Search

In this article, we estimate one-, two- and three-factor generalized Vasicek interest rate models using Japanese yield curve panel data over the important period 2000 to 2010. The state space form of the model is presented and the Kalman filter applied. The empirical results provide support for the two and three factor models and simulations of the models over the period indicate that the two and three factor models performance tracks the Japanese yield curve.

In this paper, we estimate the term structure of interest rate volatilities. It is well known that volatility is the main input for option and other fixed income derivatives valuation models. However, we find that volatility estimates depend significantly on the model used to estimate the zero coupon yield curve (Nelson and Siegel; Vasicek and Fong) and the assumption concerning the heteroskedasticity structure of errors (OLS or GLS weighted by duration). We conclude in our empirical analysis that there are significant differences between these volatility estimates in the short term (less than one year) and in the long term (more than 10 years).

A life annuity contract is an insurance instrument which pays pre-scheduled living benefits conditional on the survival of the annuitant. In order to manage the risk borne by annuity providers, one needs to take into account all sources of uncertainty that affect the value of future obligations under the contract. In this paper, we define the concept of annuity rate as the conditional expected present value random variable of future payments of the annuity, given the future dynamics of its risk factors. The annuity rate deals with the non-diversifiable systematic risk contained in the life annuity contract, and it involves mortality risk as well as investment risk. While it is plausible to assume that there is no correlation between the two risks, each affects the annuity rate through a combination of dependent random variables. In order to understand the probabilistic profile of the annuity rate, we apply comonotonicity theory to approximate its quantile function. We also derive accurate upper and lower bounds for prediction intervals for annuity rates. We use the Lee–Carter model for mortality risk and the Vasicek model for the term structure of interest rates with an annually renewable fixed-income investment policy. Different investment strategies can be handled using this framework.

We consider a portfolio optimization problem under stochastic volatility as well as stochastic interest rate on an infinite time horizon. It is assumed that risky asset prices follow geometric Brownian motion and both volatility and interest rate vary according to ergodic Markov diffusion processes and are correlated with risky asset price. We use an asymptotic method to obtain an optimal consumption and investment policy and find some characteristics of the policy depending upon the correlation between the underlying risky asset price and the stochastic interest rate.

The Edgeworth expansion of the log-likelihood ratio by the maximum likelihood method in the Vasicek model is studied in this paper. Our method provides an improvement on the accuracy of reverting speed estimation under finite time data. The error bound is also provided. The Edgeworth expansion for the distribution is helping with this improvement. Hence, the reverting speed of interest rates can achieve a better simulation result based on such modifications.

We compute the first-order corrections to marginal utility-based prices with respect to a ‘small’ number of random endowments in the framework of three incomplete financial models. They are a stoch...

Complex financial instruments with multiple state variables often have no analytical formulas and therefore must be priced by numerical methods, like lattice ones. For pricing convertible bonds and many other interest rate-sensitive products, research has focused on bivariate lattices for models with two state variables: stock price and interest rate. This paper shows that, unfortunately, when the interest rate component allows rates to grow in magnitude without bounds, those lattices generate invalid transition probabilities. As the overwhelming majority of stochastic interest rate models share this property, a solution to the problem becomes important. This paper presents the first bivariate lattice that guarantees valid probabilities. The proposed bivariate lattice grows (super)polynomially in size if the interest rate model allows rates to grow (super)polynomially. Furthermore, we show that any valid constant-degree bivariate lattice must grow superpolynomially in size with log-normal interest rate models, which form a very popular class of interest rate models. Therefore, our bivariate lattice can be said to be optimal.

We present a European option pricing when the underlying asset price dynamics is governed by a linear combination of the time-change Levy process and a stochastic interest rate which follows the Vasicek process. We obtain an explicit formula for the European call option in term of the characteristic function of the tail probabilities.

We consider an economic model with a deterministic money market account and a finite set of basic economic risks. The real-world prices of the risks are represented by continuous time stochastic processes satisfying a stochastic differential equation of diffusion type. For the simple class of log-normally distributed instantaneous rates of return, we construct an explicit state-price deflator. Since this includes the Black-Scholes and the Vasicek (Ornstein-Uhlenbeck) return models, the considered deflator is called Black-Scholes- Vasicek deflator. Besides a new elementary proof of the Black-Scholes and Margrabe option pricing formulas a validation of these in a multiple risk economy is achieved.

The paper is focused on an analysis of model fit of Dynamic Stochastic General Equilibrium (DSGE) models following New Open Economy Macroeconomics (NOEM). Unlike most of the literature on the topic, this paper does not use Bayesian posterior odds ratio to analyze model fit to data; it uses alternative tools instead. In order to compare the results of the alternative tools to the standard posterior odds ratio, this paper uses the findings of Slanicay and Vašíček (2009), who compared model fit to data of several models with the tool Bayesian posterior odds ratio. The goal of the paper is to verify the results of Slanicay and Vašíček’s (2009) model variants with different criteria than posterior odds and to compare the results with findings of their paper. The tools for the analysis are criteria based on root mean squared error (RMSE) and tools from the Global Sensitivity Analysis toolbox. Conclusions of this paper are the following: Habit persistence in consumption is found to be important and price indexation unimportant as in Slanicay and Vašíček (2009). Furthermore, model variants with foreign economy modeled as AR1 processes always perform better than the ones with structurally modeled foreign economy. This finding is in contradiction to the results of Slanicay and Vašíček (2009).

In this article, the authors develop a new method of estimating multi-parameter term structure models using panel data. This technique involves recursively estimating some parameters along the cross-sectional dimension and the rest of the parameters along the time series dimension until convergence is achieved. By breaking down the parameter estimation process into two simpler procedures along these dimensions, the authors are able to isolate and solve common problems plaguing other methods such as quasi-maximum likelihood estimation via the Kalman filter. As a demonstration, they apply this technique successfully to the one-factor Vasicek and two-factor Cox–Ingersoll–Ross models using Fama–Bliss Treasury data. Simulation results indicate that this technique yields reasonable and robust parameter estimates for these models.

The Cox–Ingersoll–Ross (CIR) model and the Vasicek model are two well-known single factor models of the interest spot rate. In this paper, we construct a mapping by means of which the price of a zero-coupon bond in the CIR model may be obtained from a corresponding price in the Vasicek model. We use symmetry analysis to construct this mapping and verify it by transforming three arbitrary solutions of the pricing equation in the Vasicek model into solutions of the corresponding equation in the CIR model. Copyright © 2010 John Wiley & Sons, Ltd.

In this paper we estimate the Generalized Vasicek term structure model using United Kingdom and Euro panel data. The model is presented in a state space form and the Kalman filter is used to estimate the unobserved state variables and the parameters of the model. One and two factor versions are estimated and the empirical results provide evidence that the two factor model provides a good description of the UK and Euro yield curves.

Under the assumption of stock price and interest rate obeying the stochastic differential equation driven by fractional Brownian motion, we establish the mathematical model for the financial market in fractional Brownian motion setting. Using the risk hedge technique, fractional stochastic analysis and PDE method, we obtain the general pricing formula for the European option with fractional stochastic interest rate. By choosing suitable Hurst index, we can calibrate the pricing model, so that the price can be used as the actual price of option and control the risk management

In this paper, we introduce a new estimator of entropy of a continuous random variable. We compare the proposed estimator with the existing estimators, namely, Vasicek [A test for normality based on sample entropy, J. Roy. Statist. Soc. Ser. B 38 (1976), pp. 54–59], van Es [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Correa [A new estimator of entropy, Commun. Statist. Theory and Methods 24 (1995), pp. 2439–2449] and Wieczorkowski-Grzegorewski [Entropy estimators improvements and comparisons, Commun. Statist. Simulation and Computation 28 (1999), pp. 541–567]. We next introduce a new test for normality. By simulation, the powers of the proposed test under various alternatives are compared with normality tests proposed by Vasicek (1976) and Esteban et al. [Monte Carlo comparison of four normality tests using different entropy estimates, Commun. Statist.–Simulation and Computation 30(4) (2001), pp. 761–785].

In this paper, a finite-state mean-reverting model for the short rate, based on the continuous-time Ehrenfest process, will be examined. Two explicit pricing formulae for zero-coupon bonds will be derived in the general and special symmetric cases. Its limiting relationship to the Vasicek model will be examined with some numerical results.

PurposeThe purpose of this paper is to research stochastic dynamic investment games with stochastic interest rate model in continuous time between two investors. The market interest rate has the dynamics of Duffie‐Kan interest rate.Design/methodology/approachRecently, there has been an increasing interest in financial market models whose key parameters, such as the bank interest rate, stocks appreciation rates, and volatility rates, are modulated by stochastic interest rate. This paper uses the Duffie‐Kan stochastic interest rate model to develop stochastic differential portfolio games. By the HJB optimality equation, a general result in optimal control for a stochastic differential game with a general utility payoff function is obtained.FindingsDerive a general result in optimal control for a stochastic differential game with a general utility payoff function. The explicit optimal strategies and value of the games are obtained for the constant relative risk aversion utility games of fixed duration.Research limitations/implicationsAccessibility and availability of stochastic interest rate data are the main limitations, which apply.Practical implicationsThe results obtained in this paper could be used as a guide to actual portfolio games.Originality/valueThis paper presents a new approach for the optimal portfolio model under compound jump processes. The paper is aimed at actual portfolio games.

In this article, we shall explore the state of art of stochastic flows to derive an exponential affine form of the bond price when the short rate process is governed by a Markovian regime-switching jump-diffusion version of the Vasicek model. We provide the flexibility that the market parameters, including the mean-reversion level, the volatility rate and the intensity of the jump component switch over time according to a continuous-time, finite-state Markov chain. The states of the chain may be interpreted as different states of an economy or different stages of a business cycle. We shall provide a representation for the exponential affine form of the bond price in terms of fundamental matrix solutions of linear matrix differential equations.

The normal distribution is among the most useful distributions in statistical applications. Accordingly, testing for normality is of fundamental importance in many fields including biopharmaceutical research. A generally powerful test for normality is the Shapiro-Wilk test, which can be derived based on estimated entropy divergence. Another well-known test for normality based on entropy divergence was proposed by Vasicek (1976) which has inspired the development of many goodness-of-fit tests for other important distributions. Despite extensive research on the subject, there still exists considerable confusion concerning the fundamental characteristics of Vasicek’s test. This article presents a unified derivation of both the Shapiro-Wilk test and Vasicek’s test based on estimated entropy divergence and clarifies some existing confusion. A comparative study of power performance for these two well-known tests for normality is presented with respect to a wide range of alternatives.

This work considers the solution of the Vasicek-type forward interest rate model. A deterministic process is adopted to model the random behavior of interest rate variation as a deterministic perturbation. It shows that the solution of the Vasicek-type forward interest rate model can be obtained by solving a nonlinear semi-infinite programming problem. A relaxed cutting plane algorithm is then proposed for solving the resulting optimization problem. The features of the proposed method are tested using a set of real data and compared with some commonly used spline fitting methods.

In this paper, we address the issue of determining the optimal contribution rate of a defined benefit pension fund. The affiliate's mortality is modelled by a jump process and the benefits paid at retirement are function of the evolution of future salaries. Assets of the fund are invested in cash, stocks, and a rolling bond. Interest rates are driven by a Vasicek model. The objective is to minimize both the quadratic spread between the contribution rate and the normal cost, and the quadratic spread between the terminal wealth and the mathematical reserve required to cover benefits. The optimization is done under a budget constraint that guarantees the actuarial equilibrium between the current asset and future contributions and benefits. The method of resolution is based on the Cox–Huang approach and on dynamic programming.

The literature in the area of interest rate modelling is extensive. Traditional term structure models, such as Vasicek (1977) and Cox et al (1985) specify the short rate process. As short-term and long-term rates are not perfectly correlated, the data are clearly inconsistent with the use of one-factor time-homogeneous models. Chan et al (1992) demonstrate the empirical difficulties of one-factor continuous-time specifications within the Vasicek and Cox-Ingersoll-Ross (CIR) class of models using the generalized methods of moments.

In this paper, we consider a stochastic diffusion process able to model the interest rate evolving with respect to time and propose a first passage time (FPT) approach through a boundary, defined as the “alert threshold”, in order to evaluate the risk of a proposed loan. Above this alert threshold, the rate is considered at the risk of usury, so new monetary policies have been adopted. Moreover, the mean FPT can be used as an indicator of the “goodness” of a loan; i.e., when an applicant is to choose between two loan offers, s/he will choose the one with a higher mean exit time from the alert boundary. An application to real data is considered by analyzing the Italian average effect global rate by means of two widely used models in finance, the Ornstein-Uhlenbeck (Vasicek) and Feller (Cox-Ingersoll-Ross) models.

We present a methodology for valuing portfolio credit derivatives under a reduced form model for which the default intensity processes of risk assets follow the one-factor Vasicek model. A closed-form solution of joint survival time distribution is obtained. The solution is applied to value credit derivatives of a credit default swap index and collateralized debt obligation. The limitation of methods using the Vasicek model is discussed. We propose that the method is valid and efficient for a portfolio with small-scale correlated risk assets, for which the acceptable size is much greater than for the traditional method. Numerical examples and parameter analysis are also presented.

Three new cluster bonds have been isseued in China as the financial innovation to conquer the SME’s borrowing constrains. SMEs issued the cluster bonds as a group to the outside investors have gained great success. However the pricing mechanism of this new financial technology is still under the research. We made the logical pricing model of credit spread by a geometrical attenuation function reflecting the unexpected default of intensities. Then, we designed a pricing model of small-medium corporate cluster bonds with the spot interest rate assumed to follow Vasicek model. Finally, we tested the pricing model with one thousand times Monte Carlo methods and offered theoretical background and empirical evidence for financial innovation as well.

To predict the exchange rate EUR / MAD & USD / MAD in Morocco we used two most answered methods in the theory: the Box-Jenkins econometric model and the stochastic model of Vasicek then the comparison of the forecasted data for the month of March 2018 of the two methods with the exchange rates actually observed allowed us to retain the econometric the autoregressive integrated moving average model ARIMA (2,1,2) for EUR / MAD and (3,1,2) for USD / MAD rather than the Vasicek model.

The changes of numeraire can be used as a very powerful mean in pricing contingent claims in the context of a complete market. We apply the method of nurmeraire changes to evaluate convertible bonds when the instantaneous growth and variance of the value of issuer and those of zero-coupon bonds follow a general adapted stochastic process in this paper. A closed-form solution is derived when the instantaneous growth and variance of the value of issuer and those of zero-coupon bonds are deterministic function of time. We also consider a special case when the asset price follows GBM (Geometric Brownian Motion) and interest rate follows Vasicek’s model.

This paper considers the effect on zero-coupon bond price valuation when short rate model has non-Gaussian dependent innovations. Higher order asymptotic theory enables us to obtain the approximate bond price formula. Some numerical examples are presented, where the process of innovations follows particular model. These examples indicate non-Gaussianity and dependency of innovations have a great influence on zero-coupon bond price.

SYNOPTIC ABSTRACTIn the financial market including n risky assets with prices following the full-observed geometric-Brownian Motion model and a bank account with stochastic interest rate, we study the problem of maximizing the probability of an investor's wealth at terminal time T meeting or exceeding the value of a contingent claim C which expires at T. In this incomplete market, using an adapted duality methodology familiar in the utility maximization literatures, we characterize the optimal probability and the corresponding portfolio strategies, in the most general framework of interest rate models and contingent claims. Then we calculate in detail the case of Vasicek model for interest rate and some specific contingent claim, via the classical theorem of Time-Changing for Martingales and a generalized Cameron-Martin formula developed in the literatures.

In this paper we expand the Box Method of Sorwar et al. (2007) to value both default free bonds and interest rate contingent claims based on one factor non-linear interest rate models. Further we propose a one-factor non-linear interest rate model that incorporates features suggested by recent research. An example shows the extended Box Method works well in practice.

We derive explicit valuation formulae for an exotic path-dependent interest rate derivative, namely an option on the composition of LIBOR rates. The formulae are based on Fourier transform methods for option pricing. We consider two models for the evolution of interest rates: an HJM-type forward rate model and a LIBOR-type forward price model. Both models are driven by a time-inhomogeneous Levy process.

We study the optimal stopping problems embedded in a typical mortgage. Despite a possible non-rational behaviour of the typical borrower of a mortgage, such problems are worth to be solved for the lender to hedge against the prepayment risk, and because many mortgage-backed securities pricing models incorporate this suboptimality via a so-called prepayment function which can depend, at time t, on whether the prepayment is optimal or not. We state the prepayment problem in the context of the optimal stopping theory and present an algorithm to solve the problem via weak convergence of computationally simple trees. Numerical results in the case of the Vasicek model and of the CIR model are also presented. The procedure is extended to the case when both the prepayment as well as the default are possible: in this case, we present a new method of building two-dimensional computationally simple trees, and we apply it to the optimal stopping problem.

This work considers the resolution of the Vasicek-type interest rate model. A deterministic process is adopted to model the random behaviour of interest rate variation as a deterministic perturbation. It shows that the solution of the Vasicek-type interest rate model can be obtained by solving a non-linear semi-infinite programming problem. A relaxed cutting plane algorithm is then proposed for solving the resulting optimization problem. The numerical results illustrate that our approach essentially generates the yield functions with minimal fitting errors and small oscillation.

In this paper, we extend the one-factor, single regime shift, affine term structure model with time-dependent regime-shift probability to a multi-factor model. We model the nominal interest rate and the expected inflation rate, and estimate the term structure of the real interest rate in the Japanese government bond market using inflation-indexed bond data under zero interest rates. Incorporating the economic structure that the Bank of Japan terminates the zero interest rate when the expected inflation rate gets out of deflationary regime, we estimate the yield curve of the real interest rate for less than 10 years, consistent with the expectation of the market participants in the Japanese government bond market, where inflation-indexed bonds are traded for only around 10 years.

We characterize the dynamics of the US short-term interest rate using a Markov regime-switching model. Using a test developed by Garcia, we show that there are two regimes in the data: In one regime ,t he short rate behaves like a random walk with low volatility; in another regime, it exhibits strong mean reversion and high volatility. In our model, the sensitivity of interest rate volatility to the level of interest rate is much lower than what is commonly found in the literature. We ��

Classic interest rate models, such as those by Vasicek-Hull-White and Heath-Jarrow-Morton, assume one or more underlying stochastic processes of a particular form, with Gaussian disturbances. Interest-dependent securities are then priced under the assumption of market equilibrium, given the rates processes. These models are theoretically elegant but they can be hard to implement in practice because the probability distributions for rates at future discrete dates are no longer Gaussian. Practitioners have largely turned to “market models,” e.g., the LIBOR Market Model (LMM) of Brace, Gatarek, and Musiela, and the Swap Market Model (SMM) of Jamshidian, that do not model the instantaneous evolution of spot rates, but rather the behavior of the forward rates for the specific future dates on which a security’s cash flows will occur. This allows the use of the basic Black model for individual caplets and easy calibration of the model to the market. In this article, Wu and Chen develop closed-form valuation equations for caps, floors, and swaps with barriers within the LMM and SMM frameworks. The key is to model the joint distribution of the rate on each future payment date and the maximum or minimum level the forward may reach over the time period up to that date. Monte Carlo simulation confirms that the closed-form equations provide very accurate pricing.

In this paper, we study the long time behaviour of two classes of stochastic interest rate models. Suppose that x ( t ) is a one-factor interest rate model with positive jumps. For a suitable constant γ > − 1 2 we prove that t − 1 − γ ∫ 0 t x ( s ) d s converges almost surely as t → ∞ . A similar result is also proved for a two-factor affine model.

This paper provides a general approach to deriving no-arbitrage Gaussian term structure models with a three-fold contribution. First, we present a general relationship between the short rate, forward rates, and futures rates, and apply it to derive no-arbitrage Gaussian term structure models. We show two examples, the extended-Vasicek model of Hull and White (1990) and the two-factor model of Hull and White (1994b), in order to demonstrate the derivation procedure. Although many results presented in this article are not new to the literature, our methodology is simple, straightforward, and provides intuitive explanations. Second, our analysis fills a gap in the understanding of the relationship between the short rate process and the forward rate process, and thus provides a linkage between Heath, Jarrow, and Morton's (1990, 1992) model and Hull and White's (1990) model. Therefore, this paper contributes pedagogical value to the term structure model literature. Third, as our models use discrete time setting, it is relatively easy to implement them with numerical procedures such as the lattice approach or Monte Carlo simulations so as to price interest rate derivatives. We give numerical examples to show how to elaborate on the numerical procedure of Hull and White (1994a) using our discrete time model. Resume Cet article fournit une approche non-arbitrage generale applicable a des modeles gaussiens de structure par terme des taux d'interet. Notre contribution est triple. Tout d'abord, nous presentons une relation generale entre les taux court, le taux a terme forward et le taux a terme futures, et l'appliquons pour obtenir des modeles gaussiens de structure par terme. Le procede de derivation est illustre par deux exemples: le modele Vasicek ameliore de Hull et White (1990) et le modele a deux facteurs de Hull et White (1994b). Un bon nombre de resultats presentes sont standards dans la litterature et notre methodologie fournit des explications intuitives. En second lieu, notre analyse comble une lacune dans la comprehension du rapport entre le processus court de taux et le processus de taux a terme, et fournit ainsi un lien entre les modeles de Heath, Jarrow, et Morton (1990, 1992) et celui de Hull et White (1990). Par consequent, cet article apporte des valeurs pedagogiques a la litterature sur la structure par terme des taux d'interet. Troisiemement, etant donne que nos modeles sont mis en oeuvre en temps discret, afin d‘evaluer des derives sur les taux d'interet, il est relativement facile de mettre ces modeles en application avec des procedures numeriques telles que l'approche par “grilles” ou les simulations Monte-Carlo. Nous donnons des exemples numeriques pour demontrer comment adapter le procede numerique de Hull et White (1994a) en utilisant notre modele a temps discret.

Over the past few years, a wide array of interest rate derivative products has been introduced. Interest rate contingent claims are probably the most complex derivative securities to value, since their prices depend on the dynamics of the whole term structure of interest rates. Several pricing models that rest on theories of the term structure have been proposed. Examples are Brennan and Schwartz (1979), Courtadon (1982), Cox, Ingersoll, and Ross (1985), Dothan (1978), Longstaff (1989), and Vasicek (1977). These models price all interest rate dependent products in a consistent fashion, but they involve several unknown parameters--including the utility-dependent market price for risk and they fail to provide a perfect fit to the initial term structure of interest rates. Arbitrage-free models avoid many of the limitations of their predecessors. These models take the current term structure as given and only admit future changes in the term structure that preclude arbitrage opportunities. This approach was first proposed by Ho and Lee (1986) in the discrete time framework. Black, Derman, and Toy (1988) extended Ho and Lee's original idea with a model that provides an exact fit to the initial term structure, as well as to the current volatility of all spot interest rates. Heath, Jarrow, and Morton (1990) introduced a, more general framework which admits one or more factors, and which includes the Ho and Lee model as a special case. Hull and White (1990) derived extensions of Vasicek's (1977) and Cox, Ingersoll, and Ross's (1985) models, making them consistent with both the current term structure of interest rates and the current term structure of volatility. Arbitrage-free models are very appealing from a conceptual point of view, because they ensure that all interest rate contingent claim prices are consistent with the current term structure of interest rates and of volatilities. However, arbitrage-free models require the estimation of the initial yield curve and of the spot and/or forward rate volatility structure, in addition to the parameter estimates of earlier models based on theories of the term structure. This approach is also extremely challenging from an implementation point of view, because most arbitrage-free models developed to date do not have closed-form solutions, and thus involve complex numerical procedures which are computationally very demanding. A simple alternative to the pricing of interest rate contingent claims, which enjoys a great deal of popularity among practitioners for its simplicity and userfriendliness, employs the bond price as the state variable. Assuming that bond prices follow a geometric diffusion process, with constant volatility and constant short rate, one obtains the Black and Scholes (1973) model. Ball and Torous (1W3) retained the constant volatility assumption, but managed to produce a price process that forces the bond to equal its face value at maturity, heir approach allows bond yields to become infinite at maturity. Schaefer and Schwartz (1987) avoided this complication by postulating a process which enables the bond's volatility to decrease over time and reach zero at maturity. Their pricing formula expresses bond volatility as a linear function of its duration. As far as we know, this paper constitutes the first empirical investigation of the Schaefer and Schwartz model. We tested this model in the Canadian context on the Government of Canada bond options traded at the Montreal Exchange. The paper is organized as follows. In the initial section, we present the model and discuss its assumptions. Subsequently, we examine biases detected in implied standard deviations from transactions prices, using our analytic approximation. In the penultimate section, we test a specific trading rule which attempts to capture potential arbitrage profits detected by the model. Then we present our conclusions. A TIME-DEPENDENT VARIANCE MODEL The Schaefer and Schwartz model allows the dynamics of the underlying bond to change as it approaches maturity, and is based on the assumption that the only state variable governing the behaviour of the option is the price of the underlying bond. …

In this paper we analyze the credit spread between Italian and German government bonds after the exchange-rate agreement in May 1998. We estimate the parameters of two mean-reverting affine models for the German term structure and the spread process—the Gaussian Vasicek and the square-root Cox-Ingersoll-Ross (CIR) model. Similar to Pearson and Sun (1994) we combine cross-sectional and time-series information of daily observations to estimate the process parameters employing a maximum likelihood method. Our empirical results show that the Vasicek and CIR model describe the German term structure dynamics equally well. Both models fail to account for all observed shapes of the credit spread structure whereas the spread residuals in the Vasicek case seem to be less volatile. Our results suggest application in the area of pricing credit-sensitive instruments such as credit derivatives or the management of credit risk, especially for European government debt. Resume Nous analysons l'etalement du credit entre les obligations d'etat italiennes et allemandes apres l'accord sur le taux d'interět de mai 1998. Nous evaluons les parametres de deux modeles de retour a la moyenne pour la structure echeanciere allemande et le processus d'etalement—le Gaussian Vasicek et le modele racine-carree Cox-Ingersoll-Ross (CIR). Similairement a Pearson et Sun (1994) nous combinons l'information echantillonnee et en serie d'observations quotidiennes afin d'evaluer les parametres employant une methode a probabilite maximum. Nos resultats empiriques demontrent que les modeles Vasicek et CIR sont incapables de considerer toutes les formes observees de la structure d'etalement du credit tandis que les soldes etales dans le cas Vasicek semblent moins volatiles. Nos resultats suggerent une application dans le domaine de la valorisation d'instruments a credit instable tels que les derives de credit ou la gestion des risques de credit, specifiquement pour la dette gouvernementale europeenne.

This paper considers parameter estimation for continuous-time diffusion processes which are commonly used to model dynamics of financial securities including interest rates. To understand why the drift parameters are more difficult to estimate than the diffusion parameter, as observed in previous studies, we first develop expansions for the bias and variance of parameter estimators for two of the most employed interest rate processes, Vasicek and CIR processes. Then, we study the first order approximate maximum likelihood estimator for linear drift processes. A parametric bootstrap procedure is proposed to correct bias for general diffusion processes with a theoretical justification. Simulation studies confirm the theoretical findings and show that the bootstrap proposal can effectively reduce both the bias and the mean square error of parameter estimates, for both univariate and multivariate processes. The advantages of using more accurate parameter estimators when calculating various option prices in finance are demonstrated by an empirical study.

For the pricing of interest rate derivatives various stochastic interest rate models are used. The shape of such a model can take very different forms, such as direct modelling of the probability distribution (e.g. a generalized beta function of second kind), a short-rate model (e.g. a Hull–White model) or a forward rate model (e.g. a LIBOR market model). This article describes the general structure of optimization in the context of interest rate derivatives. Optimization in finance finds its particular application within the context of calibration problems. In this case, calibration of the (vector-valued) state of a given stochastic model to some target state, which is determined by available relevant market data, implies a continuous optimization of the model parameters such that a global minimum of the distance between the target state and the model state is achieved. In this article, a novel numerical algorithm for the optimization of parameters of stochastic interest rate models is presented. The opt...

Recent empirical finance research has suggested the potential for interest rate series to exhibit non-linear adjustment to equilibrium. This paper examines a variety of models designed to capture these effects and compares both their in-sample and out-of-sample performance with a linear alternative. Using short- and long-term interest rates we report evidence that a logistic smooth-transition error-correction model is able to best characterize the data and provide superior out-of-sample forecasts, especially for the short rate, over both linear and non-linear alternatives. This model suggests that market dynamics differ depending on whether the deviations from long-run equilibrium are above or below the threshold value. Copyright © 2007 John Wiley & Sons, Ltd.

An equilibrium model of a credit market is proposed and examined. The credit price or the interest rate in the model is determined by the consistent interaction of two macroscopic factors: supply and demand. Methods for computing an equilibrium interest rate are suggested. The methods are interpreted as market-balancing dynamics. The convergence of the methods is proved.

A new methodology is proposed to estimate theoretical prices of financial contingent claims whose values are dependent on some other underlying financial assets. In the literature, the preferred choice of estimator is usually maximum likelihood (ML). ML has strong asymptotic justification but is not necessarily the best method in finite samples. This paper proposes a simulation-based method. When it is used in connection with ML, it can improve the finite-sample performance of the ML estimator while maintaining its good asymptotic properties. The method is implemented and evaluated here in the Black-Scholes option pricing model and in the Vasicek bond and bond option pricing model. It is especially favored when the bias in ML is large due to strong persistence in the data or strong nonlinearity in pricing functions. Monte Carlo studies show that the proposed procedures achieve bias reductions over ML estimation in pricing contingent claims when ML is biased. The bias reductions are sometimes accompanied by reductions in variance. Empirical applications to U.S. Treasury bills highlight the differences between the bond prices implied by the simulation-based approach and those delivered by ML. Some consequences for the statistical testing of contingent-claim pricing models are discussed. The Author 2009. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org., Oxford University Press.

Joint default modeling for a set of firms is crucial in the context of pricing credit derivatives. We consider here a model for defaults among multiple firms based on Vasicek or Ornstein–Uhlenbeck models for the hazard rates of the underlying companies or “names.” We analyze the impact of volatility time scales on the default distribution for the set of firms. We also consider the associated impact on a particular credit derivative contract, the so-called collateralized debt obligation (CDO). We demonstrate how correlated fluctuations in the parameters of the firm hazard rates affect the loss distribution and prices associated with the CDOs. The effect of stochastic parameter fluctuations is to change the shape of the loss distribution and cannot be captured by using averaged parameters in the original model. Our analysis assumes a separation of time scales and leads to a singular-regular perturbation problem [J.-P. Fouque, G. Papanicolaou, and R. Sircar, Derivatives in Financial Markets with Stochastic V...