Affine Processes, Arbitrage-Free Term Structures of Legendre
Polynomials, and Option Pricing
Caio Ibsen Rodrigues de Almeida∗
January 13, 2005
Abstract
Multivariate Affine term structure models have been increasingly used for pricing derivatives in fixed income markets. In these models, uncertainty of the term structure is driven by a state vector, while the short rate is an affine function of this vector. The model is characterized by a specific form for the stochastic differential equation (SDE) for the evolution of the state vector. This SDE presents restrictions on its drift term which rule out arbitrages in the market. In this paper we solve the following inverse problem: Suppose the term structure of interest rates is modeled by a linear combi-nation of Legendre polynomials with random coefficients. Is there any SDE for these coefficients which rules out arbitrages? This problem is of particular empirical interest because the Legendre model is an example of factor model with clear interpretation for each factor, in which regards movements of the term structure. Moreover, the Affine structure of the Legendre model implies knowledge of its conditional character-istic function. From the econometric perspective, we propose arbitrage-free Legendre models to describe the evolution of the term structure. From the pricing perspec-tive, we follow Duffie et al. (2000) in exploring Legendre conditional characteristic functions to obtain a computational tractable method to price fixed income deriva-tives. Closing the article, the empirical section presents precise evidence on the reward of implementing arbitrage-free parametric term structure models: The ability of ob-taining a good approximation for the state vector by simply using cross sectional data.
Keywords: Consistent Term Structure Models, Multi factor Affine Processes, Leg-endre Polynomials, Derivatives Pricing.
EFM Classification Code: 310
Research Area according to EFM C.C.: 310 or 450
∗IBMEC Business School, Av. Rio Branco 108 / 17th Floor, Centro, Rio de Janeiro, Brazil, Phone:
1
Introduction
Any acceptable model which prices interest rates derivatives must fit the term struc-ture observed today. This idea, pioneered by Ho and Lee (1986), has been explored by many researchers, from arbitrage free models of the short term rate (Black et al. (1990), Hull and White (1993) among others) to more complex models considering the evolution of the whole forward curve as an infinite system of stochastic differential equations (Heath, Jarrow and Morton (1992); HJM). In particular, these models for the forward curve expect as input a continuous initial forward rate curve. However, in the market, we just observe a discrete set of bond prices. This fact motivates one to propose parameterized families to estimate a continuous forward rate curve using the observed data. A very plausible ques-tion rises at this point: Choose a specific parametric family Θ of funcques-tions to represent the forward curves, and also an arbitrage free interest rate modelX 1. Suppose we use an initial curve that lay within Θ as input for modelX. Will this interest rate model evolve through forward curves that lay within family Θ? Motivated by this question, Bjork and Christensen (1999) define consistent pairs (Θ,X) as the ones whose answer to the above question is positive. In particular, they studied the problem of consistence between the Nelson-Siegel family (Nelson and Siegel (1987)) and any HJM interest rate model with deterministic volatility. They identified that there is no such interest model consistent with that family.
The Nelson-Siegel model is an important example of parametric family of forward rate curves because it is widely adopted by central banks in Europe (see for instance Svens-son (1994) or AnderSvens-son and Sleath (1999)). It models the forward curve as a combination of three linearly independent functions{1, e−αx, xe−αx}. Figure 1 show these three func-tions. The first term is related to long term forward rates, the second to short term rates, and the third to medium term rates. The curve shape of the forward curveG(z, .) is given by the expressionG(z, x) =z1+z2e−z4x+z3xe−z4x, wherez is a four-dimensional vector
representing the parameters andxdenotes time to maturity. Despite all its good empirical features and general acceptance by the financial community, Filipovic (1999) extended the results of Bjork and Christensen showing that there is no Itˆo process, including the ones with stochastic volatility, that is consistent with Nelson-Siegel family.
In a recent paper De Rossi (2004) applies the same consistency results obtained by Bjork and Christensen to propose a consistent exponential dynamic model for the instan-taneous forward rates curve, and estimates it using historical data on LIBOR and UK. swap rates, and a Kalman filter as the estimation tool. The main motivation behind this type of work is that whenever we obtain consistency between a parametric family of curves and an interest rate model, if that parametric family allows a good fitting of the cross sec-tional term structure of interest rates, the combined model parametric family/interest rate model will be useful to price derivatives and yet consistent with a dynamic econometric analysis of the term structure.
Burraschi and Corielli (2003) present theoretical results indicating that the use of inconsistent parametric families to obtain smooth interest rates curves, introduces
time-1
inconsistent errors that violate the standard self financing arguments of replicating strate-gies. This produces direct consequences to risk management procedures.
It appears to be of great interest to obtain parametric families to model forward curves that present good cross sectional fitting and simultaneously admit at least one consistent arbitrage free interest rate model, which will allow to consistently match the dynamics of bonds/swaps and options on these instruments.
In this paper, we show how to construct consistent interest rate processes for a family of term structures parameterized by linear combinations of Legendre polynomials. We base our results on the framework proposed in Filipovic (1999). The Legendre family has been successfully applied to estimate term structures in emerging markets both from a cross sectional perspective (Almeida et al. (1998)) and from a “time series” perspective (Almeida et al. (2003))2 and presents a natural interpretation as a factor model. Fac-tor models with well-suited loading facFac-tors are recently being explored to capture many stylized facts of term structures (see Diebold and Li (2003)). Moreover, we show that the Legendre model, under certain restrictions on the diffusion structure of the random Legendre coefficients3, can be seen as a particular Affine term structure model as first
characterized by Duffie and Kan (1996). Due to their analytical tractability, Affine mod-els have been recently used by the empirical research community to extract information about interest rates and risk premia historical behavior (for instance see Duffee (2002), Duarte (2004) and Dai and Singleton (2002) for empirical analysis of US data and Almeida (2004a) for Brazilian data) and also for pricing options (see Duffie et al. (2000), and Sin-gleton and Umantesev (2002)). Moreover, based on the theoretical results obtained in this paper, Almeida (2004b) estimates different Dynamic Legendre Models using Brazil-ian Swaps data.
The paper is organized as follows. Section 2 presents some basic relations which ap-pear in fixed income markets. In Section 3, we introduce the class of Affine Term Structure Models. Section 4 presents the parameterized Legendre family, and constructs arbitrage free interest rate models consistent with this family. In Section 5, we make a brief discus-sion of possible applications of the model, including dynamic term structure estimation and option pricing on the affine term structure setting, and show in particular how to price options using an arbitrage-free version of the Legendre model as proposed in Section 4. Section 6 is an empirical section. It presents an implementation of a multi-factor Gaus-sian Legendre Dynamic model, making use of Brazilian swaps data. There, an analysis of the term structure is performed, followed by details on the model estimation and by an interpretation of the risk premia as a function of the different sources of uncertainty which drive term structure movements. Section 7 concludes the article. The appendix presents the proof of Proposition 2.
2
In that paper, although time series for the coefficients multiplying the Legendre polynomials are obtained, no dynamic arbitrage free model is proposed. The coefficients series are obtained by sequential cross sectional estimation.
3
2
The Term Structure of Interest Rates
This section presents a brief description of basic notation in fixed income markets. Those acquainted with such notations might want to begin directly by Section 3.
Consider a complete probability space (Ω,F, P). LetW be ad-dimensional Brownian Motion constructed in this space, restricted to the interval of time [0, τ]. Fix also the standard filtration{Ft: 0≤t≤T}of W.
Assume the existence of an integrable short-rate processhrepresenting continuously compounding rate of interest on riskless securities. Define also the money market account
Dt = e
t
0hudu, defined by the value of one dollar invested on an account continuously
accumulating the short term interest rate. Investment of one unit of money on riskless securities from timetto timesyields Ds
Dt units at times.
A zero coupon bond is a fixed income instrument that pays one unit of money at its maturity time. Denote byP(t, T) the timetprice of a zero coupon with time of maturity
T. Absence of arbitrage implies the existence of a probability measureQ, equivalent toP, under which the price of any zero coupon bond, appropriately discounted by the money market account deflator, is aQ-martingale. Combining this fact with the fact that a zero coupon bond has price 1 at its maturity we can write:
P(t, T)
Dt
=EQ
P(T, T)
DT |Ft
=EQhe− 0Thudu|F
t
i
(1)
where the measurability ofDt leads us directly to the well known formula for the price of
a zero coupon bond:
P(t, T) =EQ
h
e− tThudu|Ft
i
(2)
The continuously compounded yield to maturityR(t, x) of a zero coupon bond maturing at timet+x is defined by:
R(t, x) =−log(P(t, t+x))
x (3)
R(t, .) is denominated the term structure of interest rates, and it is a function that maps maturities or terms into annualized interest rates charged for loans with duration equal to these terms.
2.1 Forward Rates
From the term structure of interest rates we can obtain forward rates, which are breakeven rates that equate the return of an investment on a long term bond to the return of an investment on two shorter term bonds with sum of maturities equal to the long term bond maturity. Denote by g(t, t1, t2) the time t forward rate for time t1 with maturity
t2−t1. It is related to the term structure by the following equation:
Usingt1 =t+x, reorganizing terms and taking the limit whent2converges tot1 we obtain
the instantaneous forward rates curve:
r(t, x) = lim
t2→t+xg(t, t+x, t2) =R(t, x) +x
∂R(t, x)
∂dx (5)
wherer(t, x) denotes the timetinstantaneous forward rate with time to maturityx. It is related to the bond price by the following formula:
P(t, T) =e− 0T−tr(t,s)ds (6)
Finally the short term rate is obtained as the limit of the instantaneous forward rates whenx→0:
ht= lim
x→0r(t, x) (7)
3
Affine Term Structure Models
Consider a complete probability space (Ω,F, P) and a n-dimensional state space processZ ={Zt}0<t<∞ driving uncertainty of the term structure of interest rates, whose
dynamics satisfies the following stochastic differential equation:
dZt=ν(Zt)dt+σ(Zt)dWt (8)
withWt being a d-dimensional Brownian Motion under P, ν :Rn → Rn and σ :Rn →
Rn×d.
Assumption of absence of arbitrage4 guarantees the existence of an Equivalent Martingale Measure Q under which the prices of bonds discounted by an appropriate deflator are
Q-martingales5 (Martingale Condition). Under the risk neutral measure Q, process Z
evolves according to:
dZt=µ(Zt)dt+σ(Zt)dWt∗ (9)
whereWt∗isd-dimensional Brownian Motion underQ, andµcan be obtained as a specific function of σ so that the Martingale Condition is satisfied. We interchange from one measure to the other by making use of Girsanov’s Theorem which relates the two Brownian Motion vectors by:
Wt∗ =Wt+
Z t
0
Λsds (10)
where vector Λ is the market price of risk, the price payed by the uncertainty generated by each coordinate of the Brownian Motion, in this continuous time setting. Making use of Equations (8)-(10) we see thatµ must satisfy:
µ(Zt) =ν(Zt)−Λtσ(Zt) (11)
4
with some extra technical conditions (see Duffie (2001), Chapter 6, Section K). 5
We denominate the interest model (8)-(11) Affine, if the short rate is an affine function ofZ and, if in addition, at instantt, the price of a zero coupon bond with time of maturityT,P(t, T) may be written as:
P(t, T) =eA(T−t)+B(T−t)Zt (12)
whereA:R → R andB :R → Rn areC1 functions. Equation (12) implies that the spot
curve can be written as a linear combination of the state space variables with coefficients dependent on the time to maturityx:
R(t, x) =−(A(x) +B(x)Zt)
x (13)
Using the fact thatDt−1P(t, T) is aQ-martingale and equation (12), Duffie and Kan obtained two important results:
1) They showed that if the functions{Bi}i=1,2,...,n and{Bi}{Bj}i=1,2,...,n;j=1,2,...,n are
lin-early independent6, the Martingale Condition constrains functions µandσσt to be affine
inZ, allowing the SDE (9) to be written in the following form:
dZt= (aZt+b)dt+ Σ
p
γ1(Zt) 0 . . . 0
0 pγ2(Zt) . . . 0
..
. ... . .. ...
0 0 . . . pγd(Zt)
dWt∗, (14)
wherea∈ Rn×n,b∈ Rn, Σ∈ Rn×d,γ
i(x) =αi+βix,αi ∈ R and βi ∈ Rn.
2) They also showed that the Feyman-kac equation for the function P gives ordinary differential equations that the functionsA(x) andB(x) must satisfy, linking these functions with the functionsµand σ defining the SDE (14) (For details see Duffie (2001)).
For a more general reference on Affine Models with applications in finance, see Duffie et al (2003).
4
The Legendre Family
4.1 Is Legendre Affine?
Almeida et al. (1998) use the Legendre family to estimate term structures of interest rates in Emerging markets. They model the term structure as a linear combination of Legendre polynomials7:
R(z, x) =
n
X
j=1
zjpj−1(2 x
l −1) (15)
6
This condition guarantees unique correspondence between functions {Bi}i=1,2,...,n and the SDE for the state vectorZ.
7
where l represents the largest maturity in the fixed income market under consideration, andpjstands for the Legendre polynomial of degree j, which can be obtained by Rodrigues’
formula8:
pj(x) =
1 2jj!
dj dxj(x
2−1)j (16)
Figure (2) presents the first four Legendre polynomials. Note how each polynomial has a clear interpretation in terms of the type of movements that they generate for the term structure. The polynomial of degree 0 generates changes in level, the polynomial of degree 1 is responsible for changes in the slope, polynomial of degree 2 for changes in the curvature, polynomial of degree 3 for more complex changes in the curvature, and so on.
It is straightforward to note that the Legendre model is in the class of Affine processes in the sense that the term structure is represented by an Affine function of the state space vector, as shown in Equation (15). Moreover, comparing Equations (13) and (15) we obtain:
AL(x) = 0
BL(x) =
−xp0(2lx −1)
−xp1(2lx −1)
.. .
−xpn−1(2lx −1)
t
(17)
The first interesting thing to be noted is that the Legendre model appears to be a par-ticular Affine Term Structure Model withthe very nice property that no ordinary
differential equation should be solved in order to obtain the functions A and
B which appear in the yield Equation (13). The functions are pre-defined and are of course directly related to the Legendre polynomials themselves as shown in Equation (17). Then, two natural questions are risen by the fact that the Legendre model appears to be a particular Affine Model: What type of dynamics (SDEs) should the Legendre state space vector follows in order to preclude arbitrages in the market? Will these dy-namics be of the type presented in Equation (14)? Well, it is easy to show that functions
{BiL}i=1,2,...,n and {BiL}{BjL}i=1,2,...,n;j=1,2,...,n are not linearly independent. For example,
functions B1L, B2L, B3L and B1LB2L are linearly dependent. This can be proved by solving the following equation:
aBL
1(x) +bBL2(x) +cB3L(x) =B1L(x)B2L(x)
−ax−bx(2lx −1)−cx2(3(2lx−1)2−1) =x2(2lx−1) (18)
whose solution is given bya=−6l,b=−2l and c=−3l.
The linear dependence of the functions above prevents us from using Duffie and Kan results to show that the SDE followed by Legendre dynamic factors would be given by Equation (14). However in the next section we show that they will follow SDEs slightly more general regarding the structure of the diffusion coefficients σ as appears in Equation (9) but more restricted regarding the dimension of the Brownian Motion vector characterizing uncertainty of the term structure.
8
4.2 Legendre Consistent Interest Rate Models
Filipovic (1999) considers the existence of a forward curve manifoldζand of a finite state space process{Zt}0≤t≤∞in (Ω,F, Q,Ft). Z evolves according to an SDE where the
drift and diffusion terms are just imposed to be progressively measurable:
Zti =Z0i +
Z t
0
bisds+
d
X
j=1
Z t
0
σsijdWs∗j, i= 1, ..., n, 0≤t≤ ∞ (19)
This is the process followed by the parameters defining the shape of the manifold ζ, in the sense that the instantaneous forward rate curve is imposed to be explicitly obtained from the finite dimensional processZ, through the application of Z on the deterministic functionG(., x):
r(t, x) =G(Zt, x) (20)
This actually defines an interest rate model. The consistence between processZ and fam-ilyζ is obtained if the discounted prices of all zero coupon bonds, discounted by the short rate processh obtained fromr using Equation (7), follow Q-martingales.
Using Ito’s formula, Filipovic proves the following proposition:
Proposition 1. Z is consistent with family ζ only if, probability almost surely, the following equation holds:
∂G(Z, x)
∂x =
n
X
i=1
bi∂G(Z, x) ∂zi +1 2 n X i,j=1 aij
∂2G(Z, x)
∂zizj2 −
2∂G(Z, x)
∂zi
Z x
0
∂G(Z, y)
∂zj
dy
(21)
wherea=σσt.
Proof. See Proposition 3.2 in Filipovic (1999).
Note that Equation (21) is necessary but not sufficient to guarantee consistence of the interest rate model. The reason is that some technical conditions should be imposed to the diffusion coefficientσto guarantee that discounted bond prices will be indeed martingales and not local martingales9. For instance, if we impose thatσ is a bounded function than
Equation (21) is also sufficient. Its sufficiency is important because it is the way we use to propose arbitrage free Legendre models.
Assuming the minimal technical conditions which guarantee sufficiency of Equation (21), we consider the Legendre family to take place as a candidate familyζ. The Legendre instantaneous forward rates can be written as a polynomial on the maturity variable x
with coefficients{z˜j}j=1,...,n obtained as linear combinations of the variables{zj}j=1,...,n:
G(˜z, x) =
n
X
j=1
˜
zjxj−1 (22)
To obtain Equation (22) use Equations (5) and (15):
G(z, x) =
n
X
j=1
zj
"
(pj−1(
2x−1
l )) +
2x l
dpj−1(y) dy y=2x−1
l
#
(23)
9
Now, collect terms matching the powers ofxin Equations (22) and (23) to obtain a linear system relating variables ˜zs and zs.
Due to the structure of the Legendre polynomials, the inverse problem of identifying the drift and diffusion of process Z consistent with the Legendre family does not present a unique solution. Nevertheless, applying Proposition 1 to the Legendre family, we are able to prove the following proposition10:
Proposition 2. Assume d > [n2], and parameterize the forward rate curve using the first n Legendre polynomials, using Equation (22).Then there is at least one non-trivial state space processZconsistent with the Legendre family satisfyingσii6= 0, i= 1,2, ...,[n2],
in Equation (19). Proof: See Appendix.
Proposition 2 proves that for any Legendre parameterization of the term structure there exist consistent processes depending on different coordinates of the original Brownian Mo-tion vector characterizing uncertainty of interest rates. This is a remarkable fact when compared for instance with the Nelson-Siegel family consistent processes which are of de-terministic nature. Of course, not everything is perfect because that fact that the Legendre dynamic model allows us to obtain prices of zero coupon bonds without solving ordinary differential equations comes with the cost that its dynamic arbitrage free models can not depend on all d coordinates of the original Brownian motion. This restriction appears when we impose that the discounted prices of the zero coupons should beQ-martingales. In the next subsection we present some examples of SDEs consistent with the Legendre family.
4.2.1 Some Examples
We present three simple examples to illustrate the result of Proposition 2: A two dimensional factor model proposed in Filipovic (2001), a three dimensional factor model explored in Almeida et. all (2003), and a four dimensional factor model which admits a bi-dimensional brownian motion driving the uncertainty in the term structure. Note that in our notation, we use interchangeably the concept of factors and state variables, as opposed to other works where factors only represent stochastic state variables.
A Two Factor Model
Filipovic (2001) showed that Equation (14) which presents the dynamics of Affine models which satisfy the condition that {Bi}i=1,2,...,n and {Bi}{Bj}i=1,2,...,n;j=1,2,...,n are
linearly independent, is not in general satisfied by all Affine processes. Filipovic pro-posed a two dimensional affine factor model, by setting in Equation (12) A(x) = 0 and
B(x) = [−x− x22]. Let W be a one dimensional Brownian Motion andβ :R2 → R be a generic function, not necessarily affine. The following diffusionZ ∈ R2+:
dZt1=Zt2dt+p
β(Zt1, Zt2)dWt∗
dZt2=β(Zt1, Zt2)dt (24)
10
was shown to be consistent with the linear polynomial parametric form for the spot curve:
R(z, x) =z1+x
2z2 (25)
The parametric form of the spot curve in Equation (25) is equivalent to the one obtained using the first two Legendre polynomials RL(z, x) = z
1 +z2(2lx −1). So we would
ex-pect that the diffusion presented in Equation (24) should be consistent with a Legendre dynamic model with two factors. This is precisely true according to the construction ar-gument given in the proof of Proposition 2 (see Appendix).
We see that the only dynamic affine two factor models consistent with a term struc-ture parameterized by a linear function on the maturity variable, are those which havea stochastic level for the term structure with a deterministic slope (conditioned on the level).
Depending on the choice of β we can obtain different (variants of) one factor mod-els that have been proposed in the past literature. For instance, if we take β constant in Equation (24), we obtain a one factor Gaussian model with deterministically varying drift
Zt2 =Z02+βt, which is exactly the Ho and Lee (1986) model. This model presents closed form formulas for both zero coupon bond prices, and zero coupon bond option prices (see James and Webber (2000)). If we setβ =−aZt2 we obtain a Gaussian model with drift and volatility varying deterministically along time. In this case there are two possibilities:
If a > 0 the deterministic slope Zt2 =Z02e−at should be always negative, asymptotically
going to zero, in order to guarantee that β is always positive. This is obtained simply setting Z02 < 0. The stochastic level Z1 will be a Gaussian process with distribution
Zt1 ∼ N(Z01 + (1−
Z2
0
a e−at),(Z02)2
Rt
0 e−2asds). On the other hand, if a < 0 then Z2
should be always positive in order to guarantee positivity of β and this is not an inter-esting case because the term structure model is explosive. Just as a simple illustration, Figure 3 presents the results of a simulation of a discrete version of Equation (24), using
a= 1, Z02 =−0.03, and Z01 = 0.28, and a term structure with hypothetical maturities of
{0.5,1,2,3,4,5,7,10} years. There we have the evolution of the stochastic level Z1, the deterministic slopeZ2, and the whole term structure obtained by using Equation (25)11.
If we further increase the level of complexity of the model by allowing β to depend onZ1then we would have the instantaneous volatility of the stochastic levelZ1depending on the level value, and its drift depending on the integral of its path. In this case, more careful conditions should be imposed to the parameters of the model in order to guarantee that the instantaneous volatility of the stochastic levelZ1 will remain non negative (see Dai and Singleton (2000)).
A clear limitation of the model though, is that it does not exhibit mean reversion on the stochastic level dynamics, a feature that even the original Vasicek (1977) model addresses. In addition, as a one stochastic factor model, it is certainly not good enough to capture the evolution of fixed income data as documented by the empirical finance re-search community. In general, two or three stochastic factors are found to be necessary to drive the dynamics of term structures (see for instance, Litterman and Scheinkman (1991) for US treasure market, and Almeida et al. (2003) for Brazilian Sovereign market).
11
Will Three Factor Dynamic Legendre Models do a Better Job?
Our second example considers the parameterization obtained for the term struc-ture of interest rates when the first three Legendre polynomials are used:
R(z, x) =P3
j=1zjpj−1(2lx−1)
=z1+z2(2lx −1) + z3
2 (3(2
x
l −1)2−1)
= (z1−z2+z3) +2l(z2−3z3)x+ l62z3x2
(26)
which directly implies the instantaneous forward curve:
G(z, x) = (z1−z2+z3) + 4
l(z2−3z3)x+
18
l2z3x
2 (27)
Making the transformation ˜z1 =z1−z2+z3, ˜z2 = 4l(z2−3z3) and ˜z3 = 18l2z3, we get an
equation in the same form as Equation (22). Now, applying Proposition 2 it is not difficult to see that the following dynamics provides a consistent state space vectorZ:
dZ˜t1= ˜Zt2dt+
q
β( ˜Zt1,Z˜t2,Z˜t3)dWt∗ dZ˜t2= 2 ˜Zt3+β( ˜Zt1,Z˜t2,Z˜t3)dt
˜
Zt3 = ˜Z03
(28)
where in this caseβ :R3 → Ris a generic function. Note that to obtain the dynamics of the original variables we just have to solve the following linear system for vector (z1, z2, z3)t:
˜
z1=z1−z2+z3
˜
z2= 4l(z2−3z3)
˜
z3= 18l2z3
(29)
which has as solutionz1 = ˜z1+ 4lz˜2+l92z˜3,z2 = 4lz˜2+l62z˜3, and z3 = 18l2z˜3.
Observe that the three factor model does not present any considerable extra feature in its structure, when compared to the previously proposed two factor model. The reasons for this are two: first, it still presents only one source of randomness (one dimensional Brownian Motion), and second, the third factor is not only a deterministic function but is also constant along time. in other words the models presented in Equations (24) and (28) are essentially equivalent. The lesson we learn from this observation is that the price that is paid to impose restrictions to the Legendre factors dynamics in order to preclude arbitrages in the market is that Dynamic Legendre models with an odd number of factors won’t play an important role in empirical analysis. Said that we move to our last example.
Genuine Bi-dimensional Uncertainty
for instance Heidari and Wu (2003) or Litterman and Scheinkman (1991)).
Consider the parameterization for the term structure using the first four Legendre polynomials:
R(z, x) =P4
j=1zjpj−1(2lx −1)
=z1+z2(2lx −1) +z3
2(3(2
x
l −1)2−1) +
z4
2(5(2
x
l −1)3−3(2
x
l −1))
= (z1−z2+z3−z4) +2l(z2−3z3+ 6z4)x+l62(z3−5z4)x2+20l3z4x3
(30)
which on its turn implies:
G(z, x) = (z1−z2+z3−z4) +
4
l(z2−3z3+ 6z4)x+
18
l2(z3−5z4)x
2+ 80
l3z4x
3 (31)
Making the transformation ˜z1 =z1−z2+z3−z4, ˜z2 = 4l(z2−3z3+ 6z4), ˜z3= 18l2(z3−5z4),
and ˜z4= 80l3z4, and again applying Proposition 2, the following dynamics provides a state
space vectorZ consistent with the term structure parameterized by Equation (30):
dZ˜t1 = ˜Zt2dt+
q
β( ˜Zt1,Z˜t2,Z˜t3,Z˜4)dWt∗1
dZ˜t2 = 2 ˜Zt3+β( ˜Zt1,Z˜t2,Z˜t3)dt+
q
γ( ˜Zt1,Z˜t2,Z˜t3,Z˜4)dWt∗2 dZ˜t3 = 3 ˜Zt4dt
dZ˜t4 = 12γ( ˜Zt1,Z˜t2,Z˜t3,Z˜4)dt
(32)
where in this case β :R4 → R and γ :R4 → R are generic functions. Again, to obtain
the dynamics of the original variablesz we just have to solve a simple linear system for vector (z1, z2, z3, z4)t as previously showed in example two:
˜
z1 =z1−z2+z3−z4
˜
z2 = 4l(z2−3z3+ 6z4)
˜
z3 = 18l2(z3−5z4)
˜
z4 = 80l3z4
(33)
Interesting to note from the System (33) is that although factors ˜Z1 and ˜Z2 are not
corre-lated, the original variables the levelZ1 and slope Z2 are, something that is usually true
in empirical studies of the term structure. As an illustration of the extra flexibility of this model when compared to the previous two factor model, Figure 4 presents four different term structure evolutions along time under the four factor Legendre Dynamic Model.
Girsanov plays a role in Flexibility
5
Applications
5.1 Econometric Analysis of the Term Structure
This consists in estimating a dynamic term structure model to fit historical term structure data. The question is, what are the purposes of fitting the term structure? The econometrician is interested in understanding the behavior of fixed income instru-ments and their relations to economic and political events. While theoretical models were proposed, like the seminal one dimensional Gaussian model by Vasicek (1977), the mul-tifactor Gaussian model by Langetieg (1980), the square root equilibrium model by Cox et al. (1985), and the general multifactor affine model by Duffie and Kan (1996), empir-ical implementations took much longer to appear. For instance, Chen and Scott (1993) propose a multifactor estimation of the Cox et al. model while Dai and Singleton (2000) estimate a multifactor gaussian model, among other affine models, for the US term struc-ture of treasure bonds. Nowadays these empirical implementations are in the center of the discussion of the research community, manifested through empirical papers in mon-etary policy (Piazzesi (2003)), combination of arbitrage free term structure models with macroeconomic variables (Rudebusch and Wu (2003)), credit derivatives valuation (Duffie et al. (2003)), among others.
We believe the Legendre dynamic (arbitrage free) model can be inserted in any of these empirical contexts. It presents one big advantage over general affine term struc-ture models regarding the risk neutral measure: There is no need for solving ordinary differential equations to obtain the price of a zero coupon bond, which is directly given by Equation (12) with Equation (17) substituted on it. For this reason, it is simpler to implement than a general affine term structure model presenting less computational costs. Steps towards the empirical direction can be observed in Almeida (2004b), where different Legendre Dynamic models are estimated using Brazilian interest rates swap data. The idea there is to show how easy interpretation of Legendre dynamic factors allows one to pick up specific qualitative characteristics of Affine processes. In particular, in order to offer further illustration on this paper, we implement in Section?? a Gaussian version of the Legendre dynamic model, with 6 polynomials parameterizing the term structure12.
The key element to understand when implementing dynamic term structure models is that one have to work back and forth between the risk neutral probability measure and the physical probability measure. That happens because although the dynamics is estimated under the physical (or historical) measure13 the pricing of the fixed income in-struments is performed under the artificially created risk neutral measure. This means that we write the likelihood function for the state space vector under the physical mea-sure while we fit the cross section of prices (or yields) under the risk neutral meamea-sure. As Equation (10) shows once one parameterizes the model under one of the measures and also proposes a parametric form for the market price of risk Λ, the whole model is specified under both measures.
12
For more operational details on the implementation see Almeida(2004b). 13
It has been recently observed by Duffee (2002) that one of14 the most general
pa-rameterizations for the market price of risk of multifactor affine models which guarantees that the dynamic of the state vector is affine under both probability measures is given by:
Λt=
p
Stλ0+
q
St−λYYt, (34)
where S is the matrix that post multiply matrix Σ in Equation (14), and S− is defined by:
Stii−=
( 1
Sii t
, if inf(αi+βitYt)>0.
0, otherwise.
)
(35)
Almeida (2004b) explores this parameterization of the market prices of risk when imple-menting Legendre Dynamic Models. In Section ??, we also use this parameterization to implement the Legendre Gaussian model.
5.2 Option Pricing with the Legendre Interest Rate Model
Suppose fixed an affine term structure modelX ∈ X, where X is the state space vector driven by an SDE of the type (14). Assume that you can represent the short-term ratert=R(Xt, t) whereR is an affine function. Duffie et al. (2000) derived a closed-form
expression for the transform:
E
exp(−
Z T
t
R(Xs, s)ds)(v0+v1.XT)eu.XT|Ft
(36)
They obtain this result by defining the function:
ψX(u, Xt, t, T) =EX
exp(−
Z T
t
R(Xs)ds)euXT|Ft
(37)
and noting that the affine structure of the term structure model guarantees that function
ψsatisfies:
ψX(u, x, t, T) =ef(t)+g(t)x (38)
where f and g satisfy certain specific complex-valued ODEs depending on the original drift and diffusion coefficients of processX, and also on functionR.
Using a slight modification of this result, they show how to price options on zero coupon bonds. We already know that the price of a zero coupon bond under an Affine model is exponential affine. Then, letting ρ(t, T) = exp(−RT
t R(Xs, s)ds), a call option
has the following schematic expression for its price:
Ct=Et
ρ(t, T)(eu.XT −K)+=E
t
ρ(t, T)eu.XT1
{δXT≥K}
−KEt
ρ(t, T)1{δXT≥K}
(39)
14
The idea for this calculation comes from noting that under Affine Models, using Fourier methods they are able to efficiently calculate the following expression:
G(y, t, u, δ) =E
exp(−
Z t
0
R(Xs, s)ds)eu.Xt1{δXt≤y}
(40)
So the cost of calculating the price of a zero coupon option under an Affine Model is the cost of solving three pairs of Ricatti equations (one to price the zero coupon bond, two to price the option), and using two times Levy’s Fourier inversion formula (see Duffie et al. (2000)) to obtain functionGfor the two terms that appear in Equation (39).
Under the derivatives pricing framework, the Legendre Dynamic Model does not bring any additional advantage when compared to other Affine Processes. The reason for this is that to price a derivative under the Legendre model one will have to solve the same type of Ricatti equations one would solve if using any Affine Model. The only advantage is that these Ricatti Equations shouldn’t be solved to price the bond. On the other hand, it shares all the advantages of other affine processes as well. Once we solve the inverse problem of proposing one consistent SDE system for the Legendre coefficients, we can apply the methodology devised by Duffie et al. (2000) to price zero coupon bond options, caps and floors, or any one of the new analytical methodologies proposed to price coupon bond options and swaptions (see Singleton and Umantesev (2003) and Collin Dufresne and Goldstein (2002)).
One particular case which is worth mentioning is that of a multifactor Legendre Dynamic Model where all factors are Gaussian or deterministic. This can be seen, for instance, in example 3 of the last subsection, by settingβandγequal to constant numbers. This would give us Z4 as a linear function Zt4 = Z04 + γ2t, Z3 a quadratic function,
Zt3 = 3Z04t+ 3γt2, and Z2 and Z1 Gaussian processes. In this case, the risk neutral distribution of the price of a zero coupon bond will be log-normal, because:
P(t, T) =e−xR(t,x) =e−xR(Zt,x) (41)
wherex=T−t, and:
R(Zt, x) =
4
X
j=1
Ztjpj−1(
2x
l −1) (42)
Then, by the fact that a linear combination of the Z variables will be Gaussian, we directly note that for any maturityT the bond price will be log-normal. At this point we can use well known results on forward measures, first obtained by Jamshidian (1989) for the Vasicek (1977) model, and later generalized by Geman et al. (1995). By Ito’s lemma:
dP(t, T) =P(t, T)(rtdt−
p
βxdWt∗1−√γx(2x
l −1)dW
∗2
t ) (43)
Letη(t, T) =qβx2+γx2(2x
l −1)2. The price of a call option with strikeK and maturity
U, on a T maturity zero coupon bond, will be given by15:
Ct=P(t, U)N(d+(t, T))−KP(t, T)N(d−(t, T))
d+,−(t, T) =lnP(t,U)
P(t,T)
−lnK+−12v2U(t, T)(vU(t, T))−1
vU2(t, T) =RT
t |η(u, T)−η(u, U)|
2du
(44)
15
6
Implementation of a 6 Factor Legendre Dynamic Model
In this Section, following exactly the same idea of the examples presented in section 4.2.1, we construct and implement an arbitrage-free model where the term structure is driven by a linear combination of the first 6 Legendre polynomials.
Data consists of historical series of Brazilian interest rates swaps for maturities 30, 60, 90, 120, 180, 270, 360 and 720 days, from August 2, 1999 to January 29, 200316. Figure
5 presents the historical evolution of the Brazilian swap data. Almeida (2004a) applied Principal Component Analysis (PCA) on the first differences of swap yields and showed that three factors account for 98.7% of the movements of the swap term structure for the period from January 2, 2001 to January 29, 2003. We apply PCA for the longer sample and confirm the fact that three factors explain the majority of the swap term structure movements (98.5%). Using this fact, for each day, we apply the cross sectional Legendre model (Almeida et al. (1998)), using the three first Legendre polynomials, constant, linear and quadratic. Figure 6 presents the fitting of the static model for four different days: August 6, 1999, April 14, 2000, October 11, 2002 and January 29, 2003. Blue points represent the swaps yields while the dashed line represents the Legendre polynomial fitted term structure. Note that for all different historical moments the polynomial fitting works well, with better performance when the term structure is less concave (smaller curvature factor). We give the denomination of Legendre coefficient of degree j to the estimated coefficient multiplying the Legendre polynomial of degreejin the fitting procedure. Figure 7 presents the time series of the three Legendre coefficients. The Legendre coefficient of degree 0 represents the level factor. It has respective mean and standard deviation values of 22.03% and 4.02%. Intuitively, from investors viewpoint, high values of the level factor indicate perception of immediate risk on lending money. The Legendre coefficient of degree 1 represents the slope factor. Its mean and standard deviations are respectively 2.88% and 2.19%. Low values of the slope factor are consistent with flat term structures while high values are consistent with steep term structures and indicate expectation of future risk in lending money for short and medium term maturities. Note on Figure 7 that the term structure has been flat during the year 2000 and very steep during 2002, year of president’s election in Brazil. In 2001 the slope factor achieves its higher value around the September 11 attack to the World Trade Center. The Legendre coefficient of degree 2 represents the torsion factor, and basically indicates the degree of concavity of the term structure. Negative values indicate concavity while positive values convexity. Its mean and standard deviations are respectively -0.66% and 0.85%. Note that along almost the whole sample path the term structure presents concave curvature. The three factors have a high degree of correlation, as usually perception of immediate risk (level factor) comes together with higher expectation of future risks (slope and curvature factors). Table 1 presents the correlation coefficients of the Legendre factors obtained by the Static model.
At this point, we know from previous examples that for any parameterization of the term structure as a linear combination of a fixed finite number of Legendre polynomials (sayn), we can always obtain a dynamic arbitrage-free model, with a very general diffusion
16
structure for the n
2
Legendre coefficients. In particular, for the implementation of a Gaussian model, we assume that their diffusion will match that of affine models, presented in Equation (14). That will imply the following schematic SDE for the Gaussian Legendre Dynamic model:
dZt=µQ(Zt)dt+ ΣZtdWt∗ (45)
Now, using Proposition 1 we obtain the exact restriction that the drift of the state variables should satisfy as a function of the diffusion. In particular, for the deterministic factors (the last three factors), we can obtain their explicit deterministic dynamics:
µQ(Zt)4 = 0.2Σ222+ 2.6751Σ233+ 8.75Zt,5−3.15Zt,6 µQ(Zt)5 = 0.2577Σ233+ 10.8Zt,6
µQ(Zt)6 = 0.1431Σ233
(46)
Finally we explicitly solve the simple ODE’s implied for these factors:
Zt,4=Z0,4+ (0.2Σ 2
22+ 2.6751Σ 2
33+ 8.75Z0,5−3.15Z0,6)t+ (1.8041Σ 2
33+ 94.5Z0,6)
t2
2 + 13.5231Σ
2 33
t3
6 (47)
Zt,5=Z0,5+ (0.2577Σ233+ 10.8Z0,6)t+ 1.5455Σ233 t2
2 (48)
Zt,6 =Z0,6+ 0.1431Σ233t (49)
At this point, we explicitly see that the dynamics of the state variablesZt,4,Zt,5 and Zt,6
are deterministic and, in addition, are completely determined by the parameters Σ22 and
Σ33, and the initial conditionsZ0,4,Z0,5 and Z0,6, which are also treated as parameters of
the model.
We use the time series of the Legendre static factors to identify which swaps should be priced without error. The time series of the residuals from the static fitting procedure performed above indicate that the residuals in fitting the swaps with maturities 60, 270 and 720 days present the smallest standard deviations. Then, we assume that these swaps are priced exactly. We estimate the model by Maximum Likelihood, with maximum value of the log-likelihood function achieved being 42.6117. For each timet, the implied stochastic factors, first 3 variables in the state vectorYt,1, Yt,2, Yt,3, are extracted using the following
linear system:
Swtexact−[p3(xexact)p4(xexact)p5(xexact)]
Zt,4 Zt,5 Zt,6
= [p0(xexact)p1(xexact)p2(xexact)]
Zt,1 Zt,2 Zt,3
(50) whereSwtexact denotes the vector of swap rates priced exactly,pi(xexact) denotes the
Leg-endre polynomial of degree i evaluated at the vector xexact, and xexact is a vector of
transformed maturitiesxexact = 2τexactl −1,τexact being the maturities of the swaps priced
exactly.
Figure 8 shows the following remarkable fact: That for the Gaussian model, the
17
stochastic Legendre factors are not very much affected if obtained by sequentially solving linear regressions using the static model, in stead of by solving the full dynamic model including the deterministic factors. It presents, for the factors attached to the first three Legendre polynomials, level, slope, and curvature, the difference between Dynamic factor obtained by Maximum Likelihood and Static factor, obtained by running independent cross sectional regressions. Note that for the three factors the differences along time are all less than 50 bp, with mean and standard deviations of: 1.2 and 6 bp for the level factor, 5.9 and 10 bp for the slope factor, and, 2.1 and 11 bp for the curvature factor. In addition, Figure 9 confirms the fact that the deterministic factors have practically no influence in the implied values of the dynamic level, slope, and curvature factors, with values of less than 10 bp for the three deterministic factors, along the whole historical sample path. A plausible explanation for this fact is that the model restricts the initial values Y0,4, Y0,5
and Y0,6, and the parameters Σ2,2 and Σ3,3 in a way that the influence of the
determin-istic factors is minimal. The reason is that determindetermin-istic factors do not capture well the stochastic behavior of the term structure, only existing for reasons of consistency of the model, imposing some restrictions on the parametric space. This analysis indicate that for the identification of qualitative properties of the dynamic model we may use the time series obtained thorough the linear regressions of the static model. This is a very interest-ing approach from the practitioner viewpoint: Direct interpretation of the stochastic factors as responsible for different types of movements of the term structure is provided by the model, with the additional advantage of allowing the analysis of the factor dynamics to be done without the actual implementation of the model.
Table 2 presents the parameters values, their standard deviations calculated by the Outer Product Method (BHHH), and the ratio std valuevalue which allow the performance of standard asymptotic tests of parameters significance. Bold ratios indicate significant parameters at a 95% confidence level, with exception of parameter ΛY(1,2), which is
in De Rossi (2004)). The average daily value for the Legendre dynamic gaussian model SSE is 3.7 10−5, approximately twice the value of SSE for De Rossi’s model. However For approximately 70% of the historical sample the Legendre model presents sum of squared residuals smaller than the two factor model.
The fact that each dynamic factor plays a role as a known movement of the term structure allows a direct interpretation of the risk premia charged by investors in the Brazilian swap market. Whenever analyzing risk premia and change of measure, for each factor on the model, at least two different effects should be considered: How much the factor dynamics is affected with the change of measure, and how much the factor itself contributes to the prices of risk of each source of uncertainty, represented by each entry in the Brownian Motion vector. In this sense, matrix ΛY indicates that investors perception
of risk is primarily related to the slope factor. First, the slope factor presents the most affected drift in the change from the physical measure to the risk neutral measure, through ΛY(2,2) and ΛY(2,3), which are the only ΛY-significant parameters at a 95% confidence
level. In addition, the slope factor has significant effect on the risk premia charged on the first and second fundamental sources of uncertainty on the term structure18 through
parameters ΛY(1,2) and ΛY(2,2). We can also see that curvature has its role on the
premia of the slope factor (ΛY(2,3)) but investors do not directly charge risk premia for
the curvature factor (ΛY(3,3) has no significance). This qualitative analysis indicate that
although the level factor is responsible for capturing the majority of the term structure movements, investors charge premia, and thus are more worried about changes in the slope and curvature of the term structure, when slope and curvature are respectively defined by the Legendre polynomials of degree one and two.
For the Gaussian model, the timet instantaneous expected excess return of a swap with maturityτ is given by19:
eit,τ = [P0(τ)P1(τ)P2(τ)]ΣΛYYt (51)
Equation (51) indicates that the instantaneous expected excess return is a linear combi-nation of some of the factors of the model, where weights on specific factors come from a combination of parameters in matrix ΛY, Σ and also Legendre polnomial terms which
are maturity dependent. However, the most important weights are the ones which come from matrix ΛY because the only role of these parameters is capture risk premia, whereas
the role of Σ is divided between fitting the cross sectional, minimizing the effect of the deterministic factors, and also capturing the dynamics of yields underP through the tran-sition densities. That is another way of understanding that ΛY gives information on which
factors are important on the premia charged by investors to hold bond positions. From Equation (51) view, we see again that investors care about the slope and curvature factors.
7
Conclusion
This work uses financial mathematics tools to justify the use of term structure mod-els parameterized by linear combinations of Legendre polynomials as a consistent model,
18
First two elements of the Brownian vectorW. 19
in the sense of presenting dynamic restrictions which prevent the existence of arbitrages in the market. The Legendre arbitrage free model is classified as belonging to the family of Affine Models proposed by Duffie and kan (1996), presenting a few more general dy-namics than Affine models in which regards the diffusion matrix of the SDEs. Possible applications are briefly discussed, and an empirical Section presents the implementation, and interpretation of a 6-Legendre Polynomial Dynamic Model where all the factors are Gaussian and/or deterministic.
Many recent works in term structure modelling study arbitrage free models with pre- parameterized maturity functions multiplying the state space vector, in a very sim-ilar setting to that of the Legendre model. Just to mention a few see Filipovic (2002), Filipovic and Sharef (2004) and De Rossi (2004). Collin-Dufresne et al. (2003) propose performing Taylor expansions on Affine term structure models to obtain models with a clear interpretation of the dynamic factors as level, slope, curvature and so on, precisely the intuitive argument in favor of the Legendre model. In addition, Pan and Wu (2003) propose a one factor quadratic term structure model driving the level of the term structure which prevents interest rates from being negative and fits well US swap and treasure data. These are some indications that the present work is an interesting step in the direction of understanding the trade-off between proposing extremely sophisticated models with usu-ally high computational costs and harder interpretation, versus simpler models yet with reasonable empirical power. Surely, much is still to be done in empirical fixed income.
8
Acknowledgements
I thank participants of the Second Brazilian Meeting of Finance, Applied Mathemat-ics Seminar at Stanford Department of MathematMathemat-ics, and Empirical Finance Meeting at Stanford Graduate School of Business, for helpful remarks. In particular, I thank Damir Filipovic for fruitful conversations on the characterization of affine term structure models. All remaining errors are, of course, of my own responsibility.
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9
Appendix
Proof of Proposition 2.
Basically, apply Proposition 1 to the Legendre forward rates provided by Equation (22). This will generate a polynomial equation in x with coefficients depending on process Z
and the processes that describe its dynamics, its driftband squared diffusiona=σσt. To the purpose of applying Proposition 1 we first obtain the following quantities:
∂G(Z,x)
∂x =
Pn
j=2(j−1)Zjxj−2
∂G(Z,x)
∂zj =x
j−1, j = 1, ..., n
∂G(Z,x)
∂zizj = 0, i, j = 1, ..., n
Rx
0
∂G(Z,y)
∂zj dy=
xj
j , j = 1, ..., n
(52)
Substituting these quantities in Equation (21) we obtain:
n
X
j=2
(j−1)Zjxj−2 =
n
X
i=1
bixi−1+1 2 n X i,j=1 aij
−2xi−1x
j
j
(53)
Note that by matching terms in thexpolynomial on both sides of Equation (53) and using the symmetry of matrixa, the following condition should be satisfied by matrixa:
aij = 0, whenever i+j > n (54)
In particular makingi=j, this restriction in aimplies:
aii=
d
X
k=1
(σik)2 = 0→σik = 0, k= 1, ..., d; i >hn
2
i
(55)
Which on its turn implies:
aij =aji =
d
X
k=1
σikσjk = 0, i >hn
2
i
, j = 1, ..., n (56)
This shows that the only possible arbitrage free processes with term structures parame-terized by the firstn Legendre polynomials are those which depend only on at mostn
2
stochastic factors.
In order to see that there is always at least one non-trivial consistent processZ just observe Equation (53) and note that the free parametersaij,i, j≤2n, andbi,i= 1, .., n
comprise a complete polynomial of degreen−1 in the variablex. Then the left-hand side polynomial of degreen−2 can be easily matched with the appropriate choices ofband σ, as functions of Z, in a Markovian setting much simpler than Equation (19) allows us to be.
The last part of the proof consists in actually constructing one particular arbitrage free Legendre term structure model. For simplicity takeσij =σji= 0, ifi6=j implying:
aij =
Plugging Equation (57) in Equation (53) we obtain:
n
X
j=2
(j−1)Zjxj−2= n
X
i=1
bixi−1−
[n 2]
X
i=1
(σii)2
x2i−1 i
(58)
Matching coefficients in thex polynomial we obtain:
b2k−1 = (2k−1)Z2k, 1≤k≤n
2
b2k= 1kσkk+ 2kZ2k+1, 1≤k≤n−21 (59)
bn=
2
nσ
nn, forneven
0, forn odd
(60)
Note that we are not obliged to choosebis andσiiwhich are respectively affine and square
Factor Level Slope
Level 1.00
Slope 0.89 1.00
Curvature -0.61 -0.72
Table 1: Correlation Structure of the Legendre Static Factors.
Parameter Value Standard Error ratio absStd Err.(Value)
κ11 0.08911 3.2940 0.027
κ12 -6.9940 6.1000 1.147
κ22 10.4200 2.1200 4.912
κ13 11.3400 11.19 1.013
κ23 28.0700 6.2960 4.458
κ33 0.5653 2.6780 0.211
Σ11 0.0918 0.0014 62.40
Σ22 0.0518 0.0009 57.43
Σ33 0.0305 0.0004 75.89
λ0(1) -3.5380 6.9400 0.510
λ0(2) 1.2760 1.6830 0.758
λ0(3) -0.0687 1.0950 0.062
λY(1,1) 0.9699 35.37 0.027
λY(1,2) -97.90 63.13 1.551
λY(2,2) 201.1 39.07 5.146
λY(1,3) 139.7 117.0 1.194
λY(2,3) 455.0 114.6 3.970
λY(3,3) 18.49 68.88 0.269
Y0,4 -0.00133 0.00016 8.157
Y0,5 0.00042 0.00003 12.03
Y0,6 -0.00018 0.000005 35.19
Table 2: Parameters and Standard Errors for the Gaussian Model Using Brazilian Data.
Parameter Mean (bp) Standard Deviation (bp) Skewness Kurtosis p-value JB test
ǫ30 -2 30 -0.95 6.28 0
ǫ60 5 19 0.74 5.16 0
ǫ120 13 27 0.87 3.02 0
ǫ180 13 23 0.51 3.45 0
ǫ360 -15 25 -0.35 3.28 0.000028
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0
0.2 0.4 0.6 0.8 1
t
NS(t)
Nelson and Siegel Exponential Functions
Figure 1: The Nelson-Siegel Exponential Functions.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
t
P(2t/5 −1)
The First four Legendre Polynomials
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.05
0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time (in Years)
Level Value
Temporal Evolution of the Stochastic Level
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.05
−0.045 −0.04 −0.035 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005
Slope Value
Time (in Years)
Temporal Evolution of the Deterministic Slope
0 2
4 6
8 10
0 0.5 1 1.5 2 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Maturity (Years)
Term Structure Evolution
Time
Interest Rates
0 2 4 6 8 10 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Maturity (Years) Term Structure Evolution
Time Interest Rates 0 2 4 6 8 10 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2
Term Structure Evolution
Interest Rates Time Maturity (Years) 0 2 4 6 8 10 0 0.5 1 1.5 2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Maturity (Years) Term Structure Evolution
Time Interest Rates 0 2 4 6 8 10 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Maturity (Years) Term Structure Evolution
Time
Interest Rates
Figure 4: Some Term Structure Shapes under a Four Factor Legendre Dynamic Model with Stochastic Level and Stochastic Slope.
0 0.5 1 1.5 2 0 200 400 600 800 10000.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
maturity (in Years) Time Evolution (in Days)
Interest Rates
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.18
0.2 0.22 0.24 0.26 0.28 0.3
Time to Maturity (Years)
Interest Rates
Fitting for August 6, 1999
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.186
0.188 0.19 0.192 0.194 0.196 0.198 0.2
Time to Maturity (Years)
Interest Rates
Fitting for April 14, 2000
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.18
0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38
Time to Maturity (Years)
Interest Rates
Fitting for October 11, 2002
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.25
0.26 0.27 0.28 0.29 0.3 0.31
Time to Maturity (Years)
Interest Rates
Fitting for January 29, 2003
01/01/00 01/01/01 01/01/02 01/01/03 0.14
0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34
Time
Coefficient Value
Time Series of The Level Legendre Coefficient (Degree 0)
01/01/00 01/01/01 01/01/02 01/01/03 −0.02
0 0.02 0.04 0.06 0.08 0.1
Time
Coefficient Value
Time Series of The Slope Legendre Coefficient (Degree 1)
01/01/00 01/01/01 01/01/02 01/01/03 −0.035
−0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01
Time
Coefficient Value
Time Series of The Torsion Legendre Coefficient (Degree 2)
0 100 200 300 400 500 600 700 800 900 −30
−20 −10 0 10 20 30 40
Difference (in bp)
Time Evolution (in Days)
Difference Between Level Factor in the Dynamic and Static Legendre Model
0 100 200 300 400 500 600 700 800 900 −30
−20 −10 0 10 20 30 40
Difference Between Slope Factor in the Dynamic and Static Legendre Model
Time Evolution (in Days)
Difference (in bp)
0 100 200 300 400 500 600 700 800 900 −60
−40 −20 0 20 40 60
Time Evolution (in Days)
Difference Between Curvature Factor in the Dynamic and Static Legendre Model
Difference (in bp)
0 100 200 300 400 500 600 700 800 900 −15
−10 −5 0 5 10 15
Factor Value (in bp)
Deterministic Factors for the Multifactor Gaussian Model
Time Evolution (in Days) Y4
Y 6
Y5
Figure 9: Conditionally Deterministic Factors in the Multifactor Gaussian Model.
0 100 200 300 400 500 600 700 800 900 0
0.5 1 1.5 2 2.5 3 3.5x 10
−4
Time Evolution (in Days)
Sum of Cross Sectional Squared Residuals
0 100 200 300 400 500 600 700 800 900 0.15
0.2 0.25 0.3 0.35 0.4
Time Evolution of 3 Brazilian Swap Yields
Time Evolution (in Days)
Interest Rate
