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The Term Structure of InterestRates as a Random Field.
Applications to Credit Risk
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Diploma thesis submitted to the
ETH ZURICH and UNIVERSITY OF ZURICH
for the degree of
MASTER OF ADVANCED STUDIES IN FINANCE
presented by
HANSJORG FURRERDr. sc. math. ETH
born 30 August 1968citizen of Schlatt ZH
supervisor
Prof. Dr. P. Schonbucher
2003
The graph on the cover depicts a simulated Gaussian random field with correlationfunction
(0.0) c(x, y) =
1
2γ−1Γ(γ)
(‖x−y‖
θ
)γ
Jγ(‖x− y‖/θ) , ‖x− y‖ 6= 0,
1 , ‖x− y‖ = 0,
where we set γ = 5 and θ = 2. The function Jγ denotes a modified Bessel functionof the third kind of order γ.
||x−y||
c(||x
−y|
|)
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Figure 0: Graph of the correlation function c in (0.0) forγ ∈ 5, 2, 1, 0.5, 0.2, from top to bottom, all with θ = 2.
Source: www.math.umd.edu/∼bnk/bak/generate.cgi
THE TERM STRUCTURE OF INTEREST RATES AS ARANDOM FIELD. APPLICATIONS TO CREDIT RISK
HANSJORG FURRER
Abstract. The principal aim of this paper is the modeling of the termstructure of interest rates as a positive-valued random field. Specialemphasis is given to chi-squared fields which can be generated from afinite number of Gaussian fields. In particular, we introduce a short ratemodel which is based on the square of an Ornstein-Uhlenbeck process.It is shown that bond prices can be expressed as an exponential-affinefunction of the short rate. We extend our approach to the credit risk areaand model the default intensities of a class of obligors as a two-parameterpositive-valued random field. Finally, random fields will be applied tothe modeling of firm values in a structural credit risk framework.
1. Introduction
Spatial patterns occur in a wide variety of scientific disciplines such as hy-drology, meteorology, climatology, neurology, geo-statistics, soil science, andmany more. To cover models of spatial variation, the theory of stochasticprocesses turns out to be useful provided the index set of the processes isdefined in a general manner. If Z(x), x ∈ T , labels topographical heightfor example, then T is commonly assumed to be a subset of R2. In theevent the dynamic aspect is of importance too, the index set may include anadditional component capturing the development in time. This leads to so-called spatio-temporal models. The extension of single-parameter stochasticprocesses to multi-parameter processes is commonly designated as randomfield. To specify a random field it suffices to give the joint distribution of anyfinite subset of (Z(x1), . . . , Z(xn)) in a consistent way. Key characteristicsof a random field are its mean and covariance function. In general, however,the distribution of a random field is not fully specified by the mean andcovariance function alone. This is only the case for an important subclassof random fields known as Gaussian fields. In most modeling strategies, aspecific parametric form for the covariance function R is assumed. Care hasto be taken in order to derive a non-negative definite function. Correlationmodels that are not non-negative definite can lead to negative variancesinconsistent with theory and intuition. For spatio-temporal processes, themain difficulty lies in specifying an appropriate space-time covariance struc-ture. One simple way would be to multiply a time covariance function witha space covariance function. Such separable covariance models, however, ne-glect any space-time interactions, and are thus mainly used for conveniencerather than for their ability to fit the data well.Random fields in the realm of term structure models were introduced byKennedy [16], [17]. He proposed a model where the instantaneous forward
1
2 HANSJORG FURRER
rates f(t, T ) are modeled as a two-parameter Gaussian random field. Thefirst parameter of the forward rate surface corresponds to the current time,the second to the maturity date. In more technical terms, Kennedy setsf(t, T ) = µ(t, T )+Z(t, T ), and derives a necessary and sufficient restrictionon the drift function µ(t, T ) to ensure that discounted zero-bond prices aremartingales under the risk-neutral measure. A consequence of this model-ing approach is that bond prices of different maturities are no longer per-fectly correlated. Indeed, the correlation structure can be chosen in anarbitrary way. Goldstein [13] concentrates on the dynamics of the forwardrates df(t, T ) = µ(t, T ) dt+σ(t, T ) dZ(t, T ), and generalizes Kennedy’s driftrestriction result to non-Gaussian fields. Collin-Dufresne and Goldstein [3]specify the bond price dynamics as a Gaussian random field, and consider ageneralized affine framework where the log-bond prices themselves are takenas state variables. Thus, in contrast to the traditional affine framework, thestate vector is of infinite dimension.
Our principal aim in this paper is the modeling of the term structure ofinterest rates as a positive-valued random field. Starting from an arbitraryGaussian field, we obtain a positive-valued field by means of a positivetransformation. Of particular importance is the Ornstein-Uhlenbeck pro-cess (OU process) with parameter β and size c > 0, characterized by thecovariance function R(s, t) = c exp−β|s − t|. Observe that the two pro-cesses one-dimensional Brownian motion and OU process are both zero-meanGaussian processes. However, only the latter is stationary. We considershort rate models which include either the square of Brownian motion orthe square of an OU process, and show that bond prices are exponential-affine functions of the short rate. Our single-parameter models will thenbe carried forward to the credit risk area, thereby extending the parameterset to include an additional spatial component. In doing so, we arrive ata model for the default intensities of a family of obligors. Alternatively, atwo-parameter spatio-temporal framework can also be used to model theobligors’ market value of assets.
The rest of this paper is organized as follows. Section 2 is of expositorynature and is devoted to a brief overview of the theory of random fields.The material in this section is mainly taken from the books of Adler [1] andMatern [19]. As a prelude to the more specific random fields, we presentin Section 2.1 a short summary of the theory of isotropic fields. These areprocesses that are, from a distributional viewpoint, invariant under rotationsabout the origin. Consequently, the correlation of an isotropic field solelydepends on the radial distance from the origin. In Section 2.2 we introducethe important class of Gaussian random fields. Because of the simplicity inworking with the multivariate normal distribution, Gaussian random fieldshave gained a lot of attraction. An important result in the realm of Gaussianfields is the Karhunen-Loeve expansion. In essence, the Karhunen-Loeveexpansion provides a representation of a Gaussian field in terms of an infiniteseries. We illustrate its use in the Brownian case by deriving the so-calledCameron-Martin formula. There is no doubt that Gaussian random fields arequite useful in modeling various phenomena. In some applications, however,the use of a Gaussian field is inappropriate because Gaussian fields can attain
THE TERM STRUCTURE AS A RANDOM FIELD 3
negative values with positive probability. The so-called chi-squared randomfields which we introduce in Section 2.3 are positive-valued fields which aregenerated from a finite number of stationary Gaussian fields.
In Section 3 we start dealing with specific short rate models. We first presenta framework where the spot rate is based on the square of standard Brow-nian motion, see Section 3.1. The square of a Brownian motion, however,does not qualify as a chi-squared field since Brownian motion is not a sta-tionary process. Notwithstanding this, the proposed model has its meritssince bond prices are expressible as exponential-affine functions of the shortrate. Spurred by the arguable lack of stationarity, we extend our search forappropriate short rate models, and present in Section 3.2 a framework wherethe random field component is the square of an OU process. Indeed, thesquare of an OU process is a chi-squared field on the real line, and yet theaffine structure remains intact. As an application, we calculate the price ofa European call option written on a zero-bond. Section 3.3 is an excursioninto the area of forward rate random field models. As such, the materialpresented here is not new, and we follow Goldstein’s [13] approach to mod-eling the forward rate dynamics as df(t, T ) = µ(t, T ) dt + σ(t, T ) dZ(t, T ),where Z(t, T ) denotes a random field with deterministic correlation struc-ture c(t;T1, T2). We restate the result and proof in which the form of thedrift function µ(t, T ) is specified in order to preclude arbitrage under therisk neutral measure. Section 3.3 must be seen in preparation for Section 4.4where we derive the drift restriction in a defaultable framework under therecovery of market value assumption.
Section 4 starts with a brief overview of the ideas and the terminologyinherent to the pricing of defaultable securities. Defaultable pricing modelsare commonly classified either as reduced form models or structural models.The former category assumes that default occurs at a random time τ whichis governed by a risk-neutral intensity process λ = λ(t) : t ≥ 0. Structuralmodels, on the other hand, are based on modeling the stochastic evolutionof the assets of the issuer with default occurring at the first time the assetsfall below a certain level. In Section 4.1 we consider reduced-form debtpricing models more thoroughly. This class of models is quite flexible in thesense that it allows for non-zero recovery. Duffie and Singleton [11] considerreduced form models under the so-called recovery of market value (RMV)assumption. This means that, if default occurs at time t, a fraction, say Lt,of the market value is lost. This leads to a pricing rule based on a default-adjusted short rate process r = r+s, where s = λL denotes the short spread.Modeling both processes r and s as in Section 3.2, thereby allowing for non-zero correlation between r and s, we show that defaultable zero-coupon bondprices are of exponential-affine form. For simplicity, we will assume L ≡ 1,implying that s = λ. In Section 4.2 we consider a family of obligors andmodel the short spread as a two-parameter random field, one parameterreferring to the current time and the other labeling the obligor. This setupcorresponds to a spatio-temporal model, and hence we are confronted withthe determination of appropriate covariance structures. Section 4.3 aimsto model the firm value of a family of obligors as a two-parameter randomfield. Special emphasis is given to the fraction of obligors who will default
4 HANSJORG FURRER
at a fixed maturity date T . Section 4.4 returns to the theme of modelingthe instantaneous forward rates as a random field. This time, however, wedeal with defaultable securities, and derive the form of the drift functionand the form of the credit spread necessary to preclude arbitrage. Section 5concludes the article.
2. An overview of random fields
Let (Ω,F ,P) be a probability space on which all random objects will be de-fined. A filtration Ft : t ≥ 0 of σ-algebras, satisfying the usual conditions,is fixed and defines the information available at each time t.
Definition 2.1. (Random field). A (real-valued ) random field is a familyof random variables Z(x) indexed by x ∈ Rd together with a collection ofdistribution functions of the form Fx1,...,xn which satisfy
Fx1,...,xn(b1, . . . , bn) = P[Z(x1) ≤ b1, . . . , Z(xn) ≤ bn] ,
b1, . . . , bn ∈ R.
The mean function of Z is m(x) = E[Z(x)] whereas the covariance functionand the correlation function are respectively defined as
R(x,y) = E[Z(x)Z(y)]−m(x)m(y)
c(x,y) =R(x,y)√
R(x,x)R(y,y).
Notice that the covariance function of a random field Z is a non-negativedefinite function on Rd ×Rd, that is if x1, . . . ,xk is any collection of pointsin Rd, and ξ1, . . . , ξk are arbitrary real constants, then
k∑`=1
k∑j=1
ξ`ξj R(x`,xj) =k∑`=1
k∑j=1
ξ`ξjE[Z(x`)Z(xj)]
= E[( k∑
j=1
ξjZ(xj))2]
≥ 0 .
Without loss of generality, we assumed m = 0. The property of non-negativedefiniteness characterizes covariance functions. Hence, given any functionm : Rd → R and a non-negative definite function R : Rd × Rd → R, it isalways possible to construct a random field for which m and R are the meanand covariance function, respectively. Non-negative definite functions canbe characterized in the following way, see Bochner [2].
Theorem 2.1. (Bochner’s Theorem ) A continuous function R from Rd
to the complex plane is non-negative definite if and only if it is the Fourier-Stieltjes transform of a measure F on Rd, that is the representation
R(x) =∫
Rd
eix·λλλ dF (λλλ)
holds for x ∈ Rd. Here x ·λλλ denotes the scalar product∑d
k=1 xkλk and F isa bounded, real valued function satisfying
∫A dF (λλλ) ≥ 0 for all measurable
A ⊂ Rd. 2
THE TERM STRUCTURE AS A RANDOM FIELD 5
Occasionally we shall consider two random fields Z1, Z2 simultaneously. Thecross covariance function is defined as
R12(x,y) = E[Z1(x)Z2(y)]−m1(x)m2(y) ,
where m1 and m2 are the respective mean functions. Obviously, R12(x,y) =R21(y,x). A family of processes Zι with ι belonging to some index set Ican be considered as a process in the product space (Rd,I ). From thisrepresentation, consistency conditions for cross covariance functions can bededuced, see Cramer [4].A central concept in the study of random fields is that of homogeneity or sta-tionarity. We call a random field homogeneous or (second-order) stationaryif E[Z(x)2] is finite for all x and
• m(x) ≡ m is independent of x ∈ Rd
• R(x,y) solely depends on the difference x− y.
Thus we may consider
R(h) = Cov(Z(x), Z(x + h)) = E[Z(x)Z(x + h)]−m2 , h ∈ Rd ,
and denote R the covariance function of Z. In this case, the followingcorrespondence exists between the covariance and correlation function, re-spectively:
c(h) =R(h)R(0)
,
i.e. c(h) ∝ R(h). For this reason, the attention is confined to either c or R.Two stationary random fields Z1, Z2 are stationarily correlated if their crosscovariance function R12(x,y) depends on the difference x−y only. The tworandom fields are uncorrelated if R12 vanishes identically.
2.1. Isotropic random fields. An interesting special class of homogeneousrandom fields that often arise in practice is the class of isotropic fields. Theseare characterized by the property that the covariance function R dependsonly on the length ‖h‖ of the vector h:
R(h) = R(‖h‖) .In many applications, random fields are considered as functions of “time”and “space”. In this case, the parameter set is most conveniently writtenas (t,x) with t ∈ R+ and x ∈ Rd. Such processes are often homogeneousin (t,x) and isotropic in x in the sense that
E[Z(t,x)Z(t+ h,x + y)] = R(h, ‖y‖) ,where R is a function from R2 into R. In such a situation, the covariancefunction can be written as
R(t, ‖x‖) =∫
R
∫ ∞
λ=0eituHd(λ‖x‖) dG(u, λ) ,
where
Hd(r) =(2r
)(d−2)/2Γ(d/2)J(d−2)/2(r)
6 HANSJORG FURRER
and Jm is the Bessel function of the first kind of order m and G is a multipleof a distribution function on the half plane (λ, u)|λ ≥ 0, u ∈ R. For aproof, see for instance Adler [1], Section 2.5.
To ensure non-negative definiteness of R one often specifies R to belong toa parametric family whose members are known to be positive definite. Oneattempt is to use separable covariances
R(t,x) = R(1)(t)R(2)(x) ,
where R(1) is positive definite on R+ and R(2) is a positive definite functionon Rd. A simple example would be R(1)(t) = exp−θ1t and R(2)(x) =exp−θ2‖x‖, yielding R(t,x) = exp−θ1t − θ2‖x‖, θ1 > 0, θ2 > 0. How-ever, the class of separable correlation functions is limited as it neglectsany space-time interactions. Cressie and Huang [5] therefore discuss classesof non-separable stationary covariance functions. See also Chapter 5 ofGneiting [12] where isotropic correlation functions in the space domain arediscussed.
Example 2.1 (Cressie and Huang [5]). Three parameter non-separable space-time stationary covariance function
(2.1) R(t,x) =σ2
(a2t2 + 1)d/2exp− b2‖x‖2
a2t2 + 1
,
where a ≥ 0 is the scaling parameter of time, b ≥ 0 is the scaling parameterof space, and σ2 = R(0,000).
spatial lag
time
lag
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0 0.4
0.5
0.6
0.7
0.80.91
Figure 1: Contour plot of the co-variance function (2.1) with a =b = σ2 = 1 and d = 1.
0
0.2
0.4
0.6
0.8
1
time lag
0
0.2
0.4
0.6
0.8
1
spatial lag
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1c
Figure 2: Perspective plot of thecovariance function (2.1) witha = b = σ2 = 1 and d = 1.
2.2. Gaussian random fields. An important special class of random fieldsis that of Gaussian fields. A Gaussian random field is a random field whereall the finite-dimensional distributions Fx1,...,xn are multivariate normal. Asa consequence, a Gaussian random field is completely determined by specify-ing the mean and covariance function, respectively. A key result in the the-ory of Gaussian fields is the so-called Karhunen-Loeve expansion. In essence,
THE TERM STRUCTURE AS A RANDOM FIELD 7
the Karhunen-Loeve expansion provides a representation of a Gaussian ran-dom field through a type of eigenfunction expansion. Before stating theresult, we introduce some terminology and notation. Let U be a compactinterval in Rd and let R be a continuous covariance function on U × U .A nonzero number λ for which there exists a function φ satisfying
(2.2)∫UR(x,y)φ(y) dy = λφ(x)
and∫U |φ(y)|2 dy < ∞ is called an eigenvalue of R and the corresponding
function φ an eigenfunction. In general, there exists a sequence (λi)i∈N ofeigenvalues and corresponding eigenfunctions (φi)i∈N fulfilling (2.2). Onecan assume that the eigenfunctions φk form an orthonormal sequence, i.e.
(2.3)∫Uφi(y)φk(y) dy =
1, i = k
0, i 6= k .
A fundamental result in the theory of integral equations is the followingtheorem, see for instance Adler [1], Theorem 3.3.1.
Theorem 2.2. (Mercer’s Theorem ). Let R(x,y) be continuous and non-negative definite on the compact interval U × U ⊂ R2d, with eigenvalues λjand eigenfunctions φj satisfying
∫U R(x,y)φ(y) dy = λφ(x). Then
R(x,y) =∞∑j=1
λjφj(x)φj(y) ,
where the series converges absolutely and uniformly on U × U . 2
The Karhunen-Loeve expansion of a Gaussian random field is given in thefollowing theorem, see Adler [1], Theorem 3.3.3 for more details.
Theorem 2.3. (Karhunen-Loeve expansion) Let Z(x) be a real-valued,zero-mean, Gaussian random field with continuous covariance function Rwhich has Mercer expansion R(x,y) =
∑j λjφj(x)φj(y). Then, under some
regularity conditions,
(2.4) Z(x) =∞∑i=1
√λi φi(x)ξi ,
in L2 and a.s., where (ξi)i∈N is a sequence of independent and identicallystandard normally distributed random variables. 2
We shall occasionally consider integrals of the type
IU =∫Uζ(y)Z(y) dy ,
where the deterministic function ζ is continuous over the subset U ⊂ Rd, andthe random field Z is stationary with covariance function R. Such integralscan be defined in analogy with the Riemann integral by considering limitsof finite linear forms of the form Z(xk). The first two moments of IU arethen given by
E[IU ] =∫Uζ(y)m(y) dy , Cov(IU , IV) =
∫U
∫Vζ(x)ζ(y)R(x− y) dxdy .
8 HANSJORG FURRER
If it is assumed that Z(x) is a Gaussian field then IU is a Gaussian ran-dom variable with mean value
∫U ζ(y)m(y) dy and variance Var(IU ) =∫∫
U×U ζ(x)ζ(y)R(x− y) dxdy.
In the following Proposition we derive the so-called Cameron-Martin for-mula. We present a proof which is based on the Karhunen-Loeve expansionfor Brownian motion W . Note that the process W 2 is not a chi-squaredfield in the sense of Definition 2.2 because W is not a stationary process.The Cameron-Martin formula can be used for bond pricing purposes whenthe short rate process is described in terms of W 2, see Section 3.1. Analternative proof of the Cameron-Martin formula can be found in Revuzand Yor [22], p. 425.
Proposition 2.1. Let W denote standard one-dimensional Brownian mo-tion. Then, for α ∈ R+,
(2.5) E[e−α
∫ T0 W 2(s) ds
]=(cosh
(T√
2α ))−1/2
,
where cosh(x) = (expx+ exp−x)/2 denotes the hyperbolic cosine func-tion.
Proof. The usefulness of the Karhunen-Loeve expansion hinges on the abil-ity to solve the integral equation (2.2). Recall that the covariance function Rin the Brownian case is given by R(s, t) = s ∧ t, where a ∧ b = mina, b.Equation (2.2) therefore reads
λφ(x) =∫ T
0(x ∧ y)φ(y) dy =
∫ x
0yφ(y) dy + x
∫ T
xφ(y) dy .
From this, it follows that φ(0) = 0. Taking the derivative with respect to x,we obtain λφ(x) =
∫ Tx φ(y) dy. Setting x = T , we get φ(T ) = 0. Taking the
derivative once more with respect to x, it follows that
(2.6) λφ(x) = −φ(x) .
The general solution of (2.6) is given by
φ(x) = A sin(x/√λ)
+B cos(x/√λ).
The initial condition φ(0) = 0 yields B = 0. On the other hand, fromφ(T ) = 0 we conclude that A cos(T/
√λ) = 0, whence T/
√λ = (2k + 1)π/2
for k ∈ N0, or equivalently
(2.7) λk =( 2T
(2k + 1)π
)2.
The value of the constant A can be obtained from the orthonormality con-dition (2.3). We have that
1 =∫ T
0φ2k(y) dy = A2
∫ T
0sin2
(y/√λk)dy
= A2
∫ T
0sin2
(yπ(2k + 1)2T
)dy
THE TERM STRUCTURE AS A RANDOM FIELD 9
=2TA2
(2k + 1)π
∫ (2k+1)π/2
0sin2(u) du = A2T/2 .
In the last equality we used∫
sin2(u) du = u/2 − sin(2u)/4. Solving for Athus yields A =
√2/T . Summarizing, we have that Brownian motion can
be defined as the infinite sum (in L2([0, T ]) and a.s.)
W (t) =∞∑k=1
2√
2Tπ(2k + 1)
sin(πt(2k + 1)
2T
)ξk , t ∈ [0, T ],
with independent standard normal random variables ξk. We are now in aposition to calculate the Laplace transform of
∫ T0 W 2(s) ds.
E[e−α
∫ T0 W 2(s) ds
]= E
[e−α
∫ T0
(∑k
√λkφk(s)ξk
)2ds]
= E[e−α
∫ T0
∑k,j
√λkλjφk(s)φj(s)ξkξj ds
]= E
[e−α
∑k,j
√λkλjξkξj
∫ T0 φk(s)φj(s) ds
].
Using the orthonormality of the sequence (φi) and the independence of thestandard normal variates ξi we get
E[e−α
∫ T0 W 2(s) ds
]= E
[e−α
∑j≥0 λjξ
2j
]=∏j≥0
E[e−αλjξ
2].
Let Z ∼ N (0, 1). Note that E[e−γZ
2]=∫
R e−γz2ϕ(z) dz =
(1 + 2γ
)−1/2,where ϕ(x) = 1/
√2π exp−x2/2, x ∈ R, denotes the density function of
the standard normal distribution. Consequently,
E[e−α
∫ T0 W 2(s) ds
]=∏j≥0
(1 + 2αλj
)−1/2 =(∏j≥0
(1 + 2αλj
))−1/2.
It remains to show that∏j≥0(1 + 2αλj) = cosh
(T√
2α). From the identity
cosh(y) = cos(iy) and the infinite product expansion of the cosine functioncos(x) =
∏n≥1
(1−4x2/(π(2n−1))2
)we find the infinite product expansion
of the hyperbolic cosine function:
cosh(y) =∏k≥0
(1 +
4y2
π2(2k + 1)2).
Now observe that, with λk as given in (2.7),∏k≥0
(1 + 2αλk
)=∏k≥0
(1 +
4θ2
π2(2k + 1)2)
= cosh(θ)
with θ = T√
2α, which completes the proof.
2.3. Chi-squared random fields. The amenities in working with Gauss-ian random fields are mainly a consequence of the relatively easy form of the(multivariate) Gaussian distribution. Nature, however, does not always pro-duce Gaussian fields. For instance, the tails of the distribution of a random
10 HANSJORG FURRER
field may be fatter than normal. From a modeling point of view, Gaussianfields may also be inappropriate because they allow for negative values. Inthe credit risk context for example it makes little sense to stipulate modelwhere an obligor’s default intensity is modeled as a Gaussian random field.This would imply that negative intensities occur with positive probability.This is inconsistent with theory, since, for a deterministic intensity λ, therisk-neutral probability of an obligor’s default during the infinitesimal timeinterval [t, t + dt], conditional on survival up to t, is given by λdt. Theso-called chi-squared fields are a possible way out. A chi-squared field is apositive-valued random field generated from a finite number of stationaryGaussian random fields. Chapter 7 of Adler [1] provides an introduction tonon-Gaussian fields, including chi-squared fields.
Definition 2.2. (Chi-squared random field). Let Z1, . . . , Zn be inde-pendent, stationary real-valued Gaussian random fields with mean func-tion m(x) = E[Zi(x)] = 0, i = 1, . . . , n, common covariance functionR(y) = E[Z(x)Z(x + y)] and variance σ2 = R(0). For x ∈ Rd, the process
Y (x) := Z21 (x) + · · ·+ Z2
n(x)
is called a chi-squared field with parameter n.
The name chi-squared field with parameter n stems from the fact that therandom variable Y (x) has a scaled one-dimensional chi-squared distributionwith n degrees of freedom. Recall that the density function of a scaled chi-squared distributed random variable Y with r degrees of freedom is given by
(2.8) fχ2(r,σ)(u) =1
Γ(r/2) (2σ)r/2ur/2−1 e−u/(2σ) , u ≥ 0.
Notice that E[Y ] = rσ2 and Var(Y ) = 2rσ4.Since a chi-squared field Y is generated from a finite number of stationary(Gaussian) fields, it follows that Y is stationary too. To derive the covari-ance function R∗ of a chi-squared field Y we need the well-known propertythat the conditional distributions of a multivariate normal distribution areagain multivariate normal. In particular, for a bivariate normal randomvector Z = (Z1, Z2)′ with mean vector µµµ = (µ1, µ2)′ and covariance matrix
Σ =(
σ21 ρσ1σ2
ρσ1σ2 σ22
), σ1, σ2 ≥ 0 , |ρ| ≤ 1 ,
the conditional distribution of Z1 given Z2 is normal with mean E[Z1|Z2] =µ1 + ρσ1σ
−12 (Z2− µ2) and variance Var(Z1|Z2) = σ2
1(1− ρ2). From this, wefind that
E[Z2
1Z22
]=∫
RE[Z2
1Z22 |Z2 = z
]ϕ(z − µ2
σ2
)/σ2 dz
=∫
Rz2 E
[Z2
1 |Z2 = z]ϕ(z − µ2
σ2
)/σ2 dz
= (σ22 + µ2
2)(µ2
1 + σ21
(1− ρ2
))+ 4µ1µ2σ1σ2 +
(ρσ1µ2
)2 + 3(ρσ1σ2
)2.
(2.9)
THE TERM STRUCTURE AS A RANDOM FIELD 11
Assume that σ2 = R(x,x) = 1. By independence of the fields Zi, Zk, i 6= k,we get
R∗(x,y) = E[Y (x)Y (y)
]−mY (x)mY (y)
= E[( n∑
i=1
Z2i (x)
)( n∑k=1
Z2k(y)
)]− n2
= E[ n∑i=1
Z2i (x)Z2
i (y) +∑i6=k
Z2i (x)Z2
k(y)]− n2
=n∑i=1
E[Z2i (x)Z2
i (y)]+∑i6=k
E[Z2i (x)
]E[Z2k(y)
]− n2
=n∑i=1
(1−R2(x,y)) + 3R2(x,y)
+ n(n− 1)− n2
= 2nR2(x,y) ,
(2.10)
where R denotes the common covariance function of the fields Zi. In thesecond-last equality we used (2.9).
3. Default-free models
The theory of interest rate modeling is commonly based on the assumptionof a specific one-dimensional dynamic for the instantaneous spot rate pro-cess r. We take as given an arbitrage free setting in which all securities arepriced in terms of r and an equivalent martingale measure Q. Modeling di-rectly such dynamics is very convenient because all fundamental quantitiessuch as forward rates or bonds are readily defined. For instance, the time-tprice of a zero-coupon bond with maturity T , t ≤ T , is characterized by
(3.1) P (t, T ) = EQ
[e−
∫ Tt rs ds
∣∣Ft] .From this expression it is clear that whenever we can determine the distri-bution of exp−
∫ Tt rs ds in terms of a specific dynamic for r, bond prices
can be calculated. In the Vasicek model, the short rate process r is mod-eled as a mean reverting Ornstein-Uhlenbeck process (OU process) underthe risk-neutral measure Q, that is drt = κ(m − rt)dt + σdW (t), where κ,m, and σ are positive constants and W denotes standard Brownian motion.The above stochastic differential equation has the solution
rt = rse−κ(t−s) +m
(1− e−κ(t−s)
)+ σ
∫ t
se−κ(t−u) dW (u) ,
and it is well-known that rt conditional on Fs, s ≤ t, has a Gaussian dis-tribution. Moreover,
∫ ts ru du is itself normally distributed, implying that
bonds can be priced by directly computing the expectation (3.1).The main drawback of the Vasicek model is that the short rate rt can at-tain negative values with positive probability. In the Cox, Ingersoll andRoss (CIR) framework, a “square root” term in the diffusion coefficient of
12 HANSJORG FURRER
the instantaneous short rate dynamics is introduced:
(3.2) drt = κ(m− rt)dt+ σ√rt dW (t) .
Not only provides the CIR model positive1 instantaneous short rates butalso remains analytically tractable.
Motivated by the general theory of random fields, we shall subsequentlyconsider short rate models where the evolution of r is specified in integralrather than in differential form. In such a model, the stochastic componentwill be represented by a random field Z. It is tempting to postulate a modelwhere the random component is a Gaussian field. The ease of analyticaltractability provoked by the normal distribution, however, comes again atthe cost of possible negative values for rt. To circumvent this problem, wewill adhere to positive-valued random fields. Observe that a positive-valuedfield can be generated from an arbitrary random field Z by means of atransformation g(Z), where g denotes some positive function. Specifically, inSection 3.1 we consider a short rate model based on the square of Brownianmotion W . Recall that one-dimensional Brownian motion is a zero-meanGaussian process with covariance function R(s, t) = s ∧ t, implying thatneither W nor W 2 are stationary processes. The full merits of this modellie in its tractability though, culminating in an affine term structure forzero-coupon bond prices, see Proposition 3.1. In Section 3.2 we advocatea short rate model where the random field component is the square of anOU process. It is well-known that the OU process is a stationary zero-meanGaussian process, and hence its square is a chi-squared field on the line.Given these model specifications, the analytical tractability and the affinestructure for the zero-coupon bond prices are preserved, see Proposition 3.2.As an application, we derive the price of a European call option written ona zero-coupon bond. We conclude the essay on default-free models with anexcursion into forward rate random field models, see Section 3.3.
3.1. The short rate process as a squared Brownian motion. Fort ≥ 0, we consider the following model setup:
rt = r0 + α(t) + σY (t)
Y (t) = W 2(t) ,W : standard Brownian motion.
(3.3)
We assume that r0 and σ are non-negative constants, and that the deter-ministic function α is positive and differentiable with α(0) = 0. Observethat the first two moments of r are given by
E[rt] = r0 + α(t) + σt
Var(rt) = 2σ2t2
Cov(rs, rt) = 2σ2(s ∧ t)2 .
1The solution r of (3.2) will never reach zero from a strictly positive initial value providedκm > σ2/2, which is sometimes known as Feller condition.
THE TERM STRUCTURE AS A RANDOM FIELD 13
In the last equality we used the fact that E[W (s)2W (t)2
]= 2(s ∧ t)2 + st.
This follows from (2.9) since Cov(W (s),W (t)
)= s ∧ t, implying that ρ =
Corr(W (s),W (t)
)= (s ∧ t)/
√st.
Proposition 3.1. The short-rate model (3.3) provides an affine term struc-ture model in the sense that zero-coupon bond prices can be written as
P (t, T ) = eA(t,T )−B(t,T )rt , where
A(t, T ) =(r0 + α(t)
)B(t, T )−
∫ T
t
(r0 + α(s)
)ds
+ log(cosh
(√2σ(T − t)
))−1/2,
B(t, T ) =1√2σ
tanh(√
2σ(T − t)).
Remark. Notice that
P (0, T ) = eA(0,T )−B(0,T )r0
= e−∫ T0
(r0+α(s)
)ds(cosh
(T√
2σ ))−1/2
,
a result which can directly be obtained from Proposition 2.1. 2
Proof. Recall that Y = W 2 is the square of a one-dimensional Bessel processstarted at 0, i.e. Y is the strong solution of the SDE
dY (t) = dt+ 2√Y (t) dW (t) , Y (0) = 0.
Equivalently, W 2(t) = 2∫ t0 W (s) dW (s)+ t. Now observe that the dynamics
of r can be written as
drt = α(t) dt+ σ dY (t)
= α(t) dt+ σ(dt+ 2
√Y (t) dW (t)
)=(α(t) + σ
)dt+ 2σ
√Y (t) dW (t)
= µ(t, r) dt+ σ(t, r) dW (t) ,
say, with µ(t, r) = α(t) + σ and σ(t, r) = 2σ√Y (t). It follows that
σ2(t, r)2
= 2σ2Y = 2σ(rt − r0 − α(t))
= σ(t) + τ(t)rt ,
say, with σ(t) = −2σ(r0 + α(t)) and τ(t) = 2σ. Similarly, we can writeµ(t, r) = µ(t) + η(t)rt, where µ(t) = α(t) + σ and η(t) = 0. It follows fromthe fundamental PDE for the zero-coupon bond price that the functionsA(t, T ) and B(t, T ) satisfy the following system of differential equations
−∂At = σ(t)B2(t, T )− µ(t)B(t, T ) , A(t, t) = 0 ,
∂Bt = τ(t)B2(t, T )− η(t)B(t, T )− 1 , B(t, t) = 0 .
14 HANSJORG FURRER
Thus, the function B satisfies ∂Bt = 2σB2(t, T ) − 1, from where it followsthat
B(t, T ) =1√2σ
tanh(√
2σ(T − t)).
For the function A we get, by means of partial integration,
A(t, T ) = −∫ T
t
(∂Bt(u, T ) + 1
)(r0 + α(u)) du−
∫ T
t
(α(u) + σ
)B(u, T ) du
=(r0 + α(t)
)B(t, T )−
∫ T
t
(r0 + α(s)
)ds− σ
∫ T
tB(u, T ) du
Since∫
tanh(x) dx = log(cosh(x)
), we conclude
σ
∫ T
tB(u, T ) du =
σ√2σ
∫ T
ttanh
(√2σ(T − u)
)du
= log(cosh
(√2σ(T − t)
))/2 ,
which completes the proof.
Remark. To generate a positive valued spot rate process r originating fromlinear Brownian motion we could also postulate a model that includes |W |instead of W 2. The default-free bond prices at time t = 0 can then be cal-culated via the formula
E[e−α
∫ 10 |W (s)| ds
]=
∞∑j=0
θje−δjα2/3
,
where δj are the positive roots of the derivative of
P (y) =√
2y(J−1/3
((2y)3/2/3
)+ J1/3
((2y)3/2/3
))/3 .
Here Jα are Bessel functions of order α and θj =(1 + 3
∫ δj0 P (y) dy
)/(3δj),
see Kac [14]. 2
3.2. The short rate process as a squared OU process. As mentionedbefore, the process W 2 is not a chi-squared process in the sense of Defini-tion 2.2. For β > 0, % > 0, we shall now consider a process Z given by
(3.4) dZ(t) = −βZ(t) dt+ % dW (t) , t ≥ 0,
whereW denotes standard linear Brownian motion. Equation (3.4) is knownas Langevin’s equation, and has the well-known solution
Z(t) = Z(0)e−βt + %
∫ t
0e−β(t−s) dW (s) , t ≥ 0.
If the initial variable Z(0) has a normal distribution with mean zero andvariance %2/(2β), then Z is called an OU process with parameter β andsize c = %2/(2β) > 0. Note that such a process is a stationary zero-meanGaussian process with covariance function R(s, t) = c exp−β|t − s|, see
THE TERM STRUCTURE AS A RANDOM FIELD 15
for example Karatzas and Shreve [15] p. 358. The eigenvalues and eigen-functions of R can be derived as in the Brownian case. Let ωk denote theroots of ω + β tan(ωT ) = 0 and δj be the roots of β − δ tan(δT ) = 0. Thenthe integral equation
∫ T0 c e−β|x−y|φ(y) dy = λφ(x) has the eigenvalues
2βcω2
1 + β2,
2βcδ21 + β2
,2βc
ω22 + β2
,2βc
δ22 + β2, . . .
and the corresponding eigenfunctions φ are given by
sin(ωkx)√T/2− sin(2Tωk)/(4ωk)
,cos(δjx)√
T/2 + sin(2Tδk)/(4δk),
see Appendix A for a derivation of this result.By analogy with (3.3), we consider the following alternative short rate ran-dom field model. For t ≥ 0, define
rt = r0 + α(t) + σY (t)
Y (t) = Z2(t) ,
Z : OU process with parameter β and size c = %2/(2β) > 0.
(3.5)
We suppose r0 and σ are non-negative constants, and the deterministicfunction α is positive and differentiable with α(0) = 0. Note that Y is achi-squared field on the real line with parameter 1 and covariance functionR∗(s, t) = 2c2 exp−2β|t− s|, see (2.10). Hence, the first two moments ofr are given by
E[rt] = r0 + α(t) + σc
Var(rt) = 2σ2c2
Cov(rs, rt) = 2σ2c2 exp−2β|t− s| .Proposition 3.2. The short-rate model (3.5) provides an affine term struc-ture model in the sense that zero-coupon bond prices can be written as
(3.6) P (t, T ) = eA(t,T )−B(t,T )rt ,
where
A(t, T ) =(r0 + α(t)
)B(t, T )−
∫ T
t
(r0 + α(s)
)ds+ β(T − t)/2
+ logcosh
(√∆(T − t) + atanh
(β/√
∆))√
1− β2/∆−1/2
,
B(t, T ) =−β +
√∆ tanh
(√∆(T − t) + atanh
(β/√
∆))
2%2σ,
with ∆ = β2 + 2%2σ.
Remark. When β = 0 and % = 1, the random field component Y is thelinear Brownian motion. In that case, the above functions A and B coincidewith the corresponding functions in Proposition 3.1. 2
16 HANSJORG FURRER
t
r
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
t
r
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Figure 3: Two simulated sample paths of the short rate process r specifiedin (3.5) with r0 = 0, α ≡ 0, and β = % = σ = 1. We used the Eulerdiscretization of (3.4) to obtain simulated versions of the process Z.
Proof. Applying Ito’s formula to the function f(x) = x2, we find that
dY (t) = 2Z(t)dZ(t) + %2 dt
= 2Z(t)(−βZ(t) dt+ %dW (t)
)+ %2 dt
=(%2 − 2βY (t)
)dt+ 2%
√Y (t) dW (t)
Y (0) = Z2(0) .
(3.7)
The dynamics of r then reads
drt = α(t) dt+ σ dY (t)
= α(t) dt+ σ(%2 − 2βY (t)
)dt+ 2%σ
√Y (t) dW (t)
=(α(t) + σ(%2 − 2βY (t))
)dt+ 2%σ
√Y (t) dW (t)
= µ(t, r) dt+ σ(t, r) dW (t) ,
say, with µ(t, r) = α(t)+σ(%2−2βY (t)
)and σ(t, r) = 2%σ
√Y (t). It follows
thatσ2(t, r)
2= 2%2σ2Y = 2%2σ(rt − r0 − α(t))
= σ(t) + τ(t)rt ,
say, with σ(t) = −2%2σ(r0 + α(t)) and τ(t) = 2%2σ. Similarly, we canwrite µ(t, r) = µ(t) + η(t)rt, where µ(t) = α(t) + σ%2 + 2β(r0 + α(t)) andη(t) = −2β. The functions A(t, T ) and B(t, T ) are the solutions to the sys-tem
THE TERM STRUCTURE AS A RANDOM FIELD 17
−∂At = σ(t)B2(t, T )− µ(t)B(t, T ) , A(t, t) = 0 ,
∂Bt = τ(t)B2(t, T )− η(t)B(t, T )− 1 , B(t, t) = 0 .
With τ and η as above, the Riccati-type equation for B reads
∂Bt = 2%2σB2(t, T ) + 2βB(t, T )− 1, B(t, t) = 0 ,
whence
B(t, T ) =−β +
√∆ tanh
(√∆(T − t) + atanh
(β/√
∆))
2%2σ,(3.8)
where ∆ = β2 + 2%2σ. For the function A we obtain
A(t, T ) = −∫ T
t
(∂Bt(u, T ) + 1
)(r0 + α(u)) du−
∫ T
t
(α(u) + %2σ
)B(u, T ) du
=(r0 + α(t)
)B(t, T )−
∫ T
t
(r0 + α(s)
)ds− %2σ
∫ T
tB(u, T ) du .
Recall that∫
tanh(x) dx = log(cosh(x)
)and that cosh(atanh(x)) =
(1− x2)−1/2, hence∫ T
tB(u, T ) du
=−β(T − t) +
√∆∫ T
ttanh
(√∆(T − u) + atanh
(β/√
∆))
du
2%2σ
=−β(T − t) + log
cosh(√
∆(T − t) + atanh(β/√
∆))√
1− β2/∆
2%2σ.
This leads to
A(t, T ) =(r0 + α(t)
)B(t, T )−
∫ T
t
(r0 + α(s)
)ds+ β(T − t)/2
+ logcosh
(√∆(T − t) + atanh
(β/√
∆))√
1− β2/∆−1/2
,
which completes the proof.
Assuming now that for all contingent claims X there exists a suitable self-financing strategy that replicates the claim, the time-t price of such a claimis given by
(3.9) πt(X) = EQ
[e−
∫ Tt rs dsX(T )
∣∣Ft] .In particular, the case of a European call option with maturity S, strikeprice K and written on a zero-coupon bond with maturity T > S leads tothe formula
18 HANSJORG FURRER
πt = EQ
[e−
∫ St ru du
(P (S, T )−K
)+∣∣Ft] .For simplicity, we consider the time-0 price of such an option. Let B(t)denote the bank account numeraire, that is B(t) = exp
∫ t0 rs ds. Then
π0 = EQ
[(P (S, T )−K
)+/B(S)
]= EQ
[(eA(S,T )−B(S,T )rS −K
)+/B(S)
]= EQ
[eA(S,T )−B(S,T )rS1rS<r∗
1B(S)
]−K EQ
[1rS<r∗
1B(S)
].
where r∗ =(A(S, T )− log(K)
)/B(S, T ). We apply the change of numeraire
technique to change from the risk neutral measure Q to the T -forward mea-sure QT . The latter is defined by the Radon-Nikodym derivative
dQT
dQ
∣∣∣∣∣Ft
=P (t, T )
P (0, T )B(t), 0 ≤ t ≤ T ,
so that the price at time 0 of the above claim is given by
(3.10) π0 = P (0, T )EQT
[1rS<r∗
]−K P (0, S)EQS
[1rS<r∗
].
In what follows, we determine the distribution of the short rate rS under theforward measures QT and QS , respectively. To begin with, recall that thebond price dynamics can easily be obtained via Ito’s formula. In an affinemodel, this leads to
dP (t, T )P (t, T )
= rt dt−B(t, T )σ(t, r) dW (t) ,
where σ(t, r) denotes the volatility coefficient of the short rate dynamics.Furthermore, under QT ,
(3.11) WT (t) = W (t)−∫ t
0λTs ds
is a QT -Brownian motion, where λTt = −B(t, T )σ(t, r). From equations(3.7) and (3.11), we obtain the dynamics of Y under the measure QT :
dY (t) =(%2 − 2Y (t)
(β + 2%2σB(t, T )
))dt+ 2%
√Y (t) dWT (t)
=(%2 − 2Y (t)β(t, T )
)dt+ 2%
√Y (t) dWT (t) ,
say, where
(3.12) β(t, T ) = β + 2%2σB(t, T ).
Under the measure QT , the process Y can be thought of as the square of aprocess Z whose dynamics is given by
(3.13) dZ(t) = −β(t, T )Z(t) + % dWT (t) .
The stochastic differential equation (3.13) has the solution
THE TERM STRUCTURE AS A RANDOM FIELD 19
Z(t) = e−(ϑ(t,T )−ϑ(s,T ))Z(s) + %
∫ t
se−(ϑ(t,T )−ϑ(u,T )) dWT (u) , 0 ≤ s ≤ t,
where ϑ(t, T ) =∫ t0 β(u, T ) du. It follows that, under the T -forward mea-
sure QT , Z(t) conditional on Z(s) is normally distributed with mean andvariance given respectively by
EQT[Z(t)|Z(s)] = Z(s)e−
∫ ts β(u,T ) du ,
VarQT(Z(t)|Z(s)) = %2
∫ t
se−2
∫ tu β(v,T ) dv du .
(3.14)
Before we proceed with the derivation of the bond option price, we no-tice that the expressions on the right hand side of (3.14) can be calculatedexplicitly. Indeed, with B(t, T ) and β(t, T ) as given in (3.8) and (3.12),respectively, a straightforward computation shows that, for θ ≥ 0,
(3.15) e−θ∫ t
s β(v,T ) dv =
(cosh
(√∆(T − t) + C
)cosh
(√∆(T − s) + C
))θ ,where we set for convenience C = atanh(β/
√∆). We obtain the conditional
variance of Z(t) given Z(s) if we set θ = 2 in (3.15) and integrate from sto t. This leads to∫ t
se−2
∫ tu β(v,T ) dv du = cosh2
(√∆(T − t) + C
)∫ t
scosh−2
(√∆(T − u) + C
)du
=cosh2
(√∆(T − t) + C
)√
∆×(
tanh(√
∆(T − s) + C)− tanh
(√∆(T − t) + C
)).
Now set x = (r∗−r0−α(S))/σ. Recall that, under both measures Q and QT ,the random variable Z(0) has a normal distribution with mean zero and vari-ance %2/(2β). Consequently, we have that
EQT
[1rS<r∗
]= QT
[rS < r∗
]= QT
[r0 + α(S) + σZ2(S) < r∗
]= QT
[Z2(S) < x
]=∫
RQT
[Z2(S) < x|Z(0) = z
]ϕ(z√
2β/%)√
2β/% dz .
Now introduce the notation
µ(U,V,W )(z) = ze−∫ V
U β(ω,W ) dω , σ2(U,V,W ) = %2
∫ V
Ue−2
∫ Vτ β(ω,W ) dω dτ .
Using this formulation, it follows from (3.14) that Z(S)|Z(0) = z has a nor-mal distribution with mean and variance given respectively by µ(0,S,T )(z)and σ2
(0,S,T ). We conclude that, under the measure QT ,
20 HANSJORG FURRER
Z2(S)|Z(0) = zL= σ2
(0,S,T ) χ2
(1,µ(0,S,T )(z)σ(0,S,T )
),
where χ2(r, δ) denotes a non-centrally chi-squared distributed random vari-able with r degrees of freedom and non-centrality parameter δ. The densityfunction of such a random variable is given by
(3.16) fχ2(x; r, δ) =∞∑j=0
e−δ/2(δ/2)j
j!fχ2(r+2j)(x) .
Here fχ2(ρ)(x) is the density function of the (central) chi-squared distri-bution with ρ degrees of freedom and scaling parameter σ = 1, see (2.8).Combining the above we deduce that
EQT
[1rS<r∗
]=∫
RFχ2
(r∗−r0−α(S)
σσ2(0,S,T )
; 1, µ(0,S,T )(z)
σ(0,S,T )
)ϕ(z√
2β/%)√
2β/% dz ,
where Fχ2(x; r, δ) is the cumulative distribution function of a non-central chi-squared variate with r degrees of freedom and non-centrality parameter δ.Similarly, we can determine the law of rS under the S-forward measure QS ,yielding
EQS
[1rS<r∗
]=∫
RFχ2
(r∗−r0−α(S)
σσ2(0,S,S)
; 1, µ(0,S,S)(z)
σ(0,S,S)
)ϕ(z√
2β/%)√
2β/% dz .
Putting things together, we arrive at the European call option price asspecified in (3.10).
Remarks. 1) It is clear that bond prices can be calculated if we can char-acterize the distribution of exp−
∫ Tt Y (s) ds, where Y = Y (t) : t ≥ 0
denotes the solution of a process with the dynamics of (3.7). Pitman andYor [20] consider the stochastic differential equation
(3.17) dh(t) =(δ + 2αh(t)
)dt+ 2
√h(t) dW (t) , t ≥ 0,
and study the law of exp−γ∫ t0 h(s)µ(ds) for non-negative measures µ(ds)
on R+.
2) The formulation in (3.5) can be generalized in the following way:
rt = r0 + α(t) + σY (t),
Y (t) =n∑i=1
Z2i (t) ,
Zi : independent OU processes with parameter βiand size ci = %2
i /(2βi) > 0.
We can write rt = x(t)+ y2(t)+ · · ·+ yn(t), where x(t) = r0 +α(t)+σZ21 (t)
and yk(t) = σZ2k(t), k = 2, . . . , n. Due to independence of the processes Zk,
we have that
THE TERM STRUCTURE AS A RANDOM FIELD 21
P (t, T ) = E[e−
∫ Tt x(s) ds
∣∣Ft] n∏k=2
E[e−
∫ Tt yk(s) ds
∣∣Ft]= P1
(t, T ;x(t),Θ1
) n∏k=2
P1
(t, T ; yk(t),Θ
),
where Θ1 = (r0, α(t)) and Θ = (0, 0). The function P1(t, T ;x(t),Θ1) is thebond price formula (3.6) with rt replaced by x(t). Analogously, the func-tion P1(t, T ; yk(t),Θ) denotes the bond price formula (3.6) with rt replacedby yk(t) and r0, α(t) replaced by Θ = (0, 0). This implies that the bondprices P (t, T ) are expressible as exponential-affine functions of the n factorsx, y2, . . . , yn. 2
3.3. Forward rate random field models. Up to now, we confined our-selves to single-parameter random fields. In Sections 3.1 and 3.2 we in-troduced short rate models which are indexed by time as their sole pa-rameter. Bond prices are then characterized by means of the fundamentalrelation (3.1). An alternative and widely used approach to the bond pricemodeling is based on an exogenous specification of a family f(t, T ) of forwardrates, where 0 ≤ t ≤ T ≤ T ∗ and T ∗ denotes a fixed horizon date. That is,f(t, T ) is the forward interest rate at date t ≤ T for instantaneous risk-freeborrowing or lending over the infinitesimal period [T, T + ∆T ]. Given sucha family f(t, T ), bond prices are then characterized by the relation
P (t, T ) = exp−∫ T
tf(t, v) dv
.
Heath, Jarrow and Morton (HJM) assumed that, for a fixed maturity T , theforward rate f(t, T ) evolves according to the diffusion process
(3.18) df(t, T ) = α(t, T ) dt+ σ(t, T ) dW (t) , f(0, T ) = fM(0, T ) ,
with T 7→ fM(0, T ) denoting the market curve at time t = 0. The advan-tage of modeling forward rates as in (3.18) is that the initial term structureof interest rates is, by construction, an input of the model. The dynam-ics in (3.18), however, is not necessarily arbitrage-free. Heath, Jarrow andMorton proved that, in order for an equivalent martingale measure to exist,the function α can not be chosen arbitrarily, but must satisfy
α(t, T ) = σ(t, T )∫ T
tσ(t, v) dv .
Kennedy [16], [17] interprets the forward rate f(t, T ) as a two-parameterrandom field. Specifically, for a Gaussian random field Z, he considersthe forward rate surface f(t, T ) = µ(t, T ) + Z(t, T ), and derives a neces-sary and sufficient restriction on the drift function µ(t, T ) to ensure thatdiscounted zero-bond prices are martingales under the risk-neutral mea-sure. Goldstein [13] concentrates on the dynamics of the forward rate,df(t, T ) = µ(t, T ) dt + σ(t, T ) dZ(t, T ), and generalizes Kennedy’s drift re-striction result to non-Gaussian fields. The purpose of this section is to set
22 HANSJORG FURRER
the scene for Section 4.4. Proposition 4.1 of that section generalizes Propo-sition 1 in Goldstein [13] to credit risk models under the recovery of marketvalue assumption.Let Z(t, T ), t ≤ T , be a random field labeled with two time indices. Thefirst parameter refers to the current time, the second to the maturity date.For all dates t, the random field describes a realization of a random functionT 7→ Z(t, T ). The correlation between the quantities Z(t1, T1) and Z(t2, T2)is expressed as Cov
(Z(t1, T1), Z(t2, T2)
)= R(t1, t2, T1, T2), where the func-
tion R is non-negative definite in (ti, Ti), i ∈ 1, 2. Specifically, we suppose
(3.19) Cov(Z(t1, T1), Z(t2, T2)
)= R(t1 ∧ t2, T1, T2) .
The fact that the covariance function R is specified as a function of t1 ∧ t2ensures that the increments of the random field in the t-direction are uncor-related. Indeed, for t, h ∈ R+ such that t+h ≤ T and u ≤ t∧S we have that
Cov(Z(t+ h, T )− Z(t, T ), Z(u, S)
)= Cov
(Z(t+ h, T ), Z(u, S)
)− Cov
(Z(t, T ), Z(u, S)
)= R(u, T, S)−R(u, T, S)= 0.
If the random field Z is Gaussian, this property is referred to as the inde-pendent increments property in the t-direction.By analogy with the Ito calculus, the correlation structure (3.19) may alsobe expressed via the cross variation
d〈Z(·, T1), Z(·, T2)〉t = c(t, T1, T2) dt
for a deterministic function c which is symmetric in T1 and T2 and non-negative definite in (t, T1) and (t, T2).
Lemma 3.1. Define the forward rate dynamics as
df(t, T ) = µ(t, T ) dt+ σ(t, T ) dZ(t, T ) ,
where Z(t, T ) is a random field with deterministic correlation structure cspecified by d〈Z(·, T1), Z(·, T2)〉t = c(t, T1, T2) dt. We suppose that µ(t, T )and σ(t, T ) satisfy the technical regularity conditions imposed by the HJMframework. Define It =
∫ Tt f(t, v) dv, whence P (t, T ) = exp−It. Then
(a) dIt = µ∗(t, T ) dt+∫ T
tdv σ(t, v) dZ(t, v)− rt dt
(b)dP (t, T )P (t, T )
= −µ∗(t, T ) dt−∫ T
tdv σ(t, v) dZ(t, v) + rt dt
+12
∫ T
tσ(t, u)σ∗(t, T, u) du dt ,
where rt = f(t, t) and
µ∗(t, T ) =∫ T
tµ(t, s) ds , σ∗(t, T, S) =
∫ T
tσ(t, v)c(t, S, v) dv.
THE TERM STRUCTURE AS A RANDOM FIELD 23
Proof. (a)
It =∫ T
tf(t, v) dv
=∫ T
t
(f(0, v) +
∫ t
0µ(s, v) ds+
∫ t
0σ(s, v) dZ(s, v)
)dv
=∫ T
tf(0, v) dv +
∫ T
t
∫ t
0µ(s, v) ds dv +
∫ T
t
∫ t
0σ(s, v) dZ(s, v) dv .
Applying the stochastic version of Fubini’s theorem, see for example Prot-ter [21], Theorem 45 p. 159, we can write
It =∫ T
0f(0, v) dv +
∫ t
0
∫ T
sµ(s, v) dv ds+
∫ t
0
∫ T
sσ(s, v) dv dZ(s, v)
−∫ t
0f(0, v) dv −
∫ t
0
∫ t
sµ(s, v) dv ds−
∫ t
0
∫ t
sσ(s, v) dv dZ(s, v)
= I0 +∫ t
0µ∗(s, T ) ds+
∫ t
0
∫ T
sdv σ(s, v) dZ(s, v)
−∫ t
0
(f(0, v) +
∫ v
0µ(s, v) ds+
∫ v
0σ(s, v) dZ(s, v)
)dv
= I0 +∫ t
0µ∗(s, T ) ds+
∫ t
0
∫ T
sdv σ(s, v) dZ(s, v)−
∫ t
0f(v, v) dv
= I0 +∫ t
0µ∗(s, T ) ds+
∫ t
0
∫ T
sdv σ(s, v) dZ(s, v)−
∫ t
0rv dv .
The differential form of It thus reads
dIt = µ∗(t, T ) dt+∫ T
tdv σ(t, v) dZ(t, v)− rt dt ,
which leaves (a).
(b) By definition, P (t, T ) = exp−It. Applying Ito’s formula to f(x) =exp−x yields
dP (t, T )P (t, T )
= −dIt +12d〈I·〉t
= −µ∗(t, T ) dt−∫ T
tdv σ(t, v) dZ(t, v) + rt dt
+12
∫ T
tσ(t, u)σ∗(t, T, u) du dt.
(3.20)
In the last summand on the right side of (3.20) we used the multiplicationformalism dZ(t, u) dZ(t, v) = c(t, u, v) dt.
Proposition 3.3. Let the dynamics of the forward rates be specified as inLemma 3.1. Then the risk-neutral drift restriction is given by
(3.21) µ(t, T ) = σ(t, T )σ∗(t, T, T ) .
24 HANSJORG FURRER
Proof. Under the measure Q, bond prices discounted at the short rate pro-cess r are Q -martingales. Set Z∗(t, T ) = P (t, T )/B(t), where B(t) de-notes the bank account numeraire, i.e. B(t) = exp
∫ t0 rs ds. Recall that
dB(t) = B(t) rt dt, hence the dynamics of Z∗(t, T ) reads
dZ∗(t, T ) =dP (t, T )B(t)
+ P (t, T ) d( 1B(t)
)=dP (t, T )B(t)
+ P (t, T )(− 1B2(t)
dB(t))
=dP (t, T )B(t)
− P (t, T )B(t)
rt dt
= Z∗(t, T )dP (t, T )P (t, T )
− Z∗(t, T ) rt dt .
Consequently, by Lemma 3.1,
dZ∗(t, T )Z∗(t, T )
=dP (t, T )P (t, T )
− rt dt
= −µ∗(t, T ) dt−∫ T
tdv σ(t, v) dZ(t, v)
+12
∫ T
tσ(t, u)σ∗(t, T, u) du dt .
In order Z∗(t, T ) be a Q-martingale, the dt-terms must vanish, whence
(3.22) µ∗(t, T ) =12
∫ T
tσ(t, u)σ∗(t, T, u) du .
Differentiating (3.22) with respect to T yields (3.21), which completes theproof.
4. Models with credit risk
When pricing default-free securities, equation (3.9) provides the fundamen-tal pricing rule. If, however, the issuer defaults before maturity T , then boththe amount and timing of the payoff to the investor are uncertain. In thiscase, it is often convenient to interpret a zero-bond as a portfolio compris-ing two securities: a security that pays one unit at maturity T if and onlyif the issuer survives up to time T and a security that pays the (random)amount Z received at default if default happens before T . Let τ denote thedefault time and let Λt = 1τ≤t be the default indicator process. Conse-quently, 1−Λt = 1τ>t is the survival indicator which has outcome 1 if theissuer has not defaulted prior to t and zero otherwise. With this notation,the price of a zero-coupon bond is then
(4.1) P (t, T ) = EQ
[e−
∫ Tt rs ds(1− ΛT )
∣∣Ft]+ EQ
[e−
∫ Tt rs dsZΛT
∣∣Ft] .Under zero recovery, i.e. Z = 0, (4.1) reduces to
THE TERM STRUCTURE AS A RANDOM FIELD 25
(4.2) P (t, T ) = EQ
[e−
∫ Tt rs ds(1− ΛT )
∣∣Ft] .4.1. Reduced form models. In a reduced-form pricing framework, it isassumed that a risk-neutral default intensity process λ = λ(t) : t ≥ 0is associated to the default time τ . That is, τ is the first jump time of acounting process with intensity λ. We suppose for the moment that there isno recovery at default whatsoever. An implication of these assumptions, asshown by Lando [18], is that the bond price (4.2) can be calculated as
(4.3) P (t, T ) = EQ
[e−
∫ Tt (rs+λ(s)) ds
∣∣Ft] ,provided that default has not occurred by time t. In the special case inwhich the default time τ and the short rate process r are independent, thebond price formula (4.3) can be decomposed into
(4.4) P (t, T ) = EQ
[e−
∫ Tt rs ds
∣∣Ft]EQ
[e−
∫ Tt λ(s) ds
∣∣Ft] .Under the assumption that default has not already occurred by time t,
(4.5) p(t, T ) = EQ[1τ>T|Ft
]= EQ
[e−
∫ Tt λ(s) ds
∣∣Ft]is the risk-neutral conditional survival probability.
The existing reduced-form models can be extended to allow for non-zero re-covery. The current bond price literature encompasses a variety of recoverymodels. By and large, they all assume that, conditional on the occurrenceof default in the next instant, the bond under consideration has a givenexpected fractional recovery. A prominent class among these models, intro-duced by Duffie and Singleton [11] and referred to as recovery of marketvalue (RMV) model, takes recovery to be a fraction of the market value ofthe bond just prior to default. To be more precise, the assumption is that,at each time t, the claim pays (1−Lt)Vt−, where Vt− = lims↑t Vs is the priceof the claim just before default, and Lt is the random variable describingthe fractional loss of market value of the claim at default. It is assumed thatthe loss process L = Lt : t ≥ 0 is bounded by 1 and predictable. Undertechnical conditions, Duffie and Singleton [11] derive the following pricingrule for a zero-coupon bond at any time t before default
(4.6) P (t, T ) = EQ
[e−
∫ Tt rs ds
∣∣Ft] ,where r denotes the default-adjusted short rate, i.e.
(4.7) rt = rt + λ(t)Lt , t ≥ 0.
From here, one can now proceed in various directions. For example, onecan model r directly or rather its components r and s, where s(t) = λ(t)Ltdenotes the short spread.
26 HANSJORG FURRER
Henceforth, we shall assume that L ≡ 1, implying s(t) = λ(t). Inspired bythe setup introduced in Section 3, we postulate a random field model forboth processes r and λ:
rt = r0 + α(t) + σrY (t) ,
λ(t) = λ0 + µ(t) + σλU(t) ,
where r0, λ0, σr, σλ are non-negative constants, α and µ positive and differ-entiable functions with α(0) = µ(0) = 0. Furthermore, we assume that Yand U are chi-squared fields with parameters n and m, respectively. Thatis, Y (t) = Z2
1 (t) + · · · + Z2n(t) and U(t) = V 2
1 (t) + · · · + V 2m(t) for indepen-
dent and stationary, zero-mean, Gaussian processes Zk, k = 1, . . . , n, andVj , j = 1, . . . ,m.
Assuming independence between the fields Y and U , the conditional survivalprobabilities (4.5) can be calculated explicitly if we use the formulation ofSection 3.2 to model the default intensity and short rate process. Empiricalevidence, however, suggest that default intensities vary with the businesscycle. During recessions, when interest rates are low, default rates tend tobe higher. A (negative) correlation between the processes r and λ can becaptured through the joint dependence of r and λ on some of the factorsZk and Vj . As an example, we consider the following setup. Let δ be a realnumber with |δ| ≤ 1. Then define
rt = r0 + α(t) + σrY (t) ,(4.8)
λ(t) = λ0 + µ(t) + σλ
(δY (t) +
√1− δ2 Y (t)
),(4.9)
where Y and Y are independent chi-squared fields with common parame-ter n. Obviously, we have that Cov
(rt, λ(t)
)= 2nδσrσλ and consequently
Corr(rt, λ(t)
)= δ. The degree of correlation between rt and λ(t) is thus
specified by the constant δ. Here we assumed that Var(Zk(t)) = RZ(0) = 1.
Next, we would like to calculate the defaultable bond price P (t, T ) in (4.6).Adding up (4.8) and (4.9), we can write
rt = rt + λ(t)
= r0 + θ(t) + ςY (t) + σY (t) ,
= x(t) + y(t) ,
say, where x(t) = r0 + θ(t) + ςY (t) and y(t) = σY (t) with r0 = r0 + λ0,θ(t) = α(t) + µ(t), ς = σr + δσλ, and σ = σλ
√1− δ2.
As an example, we concentrate again on the model specifications (3.5). Thatis, Y and Y are assumed to be the squares of independent OU processes.Due to the independence of the fields Y and Y , the price at time t of thedefaultable zero-bond is given by
P (t, T ) = EQ
[e−
∫ Tt x(s) ds
∣∣Ft]EQ
[e−
∫ Tt y(s) ds
∣∣Ft]= P1(t, T ;x(t),Θ1)P1(t, T ; y(t),Θ2) ,
(4.10)
THE TERM STRUCTURE AS A RANDOM FIELD 27
say, where Θ1 = (r0, θ(t), ς) and Θ2 = (0, 0, σ). The function P1(t, T ;x(t),Θ1)is the bond price formula (3.6) with rt replaced by x(t) and r0, α(t), σ re-placed by Θ1 = (r0, θ(t), ς). The function P1(t, T ; y(t),Θ2) is defined anal-ogously. Equation (4.10) thus tells us that defaultable bond prices are ex-pressible as exponential-affine functions of the factors x and y.
4.2. A reduced form random field model. In the following setup wetake as given a short rate process r modeled by a non-negative valued pro-cess. In addition, we consider a family of obligors indexed by ι ∈ I . Weallow the index set I be countably or uncountably infinite. We denote λ(t, ι)the default intensity at time t of obligor ι ∈ I and assume that λ(t, ι) is atwo-parameter positive-valued random field:
(4.11)rt = r0 + α(t) + σrY (t) ,
λ(t, ι) = λι0 + µ(t, ι) + σλ(ι)U(t, ι) ,
t ≥ 0,
t ≥ 0, ι ∈ I .
We suppose r0, λι0 σr, σλ(·) are non-negative constants and the deterministicfunctions α, µ are non-negative and differentiable with α(0) = µ(0, ·) = 0. Inthe sequel, we assume that I ⊆ [0, 1] and that Y , U are chi-squared randomfields with parameters n and m, respectively. The field U thus admits therepresentation U(t, ι) = V 2
1 (t, ι)+ · · ·+V 2m(t, ι), where Vk, k = 1, . . . ,m, are
independent and stationary, zero-mean space-time Gaussian random fieldswith deterministic covariance function RV :
RV (t, κ) = Cov(V (s, ι), V (s+ t, ι+ κ)
),
for ι, κ ∈ I with |ι + κ| ≤ 1. We interpret t as the temporal lag and ι asthe spatial lag of the field Vk. Observe that, for s, t ≥ 0 and ι, κ ∈ I
Cov(λ(s, ι), λ(t, κ)
)= σλ(ι)σλ(κ)Cov
(U(s, ι), U(t, κ)
)= 2mσλ(ι)σλ(κ)R
2V
(|s− t|, ‖ι− κ‖
).
Assuming that default of obligor ι ∈ I has not yet occurred by time t, therisk-neutral conditional survival probability p(t, T ; ι) is given by
(4.12) p(t, T ; ι) = EQ
[e−
∫ Tt λ(s,ι) ds
∣∣Ft] .Notice that∫ T
tλ(s, ι) ds = λι0(T − t) +
∫ T
tµ(s, ι) ds+ σλ(ι)
∫ T
tU(s, ι) ds
and that the first two moments of J(t, T ; ι) =∫ Tt U(s, ι)ds are given by
E[J(t, T ; ι)
]= mRV (0, 0)(T − t) ,
Cov(J(t, T ; ι), J(t, T ;κ)
)= 2m
∫ T
t
∫ T
tR2V
(|u− v|, ‖ι− κ‖
)du dv .
However, knowledge of the first two moments of J(t, T ; ι) in general is notenough to calculate the survival probability (4.12). To make use of the re-sults in Sections 3.2 and 4.1, we employ a separable correlation structure.
28 HANSJORG FURRER
Assume that RV is of the form
RV(|s− t|, ‖ι− κ‖
)= R
(1)V
(|s− t|
)R
(2)V
(‖ι− κ‖
).
Specifically, let m = 1 and R(1)V
(|s− t|
)= c exp−β|s− t|. Then, for each
ι ∈ I , the default intensity λ(·, ι) of obligor ι evolves as the square of anOU process. The spatial correlation can be captured by an arbitrary per-missible covariance structure, e.g. R(2)
V (‖ι− κ‖) = 1− ϑ‖ι− κ‖ with ϑ ≤ 2,or R(2)
V (‖ι− κ‖) = exp−θ‖ι− κ‖ for θ > 0. Under these assumptions, theconditional probability p(t, T ; ι) at time t that issuer ι survives to time T isof the form
(4.13) p(t, T ; ι) = P1(t, T ;λ(t, ι),Θ) ,
where P1(t, T ;λ(t, ι),Θ) denotes the bond price formula (3.6) with rt re-placed by λ(t, ι) and r0, α(t), σ replaced by Θ = (λι0, µ(t, ι), σλ(ι)). As-suming independence between the default-free short rate process r and theintensity surface λ(t, ι), the zero-coupon bond prices P (t, T ; ι) under thezero recovery assumption are given by
P (t, T ; ι) = EQ
[e−
∫ Tt rs ds
∣∣Ft] p(t, T ; ι) ,
with p(t, T ; ι) as in (4.13). If moreover r is an affine function of the variable Yas outlined in Section 3.2, we deduce that P (t, T ; ι) is an exponential-affinefunction of the factors r and λ(·, ι).
4.3. A simplified firm’s value random field model. In this section weapply the random field methodology to structural credit risk models. Indoing so, we consider a family of obligors I ⊆ [0, 1] and assume that de-fault of each obligor ι is triggered by the change in value of the assets ofits firm, see Schonbucher [25] p. 305 for more details on a simplified firm’svalue approach. Specifically, denote V (t, ι) the market value at time t of theassets of obligor ι and assume that V (t, ι) is of the form
V (t, ι) = µ(t, ι) + σ(t, ι)Y (t, ι) , t ≥ 0.
We suppose µ and σ are deterministic functions and Y is a two-parameter,spatio-temporal random field which is assumed to be homogeneous in t andisotropic in ι with covariance function R. Furthermore, we assume thatobligor ι’s default occurs at maturity date T in the event that its asset valuefalls below a prespecified deterministic barrier δ, i.e. if V (T, ι) < δ(ι).
In the following we are interested in the proportion N(T ) of obligors whoseasset value V at time T is below the barrier δ. Define
(4.14) N(t) =1
µ(I )
∫I1V (t,x)<δ(x) dx,
where µ(A) denotes the Lebesgue measure of the set A. Let I = [0, 1],hence µ(I ) = 1. For the mean value of N(T ), it is immediate that
THE TERM STRUCTURE AS A RANDOM FIELD 29
E[N(T )] =∫ 1
0E[1V (T,x)<δ(x)
]dx
=∫ 1
0P[Y (T, x) < δ(x)−µ(T,x)
σ(T,x)
]dx
=∫ 1
0P[Y (0, 0) < δ(x)−µ(T,x)
σ(T,x)
]dx .
(4.15)
The last equality follows from the homogeneity of the field Y . Similarly, wecan write down an expression for the second moment of N .
E[N2(T )
]=∫ 1
0
∫ 1
0E[1V (T,x)<δ(x)∩V (T,y)<δ(y)
]dx dy
=∫ 1
0
∫ 1
0P[Y (T, x) < δ(x)−µ(T,x)
σ(T,x) , Y (T, y) < δ(y)−µ(T,y)σ(T,y)
]dx dy .
(4.16)
Now assume that Y is a zero-mean Gaussian random field with unit varianceand covariance function R(t, x). Let us set ψ(z) =
(δ(z)− µ(T, z)
)/σ(T, z).
Equation (4.15) simplifies to
E[N(T )] =∫ 1
0Φ(ψ(x)
)dx ,
where Φ(x) =∫ x−∞ exp−u2/2du/
√2π is the standard normal distribution
function. Denoting φ(x, y; ρ) the bivariate Gaussian density function
φ(x, y; ρ) =1
2π√
1− ρ2exp−x2+2ρxy+y2
2(1−ρ2)
the second moment of N(T ) reads
E[N2(T )
]=∫ 1
0
∫ 1
0
∫ ψ(x)
−∞
∫ ψ(y)
−∞φ(v, w;R(0, |x− y|)
)dv dw dx dy .
Suppose for the moment that ψ ≡ 0. In that case, the above integral can besimplified since ∫ 0
−∞
∫ 0
−∞φ(v, w; ρ
)dv dw =
arccos(ρ)2π
.
In the general case ψ(z) 6= 0, however, it appears that closed form solutionsfor the (higher) moments of N(T ) do not exist, nor does it seem possible toobtain exact distribution results for N . Hence, the moments of N(T ) mustbe determined numerically.
Let us for the moment abandon the assumption of stationarity. Instead, weshall consider a particular separable model which is invariant in time but notstationary in space. For the spatial correlation, we recourse to the Browniancovariance structure. That is, we suppose the covariance function R of therandom field Y (t, ι) has the form:
R((t, x), (s, y)
)= R(1)(|s− t|) (x ∧ y) ,
30 HANSJORG FURRER
where R(1) is an admissible temporal covariance function on the real line.Suppose again ψ ≡ 0, whence N(T ) =
∫ 10 1Y (T,x)<0 dx. Since T is fixed,
x 7→ Y (T, x) is one-dimensional Brownian motion starting from 0 (exceptfor the scaling factor
√R(1)(0)). The distribution of N(T ) is thus specified
by Paul Levy’s Arcsine law for the occupation time of (0,∞):
P[N(T ) ≤ x] =∫ x
0
1π√u(1− u)
du =2π
arcsin(√x), 0 ≤ x ≤ 1,
see for instance Karatzas and Shreve [15], p. 273. In particular, we havethat E[N(T )] = 1/2 and Var(N(T )) = 1/8.
So far, we confined ourselves to model the obligors’ firm values as a Gaussianrandom field. The drawback of this approach is that firm values can attainnegative values with positive probability. To avoid negative outcomes wecan model the firm values as a chi-squared field:
V (t, ι) = µ(t, ι) + σ(t, ι)Y (t, ι) ,
where
Y (t, ι) = Z21 (t, ι) + · · ·+ Z2
n(t, ι) .
Here Zk, k = 1, . . . , n are independent and stationary zero-mean Gaussianrandom fields with covariance function RZ and variance σ2 = RZ(0, 0). Itis obvious that we can write down expressions for the moments of N also inthis case. The mean fraction of obligors that will default by time T is givenby
E[N(T )] =∫ 1
0Fχ2(n,σ)(ψ(x)) dx ,
where Fχ2(r,σ) denotes the cumulative distribution function of a central chi-squared distributed random variable with r degrees of freedom and scalingparameter σ, see (2.8) for the definition of the corresponding density func-tion. To derive the second moment, we suppose for simplicity n = 1. Recallfrom Section 2.3 that Z(T, x) conditional on Z(T, y) has a normal law withmean value RZ(0, |x − y|)Z(T, y)/σ2 and variance σ2 − R2
Z(0, |x − y|)/σ2,respectively. Hence
E[N2(T )
]=∫ 1
0
∫ 1
0P[Z2(T, x) < ψ(x), Z2(T, y) < ψ(y)
]dx dy
=∫ 1
0
∫ 1
0
∫ √ψ(y)
−∞P[X2 < ψ(x)
]ϕ( zσ
)/σ dz dx dy ,
say, where X is normally distributed with mean value µ = µ(x, y, z) =z RZ(0, |x− y|)/σ2 and variance σ2 = σ2(x, y) = σ2−R2
Z(0, |x− y|)/σ2. LetX denote a standard normal variate. We then have
E[N2(T )
]=∫ 1
0
∫ 1
0
∫ √ψ(y)
−∞P[(µ/σ + X
)2 ≤ ψ(x)/σ2]ϕ( zσ
)/σ dz dx dy
THE TERM STRUCTURE AS A RANDOM FIELD 31
=∫ 1
0
∫ 1
0
∫ √ψ(y)
−∞Fχ2
(ψ(x)/σ2; 1, µ/σ
)ϕ( zσ
)/σ dz dx dy.
Here Fχ2(x; r, δ) denotes the cumulative distribution function of a non-central chi-squared distributed random variable with r degrees of freedomand non-centrality parameter δ. The corresponding density function is de-fined in (3.16).
In a situation like this where the observations should be restricted to bepositive values, we could alternatively take the logarithms of the firm valuesas fundamental modeling quantities. The modeling then proceeds assum-ing that the log-transformed data have a Gaussian distribution and will bemodeled as a Gaussian random field:
V (t, ι) = eL(t,ι) ,
where, for a Gaussian random field Y ,
L(t, ι) = µ(t, ι) + σ(t, ι)Y (t, ι) .
4.4. Defaultable forward rate models. In this section we resume thesetup of forward rate random field models introduced in Section 3.3. Wesuppose the defaultable bond price prior to default is described through thefollowing equations:
P (t, T ) = exp−∫ T
tf(t, v) dv
,(4.17)
where
df(t, T ) = µ(t, T ) dt+ σ(t, T ) dZ(t, T ) .(4.18)
It is assumed that the functions µ and σ satisfy the technical regularity con-ditions imposed by HJM. Here Z(t, T ) denotes a two-parameter random fieldwith correlation structure specified by d〈Z(·, T1), Z(·, T2)〉t = c(t, T1, T2) dt,see Section 3.3. Observe that the defaultable forward rates f(t, T ) are onlydefined up to the time of default. Proposition 4.1 below shows that, with anexogenously given short spread s(t) = λ(t)Lt, one has the usual HJM driftrestriction in order to preclude arbitrage under the risk-neutral measure Q.Since λ is the intensity process associated to the default time τ , the defaultindicator process Λt = 1τ≤t which is zero before default and 1 afterwardcan be written in the form
(4.19) dΛt = (1− Λt)λ(t) dt+ dM(t) ,
where M = M(t) : t ≥ 0 is a martingale under Q. This follows from thefact that the process M(t) := Λt −
∫ t∧τ0 λ(s) ds is a martingale under Q.
32 HANSJORG FURRER
Proposition 4.1. Let Z(t, T ), t ≤ T , be a random field with determinis-tic correlation structure c specified by d〈Z(·, T1), Z(·, T2)〉t = c(t, T1, T2) dt.Suppose the defaultable zero-bond price P (t, T ) is modeled as
P (t, T ) = exp−∫ T
tf(t, v) dv
,
where
df(t, T ) = µ(t, T ) dt+ σ(t, T ) dZ(t, T ) .
We suppose µ, σ satisfy the technical regularity conditions imposed by theHJM framework. Then, with given processes λ and L for the risk-neutraldefault intensity and fractional loss quota L, respectively, we have the riskneutral drift restriction and short spread condition
µ(t, T ) = σ(t, T )σ∗(t, T, T ) ,
rt − rt = λ(t)Lt ,(4.20)
respectively, where rt = f(t, t) and σ∗(t, T, S) =∫ Tt σ(t, v)c(t, S, v) dv.
Proof. We use the fact that bond prices discounted at the risk-free shortrate process r have to be Q -martingales. Let B(t) = exp
∫ t0 rs ds denote
the bank account numeraire. Then
(4.21) Z∗(t, T ) =P (t, T )B(t)
(1− Λt) +∫ t
0(1− Ls)
P (s−, T )B(s)
dΛs ,
where Λt is the default indicator process with the dynamics specified in (4.19).Recall that dB(t)−1 = −B(t)−1rt dt. The differential form of (4.21) thusreads
dZ∗(t, T ) = d(P (t, T )B(t)
)(1− Λt)−
P (t, T )B(t)
dΛt + (1− Lt)P (t−, T )B(t)
dΛt
=(dP (t, T )
B(t)− P (t, T )
B(t)rt dt
)(1− Λt)
− P (t, T )B(t)
dΛt + (1− Lt)P (t−, T )B(t)
dΛt .
It remains to specify the dynamics of P (t, T ). By analogy with the default-free case, we define It =
∫ Tt f(t, v) dv, whence P (t, T ) = exp−It. Applying
Ito’s lemma, we obtain
dP (t, T )P (t, T )
= −dIt +12d〈I·〉t
= rt dt− µ∗(t, T ) dt−∫ T
tdv σ(t, v) dZ(t, v)
+12
∫ T
tσ(t, u)σ∗(t, T, u) du dt ,
(4.22)
where rt = f(t, t) and µ∗(t, T ) =∫ Tt µ(t, s) ds, see Lemma 3.1. In order
Z∗(t, T ) be a Q-martingale, the drift term in the dynamics of Z∗(t, T ) must
THE TERM STRUCTURE AS A RANDOM FIELD 33
be equal to zero, whence
(4.23) 0 = rt − µ∗(t, T ) +12
∫ T
tσ(t, u)σ∗(t, T, u) du− rt − λ(t)Lt .
Equation (4.23) was obtained by inserting the dynamics of P (t, T ) givenin (4.22) into the expression for dZ∗(t, T ), thereby collecting the dt-terms.Taking partial derivatives in (4.23) with respect to T leaves the drift restric-tion in (4.20).
Remark. Schmidt [24], Theorem 3.1 derives an analogous result by meansof an eigenvalue expansion of the covariance operator D of a D-Wienerprocess. 2
5. Conclusions
The theory of random fields proves successful to the modeling of both theterm structure of interest rates as well as the various quantities inherent tocredit risk models. Our focus was on non-negative valued fields. We under-stand that interest rate or default intensity models which allow for negativevalues with positive probability are inconsistent with theory and intuition.We first introduced a short rate model that is based on the square of one-dimensional Brownian motion. This setup provides an affine term structurein the sense that zero-coupon bond prices are expressible as exponential-affine functions of the short rate. The main criticism of this approach is thatneither Brownian motion nor its square are stationary processes. It is mainlyfor this reason that we then proposed a model in which (the square of) Brow-nian motion is replaced by (the square of) an OU process. The OU processis a stationary zero-mean Gaussian process. Therefore, its square is a so-called chi-squared process on the line. The corresponding short rate modelis still analytically tractable and the affine structure also remains intact.Inspired by the chi-squared processes on the line, we extended the index setof the processes by an additional spatial parameter. We interpret the spatialcomponent as a label indexing a family of obligors. The two-parameter fieldscan be used either for the modeling of default intensities in a reduced formcredit risk model or for mimicking the issuers’ firm values. The difficultywith spatio-temporal models lies in the specification of an appropriate co-variance structure. It is convenient to work with separable covariance modelssince then the single-index term structure models can nicely be integratedinto the two-parameter framework.
In this paper we dealt with simple credit risk events. Yet we believe thatthe random field theory can be exploited further to capture these and otherevents more thoroughly. For instance, we could be interested in the ’fraction’of obligors whose firm value falls below a certain level at any time t ∈ [0, T ].The object of interest would then be (t, ι)|V (t, ι) < δ ⊂ [0, T ]×I . Thisthen leads to the study of level crossings and excursion sets of random fields,see Adler [1] for a discussion of these topics. Alternatively, we can think ofthe theory of stochastic set functions as an appropriate tool for modelingcredit risk events. For a region U ∈ Rd, the quantity Z(U) would represent
34 HANSJORG FURRER
a cluster of defaulted obligors in U , where Z(U) is a stationary stochasticset function, see for instance Matern [19], Section 2.6.Before proposing more sophisticated models, however, one should attemptto calibrate the simple ones as discussed in this paper to historical data.Only then do we know whether or not these models capture the complexityof interest and credit risk markets.
Acknowledgment
I would like to thank Damir Filipovic for drawing my attention to randomfields and his constant interest in this work. I am grateful to my super-visor Philipp Schonbucher for proposing the model (4.11) and the variousdiscussions related to it.
Appendix A. Karhunen-Loeve expansion for the OU process
The derivation of the Karhunen-Loeve expansion for the OU process goesalong the same lines as the proof of Proposition 2.1. Inserting the covariancefunction R(x, y) = c exp−β|x− y| into the integral equation (2.2) yields
λφ(x) =∫ T
0ce−β|x−y|φ(y) dy
=∫ x
0ce−β(x−y)φ(y) dy +
∫ T
xce−β(y−x)φ(y) dy .
Differentiating this equation twice with respect to x we obtain
λφ(x) =(−2βc+ λβ2
)φ(x) .
This can be rewritten in the form
(A.1) φ(x) + ω2φ(x) = 0 ,
where ω2 = (2βc− λβ2)/λ. The general solution of (A.1) is given by
φ(x) = A cos(ωx) +B sin(ωx)
with the two boundary conditions
βφ(0)− φ(0) = 0
βφ(T ) + φ(T ) = 0(A.2)
whence
Aβ −Bω = 0
A(β − ω tan(ωT )
)+B
(ω + β tan(ωT )
)= 0 .
(A.3)
The system (A.3) has a non-trivial solution if and only if the determinantof (
β −ωβ − ω tan(ωT ) ω + β tan(ωT )
)
THE TERM STRUCTURE AS A RANDOM FIELD 35
is zero, whence
ω + β tan(ωT ) = 0 and β − ω tan(ωT ) = 0.
Let ωk denote the roots of the equation ω + β tan(ωT ) = 0. It follows thatA = 0 and therefore φk(x) = B sin(ωkx). The constant B can be determinedusing the orthonormality condition:
1 =∫ T
0φ2k(y) dy = B2
∫ T
0sin2(ωky) dy = B2
(T
2− sin(2Tωk)
4ωk
).
Solving for B yields B =(T/2− sin(2Tωk)/(4ωk)
)−1/2. In the same way wecan calculate the constant A when considering the roots δj of the equationβ − δ tan(δT ) = 0:
1 =∫ T
0φ2j (y) dy = A2
∫ T
0cos2(δjy) dy = A2
(T
2+
sin(2Tδj)4δj
),
yielding A = (T/2 + sin(2Tδj)/(4δj))−1/2 .
Remark. The integral equation (2.2) can be solved analytically only in spe-cial cases like the ones we considered. Where this is not possible, numericalmethods are required to determine the eigenvalues and eigenfunctions of thecovariance function R. 2
References
[1] Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, Chichester.[2] Bochner, S. (1933). Monotone Funktionen, Stieltjessche Integrale und harmonische
Analyse. Math. Ann., Vol. 108, 378-410.[3] Collin-Dufresne, P., and Goldstein, R. S. (2003). Generalizing the affine framework
to HJM and random field models. Working paper.[4] Cramer, H. (1940). On the theory of stationary random processes. Ann. Math. Vol. 41,
215-230.[5] Cressie, N., and Huang, H. (1999). Classes of non-separable spatio-temporal station-
ary covariance functions. Journal of the American Statistical Association, Vol. 94,No. 448, 1330-1340.
[6] Dai, Q., and Singleton, K. J. (2000). Specification analysis of affine term structuremodels. The Journal of Finance. Vol. LV, No. 5, 1943–1978.
[7] Da Prato, G., and Zabczyk, J. (1992). Stochastic equations in infinite dimensions.Cambridge University Press.
[8] Duffee, G. R. (1996). Estimating the price of default risk. Working paper.[9] Duffie, D., Filipovic, D., and Schachermeyer, W. (2003). Affine processes and appli-
cations in finance. The Annals of Applied Probability. Vol. 13, No. 3. 984–1053.[10] Duffie, D., and Kan, R. (1996). A yield-factor model of interest rates. Mathematical
Finance. Vol. 6, No. 4, 379-406.[11] Duffie, D., and Singleton, K. J. (1999). Modeling term structures of defaultable bonds.
The Review of Financial Studies. Special Vol. 12, No. 4, 687-720.[12] Gneiting, T. (1997). Symmetric positive definite functions with applications in spatial
statistics. Diss. University Bayreuth.[13] Goldstein, R. S. (1997). The term structure of interest rates as a random field. Work-
ing paper.[14] Kac, M. (1946). On the average of a certain Wiener functional and a related limit the-
orem in calculus of probability. Transactions of the American Mathematical Society.Vol. 59, 401-414.
36 HANSJORG FURRER
[15] Karatzas, I., and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus.Springer, New York. Second Edition.
[16] Kennedy, D. P. (1994). The term structure of interest rates as a Gaussian randomfield. Mathematical Finance. Vol. 4, No. 3, 247-258.
[17] Kennedy, D. P. (1997). Characterizing Gaussian models of the term structure ofinterest rates. Mathematical Finance. Vol. 7, No. 2, 107-118.
[18] Lando, D. (1998). Cox processes and credit-risky securities. Review of DerivativesResearch. Vol. 2, 99-120.
[19] Matern, B. (1986). Spatial Variation. Lecture Notes in Statistics, No. 36, Springer.[20] Pitman, J. and Yor, M. (1982). A decomposition of Bessel bridges. Z. Wahrschein-
lichkeitstheorie verw. Gebiete. Vol. 95, 425-457.[21] Protter, P. (1990). Stochastic Integration and Differential Equations: A New Ap-
proach. Springer, Berlin.[22] Revuz, D, and Yor, M. (1994). Continuous Martingales and Brownian Motion.
Springer, Berlin.[23] Ripley, B. D. (1981). Spatial Statistics. Wiley, New York.[24] Schmidt, T. (2003). An infinite factor model for credit risk. Submitted.[25] Schonbucher, P. J. (2003). Credit Derivatives Pricing Models. John Wiley, England.
Department of Mathematics, ETH Zurich, CH–8092 Zurich, Switzerland
E-mail address: [email protected]
