F71CM – MAF
SCHOOL OF MATHEMATICAL AND COMPUTER SCIENCES
DEPARTMENT OF ACTUARIAL MATHEMATICS AND STATISTICS
F71CM CREDIT RISK MODELLING
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Semester2 — 2019-2020 Duration: 2 hours
Candidates may attempt no more than FOUR questions.
Candidates must show all steps of the calculations and proofs in the answers.
A formula sheet may be found at the end of the exam paper.
This version created on: July 1, 2020
F71CM – MAF 2
1. In Merton’s model of the default of a single firm it is assumed that the firm’s total asset value (Vt) follows a geometric Brownian motion process described by the stochastic di↵erential equation (SDE)
dVt = μV Vtdt + V VtdWt,
where μV 2 R and V > 0 are the drift and volatility respectively. The equity and debt of the company are considered to be contingent claims on the assets of the company.
(a) What does Merton assume about the debt structure of the firm? (1)
(b) Write the pay-o↵ functions for the equity and debt of the company in terms of
the assets VT of the company at the maturity of the debt T . (1)
(c) Derive an expression for the (real-world) default probability of the firm in
Merton’s model. (2)
(d) Under the standard Black-Scholes assumptions, assume that (Vt) is a traded security and that there exists a risk-neutral pricing measure Q for contingent claims on (Vt). Derive an expression relating the real-world default probability to the risk-neutral default probability. (2)
(e) How might this formula be used in practice to estimate the probability of default? (1)
(f) Define the credit spread c(t;T) of the firm’s zero-coupon bond over a default- free zero-coupon bond in terms of the respective bond prices p1(t;T) and p0(t;T). (1)
(g) Derive a formula for the spread c(t;T) in terms of a European put option price. (Note, you do not need to write out the Black-Scholes option pricing formula and can leave your answer in terms of the put price.) (1)
(h) Comment on the Merton-implied credit spread formula and whether or not it is considered to give realistic spread values. (1)
[Total 10 marks]
F71CM – MAF 3
2. Suppose we model 100 obligors using the threshold model of default (Xi, di)1i100. Suppose for all obligors the threshold is di = 1(0.01). Let Yi = I{Xidi} denote the default indicator variable for obligor i. Let the critical variables Xi be standard normal variables (i.e. N(0,1) variables) following a 2-factor model
Xi = bi1F1 + bi2F2 + ✏i,
where F1 and F2 are independent, standard normal factors, ✏1, …, ✏100 are indepen- dent normal errors which are also independent of F1 and F2. The obligors divide into two groups of 50. For obligors in the first group bi1 = 0.8 and bi2 = 0.4, but for obligors in the second group bi1 = 0.4 and bi2 = 0.8. Suppose that all exposures are equal to one Pand all losses given default are 100% so that the portfolio loss may be written L = 100 Yi.
(a) What is the expected loss E(L) ? (1)
(b) Whatistheassetcorrelationfortwoobligorsinthesamegroupandtwoobligors in di↵erent groups? (2)
(c) Show that the model may be written as a Bernoulli mixture model where the default indicators are conditionally independent given (F1, F2) with conditional default probabilities given by pi(F1,F2), where you should derive the form of the function pi(f1, f2). (2)
(d) What distribution does pi(F1, F2) have? (1)
(e) You are given the following values for the Gaussian copula:
CGa(0.01, 0.01) ⇡ 0.00377, CGa (0.01, 0.01) ⇡ 0.0216. 0.8 0.64
Compute the default correlation for two obligors in the same group and two obligors in di↵erent groups. p (2)
(f) Hence calculate the unexpected loss var(L). (2) [Total 10 marks]
F71CM – MAF 4 3. Let Q ⇠ Beta(1, 19) be a beta-distributed mixing variable.
Assume that Y1, …, Y100 are conditionally independent Bernoulli indicator variables with conditional default probability given by P(Yi = 1|Q = q) = q. You may assume 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
Show that the theoretical variance of the importance sampling estimator in part (c) is given by
bIS MX (t)E(e tX I{X>c}) ✓2 var(✓n )= n .
(3) More generally, exponential tilting may be understood as a change of proba-
bility measure from P to Qt in which the new measure is defined by
(IA) = E (MX(t)IA),
for some random variable X. Suppose we exponentiallPy tilt the distribution of Y1, …, Ym in part (b) using the random variable L = mi=1 eiYi. 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. (2) [Total 10 marks]
F71CM – MAF 6 5. Give brief answers to these questions about credit derivatives:
(a) CDS protection is often described as taking out insurance against default risk. But in what fundamental ways does CDS protection di↵er from insurance?
(b) What is meant by the CDS spread? (1)
(c) If a CDS contract specifies physical settlement in the event of default, what happens at default? (1)
(d) An investment bank uses CDS to speculate on changes in the market’s opinion of default risk. It holds a protection buyer position with respect to some reference entity. How could the investment bank use CDSs to make a profit out of a worsening credit outlook for the reference entity? (1)
(e) Draw a picture of an asset-based CDO structure and explain all the payment flows between the di↵erent participants. (3)
(f) For the equity and senior tranches of a CDO, explain the e↵ect that increased credit correlation typically has on the value of an investor’s security. (2)
[END OF PAPER]
[Total 10 Marks]
F71CM – MAF 7 F71CM — Credit Risk Modelling — Formula Sheet
1. Bernoulli distribution.
A random variable X follows Bernoulli distribution, denoted as X ⇠
Be(p), then Pr(X = 1) = p and Pr(X = 0) = 1 p.
2. Beta distribution.
A random variable X follows Beta distribution, denoted as X ⇠ Beta(a, b), a > 0, b > 0, then the density function of X is
1 xa 1(1 x)b 1,0x1 g(x) = ⇢ (a,b) ,
0, otherwise where (a, b) is a Beta function.
3. Beta function.
Z1 a 1 b 1 (a,b) = x (1 x) dx
= (a) (b) , (a+b)
where (a) is the Gamma function.
4. Gamma function.
(a)=Z 1xa 1e xdx,a>0. 0
If a is an positive integer, then
(a) = (a 1)! = 0 ⇥ 1 ⇥ 2 ⇥ … ⇥ (a 1).
5. Standard Normal distribution.
A random variable X follows standard Normal distribution, denoted as X ⇠ N (0, 1), if its density function
fX(x)=p2⇡e 2 , 1
1 (x μ)2
fY(x)= p2⇡e 2 2 , 1