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