Vasicek, O.A. (1977) An equilibrium characterization of the term structure. J. Fin. Econ. 5, 177-188.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the problem of no short maturity credit spreads. Furthermore by allowing for stochastic interest rates one may better calibrate to existing term structures of interest rates. The model supposes that asset values V (t) follow a geometric Brownian motion while interest rates r(t) follow a Vasicek [15] process with movements that are correlated with the stock. Specically it 8 is supposed that dV = rV dt VdW V (t) dr = r) dt dW r (t) dW V dW r = dt At the rst passage time of V to K the default threshold, we have a default in which the recovery level is some constant write down of ....

Vasicek, O, (1977), \An equilibrium characterization of the term structure," The Journal of Financial Economics,5, 177-188. 21

....section except the generalised Cox Ingersoll Ross model with = 1 satisfy condition (3.8) of Lemma 3.9, hence the Poisson approximation of the upcrossings is also explicitly given for u t = a t x b t and = ln Q(x) where Q is either or . Example 3.10. The Vasicek model (Vasicek [87]) In this model the di usion coecient is (x) 0. The solution of the SDE (3.1) with X 0 = x is given by X t = d(t s) dW s ; t 0 : 18 1 0 1 2 3 4 Figure 3.11. Simulated sample path of the Vasicek model (with parameters c = d = 1) and corresponding normalising constants b t ....

Vasicek, O.A. (1977) An equilibrium characterization of the term structure. J. Fin. Econ. 5, 177-188.

....ranges of all the key parameters determining the value of rate caps and then present, describe and interpret our simulation results. MODEL SPECIFICATION General Assumptions The general setting of our model for pricing any default free contract is described by four assumptions [2] 3] 5] [8]: A.1 The spot rate of interest (r) follows a mean reverting process: dr = k( r)dt o(r)dz, 1) where is the steady state mean, k is the speed of adjustment, o(r) is the standard deviation of the spot rate, and dz is a Wiener process. A.2 The value of a default free contract is a function ....

O. Vasicek. An Equilibrium Characterization of the Term Structure. Journal of Financial Economics 5: 177-188, 1977.

....The parameter # measures the decision maker s degree of impatience. Given that we are interested here in the optimal hedge and not in the optimal distribution of the dividend in time, we will assume r = #. In the case r the distributed amount would clearly be a function of time. See Vasicek [15]. Similar modelisations of the exchange rate behavior could also be implemented. If the volatility is a deterministic function of the exchange rate, and such that #(x) # # x, one obtains a square root process, which has been used for example by Cox, Ingersoll and Ross [5] to model the evolution ....

O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5 (1977), pp. 177--188.

....brief mathematical derivations for the main functions of the interest rate model we use. II. THE INTEREST RATE MODEL Interest rate models are among the more sophisticated financial models, and their calibration is quite challenging. We are going to use the Vasicek model for interest rates [14,18] as a paradigm for employing consistency hints in the calibration of financial models. This concrete example will enable us to do a full derivation of the consistency hint equations and to illustrate the numerical results using real life data. It is fairly straightforward to adapt our method to ....

O. Vasicek, "An equilibrium characterization of term structure," Journal of Financial Economics, vol. 5, pp. 177-188, November 1977.

....GBM : dX t = X t dt X t dW t (1.2) by Osborne (1959) for modeling stock price, and VAS : dX t = 0 1 X t )dt dW t ; 1.3) CIR SR : dX t = 0 1 X t )dt p X t dW t ; 1.4) CIR VR : dX t = X 3=2 t dW t ; 1.5) CKLS : dX t = 0 1 X t )dt X t dW t ; 1. 6) by Vasicek (1977), Cox, Ingersoll, and Ross (1985) Cox, Ingersoll, and Ross (1980) and Chan, Karolyi, Longsta and Sanders (1992) for modeling interest rate dynamics, respectively. Current researches including parametric approaches to estimate ( and ( have been surveyed in Stanton (1997) To relax model ....

Vasicek, O.A. (1977), \An Equilibrium Characterization of the Term Structure," Journal of Financial Economics 5, 177-188.

....are directly related to consumer spending, corporate earnings, asset pricing, in ation and overall economy. See Mishkin (1997) for further discussions. Many useful short rate models have been proposed to explain term structure dynamics and other issues in nance. See for example Merton (1973) Vasicek (1977), Dothan (1978) Brennan and Schwartz (1979, 1980) Cox, Ingersoll and Ross (1980, 1985) Constantinides and Ingersoll (1984) Schaefer and Schwartz (1984) Feldman (1989) Longstall (1989) Hull and White (1990) Black and Karasinki (1991) Longsta and Schwartz (1992) Chan, Karolyi, Longsta ....

.... process fX t g respectively (Wong 1970, and Due 1996) Note that (t; X t ) lim 4 0 1 4 E(X t 4 X t jX t ) 2 (t; X t ) lim 4 0 1 4 E[ X t 4 X t ) 2 jX t ] 2) Examples of (1) include geometric Brownian motion (GBM) for stock prices and interest rate models of Merton (1970) Vasicek (VAS) 1977), Chan, Karolyi, Longsta and Sanders (CKLS) 1992) among others. Di erent models postulate di erent forms on and . For instance, GBM: dX t = X t dt X t dW t ; VAS: dX t = 0 1 X t ) dt dW t ; CIR: dX t = 0 1 X t ) dt p X t dW t ; CKLS: dX t = 0 1 X t ) dt ....

Vasicek, Oldrich, 1977, An equilibrium characterization of the term structure, Journal of Financial Economics 5, 177-188.

....asset pricing results on the time variation of the risk premia in the #nancial markets. Note that the one factor model of interest rates is nested within the two factor model of interest rate outlined in #12# #13#. Setting # z = # z = 0 and #xing z to be a constant, we obtain the one factor #Vasicek #1977## model of interest rates: dr#t#=# r #z#r#t## dt # r d r #t#. Imposing appropriate restrictions on the parameter vector in #27# #28#, we can similarly infer the parameters 16 of the one factor model. In Panel A and Panel B of Table 3, we present the parameter estimates and compare the ....

Vasicek, O., 1977, #An equilibrium characterization of the term structure," The Journal of Financial Economics 5, 177-188.

....more underlying factors, usually interest rates. Bond prices are assumed to be functions of these driving stochastic processes. The prices of contingent claims are then derived by imposing the condition that there are no arbitrage opportunities in the economy between bonds of different maturity. Vasicek s (1977) model is an example of a single factor no arbitrage model in which the whole term structure depends on a single stochastic variable or factor, in this case the short interest rate. It employs an arbitrage argument in relation to the expected return on bonds of differing maturity. The Brennan and ....

Vasicek, O. (1977), "An Equilibrium Characterization of the Term Structure", Journal of Financial Economics, 177-188.

....hedge in the natural instruments. Identifying the natural hedge instruments for #xed income derivatives, we can apply the exchange option framework to the hedging of bond options as well as interest rate derivatives such as caps, #oors and swaptions, using Gaussian term structure models of the Vasicek #1977# type, or lognormal interest rate #market models such as Miltersen, Sandmann and Sondermann #1997#, Brace, Gatarek and Musiela #1997# or Jamshidian #1997#. By explicitly studying the hedging strategy,we arriveatavery straightforward proof of the El Karoui, Jeanblanc Picqu#e and Shreve #1995# and ....

....we have discussed the robustness of Gaussian hedges in general terms. Wegobeyond the Black Scholes framework usually considered in the literature on misspeci#cation to showhow these results can be applied to manyinterest rate derivatives. Gaussian short rate models. In one# and multifactor Vasicek #1977##type term structure models 2 , zero coupon bonds are lognormal assets. Thus, by assuming such a model the hedger can construct Gaussian hedges for options on zero coupon bonds. We denote the price at time t of a zero coupon bond with maturity T by B#t; T #. The payo# of a call option with ....

Vasicek, O. #1977#, An Equilibrium Characterization of the Term Structure, Journal of Financial Economics 5, 177#188.

....nonparametric density estimation or nonparametric regression to study the instantaneous spot interest rate, and to test term structure models. However, little is known about the performance of these methods when applied to persistent time series, such as U.S. interest rates. This paper uses the Vasicek [1977] model to study the performance of kernel density estimates of the ergodic distribution of the instantaneous spot rate. The model s tractability allows me to analyze the MISE of the kernel estimate as a function of persistence, variance of the ergodic distribution, span of the data, sampling ....

....but is known to generate downward biased bandwidth estimates in finite samples with dependent observations. We will investigate the performance of these bandwidth selection methods in more detail for the specific case of the Vasicek model. B. Kernel estimation and the Vasicek model. Vasicek [1977] used the absence of arbitrage to solve for bond prices when interest rates evolve according to the continuous time AR(1) dr = Gamma r)dt oedw: 3) Under this process, r has ergodic distribution: r) 1 q 2 oe 2 2 exp 2 6 4 Gamma:5 0 (r Gamma ) q oe 2 2 1 A 2 3 7 5 ; ....

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Vasicek, Oldrich, 1977 "An Equilibrium Characterization of the Term Structure," Journal of Financial Economics 5 (1977): 177-188.

....that occurs very often in practical applications. Several models are explicitly considered and experimented with: 1. Ma(1) model, with a simple Ar(1) model used as auxiliary model; 2. Arma(1,1) model, with an Ar(2) auxiliary model; 3. the Ornstein Uhlenbeck stochastic di erential equation, used in Vasicek (1977) to model the short term interest rate in continuous time, with an Ar(1) model on discrete data used as auxiliary model; 4. the square root stochastic di erential equation, used in Cox, Ingersoll and Ross (1985) with an Ar(1) auxiliary model applied to discrete data (after some transformation) ....

....global variance of the simple indirect estimator. A similar result could be obtained by simple indirect estimation, using H = 30 or more, thus at much higher computational cost. 5. 3 Ornstein Uhlenbeck process As econometric model let us now consider the stochastic di erential equation employed by Vasicek (1977) to explain the behaviour of short term interest rates (Ornstein Uhlenbeck process) dy t = k(a y t )dt dW t (5:51) where y t is the spot interest rate, W t is a Wiener process. Table 3: O.U. mean estim. param. and (var. Monte Carlo var. Ind.Inf Ind.Inf Least Sqr. Par. True H=1 Cntr.Var. ....

Vasicek, O.A. (1977): \An Equilibrium Characterization of the Term Structure", Journal of Financial Economics 5, 177-88.

....with more extensive sets of timedependent parameters, which they use to match current bond yields, and possibly other asset prices, exactly. To practitioners, the logic of this choice is clear: the parsimonious models used by academics are inadequate for practical use. The four parameters of the Vasicek (1977) and Cox Ingersoll Ross (1985) models, for example, can be chosen to match five points on the yield curve (the four parameters plus the short rate) but do not reproduce the complete yield curve to the degree of accuracy required by market participants. Even complex, multifactor models cannot ....

....later, we start using a new process that is not consistent with the old one. Dybvig argued that these changes in parameter values through time implied that the framework itself was inappropriate. We examine the practitioners procedure in a relatively simple theoretical setting, a variant of Vasicek s (1977) one factor Gaussian interest rate model that we refer to as the benchmark theory. Our thought experiment is to apply models with time dependent drift and volatility parameters to asset prices generated by this theory. We judge the models to be useful, in this setting, if they are able to ....

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Vasicek, Oldrich, 1977, "An equilibrium characterization of the term structure," Journal of Financial Economics 5, 177-188.

....research area of asset pricing is modeling the term structure. Most of the recent term structure models have been developed in continuous time. There are two main approaches to modeling the term structure in continuous time: the noarbitrage approach, and the general equilibrium approach. Vasicek (1977) showed how to price zero coupon default free bonds of di#erent maturities using the no arbitrage approach used in option pricing of Black and Scholes (1973) for a given stochastic process for the spot interest rate. The approach prices all bonds on the basis of a finite number of state variables. ....

....standard Brownian motion. 4 CKLS were concerned with calibrating this general SDE econometrically to evaluate the appropriateness of these competing models for the short rate. The exact functional form of the short rate SDE is of critical importance for models of the term structure. For example, Vasicek (1977) uses an arbitrage argument to derive a partial di#erential equation for bond prices. His derivation was su#ciently general to allow for any di#usion type of SDE for the short rate and then proceeded to derive closed form bond process for the special case of an Ornstein Uhlenbeck process for the ....

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Vasicek, Oldrich, 1977, An Equilibrium Characterization of the Term Structure, Journal of Financial Economics 5, 177--188.

....payoff (as opposed to payoff at maturity) and a regular down and out put. The two options share the same barrier, #D M (t1) and the same exercise price, D M (t1) both of which are stochastic. III.D Numerical Implementations Our constant duration setup makes the mean reverting process of Vasicek (1977) a perfect candidate 9 . Alternatively, we could assume the CIR square root process (CIR, 1985) However, the local volatility of a discount bond depends on the spot interest rate, which makes the constant duration modeling difficult. Since our focus here is the extent of correlation between ....

....z r is a Wiener process whose instantaneous correlation with z D is 1. It follows that the correlation between the asset return and the interest rate is w. Assuming a zero market price of risk (without any loss of generality) the price of a discount bond can be readily calculated, as shown in Vasicek (1977). In addition, with the process in (7) the expected perpetual charter value at a future time t 1 (with a zero net growth of deposits) is obtained by taking t 2 in (5) to infinity: 8) lim t 2 ## C(t 1 )# #D F (t 1 ) e #( r # ) 2 r 2k 2 ) T D . With (4) 7) 1) the bond ....

Vasicek, O., 1977, An Equilibrium Characterization of the Term Structure, Journal of Financial Economics, 5, 177-188.

....paper are derived independently of two con currently written papers on the subject of regime shifts. Evans (1998) investigates the impact of regime switches in inflation and consumption growth to explore the relation between nominal and real interest rates. Work by Naik and Lee (1997) considers a Vasicek (1977) type term structure model with regime shifts in a continuous time setting. Relative to these papers, our derivation of the term structure, and most importantly the scope and nature of our empirical work is very di#erent. Not surprisingly, our conclusions regarding the various models also di#er ....

Vasicek, Oldrich A. (1977), "An Equilibrium Characterization of the Term Structure," Journal of Financial Economics, vol. 5, 177--188.

....Association and Wharton. Financial support from the Center for Research in Security Prices and the IBM Faculty Research Fellowship is gratefully acknowledged. All errors are mine. 1 1. Introduction The first generation of term structure models, starting with Cox, Ingersoll and Ross (1985) and Vasicek (1977), assume that the short term interest rate, as well as the other term structure factors, follow continuous time diffusions 2 (1.1) dr r dt r dZ tt tt = s where a great variety of specifications have been proposed for the functions and s 2 . The few exceptions are jump diffusion models ....

....does not satisfy (3.12) nor, naturally, 3.4) However, being Markov processes, they both satisfy (2.10) They also satisfy the conservation requirement (3.3) More realistically for interest rates, can (3. 11) distinguish between a straight diffusion, say the Ornstein Uhlenbeck specification of Vasicek (1977), and a non diffusion Markov process, such as a diffusion with jumps Consider the following two models, for which closed form transition densities are available (3.20) dr r dt dZ ttt = ba s and (3.21) dr r dt dZ J dN ttttt = ba s where N is a Poisson process with arrival rate h, ....

VASICEK, O., 1977, An Equilibrium Characterization of the Term Structure, Journal of Financial Economics, 5, 177-188.

....rate fall into the class generally referred to as constant elasticity of variance (CEV) models. The CEV model is a one factor model (i.e. Y t is a scalar) taking on the form dY t = r r Y t )dt oeY fi t dW t ; 8) where fi is a constant. Clearly, fi = 0 produces the O U process used by Vasicek (1977) as a model for the shortterm interest rate, while letting fi = 0:5 gives the squareroot process studied by Cox, Ingersoll, and Ross (1985) Plugging the drift and diffusion functions defined by (8) into (5) we get ( j Y ) n Y i=1 1 oe Y fi i Gamma1 exp 8 : Gamma 1 ....

Vasicek, O. (1977). An Equilibrium Characterization of the Term Structure. Journal of Financial Economics, 5, 177--188.

.... Squared Autoregressive Independent Variable Nominal Term Structure Model Despite its many strengths, the SAINTS (Squared Autoregressive Independent Variable Nominal Term Structure) model of Constantinides (1992) cannot be directly compared with other popular term structure models such as Vasicek (1977) and Cox, Ingersoll and Ross (1985) mainly because the stochastic discount factor of the model is exogenously specified. The primary motivation of this paper is to find an equilibrium which will support the given stochastic factor of the SAINTS model in the general equilibrium framework of Cox, ....

....to a two country framework is able to yield sign switching correlations between the interest rates of the two countries, which is consistent with the stochastic behavior of the correlations documented in the empirical literature. 1 Introduction In spite of the presence of many extensions, the Vasicek (1977) and Cox Ingersoll Ross (1985, hereafter CIR) models remain popular because of their parsimonious description of the economy. This parsimonious parameterization is particularly valuable for the implementation and empirical testing of the models. These single factor models yield an affine ....

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Vasicek, O., 1977, "An Equilibrium Characterization of the Term Structure," Journal of Financial Economics 5, 177-188..

....u. That only a rather small degrees of freedom are needed to describe most of the FRC s fluctuations was already discussed on several occasions [16, 15, 22, 7] although not exactly in the present terms. 3 Classical models 3. 1 Vasicek The simplest FRC model is a one factor model due to Vasicek [23], where the whole term structure can be ascribed to the short term interest rate, which is assumed to follow a stochastic evolution described as: dr(t) W(r 0 Gamma r(t) sdW(t) 12) where r 0 is an equilibrium reference rate, W describes the strength of the reversion towards r 0 (and is the ....

....it is awkward to account for a maximum in the volatility s(q) within such an approach (see the discussion below) However, the most obvious inconsistency of this model is the fact that the average spread should be negative. A way out is to introduce the market price of risk : as shown by Vasicek [23], the probability measure over which the average (13) is performed is not necessarily the historical average. Arbitrage arguments allow a change of measure , which in the present case simply amounts to correcting the true (i.e historical) value of r 0 to an effective value r 0 ls, where l is ....

Vasicek, O.A. (1977) "An equilibrium characterization of the term structure ", Journal of Financial Economics, 5, 177-188. 32

.... that its capital structure is more complicated than the simple capital structure assumed by models such as Merton (1974) We regard the di#erence between traditional structural models and our model to be analogous to the di#erence between one factor equilibrium models of the term structure such as Vasicek (1977) and one factor no arbitrage models of the term structure such as Hull and White (1990) The latter are non stationary in that a function of time is introduced into the drift of the short rate to make the model consistent with an exogenously specified initial term structure. Here we make the ....

Vasicek, O. A., "An Equilibrium Characterization of the Term Structure," Journal of Financial Economics, 5 (1977), 177--88.

....the model can shed light on the economics underlying the failure of the expectations hypothesis. The first main conclusion reached here is that the class of a#ne models studied most extensively to date fails at forecasting. I refer to this class, which includes multifactor generalizations of both Vasicek (1977) and Cox, Ingersoll, and Ross (1985) and is analyzed in Dai and Singleton (2000) as completely a#ne. I fit general three factor completely a#ne models to the Treasury term structure over the period 1952 through 1994. Yield forecasts produced with these estimated models are typically worse than ....

....their work by decomposing the variation in interest rate volatility into a component related to the level of short term interest rates and a stochastic volatility component. 3 I thank Rob Bliss for providing me with the yield data. 5 # t = S t # 1 . 9) This class nests multifactor versions of Vasicek (1977) and Cox et al. 1985; hereafter CIR) The main reason for the popularity of this form is that the vector S t # t is a#ne in X t . Thisimpliesa#nedynamicsforX t under both the equivalent martingale and physical measures. A#ne dynamics of X t under the physical measure allow for closedform ....

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Vasicek, Oldrich, 1977, "An equilibrium characterization of the term structure," Journal of Financial Economics 5, 177-188.

....(using the law of iterated expectations) to complete a formal proof that crediting the zero coupon bond yield corresponding to the time between resets gives cost equal to cash balance. To use (7) when interest rates are random, both the spot rate and the crediting rate are simulated using a Vasicek [1977] term structure model with means adjusted to fit today s yield curve (as described by Heath, Jarrow, and Morton [1992] or more explicitly by Dybvig [1997] Then, the appropriate coupon bond yield can be derived from the simulated forward rate curve using a formula such as those given in Table 1 ....

Vasicek, Oldrich, 1977. "An Equilibrium Characterization of the Term Structure. " Journal of Financial Economics vol. 5, 177--188.

.... can be expressed by the general model A) Let us consider the following stochastic di erential equation for the process (r t ) dr t = t) t)r t dt (t)dM t ; t 0: 3:1) We call it a generalized Hull White model because in the case of = 2 it is known as Hull White model ( 11] [18]) We assume that the functions ; and are nonrandom. Lemma 3.1. The solution of equation (3.1) can be expressed in the form (1.4) Proof. We put r t = g(t) h r 0 t Z 0 (s) g(s) ds t Z 0 (s) g(s) dM s i ; 3:2) where M is any standard symmetric stable process of the ....

Vasicek O. An equilibrium characterization of the term structure, Journal of Financial Economics, v. 5 (1977), p. 177- 188 15

.... can also extend the model to bond options by assuming that (continuously compounded) interest rates follow an autoregressive moving average process with the GARCH effects of equation (1) This leads to a family of log linear bond models, whose continuous time limits nest the diffusion models of Vasicek (1977), Heston (1990) and others. 17 Appendix: Derivation of the Generating Function and Option Formulas Proof of Proposition 2: Derivation of the Generating Function: Let x(t) log(S(t) and let f(t;T,f) be the conditional generating function of S(T) or equivalently the conditional moment ....

Vasicek, Oldrich, 1977, "An Equilibrium Characterization of the Term Structure", Journal of Financial Economics, 5, 177-188.

....error bound. Also, Luttmer (1996) focuses on consumption based asset pricing models, whereas we analyze bond pricing models and bond returns. The bond pricing models that we consider are discrete time versions of the affine yield models of Duffie and Kan (1996) This class includes the well known Vasicek (1977) and Cox, Ingersoll and Ross (CIR, 1985) models. There is by now a large literature that empirically investigates these affine yield models in frictionless markets (for example, Babbs and Nowman (1999) Backus and Zin (1994) Brown and Schaefer (1994) Chen and Scott (1993) Gibbons and Ramaswamy ....

Vasicek, O. (1977), `An Equilibrium Characterization of the Term Structure', Journal of Financial Economics, 5, 177-188.

....on the values ( # , w # ( we can predict future values of asset prices as well as estimate their bounds and find new bounds for the perturbations. In the sequel we describe the DEU approach and the corresponding optimization problems in the specific context of the classical model of Vasicek [20], where the standard Brownian motion Z underlies the stochastic di#erential equation: dr = # #r)dt #dZ, 1.14) 6 R. TICHATSCHKE et al. where r is the shortest term riskless rate of interest and the parameters #, # and # are employed to capture shifts and volatility of this rate, see [3] ....

.... approximation by solutions of Cauchy problems for non homogeneous linear ODE with unknown initial values and unknown piece wise constant right sides; Financial models of the observation of the DAX data on the basis of Vasicek s model for the shortest term riskless rate of interest (see [20]) In the cases that the feasible set K of the SIP was unbounded, the choice of the controlling parameters of the PIP method was performed in accordance with the harder conditions of Theorem 1 in [13] At the same time, in some experiments, it turns out to be possible to replace the actual ....

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Vasicek, O., An equilibrium characterization of the term structure, Journal of Financial Economics, Vo. 5, 177-188, 1977

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O.A. Vasicek. An equilibrium characterization of the term structure. J. of Financial Economics, 5:177188, 1977.

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Vasicek, O. A., 1977. An Equilibrium Characterization of the Term Structure. Journal of Financial Economics 5, 177-188. 19

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Vasicek O, (1977) `An Equilibrium Characterization of the Term Structure' Journal of Financial Economics, 5, 177-188 21

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O. Vasicek, An equilibrium characterization of the term structure, J. Finan. Econ. 5, 177-188 (1977).

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Vasicek, O. An equilibrium characterization of the term structure, 1977, Journal of Financial Economics, 5, 177--188.

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Vasicek, O.A. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177-188.

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Finance 19, 425-42. Vasicek, O.A. (1977). An equilibrium characterization of the term structure. J. Financial Economics 5, 177-88.

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Vasicek, O.: An Equilibrium Characterization of the Term Structure. Journal of Financial Economics 5, 177-188 (1977).

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