SECTION A
Answer ALL questions from Section A.
A1 (a) [Type] Provide a clear description of all of the assumptions used
in the variance-covariance method for calculating the value-at-risk
for a portfolio. [2]
(b) [Type] Provide a critique of the variance-covariance method, in-
cluding its limitations, when it may be appropriate to use, and
when it is not suitable to use. [3]
A2 A portfolio consists of three risky assets, and contains £200 of asset A,
£150 of asset B and £100 of asset C. The daily volatility of the assets
are 5%, 2% and 1% respectively.
(a) Assume the correlation between daily returns is 0.5 for assets A
and B, is 0.5 for assets B and C, and is zero for assets A and C.
Use the variance-covariance method to calculate the value-at-risk
for the portfolio over a 10 day period, at a 95% confidence level.
[3]
(b) What is the 10 day, 95% confidence level value-at-risk calculated
using the variance-covariance method if the correlation between
daily returns of assets A and B is -0.5, whilst other correlations
remain the same? [1]
(c) [Type] Explain any qualitative difference between your answer to
part (a) and your answer to part (b). [1]
Continued
A3 (a) Consider a risky company that has a time to default that follows
a hazard rate model with inhomogeneous hazard rate that is a
function of time, h(w) for T ≥ w ≥ 0. Today is time t.
Derive an expression for the probability that the company defaults
in the time period (t1, t2), where T > t2 > t1 > t > 0. [4]
(b) Consider a risky company that has a time to default that follows
a hazard rate model with a deterministic hazard rate function of:
h(t) = 0.4 for t ≤ 3;
h(t) = 0.3 for t > 3.
Today is time 0. Find the probability that the company defaults
between time t = 3 and time t = 4. [3]
A4 For this question, assume that all spreads are entirely due to default
risk, and that all defaults result in zero percent recovery rates. Assume
also that a default event for a company will result in default for all cor-
porate bonds associated with that company, and that these corporate
bonds will only default if the company defaults. Assume also that the
risk-free rate is deterministic and constant across all maturities.
(a) [Type] Explain the difference between objective default proba-
bilities and risk-neutral default probabilities. Explain also when
each should be used and why. [2]
(b) A default-free, £1,000 par, zero coupon government bond is ex-
piring in 3 years’ time and is trading at £940. Suppose that a
£1,000 par, zero coupon corporate bond, subject to credit default
risk, is also due to expire in 3 years’ time, and is currently trading
at £840. Calculate the credit spread for this corporate bond. [1]
(c) Continuing from part (b) above and assuming the information
given in (b), assume also that the risk neutral default process for
the company can be modelled using a hazard rate model with a
hazard rate function of h(u) = λu2 for a positive constant λ and
for all u ≥ 0. Calculate the price of a 4 year, zero coupon bond
that has a par of £1,000 and is issued by the same company. [3]
Turn Over
A5 (a) [Type] Explain in words what it means for a portfolio to be a self
financing portfolio. [1]
(b) [Type] Explain why self financing portfolios are important in the
derivation of no-arbitrage asset pricing results. Illustrate your ex-
planation with an example of using a self financing portfolio in
the derivation of no-arbitrage pricing results, and also with an ex-
ample using a portfolio that is not self financing. [5]
(c) [Type] Explain what an equivalent martingale probability mea-
sure is, and how it can be used in pricing derivatives. [3]
A6 [Type] A 1 year government bond with no counter-party default risk
is trading at £982, and a similar 3 year bond at £942. Both bonds
have par value £1,000. A 1 year forward rate agreement for lending
between 1 year from now and 3 years from now is being offered at 2%.
Outline clearly a trading strategy to take advantage of the arbitrage
opportunity that exists, and explain in full how the strategy provides
arbitrage. [6]
A7 Suppose a stochastic process St follows the diffusion process
dS = µdt+
σ
2
tS2dZ
where Z is a Brownian motion. Consider a function
f(x, t) = E[ek(T−t)S3T |St = x]
where k and T are constants and T > t.
Find a non-trivial partial differential equation that f follows. Your
partial differential equation must contain non-zero second order terms.
Explain all your working. [2]
Continued
SECTION B
Answer ALL questions from Section B.
B1 (a) [Type] Write a discussion of the strengths and weaknesses of using
a value-at-risk (VaR) measure for managing market risk. Where
appropriate, refer to the definition and properties of value-at-risk
in your answer, and describe other techniques that should be used
in market risk management to help compensate for weaknesses.
For the purpose of answering this question you should assume
that the most appropriate VaR technique will be applied for the
relevant portfolios, so that you should write your answer about
VaR as a measure for market risk management in general, and
not about the strengths and weaknesses of the different ways of
measuring VaR. [11]
(b) [Type] An analyst has developed two different VaR models, and
has operated them both on the same portfolio for the last year.
For the last 60 days, the analyst has recorded the daily profit
and loss on a portfolio, and has also recorded the VaR that each
model has estimated each day for a 1 day holding period at a 95%
confidence level. The results are provided in the table overleaf,
where all values are in £, and in which losses are indicated by
negative values, profit by positive values.
Comment on what this data tells you about the appropriateness of
each of the VaR models, and whether there is evidence that each
VaR model provides conservative or non-conservative estimates.
Support your answer with statistical inference where appropriate.
[5]
Turn Over
Day Profit / Loss VaR for Model 1 VaR for Model 2
1 147 330 694
2 368 417 606
3 -257 372 647
4 648 363 625
5 524 352 704
6 103 415 644
7 -22 339 661
8 -35 347 674
9 70 410 625
10 145 459 686
11 229 419 749
12 52 439 604
13 -364 474 709
14 429 378 660
15 501 376 651
16 606 349 666
17 431 379 688
18 259 449 614
19 -49 445 647
20 -87 378 610
21 146 328 679
22 156 315 665
23 172 393 664
24 -456 397 689
25 -107 383 695
26 -199 377 736
27 -299 297 606
28 -643 323 641
29 -429 422 634
30 -158 371 701
31 -349 364 660
32 12 327 638
33 305 394 721
34 -130 389 641
35 -177 390 665
36 194 351 677
37 -135 332 711
38 164 391 627
... ... ... ...
continues over page
Continued
Day Profit / Loss VaR for Model 1 VaR for Model 2
39 -478 386 613
40 268 396 695
41 802 391 675
42 550 425 634
43 279 381 693
44 -292 402 662
45 -459 379 640
46 -235 402 727
47 -120 313 601
48 911 369 684
49 316 390 770
50 810 392 622
51 78 388 677
52 -168 370 723
53 881 388 639
54 69 341 641
55 715 320 697
56 685 402 687
57 -632 434 629
58 648 332 696
59 257 314 655
60 -112 341 609
Turn Over
B2 Consider the Merton model framework for corporate debt valuation.
Assume that a firm’s capital structure consists entirely of equity and
debt, and that the debt consists only of a zero-coupon bond with
promised payment of £X, and maturity at time T in the future.
Assume also that the risk-free rate is deterministic, constant across
all maturities and is r per annum, and that the firm’s asset value fol-
lows geometric Brownian motion, with drift parameter µ and volatility
parameter σ.
(a) [Type] Describe precisely how a default event is modelled. [2]
(b) [Type] Outline the strengths, weaknesses and limitations of the
Merton model for use in credit risk measurement. [4]
Continued
B3 (a) [Type] Explain what the forward risk-neutral probability mea-
sure is and describe a key property it has relating to asset prices.
Compare and contrast both the definition and the key property
with those of the risk-neutral probability measure. [2]
(b) Assume for this question that all bonds contain no default risk.
Assume that a zero-coupon, default free discount bond that pays
out £1 in 2 years from now is currently trading at £0.8. Assume
also that the zero-coupon, default free discount bond that pays
out £1 in 1 year from now is currently trading at £0.9.
Suppose that an analyst has chosen a model such that, under
the 1 year forward risk-neutral probability measure, the price of
the 2 year bond, Yt say, follows a diffusion process given by the
stochastic differential equation
dY = 0.1Y dt+ 0.5dz
for t ≥ 0 and where z is a Brownian motion. Using the analyst’s
model, apply the martingale pricing technique to derive the price
of a European call option for the 2 year bond, with strike price
at £0.85 and maturity date 1 year from now. The call option has
payoff
YT −£0.85 if YT ≥ £0.85
0 otherwise,
where YT is the random variable indicating the price in sterling
pounds (£) of the 2 year, zero-coupon, default free discount bond
1 year from now when the option matures.
Show all steps and assumptions. [6]
(c) [Type] Provide a critique of the analyst’s model in part (b) above.
[1]
Turn Over
B4 Use the Feynman-Kac result to solve the following partial differential
equation for f(x, t), and for 0 < t ≤ T :
∂f
∂t
+ x
∂f
∂x
+
1
2
x2
∂2f
∂x2
=
f
t
with f(x, T ) = x4. [8]
Continued
B5 For this question assume that the default event for a company follows a
hazard rate model in the risk neutral world with inhomogeneous hazard
rate that is a function of time, h(w) for w ≥ 0. Assume also that the
risk-free interest rate is stochastic and may vary by time period.
(a) Consider a financial product that pays out £1 in the event of a
credit default event for a company, paid at the time of default.
Today is time t and the company is solvent. Assume that the
product expires at time T in the future with T ≥ t ≥ 0.
Derive an expression for the expected payoff of this product in
the risk-neutral world, with the expectation taken today at time
t. Formulate your expression in terms of the current price of de-
fault free, zero-coupon discount bonds and the parameters above.
This expression may involve integrals. Explain all steps and work-
ing. [3]
(b) Another, different financial product pays out a time dependent
payoff in the event that the underlying company defaults of P (u, k)
for u < T , where u is the time of default, k is the smallest positive
integer such that k ≥ u, and P (a, b) is the value at time a of a
default free, zero-coupon discount bond that matures at time b.
The payoff is paid immediatly at the time of default. The product
expires at time T in the future where T is a positive integer, and
today is time t = 0.
Suppose that a buyer of this product pays a fee that is an annual
amount of k, paid bi-annually (i.e. k/2 paid every six months)
for the life of the product. The first payment is today, and the
last payment is 6 months before the product maturity, so that
payment is made in advance. A payment adjustment is therefore
necesary in the event of a default if default does not occur exactly
on a fee payment date to adjust for protection paid for but not
used.
Derive an expression for the arbitrage free payment k set today at
time t in terms of the price today of zero-coupon discount bonds.
Simplify your expression where possible. This expression may in-
volve integrals. Explain all steps and working. [10]
(c) Consider the product described in part (b) above, but suppose
now that the buyer and seller of the product have agreed not
to incorporate a payment adjustment in the event that a default
Turn Over
does not occur exactly on a premium payment date. Find the
no-arbitrage product fee (k) in sterling pounds (£) for this prod-
uct. Assume that the product has a lifetime of two years so that
T = 2, and today is time t = 0. Assume also that h(w) = 0.4 for
0 ≤ w ≤ T , the current price of a 1 year zero coupon discount
bond is £0.92, and that current forward rate agreements (FRA)
for risk-free borrowing are as given in the table below:
FRA Current price
k0.5,1 0.95
k1,1.5 0.91
k1.5,2 0.88
where ka,b indicates the price, agreed today, that is needed to be
paid at time a to receive delivery of £1 at time b. [5]
(d) i. [Type] Would you expect the price of the product in part
(c) to be higher or lower if the one year zero coupon dis-
count bond price was lower, with everything else remaining
the same? Explain your answer. [1]
ii. [Type] Would you expect the price of the product in part (c)
to be higher or lower if the constant hazard rate was lower,
with everything else remaining the same? Explain your an-
swer. [1]
iii. [Type] What would you expect to be the price of the product
in part (c) in the limit as the constant hazard rate tends to
infinity, with everything else remaining the same? Explain
the reason for your answer in terms of the product and model
properties, not as an algebraic limit calculation. [1]
End of Paper