A1-无代写
时间:2023-04-10
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SECTION A
Answer ALL questions from Section A.
A1
[Type] Provide a critique of the variance-covariance (VCV) method for calculating the value-at-risk
for a portfolio. Include in your answer an explanation of its limitations and why these limitations
arise, and also a discussion of when it may be appropriate to use, and when it is not suitable to use.
[4]
A2
(a) Suppose that a portfolio consists of 100 units of shares in company IndustrialCo, and 50 units of
shares in company Servcore. Shares in IndustrialCo are currently trading at £1 per unit, and have an
estimated annualised volatility of 15%. Shares in Servcore are currently trading at £2 per unit, and
have an estimated annualised volatility of 35%.
An analyst calculates the 99%, 1 day value-at-risk of the portfolio as being £5.29 using the variance-
covariance (VCV) method, and in doing so assumes that there are 250 trading days in a year.
Determine what assumption the analyst has made about the correlation between the daily returns
of IndustrialCo and Servcore shares. Show all your working. [4]
(b) If the analyst had assumed that the daily returns for IndustrialCo and Servcore shares were
independent, would you have expected the analyst’s value-at-risk estimate to be more or less than
the £5.29 the analyst calculated? Explain your answer. [1]
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A3
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 corporate 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 constant across all maturities.
(a) [Type] Explain the difference between objective default probabilities and risk-neutral default
probabilities. Explain also when each should be used and why. [3]
(b) Suppose that a corporate, zero coupon, £1,000 par bond, subject to credit default risk, is due to
expire in 2 years’ time, and is currently trading at £800. A £1,000 par, default-free, zero coupon bond
issued by a government and also expiring in 2 years’ time is trading at £900. Calculate the credit
spread for this corporate bond. [2]
(c) Calculate the implied risk neutral probability that the company that issued the bond defaults
before expiration. [2]
(d) Assume that the risk neutral default process for the company can be modelled using a hazard
rate model with a constant hazard rate. Use the information in A3 (b) and (c) above to determine the
hazard rate. [2]
(e) Under this assumption of a constant hazard rate for the company default event for t > 0,
calculate the price of a 3 year, zero coupon bond that has a par of £1,000 and is issued by the same
company. [2]
A4
(a) [Type] Define what an equivalent martingale probability measure is. Include an explanation of
any terms you need to use in your definition. [2]
(b) [Type] Explain how an equivalent martingale probability measure can be used in pricing
derivatives. [2]
A5
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) = 0.05 for t ≤ 2; h(t) = 0.25 − t/10 for 2 ≤ t ≤ 2.5 and h(t) = 0
for t ≥ 2.5. Today is time 0.
(a) What is the probability that the company survives from today up until time 1? [2]
(b) Find the probability that the company has not defaulted by time t = 3. [3]
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A6
[Type] A 1 year government bond with no counter-party default risk is trading at £980, and a similar
2 year bond at £950. Both bonds have par value £1,000. A 1 year forward rate agreement for lending
between 1 year from now and 2 years from now is being offered at 3%. 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
[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 below, 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]
Day Profit / Loss VaR for Model 1 Var for Model 2
1 147 330 694
2 368 417 606
3 -380 372 647
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 -386 397 689
25 -107 383 695
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OVER
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Day Profit / Loss VaR for Model 1 Var for Model 2
26 -199 377 736
27 -299 297 606
28 -643 323 641
29 -409 422 634
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
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 -535 402 727
47 -120 313 601
48 911 369 684
49 316 390 770
50 810 392 622
51 78 388 677
52 -468 370 723
53 881 388 639
54 69 341 641
55 715 320 697
56 685 402 687
57 -632 434 669
58 648 332 696
59 257 314 655
60 -112 341 609
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SECTION B
Answer ALL questions from Section B.
B1
Use the Feynman-Kac result to solve the following partial differential equation for f(x, t), and
for t ≤ T:
∂f/∂t + x ∂f /∂x + 1/ 2 x2 ∂2 f/ ∂x2= 0
with f(x, T) = x4. [11]
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B2
Assume for this question that all bonds contain no default risk.
(a) [Type] Explain what the forward risk-neutral probability measure 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. [4]
(b) 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, in the 1 year forward risk-neutral probability
world, the change in the price of the 2 year bond over any time period of length τ<2, takes a normal
distribution (note this is a normal distribution, not a log normal distribution) with mean 0.1 × τ and
variance 0.1 × τ, where price is denoted in pounds sterling (£). Using the analyst’s model, apply the
martingale pricing technique to derive the price of a digital put option for the 2 year bond, with
strike price at £0.85 and maturity date 1 year from now.
The digital put option has payoff
£1 if Y ≤ £0.85
0 otherwise,
where Y 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. [7]
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B3
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 follows geometric Brownian motion, with drift parameter μ and volatility parameter σ.
(a) [Type] Explain how this model can be used to model credit risk, and include a clear description of
how a default event is modelled. Outline any further assumptions made in the model. [3]
(b) [Type] Outline the strengths, weaknesses and limitations of the Merton model for use in credit
risk measurement. [5]
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B4
(a) [Type] Describe the historical simulation approach to calculating value-at-risk, and provide an
outline the steps that should be taken to apply this approach. Explain also how correlation is taken
into account in this approach. [3]
(b) [Type] Describe the key strengths and weaknesses of the historical simulation value-at-risk
approach, drawing on a comparison with the variance-covariance approach and the Monte Carlo
approach where appropriate. Include a description of at least one example of a situation when the
historical simulation approach is appropriate to use, and at least one in which it may not be
appropriate to use. [7]
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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, and that tradable
default-free zero-coupon discount bonds exist for all maturities.
(a) A financial product pays out a time dependent payoff in the event that the underlying company
defaults of P(u, T) for uwith the notation that 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 immediately 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 of π that is an annual amount, but is paid bi-
annually (i.e. π/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 necessary 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 π set today at time t in terms of the price
today of default-free zero-coupon discount bonds. Simplify your expression where possible. This
expression may involve integrals. Explain all steps and working. [10]
(b) Consider the product described in part (a) 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
does not occur exactly on a premium payment date. Find the no-arbitrage product fee (π ) in
sterling pounds (£) for this product. 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) = 0.2 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
f(0.5,1) 0.97
f(1,1.5) 0.90
f(1.5,2) 0.85
where f(a,b) indicates the price, agreed today, that is needed to be paid at time a to receive
delivery of £1 at time b. [7]
(c)
(i) [Type] Would you expect the price of the product in part (b) to be higher or lower if the one
year zero-coupon discount bond price was lower, with everything else remaining the same?
Explain your answer. [1]
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(ii) [Type] Would you expect the price of the product in part (b) to be higher or lower if the
constant hazard rate was lower, with everything else remaining the same? Explain your answer.
[1]
(iii) [Type] What would you expect to be the price of the product in part (b) 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