• King's College London

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King’s College London

University Of London

This paper is part of an examination of the College counting towards the award of a degree.

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MSc and MSci Examination

7CCMFM05 Statistics in Finance

Summer 2018

Time Allowed: Two Hours

All questions carry equal marks. Full marks are awarded for correct answers

to FOUR questions. If more than four are attempted then only the best four

will count.

Within a given question, the relative weights of the different parts

are indicated by a percentage figure.

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7CCMFM05

1. a. Let X be a Binomial random variable with parameters n and p 2 (0; 1)

having probability mass function (pmf)

P(X = k) = nkpk(1 - p)n-k; (0 ≤ k ≤ n):

Compute the moment generating function (mgf) of X, and use it to compute E(X) and V AR(X). [30%]

b. Let Y = eX where X ∼ N(0; 1). Derive the probability density function

of Y . What financial quantity can Y model and why? [30%]

c. Let U have the uniform distribution on [0,1]. For what decreasing function

g(x) does the random variable T = g(U) have exponential distribution

with parameter 1? Recall that the cumulative distribution function F of

the exponential distribution with parameter λ is given by F(x) = 1-e-λx.

What can T model in credit risk? [40%]

2. a. Define the auto-covariance function of a time series. What does this function become when the time series is stationary? [20%]

b. Identify the orders p and q of an ARMA process satisfying

Xt = Xt-1 + "t - "t-1 + 1

2

"t-2:

Check causality and invertibility of this ARMA process. [40%]

c. Define volatility clustering, and give an example of a model (give name

and equation) that could be used to capture this characteristic of financial

time series. [20%]

d. Using the Spectral Representation Theorem compute the standard deviation of Xt which is taken from a time series whose spectral density satisfies:

f(!) = 225, 8! 2 [0; 1 2]. [20%