Arbitrage-free interpolation of the swap curve 1. Introduction Yield curves are constructed in ...

38 Pages·2009·0.36 MB·English

InternationalJournalofTheoreticalandAppliedFinance
(cid:13)c WorldScientificPublishingCompany
Arbitrage-free interpolation of the swap curve
MarkH.A.Davis
Department of Mathematics, Imperial College
London SW7 2BZ, England
[email protected]
VicenteMataix-Pastor
Department of Mathematics, Imperial College
London SW7 2BZ, England
[email protected]
Received(DayMonthYear)
Revised(DayMonthYear)
We suggest an arbitrage free interpolation method for pricing zero-coupon bonds of
arbitrarymaturitiesfromamodelofthemarketdatathattypicallyunderliestheswap
curve;thatisshortterm,futureandswaprates.Thisisdonefirstwithinthecontextof
theLiborortheswapmarketmodel.Wedosobyintroducinganindependentstochastic
processwhichplaystheroleofashorttermyield,inwhichcaseweobtainanapproximate
closed-form solution to the term structure while preserving a stochastic implied short
rate.Thiswillbediscontinuousbutitcanbeturnedintoacontinuousprocess(however
at the expense of closed-form solutions to bond prices). We then relax the assumption
of a complete set of initial swap rates and look at the more realistic case where the
initial data consists of fewer swap rates than tenor dates and show that a particular
interpolation of the missing swaps in the tenor structure will determine the volatility
of the resulting interpolated swaps. We give conditions under which the problem can
besolvedinclosed-formthereforeprovidingaconsistentarbitrage-freemethodforyield
curvegeneration.
Keywords:TermStructuremodelling;Liborandswapmarketmodels;HJM.
1. Introduction
Yield curves are constructed in practice from market quoted rates of simple com-
pounding with accrual periods of no less than a day. In particular, swap curves
are constructed by combinations of bootstrap and interpolation methods from the
following market data
• Short-term interest rates,
• Interest rate futures,
• Swap rates.
1
2 M. Davis & V. Mataix-Pastor
For example the GBP curve may stretch out to 52 years, and the interest rate
futures are short sterling futures, but there are only a few data points beyond 10
years (for example we may have swap rates with six month payments for 12, 15,
20, 40 and 52 years) hence the need for bootstrapping methods. Constructing the
yield curve however is a black art, covered briefly in Section 4.4 of Hull [14] but
notgenerallydescribedindetailintextbooks.Methodsincludelinearinterpolations
and cubic splines; see for example the survey by Hagan and West [12].
Ontheotherhandmarketpractitionersinterestedinpricinginterestratederiva-
tiveswillneedtospecifyanarbitrage-freemodelfortheevolutionoftheyieldcurve.
SoasBj¨orkandChristenssenrightlypointoutin[3]aquestionimmediatelyarises:
if you choose to implement a particular yield curve generator, which is constantly
being applied to newly arriving market data in order to recalibrate the parameters
of the model, will the yield curve generator be consistent with the arbitrage-free
modelspecified?Thatis,iftheoutputoftheyieldcurvegeneratorisusedasanin-
put to the arbitrage-free model, will the model then produce yield curves matching
the ones produced by the generator? We clarify this point and give an example in
the next section.
Anumberofauthors,startingfromtheworkofBj¨orkandChristenssen(see[1],
Filipovi´c [9] and Filipovi´c and Teichmann [11]), have studied this problem within
the HJM framework in an infinite-dimensional space by looking at specific classes
of functions and asking whether these functions are invariant under the HJM dy-
namics.Theyobtainsomenegativeresultsbutlaterextendedtheclassoffunctions
by using an infinite-dimensional version of the Frobenius theorem. They give con-
ditions under which these can be reduced to a finite-dimensional state vector but
don’t relate this vector to market observables. Other authors (see [7] for the most
general set up) have studied conditions on the volatility under which the HJM
modeladmitsareductiontoafinite-dimensionalMarkovianprocess.Butagainthis
Markov process is not identified to market observables.
In contrast to their work the approach of this paper is to model a finite number
ofmarketobservablesandthenextendthemodeltoawholeyieldcurvemodelinan
arbitragefreeway.Thatiswewanttofindthepriceattimet<T ofazero-coupon
bond p(t,T) with arbitrary maturity T not equal to any of the tenor dates and
starting from the dynamics of a Libor or a swap market model. To do that we need
at least the continuous time dynamics of some num´eraire asset N which would be
t
a function of market data and define p(t,T) by the following expectation formula
under the N-martingale measure
p(t,T)=N(t)EN[1/N(T)|F ]. (1.1)
t
The first attempts at modelling market observables directly was the work of
SandmannandSondermann(1989,1994)[22]andthenextendedbyMiltersen,Sand-
mannandSondermann[23][19]whofocusedtheirattentiononnominalannualrates.
Models of Libor rates were carried out by Brace, Ga¸tarek and Musiela [6], Musiela
and Rutkowski [21] and Jamshidian [17] who explicitly points out his desire to
Arbitrage-free interpolation of the swap curve 3
departfromthespotrateworld.SomemodelswereembeddedintheHJMmethod-
ology as in [19],[23], [6] and others were simply modelling a finite set of Libor rates
but then pricing products that were dependent on these given rates without any
need for interpolation, e.g. [21], [17]. Only Schl¨ogl in [24] looks at arbitrage-free
interpolations for Libor market models as the underlying model for yield curve
dynamics.
In this paper we extend his results to the case where the underlying market
consists of some short term yields and a swap market model, that is a process for
yieldsofbondsofshortmaturities,threeorsixmonths,andacollectionofobserved
spot swap rates and their volatilities.
Thecontentsofthispaperareasfollows.InSection2wegiveanexampleofhow
simpleinterpolationalgorithmscreatearbitrageopportunities.Wefindanarbitrage
free mapping from yields to the short rate and show how one could compute in
theory the trading strategy that produces arbitrage. In Section 3 we outline the
co-terminal and co-initial swap market models and introduce a novel interpolation
ofmarketratesthatallowsasimultaneoustreatmentoftheLiborandswapmarket
modelwhilepreservingthestochasticnatureoftheimpliedshortrateandproviding
anapproximateclosedformsolutionforthetermstructure.Section4constructsthe
termstructurefromamorerealisticmarketmodelwheretherearefewerswaprates
thantenordates.Itintroducesaconsistentbootstrappingprocedurethatyieldsthe
implied volatilities of the “bootstrapped” rates in closed form. This method fits in
with the interpolations carried out in chapter four and so we can then obtain the
HJM dynamics for the forward rates but driven purely by an SDE process on the
market data.
2. Yield curve generators and arbitrage opportunities
Wegiveanexampletoillustratetheconsistencyproblemtoshowhowalinearinter-
polation can introduce arbitrage opportunities. Consider the following bootstrap-
ping procedure on an arbitrage-free model for two zero-coupon bonds maturing at
times T <T . That is let y (t),y (t) be the yields, so that
1 2 1 2
p(t,T )=e−y1(t)(T1−t), p(t,T )=e−y2(t)(T2−t).
1 2
Take T ∈(T ,T ) and define the interpolated price as
1 2
p(t,T)=e−y(t)(T−t),
where
T −T T −T
y(t)= 2 y (t)+ 1 y (t)=(1−α(T))y (t)+α(T)y (t).
T −T 1 T −T 2 1 2
2 1 2 1
TakingtheT -bondasnum´eraire,absenceofarbitragedemandsthatp(t,T)/p(t,T )
2 2
be a martingale in the T -forward measure. Write
2
p(t,T)
=φ(t,y (t),y (t))=exp((β +t)y (t)+(β +β t)y (t)),
p(t,T ) 1 2 1 1 2 3 2
2
4 M. Davis & V. Mataix-Pastor
where β =−(1−α)T,β =(1−α),β =T +T ,β =−(1+α).
1 2 3 2 4
Ify(t)=(y (t),y (t))isacontinuoussemimartingalewithdecompositiony (t)=
1 2 i
M (t)+A (t), then by the Itˆo formula
i i
dφ/φ=(β y +β y )dt+(β +β t)dA +(β +β t)dA +(β +β )2d<y >
2 1 4 2 1 2 1 3 4 2 1 2 1
+(β +β t)2d<y >+2(β +β t)(β +β t)d<y ,y >+dM(t)
3 4 2 1 2 3 4 1 2
≡dA(t)+dM(t),
where M(t) is a local martingale. For absence of arbitrage, A(t) must vanish. How-
ever,thecoefficientsβ dependonT,anditisnotgenericallythecasethatA(t)≡0
i
forall T,givenafixedmodelfory(t).Thustherewillbearbitrageopportunitiesin
the model if we are prepared to trade zero-coupon bonds at interpolated prices. In
other words, a linear interpolation method to construct a yield curve is not consis-
tent with a model of the market yields. Presumably market friction in the form of
bid-ask spreads is too great to allow these opportunities to be realized in practice.
We want to explore the above example a bit further. Since the short end of the
yield curve is constructed from yields of zero-coupon bonds we model the yields
directly. Assume the following model, the strong solution to an SDE under the
P -forward measure, for the yield y(t) of a zero-coupon bond maturing at time
1
T >0,
1
dy(t)=µ(y(t))dt+σ(y(t))dw (t),
1
so that
p(t,T )=exp(−y(t)(T −t)).
1 1
We have the following arbitrage free mapping from the yield y(t) to the short rate
r(t) defined as
∂lnp(t,T)
r(t)=− lim .
T(cid:38)t ∂T
Proposition 2.1. The implied short rate r(t) for t∈[0,T ) is given by
1
r(t)=y(t)−(T −t)µ−1/2(T −t)2σ2, (2.1)
1 1
where µ is the drift of y(t) under the P -measure. For 0 < T < T the implied
1 1
risk-neutral measure P∗ is given by
ddPP∗(cid:12)(cid:12)(cid:12)(cid:12) =exp(cid:32)(cid:90) T τ1σdws1−(cid:90) T 1/2|τ1σ|2ds(cid:33). (2.2)
1 FT 0 0
Proof. The short rate is a function of y(t), r(t) = r(t,y(t)). By
num´eraire invariance we require
p(t,T)=B1(t)E1[1/B1(T)|Ft]=e−y(t)(T1−t)E1(cid:104)ey(T)(T1−T)(cid:12)(cid:12)(cid:12)Ft(cid:105)=E∗(cid:104)e−(cid:82)tTr(s)ds(cid:12)(cid:12)(cid:12)Ft(cid:105),
(2.3)
where P∗ denotes the risk-neutral measure.
Arbitrage-free interpolation of the swap curve 5
Nowletτ =T −tanddenotep(t,T)=F(t,y(t)).ByFeynman-Kacformulathe
1 1
right hand side of (2.3) is the probabilistic representation of the PDE determining
F(t,y(t)) with terminal condition F(T,y) = 1. To obtain such PDE we apply the
product rule to ey(t)(T1−t)F(t,y(t)) and we cancel the drift term
d(ey(t)(T1−t)F)=ey(t)τ1dF +Fd(ey(t)τ1)+d<F,ey(t)τ1 >
1 (cid:18)∂F ∂F ∂2F (cid:19) 1 ∂F
= +(µ+τ σ2) +1/2 σ2 dt+ σdw (t)
B (t) ∂t 1 ∂y ∂y2 B (t) ∂y 1
1 1
+F 1 (cid:2)−y(t)+τ µ+1/2τ2σ2(cid:3)dt+F 1 τ σdw (t).
B (t) 1 1 B (t) 1 1
1 1
That is
d(ey(t)(T1−t)p(t,T))=(A˜ F −(y(t)−τ µ−1/2τ2σ2)F)dt+(...)dw , (2.4)
t 1 1 1
where A˜ = ∂ +(µ+τ σ2)∂ F +1/2σ2∂2 F. This identifies simultaneously the
t t 1 y yy
short rate as r(t) = y(t)−τ µ−1/2τ2σ2 and the risk-neutral measure given by
1 1
(2.2). The “market prices” of risk are −τ σ, that is the volatility of the B (t) over
1 1
the period [0,T ).
1
In fact Eq. (2.3) defines an arbitrage free value for p(t,T) for t ≤ T ≤ T . For
1
example assuming a Gaussian process for y(t) we obtain the following
Proposition 2.2. Assume we are given positive constants a,b,σ and a Brownian
motion w(t) under the P forward-measure. Define y(t) as the strong solution to
1
dy(t)=(a−by(t))dt+σdw .
t
Then the arbitrage-free price of a zero-coupon bond maturing at T ≤T is given by
1
p(t,T)=exp(n(t,T)−m(t,T)y(t)), (2.5)
with
m(t,T)=(T −t)−(T −T)e−b(T−t),
1 1
a(cid:16) (cid:17) (T −T)2σ2
n(t,T)=(T −T) 1−e−b(T−t) + 1 (1−e−2b(T−t)).
1 b 4b
Proof. By standard results the distribution of y(T)(T −T) given y(t) with t<T
1
is N((T −T)k,(T −T)2s2) with
1 1
a
k =y(t)exp(−b(T −t))+ (1−exp(−b(T −t))),
b
σ2
s2 = (1−exp(−2b(T −t))).
2b
6 M. Davis & V. Mataix-Pastor
With these assumptions Eq. (2.3) gives us
(cid:104) (cid:12) (cid:105)
p(t,T)=e−y(t)(T1−t)E1 ey(T)(T1−T)(cid:12)F =e−y(t)(T1−t)e(T1−T)k+(T1−T)2s2/2,
(cid:12) t
giving Eq.(2.5).
In the next section we show that given a model specified under the forward
measurethereexistsauniqueshortrateindependentofmaturitywhichcorresponds
to a finite variation process representing a savings account. We use results from
Bj¨ork [2].
2.1. Self-financing trading strategies and zero-coupon bond pricing
Equation (2.3) defines an arbitrage free term structure for T < T which depends
1
on the drift and volatility of the yield y(t). We don’t need to assume any function
for the short rate. In what follows we repeat the steps in Bj¨ork’s description of
short rate models (Section 21.2 in [2]) to find an alternative derivation of the term
structure PDE in terms of self-financing trading strategies using the p(t,T) and
B (t) as traded assets which will also allow us to obtain the hedging parameters.
1
Letting p(t,T) = F(t,y(t)) and applying Itˆo to F and B (t) = exp(−y(t)(T −t))
1 1
we have
dp(t,T)=(F +µF +1/2σ2F )dt+σF dw , (2.6)
t y yy y 1
dB (t)/B (t)=(y(t)−τ µ+1/2τ2σ2)dt−στ dw . (2.7)
1 1 1 1 1 1
Denote the drift and diffusion of p by m = F +µF +1/2σ2F and b = σF ,
T t y yy T y
and similarly denote the drift and volatility of B (t) by m and b respectively.
1 1 1
Assume we are interested in pricing a zero-coupon bond with maturity T <T .
1
We form a portfolio based on T and T bonds. Let u (t) and u (t) denote the
1 T 1
proportions of total value held in bonds p(t,T) and B (t) respectively, held in a
1
self-financing portfolio at time t. The dynamics of the portfolio are given by
(cid:18) (cid:19)
dp(t,T) dB (t)
dV =V u (t) +u (t) 1 . (2.8)
T p(t,T) 1 B (t)
1
Substitute (2.6) and (2.7) into (2.8) to obtain
dV =V(u m +u m )dt+V(u b +u b )dw . (2.9)
T T 1 1 T T 1 1 1
Let the portfolio weights solve the system
u +u =1, (2.10)
T 1
u b +u b =0.
T T 1 1
Arbitrage-free interpolation of the swap curve 7
Thefirstequationistheself-financingpropertyandthesecondmakesthedw -term
1
in(2.8)vanish.ThevalueoftheSFTS(u ,u )isthesolutionto(2.10)andisgiven
T 1
by
b b
u =− 1 , u = T .
T b −b 1 b −b
T 1 T 1
Substitute the solution into (2.9) to obtain
(cid:18) (cid:19)
m b −m b
dV =V 1 T T 1 dt.
b −b
T 1
Absence of arbitrage requires to set the drift above equal to the spot rate r. That
is
m b −m b
1 T T 1 =r,
b −b
T 1
which can be rewritten as
m −r m −r
T = 1 . (2.11)
b b
T 1
The ratio above is independent of the bond and is usually known as the “market
prices of risk”. Since the risk comes from the randomness in B (t) we set the above
1
ratio equal to its volatility b = −στ . Substituting this on the right hand side of
1 1
(2.11) we obtain
r(t)=y(t)−(T −t)µ−1/2(T −t)2σ2,
i+1 i+1
so (2.11) becomes
m −r
T =b . (2.12)
b 1
T
Equation(2.12)isinfactanotherwayofwritingthePDEappearingintheproofof
Proposition2.1whichintheusualshortratemodelscorrespondstothewellknown
Vasicek PDE.
From this discussion we will show next that if p(t,T) is given by linear interpo-
lation of the yields, the implied short rate depends on the maturity date T and so
we can create an arbitrage opportunity.
2.2. Arbitrage opportunities in a linear interpolation
We now illustrate how to compute an arbitrage opportunity in a bond market
wherebondsareobtainedbylog-linearinterpolationfromasetofbenchmarkrates.
Consider again the small market introduced in the beginning of the section. Let
t < T and T ∈ [T ,T ] where T < T and let τ = T −t and τ = T −t. We are
1 1 2 1 2 2 2
given a system under the T bond forward measure P
2 2
dy =µ dt+σ dw ,
1 1,2 1 2
dy =µ dt+σ dw ,
2 2 2 2
8 M. Davis & V. Mataix-Pastor
where
µ =(y −y +τ µ +1/2σ)/τ ,
1,2 1 2 2 2 1
σ =τ σ +τ σ −2τ τ σ σ ,
1 1 2 2 1 2 1 2
so that p(t,T )/p(t,T ) is a P -martingale. Define the interpolated yield for a bond
1 2 2
maturing at time T by
y(t,T)=α(T)y (t)+(1−α(T))y (t),
1 2
where
T −T
α(T)= 2 .
T −T
2 1
From Itˆo it follows that
dy(t,T)=µ dt+σ dw (t),
y y 2
where the drift µ and volatility σ are given by
y y
µ =α(T)µ +(1−α(T))µ , σ =α(T)σ +(1−α(T))σ .
y 1,2 2 y 1 2
The new bond price is
p(t,T)=exp(−y(t,T)(T −t)).
Equations (2.6) and (2.7) now read as
dp(t,T)=(y(t,T)−µ τ +1/2(σ τ)2)dt−σ τdw (t),
y y y 2
dp(t,T )=(y (t)−µ τ +1/2(σ τ )2)dt−σ τ dw (t),
2 2 2 2 2 2 2 2 2
and we can construct a finite variation process like (2.9). Choose proportions of
wealth V in bonds B (t) and p(t,T) be
t 1
σ τ −σ τ
u =− 2 2 , u = y . (2.13)
T σ τ −σ τ 1 σ τ −σ τ
y 2 2 y 2 2
Hence the system in (2.9) is
dV −m σ τ +m σ τ
t = 2 y T 2 2dt.
V σ τ −σ τ
t 2 2 y
Letting the spot rate r(t) be
−m σ τ +m σ τ
r(t)= 2 y T 2 2
σ τ −σ τ
2 2 y
(cid:18) (cid:19)
α(T)τ σ τ (y −y ) σ τ σ τ
=y + τ (σ µ −σ µ )+σ τ (y −y )+ 2 2 1 2 − 2 2 y ,
2 σ τ −σ τ 2 1 2 2 1,2 2 2 1 2 τ τ
2 2 y
thisdependsonthematurityT.Denotebyφ(t,T)andφ(t,T )thenumberofunits
1
of bonds with maturity T and T respectively. They are related from the bond
1
Arbitrage-free interpolation of the swap curve 9
portfolios by u(t,T)=φ(t,T)V /P(t,T) and u(t,T )=φ(t,T )V /P(t,T ). Hence
T 1 1 T 1
from (2.13) these trading strategies are
(cid:82)t
T −t V(0)exp( r(s)ds)
φ(t,T)= 0 ,
T −T p(t,T)
1
(cid:82)t
t−T V(0)exp( r(s)ds)
φ(t,T )= 0 .
1 T −T p(t,T )
1 1
SincetheimpliedspotrateisasmoothfunctionofT wecanfindmaturitiesT(cid:48) and
T(cid:48)(cid:48) such that r(t,T(cid:48)) < r(t,T(cid:48)(cid:48)) and so we can borrow money at the cheaper rate
and invest in the higher rate “savings account” for as long as t < min(T(cid:48),T(cid:48)(cid:48)) to
obtain a profit.
3. Interpolation of swap market models
The data used to construct the zero curve usually from two years onwards consists
of swap rates. We now give a brief outline of the swap market model and introduce
some notation following Jamshidian [17].
3.1. Co-terminal swap market model
We are given a tenor structure 0 < T < ... < T with accrual factors θ ∈ Rn,
1 n +
θ =T −T ,sothat θ isthetimeintervalT −T expressedaccordingtosome
i i+1 i i i+1 i
day-count convention. For t ≤ T denote by B (t) the price of a zero coupon bond
i i
at time t with maturity date T and S (t) the forward swap rate starting at date T
i i i
and with reset dates T for j = i,...,n−1. Forward swap rates are related to zero
j
coupon bonds by
B (t)−B (t)
S (t)= i n , 0≤t≤T , (3.1)
i A (t) i
i
where A (t)≡(cid:80)n θ B (t), the “annuity”. We want to obtain prices of zero-
i j=i+1 j−1 j
coupon bonds from these rates. Towards that aim one can show algebraically from
(3.1) and using induction that if we let
n−1 k
(cid:88) (cid:89)
v ≡v ≡ θ (1+θ S ), (3.2)
ij ij,n k l−1 l
k=j l=i+1
v ≡v , 1≤i≤j ≤n−1, (3.3)
i ii
we can then express the ratios B /B (see [17]) for 0≤t≤T , i=1,...,n−1 as
i n i
A (t)
i =v (t), (3.4)
B (t) i
n
B (t)
i =(1+v (t)S (t)). (3.5)
B (t) i i
n
10 M. Davis & V. Mataix-Pastor
We want to be able to interpret B (t) as the price of a zero-coupon bond maturing
i
at time T , so by setting B (T )=1 we can deduce from (3.5), for i=1,...,n−1
i i i
B (T )=1/(1+v (T )S (T )). (3.6)
n i i i i i
We define the auxiliary process Y (t)
i
B (t)
Y (t):= i =(1+v (t)S (t)). (3.7)
i B (t) i i
n
These Y process will be martingales under the P forward measure. From (3.5) we
i n
have that Y /Y =1+θ L where L (t) denotes the forward Libor rate set at T
i i+1 i i i i
and paying at T and we obtain the relation between Libor and swap rates
i+1
1+v (t)S (t)
1+θ L (t)= i i . (3.8)
i i 1+v (t)S (t)
i+1 i+1
Notice that a model on the forward swap rates can generate a negative Libor rate
L (t) for some k =1,...,n−1 if it happens that
k
v (t)S (t)
S (t)< k+1 k+1 . (3.9)
k v (t)
k
See for example [8] and [18] for a more detailed discussion.
The swap market model is described by the following: a set of forward swap
rates S (0) for i = 1,...,n−1, an n−1 dimensional vector of bounded, measur-
i
able, locally Lipschitz functions ψ (t,S(t)) ∈ Rd. The swap rate follows a positive
i
martingale under the corresponding “annuity” measure, that is the measure cor-
responding to using the “annuity” A (t) as the num´eraire. More precisely, given a
i
filtered probability space (Ω,F ,P ) supporting a d-dimensional Brownian motion
t in
w , the swap rate S (t) is given by the strong solution to
in i
dS (t)=S (t)ψ (t,S(t))dw (t).
i i i in
In particular P = P , the terminal measure. The Radon-Nikodym derivative
n−1,n n
for the change of measure to the P -forward measure is given by
n
dP A (0) B (T ) v (0)
n = i n i = i . (3.10)
dP B (0) A (T ) v (T )
in n i i i i
One can use backward induction to deduce the form of the drift term for all swap
rates under the P measure. We recall from Jamshidian [17].
n
Proposition 3.1. (Jamshidian) If we are given a filtered probability space
(Ω,F ,P ),supportingaBrownianmotionw ∈Rd,anddataconsistingofann−1
t n n
dimensional vector of bounded, measurable, locally Lipschitz functions ψ (t,S)∈Rd
i
and initial forward swap rates S (0) for i=0,...,n−1, then the SDE
i
dS =−S ψ n(cid:88)−1 θj−1Sjψjt vijdt+S ψ dw , (3.11)
i i i (1+θ S ) v i i n
j−1 j i
j=i+1