RMBI 4210 – Quantitative Methods for Risk Management Final Examination, 2020
Time allowed: 3 hours Instructor: Prof. Y. K. Kwok
[points]
1. Suppose a hedger is delta-hedging a call option on an underlying stock.
(a) Explain why the delta of the call option increases when the underlying stock increases
in value. [1]
(b) Can delta assume value greater than one? Explain your answer. [1]
(c) When the stock price is gradually increasing in value when the call option comes
closer to expiry so that the call option is sufficiently deep in-the-money. The hedger
of the call option needs to purchase more units of stock in hedging. Does the cost
of hedging far exceed the call option premium collected upfront from the buyer? Explain how an appropriate delta hedging strategy can avoid catastrophic liabilities. [2]
2. Suppose a risk manager wants to hedge his exposure on an asset S using N units of hedging instrument F. The change in value of the hedged portfolio is given by
∆V =∆S+N∆F.
Recall that
Define the correlation coefficient ρSF by
and effectiveness of hedge by
3. (a) Is bond duration always an increasing function of maturity? That is, longer lived
bond always has higher duration. Prove or disprove the statement. [4]
(b) Immunization is always a dynamic procedure. Explain why. [1]
4. Suppose that each of two investments has 4% chance of a loss of $10 million, 2% chance of a loss of $1 million, and 94% chance of a profit of $1 million. These events are independent of each other.
(a) Find the VaR for one of the investments when the confidence level is 95%. [2] 1
σ2 = σ2 + 2Nσ ∆V ∆S
+ N2σ2 . ∆F
∆S,∆F
= σ∆S,∆F σ∆S σ∆F
ρS F
R=∆S ∆V,
where σ∗2 ∆V
σ2 − σ∗2 σ2
∆S
is the variance of the optimally hedged portfolio. Find R in terms of ρSF . Explain why we can achieve 100% effectiveness of hedge when ρSF tends to one. Give
your financial interpretation. [4]
(b) Find the expected shortfall when the confidence level is 95%. [2]
(c) What is the VaR for a portfolio consisting of the two investments when the confidence
level is 95%? Does VaR satisfy the subadditivity condition? [4]
(d) What is the expected shortfall for a portfolio consisting of the two investments
when the confidence level is 95%? Does expected shortfall satisfy the subadditivity condition? [4]
5. Let V be the random loss on a portfolio over a certain time horizon. We use the Pareto
distribution
( y )−1/ξ Gξ,β(y)=1− 1+ξβ
to model the random loss beyond u, where the probability that V > u + y conditional on V > u is 1 − Gξ,β (y). Let F (v) be the cumulative distribution function for the random loss variable V .
(a) Explain why the unconditional probability that V > x is given by
P[V > x] = [1−F(u)][1−Gξ,β(x−u)]. [1]
(b) Suppose 1 − F (u) is estimated by empirical data to be nu/n so that n[ξ ]−1/ξ
P [ V > x ] = nu 1 + β ( x − u ) ,
where n is the total number of scenarios and nu is the number of scenarios that the loss exceeds u. Note that x = u + y. Recall that VaR with a confidence level of q is given by
F(VaR) = q,
whereF(u)=1−P[V >u].
(i) Show the detailed derivation of finding VaR in terms of q, u, n and other u
parameters of the Pareto distribution. [1]
(ii) Recall the formula for Expected Shortfall ES, where
1∫1
ES=1−α
use this formula to find ES under this extreme value model. Show the details of
the mathematical derivation. No direct citation of the formula from the lecture
note. [4]
6. Historical default probability values reported in Moody or Standard & Poor are typically lower than the bond price based default probability values inferred from traded bond prices. Suppose the reported historical 5-year default probability is π5 = 0.57%. On the other hand, the 5-year zero-coupon bond B0,5 is traded with a credit spread of 50bps.
2
α
VaRq dq.
(a) Assume recovery rate to be zero. Find the 5-year bond price based default probability (denoted by Q5) from the credit spread. [2]
(b) We define Rd be the objective discount rate appropriate for this specific bond so that
B0,5 =(1−π5)e−5Rd. B0,5 = (1 − Q5)e−5r.
Also, we recall
where r is the riskfree interest rate. Use π5 as given in the above data to find Rd − r,
which is considered as the extra return for holding this risky bond. [3] (c) How would you compare Rd − r and credit spread of 50 bps as inferred from the
observed bond price? Explain the financial rationales behind the discrepancy. [2] 7. Suppose that:
(i) The spread on a 5-year par floater bond issued by company X is 3%
(ii) A 5-year credit default swap providing insurance against the default of company X
charges a premium rate of 250 basis points per year
How can an arbitrageur take arbitrage from the above trading opportunities? Outline
the arbitrage procedure. How does the counterparty risk of the Protection Seller limit
the arbitrage gain? [3]
8. (a)
Let S(t) denote the survival probability of a company up to time t. Let λ(t) denote the default intensity defined by
P[τdef ≤ t + ∆t|τdef > t] = λ(t)∆t, where τdef is the random default time. Recall that
S(t)=e−∫tλ(u) du. 0
Let q(t) denote the unconditional default probability density such that P[t < τdef ≤ t+∆t] = q(t)∆t.
Find the relation between q(t), S(t) and λ(t). [2]
(b) A company has issued a 2-year bond with a coupon of 4% per annum payable annually. The yield on the bond (expressed with continuous compounding) is 6% and the risk-free rate is 3% with continuous compounding. The recovery rate is 20%. Defaults can take place halfway through each year.
(i) Find the unconditional default density, assuming to be constant for the whole 2
years. [6]
(ii) Assume λ(t) to be piecewise constant over the first year and second year, where λ(t)=λ1 for0≤t≤1andλ(t)=λ2 for1
where τi is the random default time of firm i, i = 1, 2, 3. [4]
(b) The exponential survival copula associated with the survival function S(t1, t2, t3) is
defined by
For fixed i and j, show that the two-dimensional marginal survival copula is given
by
()
Cτ(ui,uj) = min uju1−θi,uiu1−θj , ij
Cτ(u ,u ,u ) = S(S−1(u ),S−1(u ),S−1(u )). 123 112233
where θi ∈ (0, 1) and θj ∈ (0, 1). In particular, find (i) Cτ (u1, u2), (ii) Cτ (u2, u3) in terms of λ1,λ2,…,λ5.
[8]
5
14. Under the CreditRisk+ framework, suppose we allocate 120 obligors into 3 bands as follows:
vj number of obligors εj μj 2 30 3 1.5 3 40 124 5 50 102
vj = common exposure in band j in units of L
εj = expected dollar loss in band j in units of L over a given time horizon μj = expected number of defaults in band j over a given time horizon
Recall εj = vjμj. For example, μ3 = 4 means the expected number of defaults among 40 obligors in band 2 is 4 over a given time horizon and the expected dollar loss in band 2 is ε3 = 3 × 4 = 12. Defaults of the obligors are assumed to be independent.
(a) Using the Poisson approximation, the probability generating function F(z) of the random number of defaults is given by
F (z) = eμ(z−1),
μj = expected number of defaults over a given time horizon from the
∑3 j=1
where μ =
whole portfolio. Let X be the random number of defaults over a given time horizon
from the whole portfolio. Find (i) P [X = 0], (ii) P [X = 3]. [3] (b) Let G(z) denote the probability generating function of the random dollar loss over
a given time horizon from the whole portfolio in units of L. Recall that G(z) = eμ[P (z)−1],
where
(i) Find P(z) in terms of the given data in the above. [2]
(ii) Write out the full analytic expression for G(z). Let D be the random number
of dollar loss in units of L over a given time horizon from the whole portfolio.
Use G(z) to find P[D = 0] and P[D = 5]. [5]
Hint: Observe the following Taylor expansion
eα2z2+α3z3+α5z5 ≈1+(α2z2+α3z3+α5z5)+12(α2z2+α3z3+α5z5)2+···. consider the coefficient of z5.
— End —
P(z) = P[V = v1]zv1 + ··· + P[V = vm]zvm.
6