A Non Parametric Calibration of the HJM Geometry[1]

7/28/2019 A Non Parametric Calibration of the HJM Geometry[1]

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A non parametric calibration of the HJM geometry: an application of It calculus to financial statistics

A non parametric calibration of the HJM geometry:

an application of It calculus to financial statistics *

Paul Malliavin

10 rue Saint Louis en l'Isle, 75004 Paris, France, e-mail:[email protected]

Maria Elvira Mancino

Universit di Firenze, Italy, e-mail: [email protected]

Maria Cristina Recchioni

Universit Politecnica delle Marche, Italy e-mail:[email protected]

1. Abstract

2. Introduction

3. An econometrically implementable method for the computation of the iterated brackets

4. Fourier estimation methodology

5. Analysis of interest rate data

6. Fortran and Matlab codes

. Abstract

e show that the geometry of the Heath-Jarrow-Morton interest rates market dynamics can be non-parametrically calibrated by the observation

a single trajectory of the market evolution. Then the hypoellipticity of the infinitesimal generator can be exactly measured. On a data set of

tual interest rates we show the prevalence of the hypoelliptic effect together with a sharp change of regime. Volatilities are computed by

plying the Fourier cross-volatility estimation methodology.

this website we show some numerical results concerning the paper [21] and we provide the FORTRAN and MATLAB software codes.

. Introduction

n empirically relevant and specific feature of the term structure of interest rates dynamics lies, from one side, in the high dimensionality of the

ate space, that is the bond values for a continuum of maturities, and, from the other side, in the low dimensionality of the variance, which is

und empirically to be influenced by no more than four independentfactors (e.g. see [10],[6]). These stylized facts lead to the formulation of the

eath-Jarrow-Morton framework (HJM hereafter) [17], in which the yield curve is driven by a low dimensional Wiener process. From the point

view of the "regularity" of the model, these two properties imply that the ellipticity assumption, which is assumed for stock market models and

icing-hedging of contingent claims in this market, is ruled out when we deal with the interest rate market models. Therefore, we are lead to

nsider non-elliptic models, and the most regular models still available are the hypoelliptic ones.

e consider a HJM model of the forward rates driven by a n-dimensional Brownian motion and we develop a methodology for exploring the

ometric properties of the forward rates evolution's space, along the lines suggested in [22]. We apply the Fourier cross-volatilities estimator

troduced in [19] to obtain a non parametric estimation of the forward rate volatility structure. Secondly, in order to estimate the iterated Lie

ackets, we prove a mathematical result which reduces the computation of the iterated Lie brackets of the driving vector fields by means of

rated volatilities.

file:///C|/Documents and Settings/ik5mic/Desktop/Sito_Mav/Sito_Mav/home.html (1 di 13)09/05/2007 12.26.55
http://dx.doi.org/10.1007/s11537-007-0666-7http://dx.doi.org/10.1007/s11537-007-0666-7


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e illustrate the capability of our methodology to numerically achieve the geometric properties of the interest rates evolution space. The

merical results confirm the hypoellipticity of the interest rate market. In fact we find that the Lie brackets of the driving vector fields are very

r from lying in the subspace generated by the first three eigenvectors. We compute the Lie brackets on an actual time series of the yield curve of

uro rates and we compute the distance of Lie brackets from the time varying subspace spanned by the three main eigenvectors. Then we

mpute the iterated second-order Lie brackets and their distances from the subspace generated by the first three eigenvectors and the first-order

ackets. Our results show that the distance of iterated Lie brackets from this subspace is large, indicating hypoellipticity. The tools devised in our

proach allow estimation of simple geometric features of the model. As an illustrating example, we compute the movements of the first

genvectors, quantified by the angle between the same eigenvector at different instants of time. We show that, in the three-years post-EMU

riod considered, the second and the third eigenvectors have a very much fluctuating behavior; on the contrary the first eigenvector has in the

er all a remarkable stability; this direction has therefore a meaning from an economic point of view. We decipher a remarkable change of this

rection during around three months corresponding to the third trimester of the year 2001. A special economic event happened during thismester (11th September 2001). We emphasize that our methodology could be implemented in real time, therefore tracking in real time the

rection of the first eigenvector could furnish to the trader an indicator on the market stability.

. An econometrically implementable method for the computation of the iteratedrackets

e consider a continuous trading economy with a trading interval [0,], for a fixed > 0. Suppose that a continuum of default free discount bondse traded with different maturities, one for each trading date T [0,]. The instantaneous forward rate at time t for maturity date T > t, f(t,T),tisfies the following equation

f(t,T)-f(0,T)

=

t

0

(s,T)ds+

d

j = 1

t

0

j(s,T)dWj(s) (2)

here (t,T) and j(t,T) are stochastic processes adapted to the filtration \cal Ft, which possibly includes all available information at time t

nerated by the term structure's evolution. We assume that the market is driven by a finite number of Brownian motions Wj (j = 1,, d). In the

rametrization used by [7], let x = T-t and

rt(x) = f(t,t+x) (3)

at is x represents the time to maturity in contrast with T which represents the maturity time and rt(x) is the instantaneous forward rate curve.

he interest rate curve takes its value in the infinite dimensional space C of continuous functions on [0,[. Nevertheless a remarkableperimental fact is that the rank n of its volatility matrix is very low n 4 (see also [6]). This result suggests that elliptic models are ruled outd hypoelliptic models are the most regular models still available (see also [22] on this point).

ow consider the family of SDE, depending upon the parameter x, describing the HJM model for the forward rate curve. As shown in [13], the

-arbitrage restriction forces the drift coefficient of the forward curve evolution to be just a function of the volatility structure, for the risk neutral

obability measure. Therefore the forward rate curve evolution has the following expression in the Musiela parametrization:

drt(x)

=

xrt(x)+

d

j = 1

x

0

j(t,x) j(t,y)

dy

dt +

d

j = 1

j(t,x) dWj(t). (4)

the literature there is a vast possible choice for the functional forms of the term's structure volatility. For our estimation we will restrict our

ention to the Markovian models, which means that we suppose the volatility function is of the form

(t,x) = ((rt))(x), (5)

here () is a smooth function. This class of volatility structure is quite general and it is considered also in [14].

herefore we suppose the following evolution for the instantaneous forward rate:



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drt(x)

=

xrt(x)+

d

j = 1

x

0

(Aj(rt))(x) (Aj(rt))(y)

dy

dt +

d

j = 1

(Aj(rt))(x) dWj(t) (6)

here Aj are "driving vector fields" defined on C, the infinite dimensional space of continuous functions on [0,[. An appropriate notion of

moothness of vector fields is a necessary hypothesis in order to prove existence and uniqueness of solutions for (6). In fact the operator r [(rt)/

x)] is an unbounded operator, therefore Banach space type differential calculus cannot be used, and a theory of differential calculus in Frchetaces is needed. We refer to [11],[12] for a precise treatment of this topic. We do not address this problem because our interest will be to work

a reasonable finite dimensional approximation, that is we will work on RM, where M is a fixed (large) number of maturities, but having an

finite dimensional" point of view in mind.

he main result of this section is the following: The brackets of the driving vector fields of the HJM diffusion can be numerically computed from a

ngle time series of market data, under the only assumption on the model of the differentiability of the infinitesimal generator's coefficients.

order to prove this result we reduce the computation of the iterated Lie brackets of the driving vector fields by means of iterated volatilities.

x M different maturities (M can be extremely large and in particular it should be d


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urther results can be found in [25].

. Fourier estimation methodology

e briefly recall the methodology which has been proposed in [19] and developed in [2], [3], [20]- [22]. Assume that the market data processes p

= (p1(t), , pn(t)) are Brownian semi-martingales satisfying the following It SDE

dpj(t)

=

d

i = 1 j

i(t) dWi + bj(t) dt j = 1,

, n

(11)

here W = (W1, , Wd) are independent Brownian motions, and ** and b* are adapted random processes. We recall now the Fourier method

r computing multivariate volatilities. From the representation (11) we define the volatility matrix, which in our hypothesis depends upon time:

j,k(t)

=

d

i = 1

ji(t)ik(t). (12)

he Fourier method reconstructs *,*(t) on a fixed time window (which we can reduce always to [0,2] by change of origin and rescaling) using

e Fourier coefficients of dp*(t). First we compute the Fourier coefficients of dpj for j = 1, , n defined by

ak(dpj)

=

1

]0,2[

cos(kt)dpj(t), bk(dpj)

=

1

]0,2[

sin(kt)dpj(t). (13)

hen we consider the Fourier coefficients of the cross-volatilities

a0(i,j)

=

1

2

]0,2[

i,j(t) dt, ak(i,j)

=

1

]0,2[

cos(kt) i,j(t) dt, (14)

bk(i,j)

=

1

]0,2[

sin(kt)i,j(t) dt. (15)

[19] it is proved the following result relating the Fourier coefficients of the variation of the process p* to the Fourier coefficients of the

latility matrix.

heorem 2Fix an integer n0>0, the Fourier coefficients of the functions i,j(t), which are the entries of the volatility matrix, are given by the

llowing limits in probability

a0(i,j)

=lim

N

N+1-n0

N

s = n0

1

2

(as(dpi)as(dp

j)+bs(dpi)bs(dp

j)) (16)

ak(i,j)

=lim

N

N+1-n0

N

s = n0

(as(dpi)as+k(dp

j)+as(dpj)as+k(dp

i)) (17)

bk(i,j)

=lim

N

N+1-n0

N

s = n0

(as(dpi)bs+k(dp

j)+as(dpj)bs+k(dp

i)). (18)



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nally using the Fourier-Fjer inversion formula it is possible to reconstruct i,j(t) from its Fourier coefficients:

i,j(t)

=lim

N(i,j)N(t) (19)

here for any t (0,2)

(i,j

)N

(t):

=

N

k = 0(1-

k

N)(ak(i,j) cos(kt)+ bk(i,j)sin(kt)). (20)

e stress the point that the computation involved in formula (10) can be realized exploiting the fact that the Fourier methodology allows us to

construct cross volatilities as a function of time (20), therefore it makes possible to iterate the procedure of computing the cross-volatilities.

. Analysis of interest rate data

e implement the described methodology on a data set of actual interest rates. To this purpose, we use a time series of Euro swap rates and

uribor rates, ranging from January 1999 (start of EMU) to December 2001, for a total of 777 days. We have Euribor rates at 3, 6 and 9 months,

d swap rates yearly from 1 to 10 years. The swap rates are bootstrapped to get the yield curve, which we define as:

y(t;T) = -

log P(t,

T)

T-t

, (21)

here P(t,T) is the price, observed at t, of one Euro payed at T. The numerical difference between y(t,T) and f(t,T) is negligible. In our data the

me to maturity x = T-t is fixed and we observe 13 maturities ranging from three months to 10 years.

e start by computing the 1313 variance-covariance matrix (t) as a function of the time. We approximate the volatility matrix (t) R1313, t[0,777/259], with the matrix N(t) obtained using the procedure described in Theorem 2, formula (19), (20). To this purpose we adopt the

urier methodology using N1 coefficients for price and N coefficients for the volatility. Note that we have H = 777 observations and the time

it is one year in the computations. The data are uniformly distributed in the time interval [0,777/259] with step-size dt = 1/259. Moreover we

oose N1 = H/2 element to reconstruct the Fourier coefficients of the process and N = H/4 Fourier coefficients of the volatility matrix and we

mooth the Fjer kernel in (20) replacing (1-k/N) with sin2( k)/(k)2, k = 1,2,N and = 2/259 and we consider a loss of resolution equal to 2 corresponding to the operation of taking the volatility. Hence the prices are given with a precision 1/N1, the covariances with a precision /

1, the first order brackets with a precision 2/N1 and the second order brackets with a precision 3/N1. Let us denote by trace(N(t)) the trace of

e matrix N(t), by i(t) R, i = 1,2,,13, 1(t) 2(t) 13(t) the eigenvalues of the matrix N(t) at time t and by Ai* R13, i = 1,2,

,13 the corresponding eigenvectors with Ai*(t)TAj

*(t) = 0, i j and ||A*i(t)|| = 1, i,j = 1,2,,13, where the superscript T denotes the transpose

erator. We define the eigenvectors Ai(t) = {i(t)}A*i(t), i = 1,2,, 13 that are orthogonal vectors with ||Ai(t)|| = {i(t)}, i = 1,2,,13.



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Figure 1: Eigenvalues versus time: different scales

s shown in Figures 1, 2, 3, the first three eigenvalues describe almost completely the spectrum of the volatility matrix. In particular Figure 1

ows the plot of the first six eigenvalues as a function of the time variable t (the unit time of the figures is the day).



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Figure 2: Percentage of variance explained by the first three eigenvectors



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Figure 3: Ratios ri(t), i = 1 (dotted line), i = 2 (dashed-dot line), i = 3 (solid line)

s previously mentioned, Figure 1 shows that only the first three eigenvalues are significant. It is a well known fact that less than four factors are

ough to account for the volatility of the whole yield curve. This is shown in Figure 2, where we plot the time-varying percentage of the variance

plained by the fist three factors, that is we show the ratios 1(t)/trace(N(t)), (1(t)+2(t))/trace(N(t)), (1(t)+2(t)+3(t))/trace(N(t)).

oreover Figure 3 shows the ratios ri(t) = i(t)/(1(t)+2(t)+3(t)), i = 1,2,3.

ow we consider the econometric reconstruction of Aj(t), j = 1,2,3 (see formulae (19), (20)). Let ti = 777 i/100, i = 1,2,,100, Figure 4 shows the

sine between Aj(ti) and Aj(ti+1), j = 1,2,3, i = 1,2,,100.

Figure 4: Variation of the first three eigenvectors

oreover we observe the surfaces generated by the cosines between the eigenvectors A1(t), A1(t). Let us denote by c1(t,t

) the cosines c1(t,t

) = A1

TA1(t)/(||A1(t)||||A1(t

)||). Figure 5 shows the surface c1(t,t

) when t = t10*i, i = 1,2,,10 t

= t10*k, k = 1,2,,10. Due to the extreme rapid

riation of the angle of the two last vectors A2, A3 it is not relevant to draw the analog of the plot 5. t = t10*i, i = 1,2,,10 t= t10*k, k = 1,2,

,10.

gure 5 shows a remarkable change of the direction of the first eigenvector during around three months corresponding to the the end of the third

mester and the beginning of the last trimester of the year 2001. We conjecture that a special economic event happened during this trimester

obably the crisis of the September 11th. We emphasize that our methodology could be implemented in real time: therefore tracking in real time

e direction of the first eigenvector could furnish to the trader an indication on the market stability.

ow we consider the first order brackets:

B1,2(t) = [A1(t),A2(t)], (22)



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B1,3(t) = [A1(t),A3(t)], (23)

B2,3(t) = [A2(t),A3(t)]. (24)

Figure 5: c1(t,t) surface, t [0,777], t [0,777] and a contour plot

e can compute these brackets (see Corollary 2.4) thank to the Fourier method proposed since, roughly speaking, they are the covariance

tween the eigenvectors and the yield curve. We focus on the time varying subspace G1,2,3(t) spanned by the first three eigenvectors A1(t), A2(t),

3(t). We compute the projection of the vectors Bi,j(t), i = 1,2, j = i+1,,3 on the subspace G1,2,3(t):

v1,2(t)

=

3

j = 1

Aj(t)TB1,2

(t)

j(t)

Aj(t), (25)

v1,3(t)

=

3

j = 1

Aj(t)TB1,3

(t)

j(t)

Aj(t), (26)



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v2,3(t)

=

3

j = 1

Aj(t)TB2,3

(t)

j(t)

Aj(t), (27)

Figure 6: Lengths of the brackets Bi,j(t), i = 1,2, j = i+1,3 versus time

d then we compute the cosine between each bracket and its projection on the subspace G1,2,3 as function of time and we denote this cosine by ci,

) = Bi,j(t)Tv1,2(t)/(||Bi,j(t)||||vi,j(t)||), i = 1,2, j = i+1,3. Figure 6 shows the lengths of the first order brackets and Figure 7 shows the cosines ci,j(t),

= 1,2, j = i+1,3 as functions of t.



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Figure 7: Cosines between the brackets Bi,j(t), i = 1,2, j = i+1,3 and their projections on G1,2,3(t) versus time

. FORTRAN and MATLAB codes

he archive Fortran_code.zip contains the Fortran code (main.f), the file containing the interest rates data (interest_data.txt) and the output files

nerated by running the code.

he code computes the eigenvalues, the eigenvectors of the volatility matrix and the first order brackets for several time values.

he Fortran code requires the use of the IMSL routine for computing the eigenvalues and the eigenvectors and it provides several output files (see

adme_fortran.txt)

he archive Matlab_code.zip contains the Matlab code (main.m), the file containing the interest rates data (interest_data.txt) and a readme file.

he Matlab code computes the volatility matrix for several time values and it plots the first six eigenvalues versus time.

eferences

]

Airault, H. and Malliavin, P. (2004) Backward regularity for some infinite dimensional hypoelliptic semi-group. Stochastic analysis and

related topics in Kyoto. In honour of K. It. Lectures given at the conference in Kyoto, Japan (2002). Math. Soc. of Japan. Advanced

Studies in Pure Math. 41: 1-11.

]

Barucci, E. and Ren, R. (2001) On measuring volatility of diffusion processes with high frequency data.Economics Letters, 74: 371-378.

]Barucci, E. and Ren, R. (2002) On measuring volatility and the GARCH forecasting performance.Journal of International Financial

Markets, Institutions and Money,12: 183-200.

]

http://www.dmd.unifi.it/persone/me.mancino/Fortran_code.ziphttp://www.dmd.unifi.it/persone/me.mancino/readme_fortran.txthttp://www.dmd.unifi.it/persone/me.mancino/Matlab_code.ziphttp://www.dmd.unifi.it/persone/me.mancino/Matlab_code.ziphttp://www.dmd.unifi.it/persone/me.mancino/readme_fortran.txthttp://www.dmd.unifi.it/persone/me.mancino/Fortran_code.zip


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Baudoin, F. and Teichmann, J. (2005) Hypoellipticity in infinite dimensions and an application in interest rate theory.Ann. Appl. Prob.,

15.

]

Bjrk, T. and Svensson, L. (2001) On the existence of finite dimensional realizations for nonlinear forward rate models.Mathematical

Finance,11: 205-243.

]

Bouchaud, J.P., Cont, R., El-Karoui, N., Potters, M. and Sagna, N. (1999). Phenomenology of the interest rate curve.Appl. Math. Finance,

6, no.3, 209-232.

]

Brace, A. and Musiela, M. (1994). A Multi-factor Gauss Markov implementation of Heath, Jarrow and Morton.Math. Finance, 4: 259-

283.

]

Carmona, R. and Tehranchi, M. (2004) A characterization of hedging portfolios for interest rate contingent claims.Ann. Appl. Probab., 14,

no.3: 1267-1294.

]

Cont, R. (2005). Modeling term structure dynamics: an infinite dimensional approach.Int. J. Theor. Appl. Finance, 8.

0]

Duffie, D. and Kan, R. (1995) Multifactor models of the term structure. in Howison, Kelly & Wilmott (Eds.)Math. Models in Finance,

London: Chapman & Hall.

1]

Filipovic, D. and Teichmann, J. (2003a) Existence of invariant manifolds for stochastic equations in infinite dimension.J. Funct. Anal., 2:

398-432.

2]

Filipovic, D. and Teichmann, J. (2003b) Regularity of finite dimensional realizations for evolution equations.J. Funct. Anal., 2: 433-446.

3]Heath, D., Jarrow, M. and Morton A. (1992) Bond pricing and the term structure of interest rates: a new methodology for contingent

claims valuation.Econometrica, 60, no.1: 77-105.

4]

Jeffrey, A., Kristensen, D., Linton, O., Nguyen, T. and Phillips, P.C.B. (2004) Nonparametric Estimation of a Multifactor Heath-Jarrow-

Morton Model: an Integrated Approach.Journal of Financial Econometrics, 2: 251-289.

5]

Malliavin, P. (1978a) Stochastic Calculus of Variations and Hypoelliptic operators. Proc. Internat. Symposium on Stochastic Differential

Equations, Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976, Wiley, New York, 1978, 195-263.

6]

Malliavin, P. (1978b) Ck-hypoellipticity with degenaracy. II, Stochastic analysis. Proc. Internat. Conf., Northwestern Univ., Evanston, Ill.,

1978, Academic Press, New York, 1978, 327-340.

7]

Malliavin, P. (1997) Stochastic analysis. A series of comprehensive studies in mathematics, vol.313. Springer-Verlag, 1997.

8]

Malliavin, P. (2006) It atlas, its application to mathematical finance and to exponentiation of infinite dimensional Lie algebras.

Proceedings of The Abel Symposium 2006 - Stochastic Analysis and Applications - A Symp. in Honour of Kiyoshi It. Springer-Verlag.

9]

Malliavin, P. and Mancino, M.E. (2002). Fourier series method for measurement of multivariate volatilities. Finance and Stochastics, 4:

49-61.

0]

Malliavin, P. and Mancino, M.E. (2005). A Fourier transform method for nonparametric estimation of volatility. Preprint Dept. Math. for



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Decisions, University of Firenze.

1]

Malliavin, P., Mancino, M.E. and Recchioni, M.C. (2007). A non parametric calibration of the HJM geometry: an application of It

calculus to financial statistics.Japanese Journal of Mathematics, 1: 55-77.

2]

Malliavin, P. and Thalmaier, A. (2005). Stochastic Calculus of Variations in Mathematical Finance, Springer Finance, 2005.

3]Musiela, M. (1993). Stochastic PDEs and term structure models.Journes Internationales de Finance, IGR-AFFI, La Baule, 1993.
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A Non Parametric Calibration of the HJM Geometry[1]

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