SCHOOL OF MATHEMATICAL AND COMPUTER SCIENCES Department of Actuarial Mathematics and Statistics
CREDIT RISK MODELLING Semester 2 2015/16
Duration: Two Hours Total marks: 60
THERE ARE 5 QUESTIONS, EACH WORTH 15 MARKS.
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ANSWER 4 QUESTIONS.
IF YOU ANSWER 5, CREDIT WILL BE GIVEN FOR THE 4 BEST ANSWERS.
“Formulae and Tables for Actuarial Examinations” and electronic calculators approved by the University may be used.
1. In this question you may assume that, in the Black-Scholes-Merton geometric Brownian motion model, the values of European call and put options written on a stock, with maturity T and strike K are given by
CBS(t,St;r, ,K,T) = St (dt,1) Ke r(T t) (dt,2) PBS(t,St;r, ,K,T) = Ke r(T t) ( dt,2) St ( dt,1),
where (St ) denotes the stock price process, is the stock volatility, r is the interest
dt,1 = and dt,2 = dt,1 pT t.
St ⌘+ 1 2(T t) K exp( r(T t)) 2
(a) In Merton’s model the debt of the firm takes the form of a zero-coupon bond with face value B and maturity T. Explain clearly why the equity and debt may be viewed as contingent claims on the total assets (Vt ) of the firm.
[3 marks] Supposethat(Vt)ismodelledbyageometricBrownianmotionwithdriftμV and
volatility V .
(b) Give a valuation formula for the equity of the firm.
(c) Show that the valuation formula for the debt of the firm is given by
Bt =p0(t,T)B (dt,2)+Vt ( dt,1),
where you should define p0(t, T ) and redefine dt,1 and dt,2 in an appropriate
way for the setting of Merton’s model.
(d) Derive a formula for the credit spread of a bond issued by the company in Merton’s model.
(e) The credit spread depends only on a measure of leverage, asset volatility and time to maturity. Define the measure of leverage and state whether the spread increases or decreases with leverage.
[Total 15 Marks]
PLEASE TURN OVER 1
2. This question concerns the pricing of a defaultable zero-coupon bond using the martingale modelling approach. The bond pays 1 unit at maturity T and the default time ⌧ of the bond is modelled using a hazard rate model under the risk- neutral measure Q. The interest rate is assumed to be given by a deterministic function r(t).
In answering the question you may use the fact that, for an integrable random
variable X ,
where Ht = ({I{⌧s} : s t}) is the information available to an investor at time t.
(a) Explain the difference between a survival claim and a payment-at-default claim.
(b) Derive a formula for the value of a payment-at-default claim in terms of the deterministic hazard function Q(t) under the risk-neutral measure and the interest rate r(t).
(c) Hence show that the risk-neutral price of the bond under the recovery-of- face-value (RF) recovery model is given by
✓✓ZT◆ZT✓Zs◆◆ p1(t,T)=I{⌧>t} exp R(s)ds +(1 ) Q(s)exp R(u)du ds
where R(t) = r(t) + Q(t) and is the loss-given-default (corrected from
(d) Simplify this formula as far as you can in the case where the risk-neutral hazard function and the interest rate are assumed to be constants.
(e) Explain why pricing a survival-at-default claim is also relevant to the problem of valuing a CDS contract.
[Total 15 Marks]
EQ(I X|H)=I EQ(I{⌧>t}X), {⌧>t} t {⌧>t} Q(⌧>t)
PLEASE TURN OVER 2
3. Let Q ⇠ Beta(a,b) be a beta-distributed mixing variable. Given Q, assume that Y1, … , Ym are conditionally independent Bernoulli indicator variables with default probability Q. You may use the fact that the density of a random variable with a Beta(a, b)-distribution is given by
g(q)= 1 qa 1(1 q)b 1, a,b,>0, 0
(a) First assume that L is a continuous random variable with a well-defined mo- ment generating function. Explain how importance sampling using exponen- tial tilting can be applied to estimate ✓.
(b) Showthatundertheaboveassumptionsthevarianceoftheimportancesam- pling estimator is given by
⇣ IS⌘ ML(t)E(e tLI{L>c}) ✓2 var ✓ˆ = .
(c) Now let L = Pmi=1 ei Yi where e1, … , em are deterministic exposures and where Y1, … , Ym are independent default indicator random variables satisfying Yi ⇠ Be(pi ) for i = 1, … , m. Derive ML(t), the moment generating function of L.
[2 Marks] (d) In the setting of part (c), suppose we make a change of probability measure
from P to Qt in which the new measure is defined by Qt P✓ etL ◆
for L = Pmi =1 ei Yi . Compute the new probability Qt(Y1 = y1,…,Ym = ym)
and show that defaults remain independent under Qt but with new default probabilities qt,i which you should derive.
in the setting of part (c). (You do not need to discuss the optimal choice of t.)
[Total 15 Marks]
Qt(A) = E (IA) = E ML(t)IA ,
(e) Write clearly the steps of an importance sampling algorithm to calculate ✓IS n
END OF PAPER 5
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