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An empirical study of the term structure of interest rates
0956-5221/93 $6.00 + 0.00 0 1993 Pergamon Press Ltd
Scmd. J. Mgm?, Vol. 9, SuppI., pp. S29446.1993 Printed in Great Britain
AN EMPIRICAL STUDY OF THE TERM STRUCTURE OF INTEREST RATES” KRISTIAN R. MILTERSEN Odense Universitet Abstract - This paper is an empirical study of the Heath-Jarrow-Morton model using General&d Method of Moments and Simulated Method of Moments on Danish bond and option prices. The paper implements a simulation approach to price contingent claims written on purely interest rate-dependent securities fulfilling the Heath-Jarrow-Morton model. This method implies simulation of solutions of stochastic differential equations since the theoretical pricing model is too complicated to give closed form pricing formulas.
Key words: Heath-Jarrow-Morton model, term structure of interest rates, default-free coupon bonds, forward rate, forward rate process, GMM, SME, implied volatility, simulation of SDEs.
1. INTRODUCTION The underlying theoretical model of this empirical study is based on the Heath-Jarrow-Morton model (Heath et al., 1992). In Miltersen (1992, Chapter 3) there is a characterization of four nested models, all based on the Heath-Jarrow-Morton model. We will make an empirical analysis of the simplest of these four models in this paper. Using this model, the so-called SDE spot rate model, it is only necessary to model the movements of the spot rate process, (X,),,,, with I denoting the time-horizon. The model uses the following stochastic differential equation (SDE) description of the spot rate process, under the equivalent martingale measure, Q, X, =At) + &(X,)*dW, + $lla(X,)l12(t-u)du,
V ~1, Q-a.e.,
(1)
wheref(.) is the initial forward rate function at date zero and Q(.) is the time-homogeneous volatility function. The purpose of this empirical study is now to investigate the possibility of estimating these two inputs, f and
*This paper has been greatly improved by discussions with Peter Ove Christensen and Bjarne Graabech Sorensen and b the discussion after the presentation at the Nordic Symposium on Contingent Claims Analysis in Finance in Turku (x bo), Finland, at the E.I.A.S.M. Doctoral Tutorial in Finance at the 19th EFA Ammal Meeting in L.&on, Portugal, and at the E.I.A.S.M. Special Anniversary Workshop on Interest Rates Derivatives in Leuven, Belgium. Comments from Simon Bemiinga, Tom Berglund, Soren Dahlgaard, Thomas Gustavsson, Osmo Jauri, Richard Stapleton and Bo Vad Steffensen, together with much advice from the two discussants, Petter Bjerksund and Hamm Kahra, were most appreciated. I am also grateful for the comments and suggestions received from Kenneth J. Singleton when an earlier draft of this paper was submitted as a research proposal at Stanford University. S29
s30
K. R. MILTERSEN
initial forward rate function not appropriate for the purpose of our model (cf. Poirier, 1976; Litzenberger and Rolfo, 1984; Shea, 1984; Tanggaard and Jakobsen, 1988; Sorensen, 1991, for references about estimating cubic spline functions). That is, to be able to use the SDE spot rate model to price contingent claims written on bonds, the initial forward rate function has to be at least six times continuously differentiable which is not possible with cubic spline functions. We therefore instead set up a simple functional form of the forward rate function,f(.), which is infinitely often differentiable. We then estimate the parameters included in this specific functional form. We use the Generalized Method of Moments (GMM) estimation technique to estimate the parameters. We also set up a statistical test for whether the specified functional form of the forward rate function can be supported by data. The data are from the Danish bond market. Secondly, we are interested in how to estimate the volatility function, (T(.), of the stochastic differential equation (1) in the SDE spot rate model at a specific point in time. We set up a simple functional form of the volatility function, given some technical requirements, developed in Miltersen (1992, Chapter 3), to ensure existence and non-negativity of the spot rate process, {Xl,,,. There are at least two ways of estimating the parameters of the volatility function, either using a time-series of bond price observations or using implied volatility calculations observing one moment of bond option prices. In this paper, we will concentrate on the last method which involves derivation of option prices from the SDE spot rate model (cf. Engle, 1982; Engle et al., 1987; Bollerslev et al., 1988; Engle et al., 1990, for references about estimating variances of bond returns using models with autoregressive conditional heteroscedasticity (ARCH) covariance structures in a time-series set-up). Unfortunately, the contingent claim prices cannot be derived as closed form expressions using the SDE spot rate model except in special cases. For instance, if the volatility function is assumed to be constant, a closed form expression is derived in Heath et al. (1992, Section 6, first example, pp. 90-91). That special case of the Heath-Jarrow-Morton model is empirically tested and compared to the Black-Scholes model in Rindell and Sand& (1991). In our empirical analysis we will study the more general model of non-constant volatility functions. Therefore, the contingent claims prices must be derived using simulations of solutions of the SDE spot rate process and calculation of sample expectations. We estimate the parameters included in the specific functional form of the volatility function, o(.) , given the estimated parameters of the initial forward rate functionA.). The estimation technique used to estimate the volatility function has to be extended from the usual GMM estimator to the so-called Simulated Moments Estimator (SME). This estimator takes into account that the moment conditions are simulated using a simulation algorithm that also depends on the unknown parameters. We also test whether the Heath-Jatrow-Morton model with the specified functional forms of both the initial forward rate function and the volatility function can be supported by the data. The option data are from the Danish Futop market.
2. THE FORWARD RATE FUNCTION We start by specifying the functional form of the initial forward rate function. We denote the price at date zero of a default-free bond expiring at date T as P(0, 2). IfJI.) denotes the forward rate function, at date zero, then the following equation holds: P(0, 7) = e-gAs)ds, t/ T&I.
(2)
THE TERM STRUCTURE
OF INTERESTRATES
s31
The time-horizon is I := [0, r], where I is the maturity of the bond in the financial bond market with the longest maturity; normally we would just set I = 00. Of course, bonds are not just pure default-free bonds, but all default-free coupon bonds are just linear combinations of pure default-free bonds. For instance, if we have a coupon bond paying out d,, units of account at times t,, for n = 1, . . . , N, with N&IN,then the price at date zero of that bond must be
Otherwise, there would be obvious arbitrage opportunities in the bond market.* We will refer to {(d,,t,))~= , as the payout touple of the coupon bond. We specify the initial forward rate function as fit) := y + (la2 + Ut + 6)eH,
(4)
with unknown parameters (y, 6, v, K, u) E 0, := IR, x IR x IR, x IR x IR. Since the underlying theoretical model to be tested determines the evolution of the forward rate function over time, we specify the initial forward rate functionfi.), to be analytical,t which implies that it is infinitely often differentiable. Moreover, the pure default-free bond prices, from Equation (2), with the specific forward rate function, from Equation (4), are derived as
P(0, r) = e
(exp(-@KC+
2
+u) L - i V
V
(l-exp(-VT))($ + ” + G)-y)T V
)
v
T&I.
(5)
3. PRICING OF CONTINGENT CLAIMS As mentioned in Miltersen (1992, Chapter 2), the real problem concerning pricing of contingent claims is to specify a consistent price process of the underlying securities under an equivalent martingale measure. When that is done, the problem concerning pricing of European contingent claims* is solved by Harrison and Kreps (1979, Corollary to Theorem 3, p. 396). The price of the contingent claim, Y, is simply
where EQ denotes the expectation under the equivalent martingale measure, Q, and where is the spot rate process, i.e. a solution to the SDE (1). K1fEtO,lO1
*Ignoring all institutional details and “market imperfections” such as taxes, transactions costs, etc. tAnalytical means that the function can be expanded to a power series in an arbitrary point in the domain of the function. This implies, for instance, that the function is equal to the Taylor expansion of the function itself in any point. This use of the term “analytical” is not to be confused with the term an “analytical expression” or a closed form expression which simply means that the expression can be written using standard mathematical functions. $Detined as a non-negative ybr,-measurable stochastic variable, Y, specifying the payout of the contingent claim at the expiration date, rW
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K. R. MILTERSEN
If, for instance, the contingent claim is a call option expiring at to with exercise price 2 on a default-free coupon bond with payout touple { (di, q>}l= ,, then Ycd,:=(c(t,, {(d.,’
T.)]”=mn J
,
(hE(l..
.,JW@J - -a+,
which can also be written, using Equation (3), as
To simplify the notation, we will define Jo&) := min{ h&{1, . . . , J} It,,cT,}. Combining Equations (6) and (7) gives the pricing formula
Unfortunately, the specification of the spot rate process,{Xt},,, from Equation (1) makes it impossible to derive closed form expressions for the option prices, except for trivial cases where the volatility function is constant as in Heath et al. (1992, Section 6, first example, pp. 90-91). However, Miltersen (1992, Theorem 3.3.14) implies that the assumption of a constant volatility is not a fruitful way to proceed, since we cannot assure positive spot rates with a constant volatility. In fact, it is easy to show that a strictly positive constant volatility will give strictly negative spot rates with strictly positive probabilities. Therefore, we are forced to use simulation of the spot rate process under the equivalent martingale measure in order to price contingent claims.* Hence, we are going to simulate Equation (1) in the life-time, [O, t,], of the option we are going to price. By combining the theoretical Heath-Jarrow-Morton model, a simulation algorithm from Pardoux and Talay (1985),t and the pricing formula (8) we can calculate prices of European options eJxpiring at date to with exercise price Z written on a coupon bond with payout touple { (dj, q))j = 1. For notational ease we will call this theoretical option price rt(fO,Z, b), where &e[O,TJ) is the expiration date, Z&IR+is the exercise price, and be{ call, put] is a Boolean variable (or a dummy variable) that keeps track of whether the option is a call or a put option.
4. THE VOLATILITY FUNCTION From Miltersen (1992, Chapter 3) there are some apriori requirements for the volatility function. We have shown the existence and uniqueness of a solution to the SDE (1) and that the spot rate process, {X,lti, will continue to be positive under some restrictions of the functions cr(.). A function fulfilling these requirements is
*The method of deriving a partial differential equation (PDE) using a no-arbitrage argument as in the derivation of, for instance, the Cox-Ingersoll-Ross model (cf. Cox et nl., 1981) and finding the contingent claims prices by numerical solutions of this PDE is not trackable in this case either (cf. Duffie, 1992, Chauter 10, Section G; Miltersen. 1992, Subsection 5.2.1.). tPardoux and Talay (1985, Equation 3.11) present a simulation algorithm specifically designed to calculate the expectation of a function of a solution to an SDE of the time homogenous Markov type. To bring the system of SDEs of which the solution is stated in Equation (8) on the Markov form we need an enhanced state process. We will denote this enhanced state process by (I’,],,. A detailed description of how this algorithm is implemented to perform the simulation is presented in Miltersen (1992, Subsection 5.2.1).
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THE TERM STRUCTURE OF INTEREST RATES
Ml - es19
K=l
f
(9)
with unknown parameters (pl, p,, p2, p2) E IR, x IR+ x IR+ x IR,. K denotes the number of independent uncertainty sources, i.e. the dimension, K, of the Wiener process in the specification of the forward rate process (1).
5. ECONOMETRIC ANALYSIS In this section we will first set up the econometric model to estimate the initial forward rate function,fl.); secondly, we will extend the model so that the volatility function, cr(.), can also be estimated. 5.1. The initial forward rate function Now, we set up an econometric model to estimate the five parameters (r, 6, v, K, u) in the initial forward rate function from Equation (4) at date zero. Our data set consists of prices of Danish government bonds at date zero; i.e. default-free coupon bonds. By combining Equations (3) and (5) we can derive the price of a coupon bond with payout touple, {(d,, t,)):, ,, at date zero as
-+u) V
-v
-(l-exp(-vQ)(-+ v
-+ v2v
SjY”
.
(10)
Our estimation technique follows Hansen (1982) and Hansen and Singleton (1982). Here the technique is used in a cross sectional set-up. All data should be collected at one specific point in time for estimating the forward rate function. We are neither concerned with the change in the functional form nor the change of the estimated parameters as calendar-time passes by. Instead, we will in Subsection 5.2 estimate the volatility of the possible changes* of the forward rate function when calendar-time passes by. 5.1.1. The error term. Our data set is a set of prices of default-free coupon bonds observed at date zero. Say, we observe M bond prices, c”, for m = 1, . . . , M. For each bond there is a corresponding payout touple. Using the model developed in Section 2, with the function, C(0, .), as defined in Equation (lo), our econometric model is Cm=C(O,((d~,t~)}~l)+~m,
V m=l,...,
M,
(11)
with:
*By possible changes, we mean changes in the forward rate function which do not generate arbitrage opportunities between bonds with different maturities and between bonds and the savings account induced by the spot rate process.
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K. R. MILTERSEN
(i) {P)~=, independent and identically distributed, * (with finite second moments); (ii) E[PI( (d,“, t,“)): ,I = 0, for all m = 1, ...,M. We will now set up the moment conditions for our estimation problem. A consequence of condition (ii) is that we can use E[amg(l(dr, t,“)lE,>] = 0, m = 1,. . . , M, for any function g as moment conditions. We are estimating five parameters, so we need at least five moment conditions, i.e. g should map into IR’, for some I 2 5. However, since we want to use the overidentifying restrictions mentioned in Hansen (1982, Section 4) to test whether our model is supported by the data, we will use 10 moment conditions, i.e. r = 10. The following moment conditions will be used: EIEmg,&l(&‘, r,“))~,)]
= 0, m = 1, . . . , M
(12)
where the function g, is picked from (1, d,, e-‘1, d,e-‘1, d12, 6,,
d,2e-2’,, d ,3, e-3’1,d,3e-3’1)’
N=l
(1, d,, e-‘1,d,e-‘1, d2, e-Q,d2e-‘2,dld2, e-+2, dld2e-+‘2)’
N=2
(1, d,, d,e-‘1, d2, d@, d3e-‘3,d,d&-‘2, d2d@-‘3, d,d2d3e41-‘2-‘3)’
N=3
dld3e-II-Q,
(1, d,e-‘1, d2e42, d&3, d&4, dld2e-‘]-‘2,d,d@-‘3, d,dg-V4,
d2d4e-‘2-‘4,d3d4e-Y4)’
(1, d,e-‘1, d2e-‘2, d,e-‘3, d4e-‘4, d&s,
N=4 d,d2e-‘I-Q, d2d3e-12-‘3,
gp,(((d,, &,)I;= ,) :=
(13) N=5
d3d4e-+‘4, d4d5e-W5)’ ( 1, d,e-‘1, d2e+, d3e+, d4e-‘4, d&s, d6e-‘6, d,d4e+-‘4, d2d5e-‘2-3,d,d,e-Y6)’
N=6
(1, d,e-ll, d,e-‘2, d3e-13,d4e+, d&s, d&6, d,e+, d,d5e-fl-t5, d3d7e-W7)
(1, d,e-‘1, d2e-‘2, d@, d,d8e-r,-%)’
N=l d4&4, d&s, d&6, d7e+, $e-%,
N=8
(1, d,e-‘1, d2e-‘2,d3e-‘3,d4e-‘4,d,e-‘5, d6e-%,d7e-‘7,$e-‘8, d@-‘N)’
N29.
5.1.2. GMM estimation. To formalize the GMM estimation for our analysis let h(Crn,N”‘, I@,“, tz)Jz ,; (‘y,6, v, K, u)) := amgp (((d,“, t,“)}:,),
m = 1, . . . , M,
(14)
and define
*This assumption ensures that the sequence,(&m]z= ,, is stationary and ergodic. In general, ergodicity and stationarity a weaker assumption than independence and identical distribution. The assumption of ergodicity and stationarity specially designed for time-series analysis; however, in a cross-sectional set-up this relaxation is of no use.
is is
s35
STRUCTURFi OF INTEREST RATES
THE TERM
MY, 6,v, K u) := E[h(C”, N”, ((d,“,t,“>f ‘; (y, 6, v,
K,
IQ)].
(15)
Then the method of moments estimator of the function k, is k&y, 6, v,
K,
u) :=
L 5 h(C”‘,IL’“, I@,“,t,“>lE,;(Y,h Vv K, u)>M
(16)
m=l
If we call the “true” parameters (r, 6, v, K, u),, then, if our model is correct, according to Equations (12), (14) and (15), k, ((“1.6, v, K, II),) = 0. So we will choose our GMM estimator, (X 6, v, KYu)zMM.of the parameters, (“I,6, v, K, u), such that the moments estimator, kiM((y, 6, v, K U)zMM),is as close to zero as possible. Since h is continuous in (‘y,6, v, K, u), which follows from Equations (11) and (lo), we can, for instance, minimize the quadratic form J&
6, v, K, U) := k,&, 6, V, K u)‘W,&,(y, 6, V, K U),
(17)
where W, is a symmetric positive definite distance matrix, determining in which sense k,((y, 6, is as close to zero as possible. This is in accordance with Hansen and Singleton (1982, p. 1274 ff). In order to get an estimator with the “smallest” asymptotic covariance matrix,* the optimal W,,,can be chosen as follows
V, K u>;w>
w; :=
(t,Elh(Q”, N”, {Cd,“, t,“>,:,; (y, 6,
V, K,
u)&(C”, N”, {(d,“,t,“>j:,;(y, 6, V, K, u>,>‘)-‘3 (18)
where (“I,6, v, K, u)i is just any COnSiStCnt MitIIatOr of the true parameter Set (y, 6, v, K, u),. This means that the estimation procedure is iterative. There are no indications in Hansen and Singleton (1982) of whether it is optimal to do this as a two-step estimation (i.e. estimate the five parameters using an inoptimal W,‘, use this consistent parameter estimate to calculate an optimal W,& and then finally estimate the parameters with the smallest covariance matrix) or whether it is better to keep re-estimating the parameters and re-calculating W,& using the estimators of the last iteration until they eventually converge in an infinite fix-point search. It follows from Hansen (1982, Theorem 3.1) that this iterative GMM estimator, (‘y,6, v, K, U);MM, is unbiased, consistent, and asymptotically efficient. The estimated asymptotic covariante matrix for the estimator is
c&f:=(D,Wgl’,)-1, (19) with
*Here “smallest” means that no other (positive semi-definite) covariance ma&ix exists that, added to a positive definite matrix, can give the matrix that we considered to be the “smallest” covariance matrix in the first place (see also Hansen and Singleton, 1982, pp. 1276-1278; Hansen, 1982, Theorem 3.2).
K. R. MILTERSEN
S36
More specifically, v%(Y, 6, v, K u),“““’ - (‘y,6, v, K, u>;> G NO, &J. 5.1.3. Testing the GMM model. If the model specification (10) is correct MJ,(y, 6, v, K, u) = Mk&fCy, 6, v, K, u)‘W&(y, 6, v, K, u)
(21)
is asymptotically X*-distributed with r-q degrees of freedom, where q = 5 is the number of parameters to estimate (cf. Hansen, 1982, Lemma 4.2). So this is a test of whether the bond data can support the model specified or whether Equation (10) is a mis-specification. 5.2. The Volatilityfinction The purpose of part two of the econometric analysis is to estimate the function rs from Equation (9). We are using implied volatility of European interest rate options to estimate the volatility. This means that we will only need the bond prices at date zero and some prices of European interest rate options at date zero, thereby avoiding, a priori, the assumption of stationarity.* 5.2.1. The error term. Our data set for this analysis consists of prices on Danish European call and put options written on one of the default-free government bonds the price of which we also used as data in estimation of the initial forward rate function. This means that we can price these options by the function K(.,.,.) defined in Section 3. The data set consists of M option prices, P’, for m = 1, . . . , M, with corresponding option touple (t,“, Z”, b”), meaning that option m is a bm option with price nm, exercise price P, and expiration date tr. This introduces the error term em:=7Cm-7r(t~,P,bm),
V m=l,...,
M,
(22)
and the econometric model: (i) {am)!= I independent and identically distributed (with finite second moments); (ii) E[E”lt,“,Z”,Er”]= 0, for all m = 1, . . . , M. Since the number of moment conditions has to be at least as great as the number of parameters to estimate? we set up five moments conditions, i.e. C
(1, Z, e-Q,Ze-‘0,Z&Aog -)’ Z At,,, Z, b) :=
Z (1, Z, e-Q,Z&o, Ze-6log c”
ifb=call
if b = put,
*It is, however, an assumption of the theoretical Heath-Jarrow-Morton model that the volatility function is the same stationary function as time passes. Anyway, this simplification of the theoretical model was made only to simplify the estimation, the volatility function in the SDE spot rate model, u(.), could be allowed to change in a deterministic way as calendar-time passes. Hence, it is a true restriction if the estimation method requires the volatility function to be stationary, a pn’oti. ?Furthermore, if we are going to use the over-identifying restrictions as in Subsubsection 51.3 as a test for the misspecification of our theoretical model, we need the number of moment conditions to be strictly greater than the number of parameters to estimate.
THE TERh4
STRUCTUREOF INTEREST
s37
RATES
where C is the price of the underlying default-free coupon bond. This gives the moment conditions for the volatility estimation E[Vg(tr, Zm,bm)]= 0, V m = 1, . . . , M.
(23)
5.2.2. Simulated moment estimation. For this estimation the GMM estimation technique described in Subsubsection 51.2 has to be extended in order to take into account the fact that the estimated prices from our theoretical model are calculated using a simulated state process which itself depends on the unknown parameters. This is called simulated moments estimation. As the expectation of a function of the solution to the SDE is calculated by using this simulated state process, the distributional properties of the estimators, the simulated moment estimator (SME), and the statistics of the over-identifying restrictions are modified. In this analysis we are following Duffie and Singleton (1993), which studies the problem in a time-series set-up. In this subsubsection, we will apply these results in a cross-sectional set-up. A normal GMM estimation of cc := (pl, pl, h, p2)’E O,, a compact subset of IP:, would be based on the moments &M(W”, $, p, b”); a) := (7c”- n,(t,m,zm, b”; I@&:,
zi”, b”),
where rc&,, Z, b; Y,,,) is defined from Equation (6) and (Y,},Io, q l denotes the enhanced state process including the price processes of the underlying pure default-free bonds. Unfortunately, there is no way of calculating rc,(.,.,.; Y$. 5.2.2.1. First modification. To remedy this, our state process, the enhanced state process, 1 %&[O, tJ’ is discretisized to ( Y,“}“,=, using the discretisizing algorithm from Pardoux and Talay (1985), the updating procedure of this discretisizing algorithm can be written as J$ := YO, rk+l:=H(rkrSk+l;Cb),
k=O,...,n-1,
where I&J”,=, is an independent sequence of random disturbances and a,, is the vector of the true parameters. Estimation of a is now based on the moments hDis((7cm, t’6:P, bm);a) := (v - x,(t;, z?, b”; Y”“))&, P, by. Duffie and Singleton (1993) prove that this modification of the moment conditions does not change the asymptotic distribution of the estimator, a:, provided that M - + 0, as (M, n) + 00. n4 That is, if we are following the GMM estimation of subsubsection 5.1.2., substituting h from Equation (14) with hDigabove, the estimator will be unbiased, consistent, and asymptotically efficient and WC%?
- ok) -No,
C&f),
with &, defined the usual way as in Equation (19). 5.2.2.2. Second mod$icution. Still, even if it is now possible, in theory, to calculate the expectation in x, by evaluating n iterated integrals, we make a second modification. Instead of evaluating the expectation, we take the average of a large sample of realizations of the enhanced state
S38
K. R. MILTERSEN
process, i.e. we simulate the solution. Let ( (I”,]“,=,}y_, be Nrealizations of the discretisized state process generated from Y; := Y,,, i = 1, . . . . , N, ~+,,i:=H(rki,Sk+I,i;ao), k=O ,...,
n-l,
i=l,....,
N,
where ( (&}“,=,}y=I are independent realizations of the random disturbances. This time, estimation of a is based on &ji,((V, t’d:P”, b”); a) := (7tm- 7t(t’d:P”, b”; { y”y)Yz,>>SCtr?pv @I7 where rt is the function defined in Section 3 augmented with the realizations of the discretisized enhanced state process. 5.2.2.3. Third modification. Unfortunately, the updating algorithm, H, that generates the discretisized version of the enhanced state process, (I$}:= ,, is dependent of the unknown parameter set, %, such that we are really generating realizations of the simulated enhanced state process, I y,a”)lt=,, using Yy := Y,, Y~~,:=H(~w,&+,;a),
k=O ,...,
n-l.
It is important to note that the starting point, Y,, is independent of the parameter, a. We have N samples of the simulated discretisized enhanced state process, { ( Yky}“,=1}y=,, and the estimation is finally based on the moments h&&x”, t;, Zm, b”); a) := (IV’- 7c(t;Il, Z”, b’“;{ Yf’):= ,))g(t;, Z”, b”). This final modification of the moment conditions does change the asymptotic distribution of the estimators, ccrE. Duffie and Singleton (1993, Theorem 4, p. 945) prove that we can still follow the GMM estimation procedure of Subsubsection 51.2 substituting h from Equation (14) with h&E. Define the method of moments estimator as kLME(a) := ;
4, h&((v’, m
t:, Z”‘,b”); a)
and the quadratic form s’,““(a) := kLME(a)‘W&jME (a),
(24)
where IV, is a distance matrix. The optimal distance matrix W$ is derived as in Equation (18). The estimator, c$zE, derived from minimizing s’,“” in Equation (24) will still be unbiased, consistent and asymptotically efficient but the asymptotic distribution will be changed such that V%((dS,ME - a,,) =+N(O,(l + ;&). 5.2.3. Testing the simulated model. The above modifications also change the statistics of the over-identifying restrictions, [cf. Equation (21)] such that
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THE TERM STRUCTURE OF INTEREST RATES
Table 1. Parameter estimates Parameters No. par.
4 3 2
1
6
Y 1.053 (58.97) 0.09052 (0.001912) 0.09130 (0.0002449) 0.09276 (0.0001538)
-0.9586 (58.97) 0.004310 (0.001392) 0.003938 (0.0005942) 0
V 0.01916 (0.5709) 0.5892 (0.6020) -! I -
K
0 -!0 0 -
U -0.02017 (0.5807) 0 0 0 -
is now asymptotically X*-distributed with r-q degrees of freedom, where r = 5 is the number of moment conditions and q = 4 is the number of parameters to estimate.
6. ECONOMETRIC ESTIMATION This section presents the results of the estimation of the parameters in the initial forward rate function and the volatility function. 6.1. Estimation results of the initial forward rate function The results from the estimation of the initial forward rate function,f, from Equation (4) are presented in Table 1. Numbers in parentheses are tbe asymptotic standard deviations of the estimators. That is, the estimates of the standard deviations are not the unbiased estimators where we are dividing by M- q. It was impossible to get the GMM algorithm to converge* with all the original five parameters “free”. By forcing one of the parameters, K, to zero the algorithm coverged in around 500 iterations. But as the correlation matrix of the estimafors indicates, there is a serious multicollinearity problem,
This, of course, also explains why the algorithm would not converge with all five parameters “free”, since with almost perfect multicollinearity the estimation algorithm can go into an infinite loop moving the perfectly correlated estimates around without improving the objective function.
*The quadratic form given in Equation (17) is minimized using a numerical procedure called the GMM algorithm. It is a simple modification and extension of the Newton-Raphson method. The algorithm is described in Miltersen (1992, Section 4.A).
s40
K. R. MILTERSEN
Table 2. Test of the models No. par. Over-id. res. D.-f. Test prob.
4
3
2
7.191 6 0.3035
7.989 7 0.3336
7.948 8 0.4385
I 12.49 9 0.1868
The following correlation matrix shows that by forcing one more parameter, 2), to zero the multicollinearity problem is fading, con-, (y, 6, v, 0,O) =
1.000 -0.958 ( 0.987
-0.958 1.000 -0.901
0.987 -0.901 1.000 1
.
This time the algorithm converged in less than 50 iterations.* For the sake of simplicity we will also check the even simpler model where we are also forcing v to one, i.e. fixing the curvature of the forward rate curve (cf. Figs 2 and 3). The correlation matrix for the estimators is in this case
corn,,› (y, 6, 1, 0, 0) =
1.000 -0.961
-0.961
and we are down to 20 iterations. Finally, we estimate the simplest possible model with a flat forward rate curve, i.e. 6 = 0. This very simple estimation converges in less than 10 iterations. All four models have been tested for the over-identifying restrictions. Table 2 shows that there are no indications in these numbers to reject any of the four models at a test level of say 5%. We hereby conclude that we cannot reject that the forward rate function with only two parameters, the constant term, ‘y,and the coefficient to the exponential function of minus the time parameter, 6, free is rich enough to explain the initial forward rate function of the chosen date. Even the one-parameter forward rate function implying a completely flat term structure of interest rate at level y cannot be rejected at a 5% test level. In Figs l-3, we have drawn the forward rate curves and the yield curves corresponding to the three non-trivial models of the four models. 6.2. Estimution results of the volatility function In this subsection we present the results of the estimation of the volatility function, o, from Equation (9). 6.2.1. Pure volatility estimation. The parameter estimates are outlined in Table 3,t numbers in parentheses are still the asymptotic standard deviations of the estimators. The model number refers to the number of independent uncertainty sources, K. The initial forward rate function used in this estimation is the three-parameter model of Subsection 6.1, Table 1. *My simple solution to the multicollinearity problem may have to be discussed. Since multicollinearity is a question of perfectly linearly related parameters, we simply fix one of these linearly related parameters to a fixed value which makes economical sense, and we give a short interpretation of the implications of the curvature of the initial forward rate curve. Alternatively, the multicollinearity problem could have been solved by using a ridge estimator (cf. Amemiya, 1985, Section 2.2, pp. 55-69). The ridge estimator, however, introduces a small bias in the estimator. tFor the estimates to be stable, it was necessary to increase the number of steps in the two-step estimation method presented in Subsubsection 5.1.2 to four steps.
THE TERM
STF3JCTURE OF INTERESTRATES
10.610.410.2I ii
;",.
g. 9.6-
9.29m 0
2
4
6
6
10
lJfn6 (y-1
I-
Forwardratewrve -
Yield curva
I
Fig. 1. The forward rate curve and the yield curve of the four-parameter model.
Time (years) -Fawardratecurw--Yieldyieldarlve Fig. 2. The forward rate curve and the yield curve of the three-parameter model.
S42
K. R. MILTERSEN
aim
0
2
4
8
8
10
Time(years)
I-
Foiward rate ame
-
Yield curve
I
Fig. 3. The forward rate curve and the yield curve of the two-parameter
model.
The correlation matrices of the estimators under the two models are
con;, (CL,,PI, c12,PJ =
1.000 0.909 -0.916 -0.974
0.909 1.000 -0.907 -0.867
-0.916 -0.907 1.000 0.836
-0.974 -0.867 0.836 1.000
(25b) I
Note that in Model 1 p, and p, ‘are strongly negatively correlated, again; indicating a serious multicollinearity problem. Finally, the two models are tested for the over-identifying restrictions in Table 4. On a 5% test level we can accept Model 1, whereas we have to reject Model 2. This is not satisfactory since Model 1 is a special case of Model 2, where the parameter set of Model 1 is a subset of the parameter set of Model 2. This could be due to the fact that the underlying bond on which the options are written is the one that has the largest residual of all bonds with respect to our model of the Table 3. Volatility parameter estimates Parameters Model no.
PI
PI
I
O.OO!W18 (0.003768)
4.992007 (2.208)
2
0.011321 (0.009777)
4.996207 (2.019)
k2 0 0.011242 (0.003894)
I% 0 4.992246 (2.264)
THE TERM STRUCTURE
s43
OF INTEREST RATES
Table4. Testof theoptionmodels Modelno.
2
1
Over-id.res.
6.7970
6.1891
D.f. Test prob.
3 0.0783
1 0.0129
initial forward rate function. That is, by comparing the observed bond prices with the estimated bond prices using the estimation of the initial forward rate function, the theoretical bond pricing model under-prices this bond significantly. This means that our theoretical option pricing model has a tendency to over-price the put options and under-price the call options written on this bond, ceteris paribus.* 6.2.2. Modified volatifiry estimation. To compensate for this, we have re-estimated the parameters of the initial forward rate function such that the bond on which the options are written is priced exactly right and such that the spot rate is unchanged. This gives the following parameter estimates for the initial forward rate function (“I,6, U) = (0.08814,0.006690,0.5892). Re-estimation of the volatility parameters gives much better results (Table 5). Models 1 and 2 refer to the same models as in the previous estimation. The modified correlatiori matrices of the estimators under these two models are
corr, (CL,,p,, O,O)=
co%4 (P,, PIPc12,
P2)
=
1.000 -0.815 1.000 -0.887 0.850 -0.745
-0.815 1.000 > ’ -0.887 1.000 -0.849 0.445
GW 0.850 -0.849 1.000 -0.751
-0.745 0.445 -0.751 1.000
*
(26b)
It seems as if the minimization algorithm has difficulties with estimating the P-parameters, since all p-estimates stay very close to the initial parameter,? also the correlation matrices, especially Equations (25a) and (26a) emphasize this, i.e. ~1,and p, are almost perfectly negatively correlated. For that reason we “fixed” the P-parameters in the following six models, labelled Models 3-8. The three non-trivial correlation matrices of these six models are cofl,
(CL,,19CL*,1) =
1.000 -0.713
-0.713 1.000~> ’
1.000 con;, (II,, 5, )-Lz, 5) = { -0.919 con;, (p,, 25, fi, 25) =
1.000 -0.400
-0.919 1.000 > ’ -0.400 1.000 > ’
*One explanation of this phenomenon could be tax considerations. tThis has been verified by choosing different initial values for p, and&.
W’b) (27~)
K. R. MILTERSEN Table 5. Modified volatility parameterestimates Parameter
0.015769 (0.001343) 0.015410 (0.001782) 0.070329 (0.001409) 0.027132 (0.001023) 0.017387 (0.000117) 0.012565 (0.000239) 0.008220 (0.000024) O.OOOOOO (0.OOOOO2)
4.945213 (0.1446) 5.006804 (0.8922) 1 -
-
0
-
0.007480 (0.000274) 0 -
1
4.960989 (0.3444) 0
-
0.050610 (0.002634) 0 -
5 5
-
25 -
0.012153 (0.000621) 0 -
-
25 -
0.007785 (0.000185)
-
-
0
-
1 0 5 0
25
which gives the same pattern, i.e. that ~1,and h are negatively correlated, which is also intuitively understandable, since the two uncertainty sources should be supplementary to each other. Comparison between the estimated option prices and the observed option prices are found in Table 6. These results look much more promising relative to the results we arrived at in the pure volatility estimation section. Finally, we have also calculated the over-identifying restrictions for these eight models in Table 7. The test probabilities of the Models 2,5 and 8 lie in a region where the obvious conclusion is that we need a larger data set to be able to either reject that the data can support the theoretical option pricing model or to conclude that we cannot reject that the data can support the theoretical model. But since the test probabilities of the rest of the models indicate that there is no reason to reject that the data can support these models, the conclusion must be that, with the data collected, we cannot reject that the data can support the theoretical model. It cannot be rejected either that the reason for the low test probabilities is due to the problems of estimating the P-parameters and the multicollinearity between the two It-parameters. The fact that one factor, i.e. K = 1, seems to be enough to model the future movements of the whole term structure of interest rates could be due to the fact that all options in our sample are written on the same underlying bond.
Table 6. Comparison between observed and estimated option prices Estimated prices for Model no. ID code
Obs. prices
1
2
3
4
5
6
7
8
2005905 2006928 2006936 2005921 2005948 2006952 2006960
0.25 0.90 0.30 0.10 0.70 1.05 2.50
0.40 0.82 0.21 0.05 0.62 0.90 2.22
0.43 0.93 0.28 0.06 0.66 0.98 2.25
0.43 0.90 0.25 0.07 0.67 0.95 2.24
0.33 0.72 0.14 0.02 0.55 0.77 2.12
0.44 0.94 0.28 0.07 0.68 0.98 2.25
0.44 0.92 0.27 0.07 0.68 1.00 2.28
0.54 1.08 0.38 0.09 0.72 1.08 2.31
0.50 1.03 0.34 0.08 0.71 1.03 2.27
s45
THE TERM STENJcTuRE OF INTEREST RATES
Table 7. Test of the modified option models Model no. Over-id. res. D.f. Test prob.
1
2
3
4
5
6
7
8
3.5905 3 0.3092
3.5911 1 0.0581
3.2874 4 0.5109
4.6975 3 0.1953
10.8213 4 0.0286
2.7650 3 0.4293
6.0260 4 0.1972
6.8315 3 0.0775
7. CONCLUSION Firstly, this paper proposed a simple functional form of the initial forward rate function. The functional form was specifically chosen to fit into the no-arbitrage Heath-Jatrow-Morton model of the future movements of the term structure of interest rates, the SDE spot rate model. One of the desired features of the initial forward rate function was that it was analytical. It differed from the traditional cubic spline functions used to estimate forward rate functions on that point. A method was then presented in order to estimate the parameters included in this specific functional form using a non-linear GMM estimation algorithm. The estimation of these parameters was performed with reasonable results. We tested whether the specified functional form of the initial forward rate function could be supported by the data in the bond market. None of our tests reject that the functional form could be supported by the data of that particular chosen date. Secondly, we were interested in estimating the volatility function of the SDE spot rate model. We specified a functional form of the volatility function with some numerical parameters which we estimated by using implied volatilities of option prices on bonds. Pricing of contingent claims using the SDE spot rate model implied simulation because it was not possible to derive the expectation of a function of a solution to the SDE that described the evolution of the bond prices as a closed form expression. Because the updating algorithm used to perform the simulation also depended on the unknown parameters that we were going to estimate, a normal GMM estimation technique could not be used without qualification. We verified that in our situation we could use the GMM estimation algorithm, although the distributions of the estimators were changed and the mis-specification test was also changed. The results of the estimation did not look very promising if the objective was to get small residuals. An ad hoc change of the way the initial forward rate function was estimated made the residuals much smaller, thereby correcting this problem. Acknowledgement - I gratefully acknowledge financial support from the Danish Research Academy, one-year stay at Stanford University, the Graduate School of Business, possible.
which made my
REFERENCES Amemiya, T., Advanced Econometrics (Cambridge, MA: Harvard University Press, 1985). Bollerslev, T., Engle, R. F. and Wooldridge, J. M., A capital asset pricing model with time-varying covariances, Joumai ofPolifical Economy (1988), Vol. 96, pp. 116-131. Cox, J. C., Ingersoll, J. E. and Ross, S. A., A re-examination of traditional hypotheses about the term structure of interest rates, The Journal ofFinance (1981), Vol. KKKVI, pp. 769-799. Duffie, J. D., Dynamic Asset Pricing Theory (Princeton, NJ: Princeton University Press, 1992). Duffie, J. D. and Singleton, K. J., Simulated moments estimation of markov models of asset prices, Econometrica (1993), Vol. 61, pp. 929-952. Engle, R. F., Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation, Econometrica (1982). Vol. 50, pp. 987-1008. Engle, R. F., Lilien, D. M. and Robins, R. P., Estimating time varying risk premia in the term structure: the ARCH-M model, Econometrica (1987). Vol. 55, pp, 391407.
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K.R.MILTERSEN
Engle, R. F., Ng, V. K. and Rothschild, M., Asset pricing with a FACTOR-ARCH covariance structure: empirical estimates for treasury bills, Journal of Econometrics (1990), Vol. 45, pp. 213-237. Hansen, L. P., Large sample properties of generalized method of moments estimators, Economefrica (1982), Vol. 50, pp. 1029-1054. Hansen, L. P. and Singleton, K. J., Generalized instrumental variables estimation of nonlinear rational expectations models, Econometrica (1982), Vol. 50, pp. 1269-1286. Harrison, M. J. and Kreps, D. M., Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory (1979). Vol. 20, pp. 381408. Heath, D., Jarrow, R. and Morton, A. J., Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation, Econometrica (1992), Vol. 60, pp. 77-105. Litzenberger, R. H. and Rolfo, J., An international study of tax effects on government bonds, The Journal of Finance (1984) Vol. XXXIX, pp. 393406. Miltersen, K. R., A Model of the Term Structure of Interest Rates, (Ph.D. thesis, Department of Management, Odense Universitet, Campusvej 55, DK-5230 Odense M, Denmark, 1992. Pardoux, E. and Talay, D., Discretization and simulation of stochastic differential equations, Actu Applicundae Marhematicae (1985), Vol. 3, pp. 2347. Poirier, D. J., The Econometrics of Structural Change - With Special Emphasis on Spline Functions, Vol. 97 of Contributions to Economic Analysis (Amsterdam, The Netherlands: North Holland, 1976). Rindell, K. and Sand&, P., An empirical examination of the pricing of European bond options, Journal of Banking and Finance (1991), Vol. 15, pp. 521-533. Shea, G. S., Pitfalls in smoothing interest rate term structure data: equilibrium models and spline approximations, JournaZ of Financial and Quantitative Analysis (1984), Vol. 19, pp. 253-269. Sorensen, B. G., Modeller til identifiition af over- og undervurderede obligationer. Publications from Department of Management, Odense Universitet 2/1991, Department of Management, Odense Universitet, Campusvej 55, DK-5230 Odense M, Denmark, 1991. Tanggaard, C. and Jakobsen, S., Estimating the term structure of interest rates using cubic splines (working paper), The Aarhus School of Business, Arhus, Denmark, 1988.
Scmd. J. Mgm?, Vol. 9, SuppI., pp. S29446.1993 Printed in Great Britain
AN EMPIRICAL STUDY OF THE TERM STRUCTURE OF INTEREST RATES” KRISTIAN R. MILTERSEN Odense Universitet Abstract - This paper is an empirical study of the Heath-Jarrow-Morton model using General&d Method of Moments and Simulated Method of Moments on Danish bond and option prices. The paper implements a simulation approach to price contingent claims written on purely interest rate-dependent securities fulfilling the Heath-Jarrow-Morton model. This method implies simulation of solutions of stochastic differential equations since the theoretical pricing model is too complicated to give closed form pricing formulas.
Key words: Heath-Jarrow-Morton model, term structure of interest rates, default-free coupon bonds, forward rate, forward rate process, GMM, SME, implied volatility, simulation of SDEs.
1. INTRODUCTION The underlying theoretical model of this empirical study is based on the Heath-Jarrow-Morton model (Heath et al., 1992). In Miltersen (1992, Chapter 3) there is a characterization of four nested models, all based on the Heath-Jarrow-Morton model. We will make an empirical analysis of the simplest of these four models in this paper. Using this model, the so-called SDE spot rate model, it is only necessary to model the movements of the spot rate process, (X,),,,, with I denoting the time-horizon. The model uses the following stochastic differential equation (SDE) description of the spot rate process, under the equivalent martingale measure, Q, X, =At) + &(X,)*dW, + $lla(X,)l12(t-u)du,
V ~1, Q-a.e.,
(1)
wheref(.) is the initial forward rate function at date zero and Q(.) is the time-homogeneous volatility function. The purpose of this empirical study is now to investigate the possibility of estimating these two inputs, f and
*This paper has been greatly improved by discussions with Peter Ove Christensen and Bjarne Graabech Sorensen and b the discussion after the presentation at the Nordic Symposium on Contingent Claims Analysis in Finance in Turku (x bo), Finland, at the E.I.A.S.M. Doctoral Tutorial in Finance at the 19th EFA Ammal Meeting in L.&on, Portugal, and at the E.I.A.S.M. Special Anniversary Workshop on Interest Rates Derivatives in Leuven, Belgium. Comments from Simon Bemiinga, Tom Berglund, Soren Dahlgaard, Thomas Gustavsson, Osmo Jauri, Richard Stapleton and Bo Vad Steffensen, together with much advice from the two discussants, Petter Bjerksund and Hamm Kahra, were most appreciated. I am also grateful for the comments and suggestions received from Kenneth J. Singleton when an earlier draft of this paper was submitted as a research proposal at Stanford University. S29
s30
K. R. MILTERSEN
initial forward rate function not appropriate for the purpose of our model (cf. Poirier, 1976; Litzenberger and Rolfo, 1984; Shea, 1984; Tanggaard and Jakobsen, 1988; Sorensen, 1991, for references about estimating cubic spline functions). That is, to be able to use the SDE spot rate model to price contingent claims written on bonds, the initial forward rate function has to be at least six times continuously differentiable which is not possible with cubic spline functions. We therefore instead set up a simple functional form of the forward rate function,f(.), which is infinitely often differentiable. We then estimate the parameters included in this specific functional form. We use the Generalized Method of Moments (GMM) estimation technique to estimate the parameters. We also set up a statistical test for whether the specified functional form of the forward rate function can be supported by data. The data are from the Danish bond market. Secondly, we are interested in how to estimate the volatility function, (T(.), of the stochastic differential equation (1) in the SDE spot rate model at a specific point in time. We set up a simple functional form of the volatility function, given some technical requirements, developed in Miltersen (1992, Chapter 3), to ensure existence and non-negativity of the spot rate process, {Xl,,,. There are at least two ways of estimating the parameters of the volatility function, either using a time-series of bond price observations or using implied volatility calculations observing one moment of bond option prices. In this paper, we will concentrate on the last method which involves derivation of option prices from the SDE spot rate model (cf. Engle, 1982; Engle et al., 1987; Bollerslev et al., 1988; Engle et al., 1990, for references about estimating variances of bond returns using models with autoregressive conditional heteroscedasticity (ARCH) covariance structures in a time-series set-up). Unfortunately, the contingent claim prices cannot be derived as closed form expressions using the SDE spot rate model except in special cases. For instance, if the volatility function is assumed to be constant, a closed form expression is derived in Heath et al. (1992, Section 6, first example, pp. 90-91). That special case of the Heath-Jarrow-Morton model is empirically tested and compared to the Black-Scholes model in Rindell and Sand& (1991). In our empirical analysis we will study the more general model of non-constant volatility functions. Therefore, the contingent claims prices must be derived using simulations of solutions of the SDE spot rate process and calculation of sample expectations. We estimate the parameters included in the specific functional form of the volatility function, o(.) , given the estimated parameters of the initial forward rate functionA.). The estimation technique used to estimate the volatility function has to be extended from the usual GMM estimator to the so-called Simulated Moments Estimator (SME). This estimator takes into account that the moment conditions are simulated using a simulation algorithm that also depends on the unknown parameters. We also test whether the Heath-Jatrow-Morton model with the specified functional forms of both the initial forward rate function and the volatility function can be supported by the data. The option data are from the Danish Futop market.
2. THE FORWARD RATE FUNCTION We start by specifying the functional form of the initial forward rate function. We denote the price at date zero of a default-free bond expiring at date T as P(0, 2). IfJI.) denotes the forward rate function, at date zero, then the following equation holds: P(0, 7) = e-gAs)ds, t/ T&I.
(2)
THE TERM STRUCTURE
OF INTERESTRATES
s31
The time-horizon is I := [0, r], where I is the maturity of the bond in the financial bond market with the longest maturity; normally we would just set I = 00. Of course, bonds are not just pure default-free bonds, but all default-free coupon bonds are just linear combinations of pure default-free bonds. For instance, if we have a coupon bond paying out d,, units of account at times t,, for n = 1, . . . , N, with N&IN,then the price at date zero of that bond must be
Otherwise, there would be obvious arbitrage opportunities in the bond market.* We will refer to {(d,,t,))~= , as the payout touple of the coupon bond. We specify the initial forward rate function as fit) := y + (la2 + Ut + 6)eH,
(4)
with unknown parameters (y, 6, v, K, u) E 0, := IR, x IR x IR, x IR x IR. Since the underlying theoretical model to be tested determines the evolution of the forward rate function over time, we specify the initial forward rate functionfi.), to be analytical,t which implies that it is infinitely often differentiable. Moreover, the pure default-free bond prices, from Equation (2), with the specific forward rate function, from Equation (4), are derived as
P(0, r) = e
(exp(-@KC+
2
+u) L - i V
V
(l-exp(-VT))($ + ” + G)-y)T V
)
v
T&I.
(5)
3. PRICING OF CONTINGENT CLAIMS As mentioned in Miltersen (1992, Chapter 2), the real problem concerning pricing of contingent claims is to specify a consistent price process of the underlying securities under an equivalent martingale measure. When that is done, the problem concerning pricing of European contingent claims* is solved by Harrison and Kreps (1979, Corollary to Theorem 3, p. 396). The price of the contingent claim, Y, is simply
where EQ denotes the expectation under the equivalent martingale measure, Q, and where is the spot rate process, i.e. a solution to the SDE (1). K1fEtO,lO1
*Ignoring all institutional details and “market imperfections” such as taxes, transactions costs, etc. tAnalytical means that the function can be expanded to a power series in an arbitrary point in the domain of the function. This implies, for instance, that the function is equal to the Taylor expansion of the function itself in any point. This use of the term “analytical” is not to be confused with the term an “analytical expression” or a closed form expression which simply means that the expression can be written using standard mathematical functions. $Detined as a non-negative ybr,-measurable stochastic variable, Y, specifying the payout of the contingent claim at the expiration date, rW
S32
K. R. MILTERSEN
If, for instance, the contingent claim is a call option expiring at to with exercise price 2 on a default-free coupon bond with payout touple { (di, q>}l= ,, then Ycd,:=(c(t,, {(d.,’
T.)]”=mn J
,
(hE(l..
.,JW@J - -a+,
which can also be written, using Equation (3), as
To simplify the notation, we will define Jo&) := min{ h&{1, . . . , J} It,,cT,}. Combining Equations (6) and (7) gives the pricing formula
Unfortunately, the specification of the spot rate process,{Xt},,, from Equation (1) makes it impossible to derive closed form expressions for the option prices, except for trivial cases where the volatility function is constant as in Heath et al. (1992, Section 6, first example, pp. 90-91). However, Miltersen (1992, Theorem 3.3.14) implies that the assumption of a constant volatility is not a fruitful way to proceed, since we cannot assure positive spot rates with a constant volatility. In fact, it is easy to show that a strictly positive constant volatility will give strictly negative spot rates with strictly positive probabilities. Therefore, we are forced to use simulation of the spot rate process under the equivalent martingale measure in order to price contingent claims.* Hence, we are going to simulate Equation (1) in the life-time, [O, t,], of the option we are going to price. By combining the theoretical Heath-Jarrow-Morton model, a simulation algorithm from Pardoux and Talay (1985),t and the pricing formula (8) we can calculate prices of European options eJxpiring at date to with exercise price Z written on a coupon bond with payout touple { (dj, q))j = 1. For notational ease we will call this theoretical option price rt(fO,Z, b), where &e[O,TJ) is the expiration date, Z&IR+is the exercise price, and be{ call, put] is a Boolean variable (or a dummy variable) that keeps track of whether the option is a call or a put option.
4. THE VOLATILITY FUNCTION From Miltersen (1992, Chapter 3) there are some apriori requirements for the volatility function. We have shown the existence and uniqueness of a solution to the SDE (1) and that the spot rate process, {X,lti, will continue to be positive under some restrictions of the functions cr(.). A function fulfilling these requirements is
*The method of deriving a partial differential equation (PDE) using a no-arbitrage argument as in the derivation of, for instance, the Cox-Ingersoll-Ross model (cf. Cox et nl., 1981) and finding the contingent claims prices by numerical solutions of this PDE is not trackable in this case either (cf. Duffie, 1992, Chauter 10, Section G; Miltersen. 1992, Subsection 5.2.1.). tPardoux and Talay (1985, Equation 3.11) present a simulation algorithm specifically designed to calculate the expectation of a function of a solution to an SDE of the time homogenous Markov type. To bring the system of SDEs of which the solution is stated in Equation (8) on the Markov form we need an enhanced state process. We will denote this enhanced state process by (I’,],,. A detailed description of how this algorithm is implemented to perform the simulation is presented in Miltersen (1992, Subsection 5.2.1).
s33
THE TERM STRUCTURE OF INTEREST RATES
Ml - es19
K=l
f
(9)
with unknown parameters (pl, p,, p2, p2) E IR, x IR+ x IR+ x IR,. K denotes the number of independent uncertainty sources, i.e. the dimension, K, of the Wiener process in the specification of the forward rate process (1).
5. ECONOMETRIC ANALYSIS In this section we will first set up the econometric model to estimate the initial forward rate function,fl.); secondly, we will extend the model so that the volatility function, cr(.), can also be estimated. 5.1. The initial forward rate function Now, we set up an econometric model to estimate the five parameters (r, 6, v, K, u) in the initial forward rate function from Equation (4) at date zero. Our data set consists of prices of Danish government bonds at date zero; i.e. default-free coupon bonds. By combining Equations (3) and (5) we can derive the price of a coupon bond with payout touple, {(d,, t,)):, ,, at date zero as
-+u) V
-v
-(l-exp(-vQ)(-+ v
-+ v2v
SjY”
.
(10)
Our estimation technique follows Hansen (1982) and Hansen and Singleton (1982). Here the technique is used in a cross sectional set-up. All data should be collected at one specific point in time for estimating the forward rate function. We are neither concerned with the change in the functional form nor the change of the estimated parameters as calendar-time passes by. Instead, we will in Subsection 5.2 estimate the volatility of the possible changes* of the forward rate function when calendar-time passes by. 5.1.1. The error term. Our data set is a set of prices of default-free coupon bonds observed at date zero. Say, we observe M bond prices, c”, for m = 1, . . . , M. For each bond there is a corresponding payout touple. Using the model developed in Section 2, with the function, C(0, .), as defined in Equation (lo), our econometric model is Cm=C(O,((d~,t~)}~l)+~m,
V m=l,...,
M,
(11)
with:
*By possible changes, we mean changes in the forward rate function which do not generate arbitrage opportunities between bonds with different maturities and between bonds and the savings account induced by the spot rate process.
s34
K. R. MILTERSEN
(i) {P)~=, independent and identically distributed, * (with finite second moments); (ii) E[PI( (d,“, t,“)): ,I = 0, for all m = 1, ...,M. We will now set up the moment conditions for our estimation problem. A consequence of condition (ii) is that we can use E[amg(l(dr, t,“)lE,>] = 0, m = 1,. . . , M, for any function g as moment conditions. We are estimating five parameters, so we need at least five moment conditions, i.e. g should map into IR’, for some I 2 5. However, since we want to use the overidentifying restrictions mentioned in Hansen (1982, Section 4) to test whether our model is supported by the data, we will use 10 moment conditions, i.e. r = 10. The following moment conditions will be used: EIEmg,&l(&‘, r,“))~,)]
= 0, m = 1, . . . , M
(12)
where the function g, is picked from (1, d,, e-‘1, d,e-‘1, d12, 6,,
d,2e-2’,, d ,3, e-3’1,d,3e-3’1)’
N=l
(1, d,, e-‘1,d,e-‘1, d2, e-Q,d2e-‘2,dld2, e-+2, dld2e-+‘2)’
N=2
(1, d,, d,e-‘1, d2, d@, d3e-‘3,d,d&-‘2, d2d@-‘3, d,d2d3e41-‘2-‘3)’
N=3
dld3e-II-Q,
(1, d,e-‘1, d2e42, d&3, d&4, dld2e-‘]-‘2,d,d@-‘3, d,dg-V4,
d2d4e-‘2-‘4,d3d4e-Y4)’
(1, d,e-‘1, d2e-‘2, d,e-‘3, d4e-‘4, d&s,
N=4 d,d2e-‘I-Q, d2d3e-12-‘3,
gp,(((d,, &,)I;= ,) :=
(13) N=5
d3d4e-+‘4, d4d5e-W5)’ ( 1, d,e-‘1, d2e+, d3e+, d4e-‘4, d&s, d6e-‘6, d,d4e+-‘4, d2d5e-‘2-3,d,d,e-Y6)’
N=6
(1, d,e-ll, d,e-‘2, d3e-13,d4e+, d&s, d&6, d,e+, d,d5e-fl-t5, d3d7e-W7)
(1, d,e-‘1, d2e-‘2, d@, d,d8e-r,-%)’
N=l d4&4, d&s, d&6, d7e+, $e-%,
N=8
(1, d,e-‘1, d2e-‘2,d3e-‘3,d4e-‘4,d,e-‘5, d6e-%,d7e-‘7,$e-‘8, d@-‘N)’
N29.
5.1.2. GMM estimation. To formalize the GMM estimation for our analysis let h(Crn,N”‘, I@,“, tz)Jz ,; (‘y,6, v, K, u)) := amgp (((d,“, t,“)}:,),
m = 1, . . . , M,
(14)
and define
*This assumption ensures that the sequence,(&m]z= ,, is stationary and ergodic. In general, ergodicity and stationarity a weaker assumption than independence and identical distribution. The assumption of ergodicity and stationarity specially designed for time-series analysis; however, in a cross-sectional set-up this relaxation is of no use.
is is
s35
STRUCTURFi OF INTEREST RATES
THE TERM
MY, 6,v, K u) := E[h(C”, N”, ((d,“,t,“>f ‘; (y, 6, v,
K,
IQ)].
(15)
Then the method of moments estimator of the function k, is k&y, 6, v,
K,
u) :=
L 5 h(C”‘,IL’“, I@,“,t,“>lE,;(Y,h Vv K, u)>M
(16)
m=l
If we call the “true” parameters (r, 6, v, K, u),, then, if our model is correct, according to Equations (12), (14) and (15), k, ((“1.6, v, K, II),) = 0. So we will choose our GMM estimator, (X 6, v, KYu)zMM.of the parameters, (“I,6, v, K, u), such that the moments estimator, kiM((y, 6, v, K U)zMM),is as close to zero as possible. Since h is continuous in (‘y,6, v, K, u), which follows from Equations (11) and (lo), we can, for instance, minimize the quadratic form J&
6, v, K, U) := k,&, 6, V, K u)‘W,&,(y, 6, V, K U),
(17)
where W, is a symmetric positive definite distance matrix, determining in which sense k,((y, 6, is as close to zero as possible. This is in accordance with Hansen and Singleton (1982, p. 1274 ff). In order to get an estimator with the “smallest” asymptotic covariance matrix,* the optimal W,,,can be chosen as follows
V, K u>;w>
w; :=
(t,Elh(Q”, N”, {Cd,“, t,“>,:,; (y, 6,
V, K,
u)&(C”, N”, {(d,“,t,“>j:,;(y, 6, V, K, u>,>‘)-‘3 (18)
where (“I,6, v, K, u)i is just any COnSiStCnt MitIIatOr of the true parameter Set (y, 6, v, K, u),. This means that the estimation procedure is iterative. There are no indications in Hansen and Singleton (1982) of whether it is optimal to do this as a two-step estimation (i.e. estimate the five parameters using an inoptimal W,‘, use this consistent parameter estimate to calculate an optimal W,& and then finally estimate the parameters with the smallest covariance matrix) or whether it is better to keep re-estimating the parameters and re-calculating W,& using the estimators of the last iteration until they eventually converge in an infinite fix-point search. It follows from Hansen (1982, Theorem 3.1) that this iterative GMM estimator, (‘y,6, v, K, U);MM, is unbiased, consistent, and asymptotically efficient. The estimated asymptotic covariante matrix for the estimator is
c&f:=(D,Wgl’,)-1, (19) with
*Here “smallest” means that no other (positive semi-definite) covariance ma&ix exists that, added to a positive definite matrix, can give the matrix that we considered to be the “smallest” covariance matrix in the first place (see also Hansen and Singleton, 1982, pp. 1276-1278; Hansen, 1982, Theorem 3.2).
K. R. MILTERSEN
S36
More specifically, v%(Y, 6, v, K u),“““’ - (‘y,6, v, K, u>;> G NO, &J. 5.1.3. Testing the GMM model. If the model specification (10) is correct MJ,(y, 6, v, K, u) = Mk&fCy, 6, v, K, u)‘W&(y, 6, v, K, u)
(21)
is asymptotically X*-distributed with r-q degrees of freedom, where q = 5 is the number of parameters to estimate (cf. Hansen, 1982, Lemma 4.2). So this is a test of whether the bond data can support the model specified or whether Equation (10) is a mis-specification. 5.2. The Volatilityfinction The purpose of part two of the econometric analysis is to estimate the function rs from Equation (9). We are using implied volatility of European interest rate options to estimate the volatility. This means that we will only need the bond prices at date zero and some prices of European interest rate options at date zero, thereby avoiding, a priori, the assumption of stationarity.* 5.2.1. The error term. Our data set for this analysis consists of prices on Danish European call and put options written on one of the default-free government bonds the price of which we also used as data in estimation of the initial forward rate function. This means that we can price these options by the function K(.,.,.) defined in Section 3. The data set consists of M option prices, P’, for m = 1, . . . , M, with corresponding option touple (t,“, Z”, b”), meaning that option m is a bm option with price nm, exercise price P, and expiration date tr. This introduces the error term em:=7Cm-7r(t~,P,bm),
V m=l,...,
M,
(22)
and the econometric model: (i) {am)!= I independent and identically distributed (with finite second moments); (ii) E[E”lt,“,Z”,Er”]= 0, for all m = 1, . . . , M. Since the number of moment conditions has to be at least as great as the number of parameters to estimate? we set up five moments conditions, i.e. C
(1, Z, e-Q,Ze-‘0,Z&Aog -)’ Z At,,, Z, b) :=
Z (1, Z, e-Q,Z&o, Ze-6log c”
ifb=call
if b = put,
*It is, however, an assumption of the theoretical Heath-Jarrow-Morton model that the volatility function is the same stationary function as time passes. Anyway, this simplification of the theoretical model was made only to simplify the estimation, the volatility function in the SDE spot rate model, u(.), could be allowed to change in a deterministic way as calendar-time passes. Hence, it is a true restriction if the estimation method requires the volatility function to be stationary, a pn’oti. ?Furthermore, if we are going to use the over-identifying restrictions as in Subsubsection 51.3 as a test for the misspecification of our theoretical model, we need the number of moment conditions to be strictly greater than the number of parameters to estimate.
THE TERh4
STRUCTUREOF INTEREST
s37
RATES
where C is the price of the underlying default-free coupon bond. This gives the moment conditions for the volatility estimation E[Vg(tr, Zm,bm)]= 0, V m = 1, . . . , M.
(23)
5.2.2. Simulated moment estimation. For this estimation the GMM estimation technique described in Subsubsection 51.2 has to be extended in order to take into account the fact that the estimated prices from our theoretical model are calculated using a simulated state process which itself depends on the unknown parameters. This is called simulated moments estimation. As the expectation of a function of the solution to the SDE is calculated by using this simulated state process, the distributional properties of the estimators, the simulated moment estimator (SME), and the statistics of the over-identifying restrictions are modified. In this analysis we are following Duffie and Singleton (1993), which studies the problem in a time-series set-up. In this subsubsection, we will apply these results in a cross-sectional set-up. A normal GMM estimation of cc := (pl, pl, h, p2)’E O,, a compact subset of IP:, would be based on the moments &M(W”, $, p, b”); a) := (7c”- n,(t,m,zm, b”; I@&:,
zi”, b”),
where rc&,, Z, b; Y,,,) is defined from Equation (6) and (Y,},Io, q l denotes the enhanced state process including the price processes of the underlying pure default-free bonds. Unfortunately, there is no way of calculating rc,(.,.,.; Y$. 5.2.2.1. First modification. To remedy this, our state process, the enhanced state process, 1 %&[O, tJ’ is discretisized to ( Y,“}“,=, using the discretisizing algorithm from Pardoux and Talay (1985), the updating procedure of this discretisizing algorithm can be written as J$ := YO, rk+l:=H(rkrSk+l;Cb),
k=O,...,n-1,
where I&J”,=, is an independent sequence of random disturbances and a,, is the vector of the true parameters. Estimation of a is now based on the moments hDis((7cm, t’6:P, bm);a) := (v - x,(t;, z?, b”; Y”“))&, P, by. Duffie and Singleton (1993) prove that this modification of the moment conditions does not change the asymptotic distribution of the estimator, a:, provided that M - + 0, as (M, n) + 00. n4 That is, if we are following the GMM estimation of subsubsection 5.1.2., substituting h from Equation (14) with hDigabove, the estimator will be unbiased, consistent, and asymptotically efficient and WC%?
- ok) -No,
C&f),
with &, defined the usual way as in Equation (19). 5.2.2.2. Second mod$icution. Still, even if it is now possible, in theory, to calculate the expectation in x, by evaluating n iterated integrals, we make a second modification. Instead of evaluating the expectation, we take the average of a large sample of realizations of the enhanced state
S38
K. R. MILTERSEN
process, i.e. we simulate the solution. Let ( (I”,]“,=,}y_, be Nrealizations of the discretisized state process generated from Y; := Y,,, i = 1, . . . . , N, ~+,,i:=H(rki,Sk+I,i;ao), k=O ,...,
n-l,
i=l,....,
N,
where ( (&}“,=,}y=I are independent realizations of the random disturbances. This time, estimation of a is based on &ji,((V, t’d:P”, b”); a) := (7tm- 7t(t’d:P”, b”; { y”y)Yz,>>SCtr?pv @I7 where rt is the function defined in Section 3 augmented with the realizations of the discretisized enhanced state process. 5.2.2.3. Third modification. Unfortunately, the updating algorithm, H, that generates the discretisized version of the enhanced state process, (I$}:= ,, is dependent of the unknown parameter set, %, such that we are really generating realizations of the simulated enhanced state process, I y,a”)lt=,, using Yy := Y,, Y~~,:=H(~w,&+,;a),
k=O ,...,
n-l.
It is important to note that the starting point, Y,, is independent of the parameter, a. We have N samples of the simulated discretisized enhanced state process, { ( Yky}“,=1}y=,, and the estimation is finally based on the moments h&&x”, t;, Zm, b”); a) := (IV’- 7c(t;Il, Z”, b’“;{ Yf’):= ,))g(t;, Z”, b”). This final modification of the moment conditions does change the asymptotic distribution of the estimators, ccrE. Duffie and Singleton (1993, Theorem 4, p. 945) prove that we can still follow the GMM estimation procedure of Subsubsection 51.2 substituting h from Equation (14) with h&E. Define the method of moments estimator as kLME(a) := ;
4, h&((v’, m
t:, Z”‘,b”); a)
and the quadratic form s’,““(a) := kLME(a)‘W&jME (a),
(24)
where IV, is a distance matrix. The optimal distance matrix W$ is derived as in Equation (18). The estimator, c$zE, derived from minimizing s’,“” in Equation (24) will still be unbiased, consistent and asymptotically efficient but the asymptotic distribution will be changed such that V%((dS,ME - a,,) =+N(O,(l + ;&). 5.2.3. Testing the simulated model. The above modifications also change the statistics of the over-identifying restrictions, [cf. Equation (21)] such that
s39
THE TERM STRUCTURE OF INTEREST RATES
Table 1. Parameter estimates Parameters No. par.
4 3 2
1
6
Y 1.053 (58.97) 0.09052 (0.001912) 0.09130 (0.0002449) 0.09276 (0.0001538)
-0.9586 (58.97) 0.004310 (0.001392) 0.003938 (0.0005942) 0
V 0.01916 (0.5709) 0.5892 (0.6020) -! I -
K
0 -!0 0 -
U -0.02017 (0.5807) 0 0 0 -
is now asymptotically X*-distributed with r-q degrees of freedom, where r = 5 is the number of moment conditions and q = 4 is the number of parameters to estimate.
6. ECONOMETRIC ESTIMATION This section presents the results of the estimation of the parameters in the initial forward rate function and the volatility function. 6.1. Estimation results of the initial forward rate function The results from the estimation of the initial forward rate function,f, from Equation (4) are presented in Table 1. Numbers in parentheses are tbe asymptotic standard deviations of the estimators. That is, the estimates of the standard deviations are not the unbiased estimators where we are dividing by M- q. It was impossible to get the GMM algorithm to converge* with all the original five parameters “free”. By forcing one of the parameters, K, to zero the algorithm coverged in around 500 iterations. But as the correlation matrix of the estimafors indicates, there is a serious multicollinearity problem,
This, of course, also explains why the algorithm would not converge with all five parameters “free”, since with almost perfect multicollinearity the estimation algorithm can go into an infinite loop moving the perfectly correlated estimates around without improving the objective function.
*The quadratic form given in Equation (17) is minimized using a numerical procedure called the GMM algorithm. It is a simple modification and extension of the Newton-Raphson method. The algorithm is described in Miltersen (1992, Section 4.A).
s40
K. R. MILTERSEN
Table 2. Test of the models No. par. Over-id. res. D.-f. Test prob.
4
3
2
7.191 6 0.3035
7.989 7 0.3336
7.948 8 0.4385
I 12.49 9 0.1868
The following correlation matrix shows that by forcing one more parameter, 2), to zero the multicollinearity problem is fading, con-, (y, 6, v, 0,O) =
1.000 -0.958 ( 0.987
-0.958 1.000 -0.901
0.987 -0.901 1.000 1
.
This time the algorithm converged in less than 50 iterations.* For the sake of simplicity we will also check the even simpler model where we are also forcing v to one, i.e. fixing the curvature of the forward rate curve (cf. Figs 2 and 3). The correlation matrix for the estimators is in this case
corn,,› (y, 6, 1, 0, 0) =
1.000 -0.961
-0.961
and we are down to 20 iterations. Finally, we estimate the simplest possible model with a flat forward rate curve, i.e. 6 = 0. This very simple estimation converges in less than 10 iterations. All four models have been tested for the over-identifying restrictions. Table 2 shows that there are no indications in these numbers to reject any of the four models at a test level of say 5%. We hereby conclude that we cannot reject that the forward rate function with only two parameters, the constant term, ‘y,and the coefficient to the exponential function of minus the time parameter, 6, free is rich enough to explain the initial forward rate function of the chosen date. Even the one-parameter forward rate function implying a completely flat term structure of interest rate at level y cannot be rejected at a 5% test level. In Figs l-3, we have drawn the forward rate curves and the yield curves corresponding to the three non-trivial models of the four models. 6.2. Estimution results of the volatility function In this subsection we present the results of the estimation of the volatility function, o, from Equation (9). 6.2.1. Pure volatility estimation. The parameter estimates are outlined in Table 3,t numbers in parentheses are still the asymptotic standard deviations of the estimators. The model number refers to the number of independent uncertainty sources, K. The initial forward rate function used in this estimation is the three-parameter model of Subsection 6.1, Table 1. *My simple solution to the multicollinearity problem may have to be discussed. Since multicollinearity is a question of perfectly linearly related parameters, we simply fix one of these linearly related parameters to a fixed value which makes economical sense, and we give a short interpretation of the implications of the curvature of the initial forward rate curve. Alternatively, the multicollinearity problem could have been solved by using a ridge estimator (cf. Amemiya, 1985, Section 2.2, pp. 55-69). The ridge estimator, however, introduces a small bias in the estimator. tFor the estimates to be stable, it was necessary to increase the number of steps in the two-step estimation method presented in Subsubsection 5.1.2 to four steps.
THE TERM
STF3JCTURE OF INTERESTRATES
10.610.410.2I ii
;",.
g. 9.6-
9.29m 0
2
4
6
6
10
lJfn6 (y-1
I-
Forwardratewrve -
Yield curva
I
Fig. 1. The forward rate curve and the yield curve of the four-parameter model.
Time (years) -Fawardratecurw--Yieldyieldarlve Fig. 2. The forward rate curve and the yield curve of the three-parameter model.
S42
K. R. MILTERSEN
aim
0
2
4
8
8
10
Time(years)
I-
Foiward rate ame
-
Yield curve
I
Fig. 3. The forward rate curve and the yield curve of the two-parameter
model.
The correlation matrices of the estimators under the two models are
con;, (CL,,PI, c12,PJ =
1.000 0.909 -0.916 -0.974
0.909 1.000 -0.907 -0.867
-0.916 -0.907 1.000 0.836
-0.974 -0.867 0.836 1.000
(25b) I
Note that in Model 1 p, and p, ‘are strongly negatively correlated, again; indicating a serious multicollinearity problem. Finally, the two models are tested for the over-identifying restrictions in Table 4. On a 5% test level we can accept Model 1, whereas we have to reject Model 2. This is not satisfactory since Model 1 is a special case of Model 2, where the parameter set of Model 1 is a subset of the parameter set of Model 2. This could be due to the fact that the underlying bond on which the options are written is the one that has the largest residual of all bonds with respect to our model of the Table 3. Volatility parameter estimates Parameters Model no.
PI
PI
I
O.OO!W18 (0.003768)
4.992007 (2.208)
2
0.011321 (0.009777)
4.996207 (2.019)
k2 0 0.011242 (0.003894)
I% 0 4.992246 (2.264)
THE TERM STRUCTURE
s43
OF INTEREST RATES
Table4. Testof theoptionmodels Modelno.
2
1
Over-id.res.
6.7970
6.1891
D.f. Test prob.
3 0.0783
1 0.0129
initial forward rate function. That is, by comparing the observed bond prices with the estimated bond prices using the estimation of the initial forward rate function, the theoretical bond pricing model under-prices this bond significantly. This means that our theoretical option pricing model has a tendency to over-price the put options and under-price the call options written on this bond, ceteris paribus.* 6.2.2. Modified volatifiry estimation. To compensate for this, we have re-estimated the parameters of the initial forward rate function such that the bond on which the options are written is priced exactly right and such that the spot rate is unchanged. This gives the following parameter estimates for the initial forward rate function (“I,6, U) = (0.08814,0.006690,0.5892). Re-estimation of the volatility parameters gives much better results (Table 5). Models 1 and 2 refer to the same models as in the previous estimation. The modified correlatiori matrices of the estimators under these two models are
corr, (CL,,p,, O,O)=
co%4 (P,, PIPc12,
P2)
=
1.000 -0.815 1.000 -0.887 0.850 -0.745
-0.815 1.000 > ’ -0.887 1.000 -0.849 0.445
GW 0.850 -0.849 1.000 -0.751
-0.745 0.445 -0.751 1.000
*
(26b)
It seems as if the minimization algorithm has difficulties with estimating the P-parameters, since all p-estimates stay very close to the initial parameter,? also the correlation matrices, especially Equations (25a) and (26a) emphasize this, i.e. ~1,and p, are almost perfectly negatively correlated. For that reason we “fixed” the P-parameters in the following six models, labelled Models 3-8. The three non-trivial correlation matrices of these six models are cofl,
(CL,,19CL*,1) =
1.000 -0.713
-0.713 1.000~> ’
1.000 con;, (II,, 5, )-Lz, 5) = { -0.919 con;, (p,, 25, fi, 25) =
1.000 -0.400
-0.919 1.000 > ’ -0.400 1.000 > ’
*One explanation of this phenomenon could be tax considerations. tThis has been verified by choosing different initial values for p, and&.
W’b) (27~)
K. R. MILTERSEN Table 5. Modified volatility parameterestimates Parameter
0.015769 (0.001343) 0.015410 (0.001782) 0.070329 (0.001409) 0.027132 (0.001023) 0.017387 (0.000117) 0.012565 (0.000239) 0.008220 (0.000024) O.OOOOOO (0.OOOOO2)
4.945213 (0.1446) 5.006804 (0.8922) 1 -
-
0
-
0.007480 (0.000274) 0 -
1
4.960989 (0.3444) 0
-
0.050610 (0.002634) 0 -
5 5
-
25 -
0.012153 (0.000621) 0 -
-
25 -
0.007785 (0.000185)
-
-
0
-
1 0 5 0
25
which gives the same pattern, i.e. that ~1,and h are negatively correlated, which is also intuitively understandable, since the two uncertainty sources should be supplementary to each other. Comparison between the estimated option prices and the observed option prices are found in Table 6. These results look much more promising relative to the results we arrived at in the pure volatility estimation section. Finally, we have also calculated the over-identifying restrictions for these eight models in Table 7. The test probabilities of the Models 2,5 and 8 lie in a region where the obvious conclusion is that we need a larger data set to be able to either reject that the data can support the theoretical option pricing model or to conclude that we cannot reject that the data can support the theoretical model. But since the test probabilities of the rest of the models indicate that there is no reason to reject that the data can support these models, the conclusion must be that, with the data collected, we cannot reject that the data can support the theoretical model. It cannot be rejected either that the reason for the low test probabilities is due to the problems of estimating the P-parameters and the multicollinearity between the two It-parameters. The fact that one factor, i.e. K = 1, seems to be enough to model the future movements of the whole term structure of interest rates could be due to the fact that all options in our sample are written on the same underlying bond.
Table 6. Comparison between observed and estimated option prices Estimated prices for Model no. ID code
Obs. prices
1
2
3
4
5
6
7
8
2005905 2006928 2006936 2005921 2005948 2006952 2006960
0.25 0.90 0.30 0.10 0.70 1.05 2.50
0.40 0.82 0.21 0.05 0.62 0.90 2.22
0.43 0.93 0.28 0.06 0.66 0.98 2.25
0.43 0.90 0.25 0.07 0.67 0.95 2.24
0.33 0.72 0.14 0.02 0.55 0.77 2.12
0.44 0.94 0.28 0.07 0.68 0.98 2.25
0.44 0.92 0.27 0.07 0.68 1.00 2.28
0.54 1.08 0.38 0.09 0.72 1.08 2.31
0.50 1.03 0.34 0.08 0.71 1.03 2.27
s45
THE TERM STENJcTuRE OF INTEREST RATES
Table 7. Test of the modified option models Model no. Over-id. res. D.f. Test prob.
1
2
3
4
5
6
7
8
3.5905 3 0.3092
3.5911 1 0.0581
3.2874 4 0.5109
4.6975 3 0.1953
10.8213 4 0.0286
2.7650 3 0.4293
6.0260 4 0.1972
6.8315 3 0.0775
7. CONCLUSION Firstly, this paper proposed a simple functional form of the initial forward rate function. The functional form was specifically chosen to fit into the no-arbitrage Heath-Jatrow-Morton model of the future movements of the term structure of interest rates, the SDE spot rate model. One of the desired features of the initial forward rate function was that it was analytical. It differed from the traditional cubic spline functions used to estimate forward rate functions on that point. A method was then presented in order to estimate the parameters included in this specific functional form using a non-linear GMM estimation algorithm. The estimation of these parameters was performed with reasonable results. We tested whether the specified functional form of the initial forward rate function could be supported by the data in the bond market. None of our tests reject that the functional form could be supported by the data of that particular chosen date. Secondly, we were interested in estimating the volatility function of the SDE spot rate model. We specified a functional form of the volatility function with some numerical parameters which we estimated by using implied volatilities of option prices on bonds. Pricing of contingent claims using the SDE spot rate model implied simulation because it was not possible to derive the expectation of a function of a solution to the SDE that described the evolution of the bond prices as a closed form expression. Because the updating algorithm used to perform the simulation also depended on the unknown parameters that we were going to estimate, a normal GMM estimation technique could not be used without qualification. We verified that in our situation we could use the GMM estimation algorithm, although the distributions of the estimators were changed and the mis-specification test was also changed. The results of the estimation did not look very promising if the objective was to get small residuals. An ad hoc change of the way the initial forward rate function was estimated made the residuals much smaller, thereby correcting this problem. Acknowledgement - I gratefully acknowledge financial support from the Danish Research Academy, one-year stay at Stanford University, the Graduate School of Business, possible.
which made my
REFERENCES Amemiya, T., Advanced Econometrics (Cambridge, MA: Harvard University Press, 1985). Bollerslev, T., Engle, R. F. and Wooldridge, J. M., A capital asset pricing model with time-varying covariances, Joumai ofPolifical Economy (1988), Vol. 96, pp. 116-131. Cox, J. C., Ingersoll, J. E. and Ross, S. A., A re-examination of traditional hypotheses about the term structure of interest rates, The Journal ofFinance (1981), Vol. KKKVI, pp. 769-799. Duffie, J. D., Dynamic Asset Pricing Theory (Princeton, NJ: Princeton University Press, 1992). Duffie, J. D. and Singleton, K. J., Simulated moments estimation of markov models of asset prices, Econometrica (1993), Vol. 61, pp. 929-952. Engle, R. F., Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation, Econometrica (1982). Vol. 50, pp. 987-1008. Engle, R. F., Lilien, D. M. and Robins, R. P., Estimating time varying risk premia in the term structure: the ARCH-M model, Econometrica (1987). Vol. 55, pp, 391407.
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