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程序代写案例-A17470W1

时间：2021-04-14

A17470W1

DEGREE OF MASTER OF SCIENCE

Mathematical and Computational Finance

Fixed Income and Credit, Stochastic Control and

Quantitative Risk Management

Trinity TERM 2021

WEDNESDAY, 14 April 2021

Opening Time: 9:30am GMT

This exam should only take you 3 hours to complete plus 1 hour technical time.

Candidates must attempt the following

• TWO questions from Part A: Fixed Income and Credit

• ONE question from Part B: Stochastic Control

• ONE question from Part C: Quantitative Risk Management

You may attempt as many questions as you wish. The best two answers from Part A and the

best answer from Parts B and C, will count toward the total mark.

Page 1 of 8

1 Part A: Fixed Income and Credit

1. Assume that there is an arbitrage free bond market with a family of risk free bonds {P (t, T ), 0 6

t 6 T < ∞}. We consider a set of dates T0, T1, . . . , Tn at which interest payments must be

made. We write δi = Ti − Ti−1 for the times between payments.

(a) [4 marks] A payer swap is a contract in which two parties exchange interest rate payments

with the payer paying fixed and receiving floating interest rate payments at LIBOR over

the lifetime of the contract. If the swap sets at T0 and finishes at Tn find the value of the

fixed rate to give the swap 0 value at initiation t < T0.

Why would an investor be interested in entering a payer swap contract?

(b) [5 marks] A swap market model enables the prices of individual swaps to be modelled via

a log normal diffusion in that if yi,k denotes the swap rate for a swap running from time

Ti to Tk, where Ti is the first date when the LIBOR rate is set, then

dyi,kt = σi,k(t)y

i,k

t dW

i,k

t ,

where σi,k(t) is a deterministic volatility function and W i,k is a Brownian motion under

a suitable measure.

Explain briefly how a swap market model for the swaps yi,n, i = 1, . . . , n can be con-

structed to enable all the individual swaps to satisfy such a log-normal diffusion. There

is no need to give a detailed derivation.

(c) [6 marks] A digital call swaption pays a unit amount if the swap rate y1,n is above a strike

K at a maturity time T . Find the time t < T0 value of this option in the swap market

model when T = T0.

(d) [8 marks] Let Li(t) denote the forward LIBOR rate at time t 6 Ti−1 for simple interest

from Ti−1 to Ti. Consider a back swap in which the LIBOR payments at time Ti in the

floating leg of the swap are made at the rate prevailing at the payment time, Li+1(Ti),

rather than at the rate Li(Ti−1), determined at the previous reset date Ti−1. Show that

the payoff from the floating leg is

n∑

i=1

δiLi+1(Ti)(1 + δiLi+1(Ti))P (t, Ti+1)

By assuming a lognormal model for the LIBOR rate Li+1(t), with volatility σi+1(t), show

that the value of the fixed rate which makes this swap have 0 value at initiation is given

by

y =

∑n

i=1

(

δiLi+1(t) + δ

2

i Li+1(t)

2 exp(

∫ Ti

t σi+1(s)

2ds)

)

P (t, Ti+1)∑n

i=1 δiP (t, Ti)

.

(e) [2 marks] Indicate briefly how could we determine the forward volatility of the LIBOR

in this model.

A17470W1 Page 2 of 8

2. We consider a general short rate model for interest rates in the form

drt = µ(t, rt)dt+ σ(t, rt)dWt,

where µ, σ are measurable functions such that the solution to the SDE exists and W is a

one-dimensional Brownian motion under a measure P.

(a) [8 marks] We assume the bond prices in this model can be written as a smooth function

F T of the short rate and time in that P (t, T ) = F T (t, r). Show that the term structure

equation for the bond prices in this model is given by

∂F T

∂t

+ ν(t, r)

∂F T

∂r

+

1

2

η(t, r)

∂2F T

∂r2

− rF T = 0,

F T (T, r) = 1,

for suitable functions ν(t, r) and η(t, r) which you should determine.

Comment on the market price of risk and how it arises when considering the evolution of

the short rate in the physical measure P and its value under a martingale measure Q.

(b) [4 marks] Assume that we now model under the martingale measure Q and derive the

ODEs for the bond prices to have an affine term structure F T (t, r) = exp(A(t, T ) −

rB(t, T )) when µ and σ2 are independent of t and linear functions of r.

(c) [13 marks] We now consider a model with a constant drift µ = α, and σ =

√

2r under

the martingale measure Q.

(i) Show that in this case the affine term structure is soluble and that the bond prices

are given explicitly as

P (t, T ) = (cosh(T − t))−α exp(−r tanh(T − t)).

You may like to note that ∫

1

x2 − 1dx = −arctanh(x),

and

tanh(x) =

sinh(x)

cosh(x)

, cosh2(x)− sinh2(x) = 1, d

dx

tanh(x) = sech2(x).

(ii) Find the evolution of the forward rates in this model and show that they satisfy the

HJM drift condition.

(iii) Find the volatility of the bond prices and comment on whether or not this is a good

model for the long term bond prices.

A17470W1 Page 3 of 8

3. (a) [5 marks] Consider a simple reduced form model for a credit event where we have default

occuring as a Poisson process with a time varying intensity λt.

(i) Find an expression for the probability of default before some maturity time T .

(ii) If the intensity is generated by a random process, so that default occurs as a Cox

process, find the price of a defaultable T -bond.

(b) [10 marks] Let τi denote the default time of entity i for i = 1, 2.

(i) Show that the correlation between defaults at time t can be written as

Corrt :=

P(τ1 < t, τ2 < t)− p1p2√

p1(1− p1)p2(1− p2)

=

P(τ1 > t, τ2 > t)− P(τ1 > t)P(τ2 > t)√

p1(1− p1)p2(1− p2)

,

where pi = P(τi < t) is the probability of default of i by time t.

(ii) Find an expression for the correlation between defaults in a model where there are

two entities with their defaults occuring as two Cox processes.

(iii) Let N1, N2, N12 be three independent Poisson processes of constant rate λ1, λ2, λ12.

Let N1, N2 represent the defaults of bank one and bank two respectively and let N12

represent the process of joint defaults, where both banks default at the same time.

Find an expression for the default correlation at time t and show that we can have

high default correlation by taking λ1, λ2 small compared to λ12.

(c) [10 marks] (i) A credit default swap gives a party A, with a cash flow from a counterparty

C, protection from default by C. We define a unit CDS as one that pays A one unit

of currency if C defaults in the period up to a time Tn. For the protection A pays

premiums at a rate s on a set of dates T1, . . . , Tn. Show that, if τ is the default time

of C, the rate which should be agreed at the initiation of the unit CDS in order that

it has 0 value at time T0 = 0 is given by

s0 =

EQ

(

e−

∫ τ

0 ruduIτ)

∑n

i=1(Ti − Ti−1)EQ

(

e−

∫ Ti

0 ruduIτ>Ti

) ,

where r is the stochastic interest rate process and Q is the risk neutral measure.

Find an expression for s0 in the case where default is modelled as a constant rate

Poisson process, rate λ, and the interest rate is a constant.

(ii) A first to default unit CDS is a credit derivative in which pays one unit of currency

if any one of n entities defaults before time T . If we assume that all entities are

independent and modelled as constant rate Poisson processes with rate λ, find the rate

that should be agreed for this first to default unit CDS to have 0 value at initiation.

[Hint: Recall that, if X1, . . . , Xm are random variables, then P(min(X1, . . . , Xm) >

t) = P(X1 > t, . . . ,Xm > t)]

(iii) Explain why, in general, the rate of this first to default unit CDS can be bounded

below by the highest individual unit CDS rate and bounded above by the sum of the

unit CDS rates on the individual entities. Are there scenarios where these bounds

are attained?

A17470W1 Page 4 of 8

2 Part B:Stochastic Control

You may assume throughout this section that all stochastic differential equations (SDEs) have

unique solutions, all stochastic integrals are martingales, and that all functions are smooth enough

for application of the Itoˆ formula.

1. On a stochastic basis (Ω,F ,F = (Ft)t∈[0,T ],P) over a fixed time horizon [0, T ], a non-negative

controlled process Y satisfies the SDE

dYt = ηYt ((θ − κψt) dt+ dWt) , (1)

where W is a Brownian motion, θ, κ, η are constants, with η > 0, and ψ is an adapted process

such that

∫ T

0 ψ

2

t dt < ∞ almost surely. In this scenario, and with α > 0 a strictly positive

constant, define the control problem with value function p : [0, T ]× R+ → R+ given by

p(t, y) := sup

ψ∈Ψ

E

[

h(YT )− 1

2α

∫ T

t

ψ2s ds

∣∣∣∣Yt = y] , (t, y) ∈ [0, T ]× R+,

where h : R+ → R+ is a bounded non-negative function, and Ψ denotes a set of admissible

controls such that the value function is finite.

(a) [4 marks] State the dynamic programming principle in terms of a martingale and super-

martingale property of an appropriate value process, and hence write down the Hamilton-

Jacobi-Bellman (HJB) equation for the value function, including a terminal condition.

(b) [6 marks] Derive an expression for the optimal feedback control function ψ̂ : [0, T ]×R+ →

R, state carefully how this function is related to the optimal control process (ψ̂t)t∈[0,T ],

and show that the resulting HJB equation is of the form

∂p

∂t

(t, y) +A0p(t, y) + 1

2

Ky2p2y(t, y) = 0,

subject to a terminal condition, where py(·, ·) denotes the partial derivative with respect

to y, A0 is a differential operator that you should give an expression for, and K is a

constant that you should identify.

(c) [6 marks] Interpret the operator A0 as the generator of the process Y in (1) under some

measure P0 ∼ P which you should identify, by giving an expression for the Radon-Nikodym

derivative of P with respect to P0, in the form

dP

dP0

= E(−M)T , (2)

for some P0 -martingale M which you should identify. Write down the P0-dynamics for

Y and show that with the measure change in (2) we recover the P-dynamics in (1).

(d) [9 marks] Look for a solution to the HJB equation of the form

p(t, y) =

1

δ

logF (t, y),

for some constant δ and function F : [0, T ] × R+ → R+, and find the value of δ such

that F (·, ·) satisfies a linear PDE. Hence, via the Feynman-Kac theorem, write down an

expectation representation for F (·, ·), and thus give the resulting solution for the value

function p(·, ·).

A17470W1 Page 5 of 8

2. On an infinite horizon stochastic basis (Ω,F ,F = (Ft)t>0,P), a stock price and its positive

stochastic volatility process Y satisfy, in a market with zero interest rate, the SDEs

dSt = YtSt(λ(Yt) dt+ dWt), dYt = a(Yt) dt+ η(ρ dWt + ρdW

⊥

t ),

for suitable functions λ(·), a(·), with ρ ∈ (−1, 1) a constant correlation and ρ :=

√

1− ρ2,

η > 0 a positive constant, and W,W⊥ are independent Brownian motions.

An agent with initial capital x > 0 trades the stock and cash and consumes at a non-negative

adapted consumption rate c, generating wealth process X. With δ > 0 a positive impatience

parameter, and U : R+ → R a utility function, the objective is to maximise utility from

consumption. The value function is defined by

u(x, y) := sup

(pi,c)∈A(x,y)

E

[∫ ∞

0

e−δtU(ct) dt

∣∣∣∣ (X0, Y0) = (x, y)] ,

with A(x, y) denoting admissible investment-consumption strategies, and pi is the process for

wealth in the stock.

(a) [2 marks] Formulate the wealth process SDE, state the dynamic programming principle

in terms of a supermartingale and martingale property of an appropriate value process,

and hence write down the HJB equation for the value function.

(b) [5 marks] Compute the optimal feedback control functions pi(·, ·), ĉ(·, ·), state carefully

how these are related to the optimal control processes (pit, ĉt)t>0, and show that the HJB

equation converts to

V (ux(x, y)) + Gu(x, y)− 1

2uxx

(λ(y)ux(x, y) + ρηuxy(x, y))

2 − δu(x, y) = 0,

where G is a differential operator you should interpret as a generator of a process, and for

which you should give an expression, and V : R+ → R is the convex conjugate of U(·),

satisfying V (y) = U(I(y)) − yI(y), y > 0, with I(·) ≡ (U ′(·))−1 the inverse of marginal

utility.

(c) [5 marks] Suppose U(c) = cp/p, p < 1, p 6= 0 is a power utility function. Assume a

separable solution to the HJB equation of the form u(x, y) = U(x)f(y) for some function

f(·), and show that the optimal control processes are then given by

ĉt = F (Yt)X̂t, pit =

G(Yt)

Yt(1− p)X̂t, t > 0,

where X̂ denotes the optimal wealth process and the functions F (·), G(·) are given by

F (y) := (f(y))−1/(1−p), G(y) := λ(y) + ρη

f ′(y)

f(y)

.

[You are not required to solve for f(·), only to show that the optimal controls will take the

forms stated.]

(d) [5 marks] Show that the optimal wealth process is given by

X̂t = x exp

(∫ t

0

(

G(Ys)

1− p λ(Ys)− F (Ys)

)

ds

)

E

(

G(Y )

1− p ·W

)

t

, t > 0.

(e) [6 marks] State the transversality condition that the value function is expected to satisfy

and, assuming that f(Yt) is bounded for all t > 0, deduce that the transversality condition

will hold provided that

δt >

∫ t

0

(

1

2

qG2(Ys)− qG(Ys)λ(Ys)− pF (Ys)

)

ds, t > 0,

where q := −p/(1− p) denotes the conjugate exponent to p.

A17470W1 Page 6 of 8

3 Part C: Quantitative Risk Management

1. (a) [8 marks] Let X,Y be two random variable representing losses from two portfolios.

(i) Give the definition of V@Rα(X), the Value-at-Risk at level α ∈ (0, 1). Prove or

disprove the validity of the following statements for arbitrary λ > 0 and c ∈ R:

• V@Rα(λX) = λV@Rα(X)

• V@Rα(X + c) = V@Rα(X) + c

• V@Rα(X + Y ) = V@Rα(X) + V@Rα(Y )

(ii) A bank holds a portfolio of N defaultable zero-coupon bonds. The bonds were issued

by different counterparties but have a common maturity T . For each counterparty,

using their CDS quotes or otherwise, the bank has a good estimate of their probability

of default. Discuss briefly the challenges in risk managing this portfolio. What are

the possible pitfalls of using V@Rα to risk-manage this portfolio? Would your risk

assessment depend continuously on α?

(b) [17 marks] Suppose that C1, . . . , Ck are d dimensional copulas and λ = (λ1, . . . , λk) has

non-negative entries with

∑k

i=1 λi = 1. Define

C(u) =

k∑

i=1

λiCi(u), u ∈ [0, 1]d.

(i) Using the characterisation of a copula function, or otherwise, show that C is a d

dimensional copula.

(ii) Suppose we can sample efficiently from each of Ci, i = 1, . . . , d. Write down an

algorithm to sample efficiently from C. Justify your answer.

(iii) Suppose d = 2. State the definition of Spearman’s rho. Let ρiS denote Spearman’s

rho for Ci and ρS denote Spearman’s rho for C. Show that

ρS =

k∑

i=1

λiρ

i

S .

You may use the fact that the variance of a uniform random variable on [0, 1] is 112 .

(iv) Suppose d = 2 and recall that Kendall’s tau for C can be computed as

ρτ (C) = 4E[C(U1, U2)]− 1,

where (U1, U2) is sampled from C. Assume now that k = 2, C1 = M is the co-

monotone copula and C2 = Π is the independence copula. Compute ρτ (M) and

ρτ (Π) and comment on the interpretation of these values. Let λ1 = λ ∈ (0, 1) so that

λ2 = 1− λ. Show that

ρτ (C) =

1

3

λ(2 + λ).

Is it true that

ρτ (C) = λρτ (C1) + (1− λ)ρτ (C2)?

A17470W1 Page 7 of 8

2. (a) [5 marks] Suppose dataX1, X2, . . . is i.i.d. from some distribution F . LetMn = max{X1, . . . , Xn}.

Recall that F is in the maximum domain of attraction of H, for some non-trivial distri-

bution function H, if (Mn − dn)/cn converges in distribution to H, for some sequences

(cn), (dn).

Why is it of interest to fit such an H when past losses are sampled from F? Can different

choices of (cn), (dn) lead to H with different tail behaviours?

Describe briefly the difference between the block maxima method and the peaks-over-

threshold method. Which one would you prefer and why?

(b) [8 marks] Suppose now X1, X2, . . . are i.i.d. exponential with mean 1/β, for some β > 0,

i.e., F (x) = 1− exp(−βx), x > 0. Recall that F is in the maximum domain of attraction

of the Gumbel distribution.

(i) Let Yn = max{X3n, X3n+1, X3n+2} and MYn = max{Y1, . . . , Yn}, n > 1. Taking

cn = 1/β and dn =

logn

β , show that (M

Y

n − dn)/cn converges in distribution to some

HY which you should identify.

(ii) Let Zn = max{Xn, Xn+1, Xn+2} and MZn = max{Z1, . . . , Zn}, n > 1. Using the same

(cn), (dn) as in (i), show that (M

Z

n − dn)/cn converges in distribution to some HZ

which you should identify.

(iii) Show that Yn and Zn have the same distribution. What is the relationship between

HY and HZ? Relate your answer to serial dependence in the data in each of the two

cases.

(c) [12 marks] The durations X1, X2, . . . of calls handled by a call centre are independent and

generalized Pareto distributed, i.e.,

P(X1 > x) =

(

1 +

ξx

β

)−1/ξ

, x > 0,

for some β > 0 and ξ ∈ (0, 1).

(i) The parameter ξ needs to be estimated. Comment on its relevance for the variance of

the call lengths. Compute the excess distribution Fu of call length over the threshold

u, for a given u > 0.

(ii) Compute V@Rα(X1), the Value-at-Risk at level α ∈ (0, 1) and ESα(X1), the expected

shortfall at level α. Deduce the value of e(V@Rα(X1)), the mean excess function for

the threshold u = V@Rα(X1).

(iii) Suppose that ξ < 0.5 and β = ξ. Show that V@Rα(X1) 6 ESα(X1) 6 2V@Rα(X1) +

1. What can you deduce about using value-at-risk or expected shortfall when man-

aging the risk of excessively long calls clogging up the call centre?

A17470W1 Page 8 of 8

DEGREE OF MASTER OF SCIENCE

Mathematical and Computational Finance

Fixed Income and Credit, Stochastic Control and

Quantitative Risk Management

Trinity TERM 2021

WEDNESDAY, 14 April 2021

Opening Time: 9:30am GMT

This exam should only take you 3 hours to complete plus 1 hour technical time.

Candidates must attempt the following

• TWO questions from Part A: Fixed Income and Credit

• ONE question from Part B: Stochastic Control

• ONE question from Part C: Quantitative Risk Management

You may attempt as many questions as you wish. The best two answers from Part A and the

best answer from Parts B and C, will count toward the total mark.

Page 1 of 8

1 Part A: Fixed Income and Credit

1. Assume that there is an arbitrage free bond market with a family of risk free bonds {P (t, T ), 0 6

t 6 T < ∞}. We consider a set of dates T0, T1, . . . , Tn at which interest payments must be

made. We write δi = Ti − Ti−1 for the times between payments.

(a) [4 marks] A payer swap is a contract in which two parties exchange interest rate payments

with the payer paying fixed and receiving floating interest rate payments at LIBOR over

the lifetime of the contract. If the swap sets at T0 and finishes at Tn find the value of the

fixed rate to give the swap 0 value at initiation t < T0.

Why would an investor be interested in entering a payer swap contract?

(b) [5 marks] A swap market model enables the prices of individual swaps to be modelled via

a log normal diffusion in that if yi,k denotes the swap rate for a swap running from time

Ti to Tk, where Ti is the first date when the LIBOR rate is set, then

dyi,kt = σi,k(t)y

i,k

t dW

i,k

t ,

where σi,k(t) is a deterministic volatility function and W i,k is a Brownian motion under

a suitable measure.

Explain briefly how a swap market model for the swaps yi,n, i = 1, . . . , n can be con-

structed to enable all the individual swaps to satisfy such a log-normal diffusion. There

is no need to give a detailed derivation.

(c) [6 marks] A digital call swaption pays a unit amount if the swap rate y1,n is above a strike

K at a maturity time T . Find the time t < T0 value of this option in the swap market

model when T = T0.

(d) [8 marks] Let Li(t) denote the forward LIBOR rate at time t 6 Ti−1 for simple interest

from Ti−1 to Ti. Consider a back swap in which the LIBOR payments at time Ti in the

floating leg of the swap are made at the rate prevailing at the payment time, Li+1(Ti),

rather than at the rate Li(Ti−1), determined at the previous reset date Ti−1. Show that

the payoff from the floating leg is

n∑

i=1

δiLi+1(Ti)(1 + δiLi+1(Ti))P (t, Ti+1)

By assuming a lognormal model for the LIBOR rate Li+1(t), with volatility σi+1(t), show

that the value of the fixed rate which makes this swap have 0 value at initiation is given

by

y =

∑n

i=1

(

δiLi+1(t) + δ

2

i Li+1(t)

2 exp(

∫ Ti

t σi+1(s)

2ds)

)

P (t, Ti+1)∑n

i=1 δiP (t, Ti)

.

(e) [2 marks] Indicate briefly how could we determine the forward volatility of the LIBOR

in this model.

A17470W1 Page 2 of 8

2. We consider a general short rate model for interest rates in the form

drt = µ(t, rt)dt+ σ(t, rt)dWt,

where µ, σ are measurable functions such that the solution to the SDE exists and W is a

one-dimensional Brownian motion under a measure P.

(a) [8 marks] We assume the bond prices in this model can be written as a smooth function

F T of the short rate and time in that P (t, T ) = F T (t, r). Show that the term structure

equation for the bond prices in this model is given by

∂F T

∂t

+ ν(t, r)

∂F T

∂r

+

1

2

η(t, r)

∂2F T

∂r2

− rF T = 0,

F T (T, r) = 1,

for suitable functions ν(t, r) and η(t, r) which you should determine.

Comment on the market price of risk and how it arises when considering the evolution of

the short rate in the physical measure P and its value under a martingale measure Q.

(b) [4 marks] Assume that we now model under the martingale measure Q and derive the

ODEs for the bond prices to have an affine term structure F T (t, r) = exp(A(t, T ) −

rB(t, T )) when µ and σ2 are independent of t and linear functions of r.

(c) [13 marks] We now consider a model with a constant drift µ = α, and σ =

√

2r under

the martingale measure Q.

(i) Show that in this case the affine term structure is soluble and that the bond prices

are given explicitly as

P (t, T ) = (cosh(T − t))−α exp(−r tanh(T − t)).

You may like to note that ∫

1

x2 − 1dx = −arctanh(x),

and

tanh(x) =

sinh(x)

cosh(x)

, cosh2(x)− sinh2(x) = 1, d

dx

tanh(x) = sech2(x).

(ii) Find the evolution of the forward rates in this model and show that they satisfy the

HJM drift condition.

(iii) Find the volatility of the bond prices and comment on whether or not this is a good

model for the long term bond prices.

A17470W1 Page 3 of 8

3. (a) [5 marks] Consider a simple reduced form model for a credit event where we have default

occuring as a Poisson process with a time varying intensity λt.

(i) Find an expression for the probability of default before some maturity time T .

(ii) If the intensity is generated by a random process, so that default occurs as a Cox

process, find the price of a defaultable T -bond.

(b) [10 marks] Let τi denote the default time of entity i for i = 1, 2.

(i) Show that the correlation between defaults at time t can be written as

Corrt :=

P(τ1 < t, τ2 < t)− p1p2√

p1(1− p1)p2(1− p2)

=

P(τ1 > t, τ2 > t)− P(τ1 > t)P(τ2 > t)√

p1(1− p1)p2(1− p2)

,

where pi = P(τi < t) is the probability of default of i by time t.

(ii) Find an expression for the correlation between defaults in a model where there are

two entities with their defaults occuring as two Cox processes.

(iii) Let N1, N2, N12 be three independent Poisson processes of constant rate λ1, λ2, λ12.

Let N1, N2 represent the defaults of bank one and bank two respectively and let N12

represent the process of joint defaults, where both banks default at the same time.

Find an expression for the default correlation at time t and show that we can have

high default correlation by taking λ1, λ2 small compared to λ12.

(c) [10 marks] (i) A credit default swap gives a party A, with a cash flow from a counterparty

C, protection from default by C. We define a unit CDS as one that pays A one unit

of currency if C defaults in the period up to a time Tn. For the protection A pays

premiums at a rate s on a set of dates T1, . . . , Tn. Show that, if τ is the default time

of C, the rate which should be agreed at the initiation of the unit CDS in order that

it has 0 value at time T0 = 0 is given by

s0 =

EQ

(

e−

∫ τ

0 ruduIτ

∑n

i=1(Ti − Ti−1)EQ

(

e−

∫ Ti

0 ruduIτ>Ti

) ,

where r is the stochastic interest rate process and Q is the risk neutral measure.

Find an expression for s0 in the case where default is modelled as a constant rate

Poisson process, rate λ, and the interest rate is a constant.

(ii) A first to default unit CDS is a credit derivative in which pays one unit of currency

if any one of n entities defaults before time T . If we assume that all entities are

independent and modelled as constant rate Poisson processes with rate λ, find the rate

that should be agreed for this first to default unit CDS to have 0 value at initiation.

[Hint: Recall that, if X1, . . . , Xm are random variables, then P(min(X1, . . . , Xm) >

t) = P(X1 > t, . . . ,Xm > t)]

(iii) Explain why, in general, the rate of this first to default unit CDS can be bounded

below by the highest individual unit CDS rate and bounded above by the sum of the

unit CDS rates on the individual entities. Are there scenarios where these bounds

are attained?

A17470W1 Page 4 of 8

2 Part B:Stochastic Control

You may assume throughout this section that all stochastic differential equations (SDEs) have

unique solutions, all stochastic integrals are martingales, and that all functions are smooth enough

for application of the Itoˆ formula.

1. On a stochastic basis (Ω,F ,F = (Ft)t∈[0,T ],P) over a fixed time horizon [0, T ], a non-negative

controlled process Y satisfies the SDE

dYt = ηYt ((θ − κψt) dt+ dWt) , (1)

where W is a Brownian motion, θ, κ, η are constants, with η > 0, and ψ is an adapted process

such that

∫ T

0 ψ

2

t dt < ∞ almost surely. In this scenario, and with α > 0 a strictly positive

constant, define the control problem with value function p : [0, T ]× R+ → R+ given by

p(t, y) := sup

ψ∈Ψ

E

[

h(YT )− 1

2α

∫ T

t

ψ2s ds

∣∣∣∣Yt = y] , (t, y) ∈ [0, T ]× R+,

where h : R+ → R+ is a bounded non-negative function, and Ψ denotes a set of admissible

controls such that the value function is finite.

(a) [4 marks] State the dynamic programming principle in terms of a martingale and super-

martingale property of an appropriate value process, and hence write down the Hamilton-

Jacobi-Bellman (HJB) equation for the value function, including a terminal condition.

(b) [6 marks] Derive an expression for the optimal feedback control function ψ̂ : [0, T ]×R+ →

R, state carefully how this function is related to the optimal control process (ψ̂t)t∈[0,T ],

and show that the resulting HJB equation is of the form

∂p

∂t

(t, y) +A0p(t, y) + 1

2

Ky2p2y(t, y) = 0,

subject to a terminal condition, where py(·, ·) denotes the partial derivative with respect

to y, A0 is a differential operator that you should give an expression for, and K is a

constant that you should identify.

(c) [6 marks] Interpret the operator A0 as the generator of the process Y in (1) under some

measure P0 ∼ P which you should identify, by giving an expression for the Radon-Nikodym

derivative of P with respect to P0, in the form

dP

dP0

= E(−M)T , (2)

for some P0 -martingale M which you should identify. Write down the P0-dynamics for

Y and show that with the measure change in (2) we recover the P-dynamics in (1).

(d) [9 marks] Look for a solution to the HJB equation of the form

p(t, y) =

1

δ

logF (t, y),

for some constant δ and function F : [0, T ] × R+ → R+, and find the value of δ such

that F (·, ·) satisfies a linear PDE. Hence, via the Feynman-Kac theorem, write down an

expectation representation for F (·, ·), and thus give the resulting solution for the value

function p(·, ·).

A17470W1 Page 5 of 8

2. On an infinite horizon stochastic basis (Ω,F ,F = (Ft)t>0,P), a stock price and its positive

stochastic volatility process Y satisfy, in a market with zero interest rate, the SDEs

dSt = YtSt(λ(Yt) dt+ dWt), dYt = a(Yt) dt+ η(ρ dWt + ρdW

⊥

t ),

for suitable functions λ(·), a(·), with ρ ∈ (−1, 1) a constant correlation and ρ :=

√

1− ρ2,

η > 0 a positive constant, and W,W⊥ are independent Brownian motions.

An agent with initial capital x > 0 trades the stock and cash and consumes at a non-negative

adapted consumption rate c, generating wealth process X. With δ > 0 a positive impatience

parameter, and U : R+ → R a utility function, the objective is to maximise utility from

consumption. The value function is defined by

u(x, y) := sup

(pi,c)∈A(x,y)

E

[∫ ∞

0

e−δtU(ct) dt

∣∣∣∣ (X0, Y0) = (x, y)] ,

with A(x, y) denoting admissible investment-consumption strategies, and pi is the process for

wealth in the stock.

(a) [2 marks] Formulate the wealth process SDE, state the dynamic programming principle

in terms of a supermartingale and martingale property of an appropriate value process,

and hence write down the HJB equation for the value function.

(b) [5 marks] Compute the optimal feedback control functions pi(·, ·), ĉ(·, ·), state carefully

how these are related to the optimal control processes (pit, ĉt)t>0, and show that the HJB

equation converts to

V (ux(x, y)) + Gu(x, y)− 1

2uxx

(λ(y)ux(x, y) + ρηuxy(x, y))

2 − δu(x, y) = 0,

where G is a differential operator you should interpret as a generator of a process, and for

which you should give an expression, and V : R+ → R is the convex conjugate of U(·),

satisfying V (y) = U(I(y)) − yI(y), y > 0, with I(·) ≡ (U ′(·))−1 the inverse of marginal

utility.

(c) [5 marks] Suppose U(c) = cp/p, p < 1, p 6= 0 is a power utility function. Assume a

separable solution to the HJB equation of the form u(x, y) = U(x)f(y) for some function

f(·), and show that the optimal control processes are then given by

ĉt = F (Yt)X̂t, pit =

G(Yt)

Yt(1− p)X̂t, t > 0,

where X̂ denotes the optimal wealth process and the functions F (·), G(·) are given by

F (y) := (f(y))−1/(1−p), G(y) := λ(y) + ρη

f ′(y)

f(y)

.

[You are not required to solve for f(·), only to show that the optimal controls will take the

forms stated.]

(d) [5 marks] Show that the optimal wealth process is given by

X̂t = x exp

(∫ t

0

(

G(Ys)

1− p λ(Ys)− F (Ys)

)

ds

)

E

(

G(Y )

1− p ·W

)

t

, t > 0.

(e) [6 marks] State the transversality condition that the value function is expected to satisfy

and, assuming that f(Yt) is bounded for all t > 0, deduce that the transversality condition

will hold provided that

δt >

∫ t

0

(

1

2

qG2(Ys)− qG(Ys)λ(Ys)− pF (Ys)

)

ds, t > 0,

where q := −p/(1− p) denotes the conjugate exponent to p.

A17470W1 Page 6 of 8

3 Part C: Quantitative Risk Management

1. (a) [8 marks] Let X,Y be two random variable representing losses from two portfolios.

(i) Give the definition of V@Rα(X), the Value-at-Risk at level α ∈ (0, 1). Prove or

disprove the validity of the following statements for arbitrary λ > 0 and c ∈ R:

• V@Rα(λX) = λV@Rα(X)

• V@Rα(X + c) = V@Rα(X) + c

• V@Rα(X + Y ) = V@Rα(X) + V@Rα(Y )

(ii) A bank holds a portfolio of N defaultable zero-coupon bonds. The bonds were issued

by different counterparties but have a common maturity T . For each counterparty,

using their CDS quotes or otherwise, the bank has a good estimate of their probability

of default. Discuss briefly the challenges in risk managing this portfolio. What are

the possible pitfalls of using V@Rα to risk-manage this portfolio? Would your risk

assessment depend continuously on α?

(b) [17 marks] Suppose that C1, . . . , Ck are d dimensional copulas and λ = (λ1, . . . , λk) has

non-negative entries with

∑k

i=1 λi = 1. Define

C(u) =

k∑

i=1

λiCi(u), u ∈ [0, 1]d.

(i) Using the characterisation of a copula function, or otherwise, show that C is a d

dimensional copula.

(ii) Suppose we can sample efficiently from each of Ci, i = 1, . . . , d. Write down an

algorithm to sample efficiently from C. Justify your answer.

(iii) Suppose d = 2. State the definition of Spearman’s rho. Let ρiS denote Spearman’s

rho for Ci and ρS denote Spearman’s rho for C. Show that

ρS =

k∑

i=1

λiρ

i

S .

You may use the fact that the variance of a uniform random variable on [0, 1] is 112 .

(iv) Suppose d = 2 and recall that Kendall’s tau for C can be computed as

ρτ (C) = 4E[C(U1, U2)]− 1,

where (U1, U2) is sampled from C. Assume now that k = 2, C1 = M is the co-

monotone copula and C2 = Π is the independence copula. Compute ρτ (M) and

ρτ (Π) and comment on the interpretation of these values. Let λ1 = λ ∈ (0, 1) so that

λ2 = 1− λ. Show that

ρτ (C) =

1

3

λ(2 + λ).

Is it true that

ρτ (C) = λρτ (C1) + (1− λ)ρτ (C2)?

A17470W1 Page 7 of 8

2. (a) [5 marks] Suppose dataX1, X2, . . . is i.i.d. from some distribution F . LetMn = max{X1, . . . , Xn}.

Recall that F is in the maximum domain of attraction of H, for some non-trivial distri-

bution function H, if (Mn − dn)/cn converges in distribution to H, for some sequences

(cn), (dn).

Why is it of interest to fit such an H when past losses are sampled from F? Can different

choices of (cn), (dn) lead to H with different tail behaviours?

Describe briefly the difference between the block maxima method and the peaks-over-

threshold method. Which one would you prefer and why?

(b) [8 marks] Suppose now X1, X2, . . . are i.i.d. exponential with mean 1/β, for some β > 0,

i.e., F (x) = 1− exp(−βx), x > 0. Recall that F is in the maximum domain of attraction

of the Gumbel distribution.

(i) Let Yn = max{X3n, X3n+1, X3n+2} and MYn = max{Y1, . . . , Yn}, n > 1. Taking

cn = 1/β and dn =

logn

β , show that (M

Y

n − dn)/cn converges in distribution to some

HY which you should identify.

(ii) Let Zn = max{Xn, Xn+1, Xn+2} and MZn = max{Z1, . . . , Zn}, n > 1. Using the same

(cn), (dn) as in (i), show that (M

Z

n − dn)/cn converges in distribution to some HZ

which you should identify.

(iii) Show that Yn and Zn have the same distribution. What is the relationship between

HY and HZ? Relate your answer to serial dependence in the data in each of the two

cases.

(c) [12 marks] The durations X1, X2, . . . of calls handled by a call centre are independent and

generalized Pareto distributed, i.e.,

P(X1 > x) =

(

1 +

ξx

β

)−1/ξ

, x > 0,

for some β > 0 and ξ ∈ (0, 1).

(i) The parameter ξ needs to be estimated. Comment on its relevance for the variance of

the call lengths. Compute the excess distribution Fu of call length over the threshold

u, for a given u > 0.

(ii) Compute V@Rα(X1), the Value-at-Risk at level α ∈ (0, 1) and ESα(X1), the expected

shortfall at level α. Deduce the value of e(V@Rα(X1)), the mean excess function for

the threshold u = V@Rα(X1).

(iii) Suppose that ξ < 0.5 and β = ξ. Show that V@Rα(X1) 6 ESα(X1) 6 2V@Rα(X1) +

1. What can you deduce about using value-at-risk or expected shortfall when man-

aging the risk of excessively long calls clogging up the call centre?

A17470W1 Page 8 of 8