contains 3 questions assessing concepts related to portfolio investment, Wiener processes, and modeling stock prices and market crashes as stochastic processes. Question 1 involves calculating the expected return and variance of a portfolio invested in riskless and risky assets. Question 2 covers properties of Wiener processes and concepts like value at risk. Question 3 examines modeling a stock price as an exponential distribution and options pricing, as well as modeling market crashes as a Poisson process.

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Shihab Hasan
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35 views3 pages

2015 Examinations

This document outlines an examination for a course in financial stochastic processes. It contains 3 questions assessing concepts related to portfolio investment, Wiener processes, and modeling stock prices and market crashes as stochastic processes. Question 1 involves calculating the expected return and variance of a portfolio invested in riskless and risky assets. Question 2 covers properties of Wiener processes and concepts like value at risk. Question 3 examines modeling a stock price as an exponential distribution and options pricing, as well as modeling market crashes as a Poisson process.

Uploaded by

Shihab Hasan
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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2015 EXAMINATIONS

PART II
MATHEMATICS & STATISTICS
MSc Quantitative Finance
Math 580 : Financial Stochastic Processes 2 Hours

ALL questions should be attempted. Marks total 50.

Question 1.

Consider a portfolio consisting of a riskless asset and two stocks. An amount y invested in the
riskless asset will be worth 1.1y. If amounts x1 and x2 are invested in the two stocks respec-
tively, these will be worth x1 R1 and x2 R2 where the dependent random variables (R1 , R2 )
satisfy !
0.01 0.01
E[(R1 , R2 )] = (1.2, 1.4), and Var[(R1 , R2 )] = .
0.01 0.09

(a) Consider an investment of y in the riskless asset, and x1 and x2 in the two stocks. The
return will be X = 1.1y + x1 R1 + x2 R2 . Write down the expected return, E(X) and the
variance of the return, Var(X). [3]
(b) Assume the initial investment satisfies y + x1 + x2 = 1. Calculate the choice of y, x1 and
x2 for which Var(X) is smallest, subject to E(X) = 1.2. [7]

please turn over

1
Math 580 Financial Stochastic Processes continued

Question 2.

Let Xt be a standard Wiener process. For a fixed value of t, Xt has a normal distribution
with mean 0 and variance t:
Xt ∼ N (0, t).

Furthermore, for s < t we have that Xs and Xt − Xs are independent and

Xt − Xs ∼ N (0, t − s).

The future value (in pounds) of a stock at time t is Yt = exp{Xt }.

(a) Calculate the mean of Yt . [3]


(b) Write down the median and the 0.01 quantile for Xt . [3]
(c) Define the value at risk of an investment at the (1 − α) level. Calculate, for time t, the
value at risk at the 99% level of £1, 000 invested in the stock. [4]
(d) Calculate the probability density function of Yt . [6]
(e) If s < t, calculate E(Yt |Ys = y). [4]

You may use without proof that if Z ∼ N (µ, σ 2 ), with σ > 0, then Z has probability density
function
(z − µ)2
 
1
fZ (z) = √ exp − for − ∞ < z < ∞,
2πσ 2 2σ 2
and Z has moment generating function
 
1 2 2
MZ (t) = E(exp{tZ}) = exp µt + σ t .
2

Furthermore if Z ∼ N (0, 1) then then 0.01 quantile of Z is -2.3.

please turn over

2
Math 580 Financial Stochastic Processes continued

Question 3.

(a) We model the future value of a stock as a random variable X, where X has cumulative
distribution function
FX (x) = 1 − exp{−λx}, for x > 0.

(i) Calculate the probability density function for X. [2]


(ii) A European call option on the stock with strike price c is a contract that gives the
buyer the right (but not the obligation) to buy one unit of the stock at price c,
where c > 0. The buyer will only exercise this option if X > c, in which case the
payoff is given by C = (X − c)+ = max{X − c, 0}.
Calculate E(C). [4]
(b) Crashes of the stock market are modelled as a Poisson process with rate 0.1 per year.
If Nt is the number of crashes by time t, then Nt has a Poisson distribution with mean
0.1t; so for r = 0, 1, 2, . . . ,:

(0.1t)r
P(Nt = r) = exp{−0.1t}.
r!
(i) Calculate the probability density function for the time until the first crash. [3]
(ii) What is the probability of precisely one crash in each of the next two years? [2]
(iii) Calculate P(N10 = 1|N10 ≥ 1). [3]
(iv) For 0 < s < t, calculate P(Ns = 0|Nt = 1). What is the distribution of the time
until the first crash given Nt = 1? [6]

end of exam

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