Two Closed-Form Formulas for the Futures Price in the Presence of a Quality Option

The paper derives closed-form formulas for the futures pricein the presence of a multi-asset quality option. This is donefor two cases: In the first one the underlying assets are zerocoupon bonds with different maturities in the single-factorVasicek model. In the second one these are commodities in amulti-factor setting, again with Vasicek interest rate uncertainty.     

Estimation for factor models of term structure of interest rates with jumps: the case of the Taiwanese government bond market

This paper examines the Ornstein–Uhlenbeck (O–U) process used by Vasicek, J. Financial Econ. 5 (1977) 177, and a jump-diffusion process used by Baz and Das, J. Fixed Income (Jnue, 1996) 78, for the Taiwanese Government Bond (TGB) term structure of interest rates. We first obtain the TGB term structures by applying the B-spline approximation, and then use the estimated interest rates to estimate parameters for the one-factor and two-factor Vasicek and jump-diffusion models. The results show that both the one-factor and two-factor Vasicek and jump-diffusion models are statistically significant, with the two-factor models fitting better. For two-factor models, compared with the second factor, the first factor exhibits characteristics of stronger mean reversion, higher volatility, and more frequent and significant jumps in the case of the jump-diffusion process. This is because the first factor is more associated with short-term interest rates, and the second factor is associated with both short-term and long-term interest rates. The jump-diffusion model, which can incorporate jump risks, provides more insight in explaining the term structure as well as the pricing of interest rate derivatives.     

Junp-Diffusion Interest Rate Process: An Empirical Examination

We investigate a jump-diffusion process, which is a mixture of an O-U process used by Vasicek (1977) and a compound Poisson jump process, for the term structure of interest rates. We develop a methodology for estimating the jump-diffusion model and complete an empirical study in comparing the model with the Vasicek model, for the US money market interest rates. The results show that when the short-term interest rate is low, both models predict an upward sloping term structure, with the jump-diffusion model fitting the actual term structure quite well and the Vasicek model overestimating significantly. When the short-term interest rate is high, both models predict a downward sloping term structure, with the jump-diffusion model underestimating the actual term structure more significantly than the Vasicek model.     

An Examination of Blume and Vasicek Betas

This paper examines the Vasicek and Blume methods for correcting OLS betas. The primary conclusions are that typical applications of Vasicek's method seem to mistakenly equate the prior distribution with the cross-sectional distribution of estimated rather than true betas, that Blume's implicit forecast of any tendency for true betas to regress towards one may not be desirable, that preliminary partitioning of firms into industry type groups (as is typical for Vasicek) is desirable, and that conversion of OLS equity betas to asset betas before applying the correction process is also desirable.     

Valuation of Portfolio Credit Derivatives with Default Intensities Using the Vasicek Model

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.     

Pricing interest-rate-derivative securities

This article shows that the one-state-variable interest-ratemodels of Vasicek (1977) and Cox, Ingersoll, and Ross (1985b)can be extended so that they are consistent with both the currentterm structure of interest rates and either the current volatilitiesof all spot interest rates or the current volatilities of allforward interest rates. The extended Vasicek model is shownto be very tractable analytically. The article compares optionprices obtained using the extended Vasicek model with thoseobtained using a number of other models.     

Derivative prices from interest rate models: results for Canada, Hong Kong, and United States

In this paper, we compute implied bond and contingent claim prices from the CKLS, Vasicek, CIR, and BS interest rate models using historical estimates for Canada, Hong Kong, and the United States. We find that default-free bond prices and contingent claim prices are sensitive to the assumed model used for these currencies, and that for Canada the CIR is the best, for Hong Kong the Vasicek and CIR models, and for the US the BS model.     

An alternative approach to the calibration of the Vasicek and CIR interest rate models via generating functions

We propose a new method to calibrate the Vasicek and Cox--Ingersoll--Ross interest rate models from bond prices. We define an appropriate generating function and derive recursive relations between the derivatives of the generating function and the bond prices. The parameters of the Vasicek and CIR models are then obtained by solving a system of linearly independent equations arising from the recursive relations. We include numerical results that show the method’s accuracy when bond prices generated from the exact formulas are used.     

Granularity adjustment for default risk factor model with cohorts

This paper examines granularity adjustments to parameter estimators in a default risk model with cohorts. The model is an extension of the Vasicek model (Vasicek, 1991) and includes a general factor and cohort specific factors. The granularity adjustments derived in the paper concern the mean and/or the variance of observed default frequencies and are easy to implement in practice. For illustration, the method is applied to the S&P corporate ratings. The Granularity Adjusted (GA) estimators are compared to the unadjusted estimators in terms of their asymptotic properties and in finite sample.     

Standard approaches to asset & liability risk** **Supported by the Swedish Insurance Federation.

We compare two different models for assets and liabilities for an insurance company that can be considered in the standard approach to solvency assessment and in particular, in determining the required target capital. The first model is suggested by a joint working party by members in CEA, Comité Européen des Assurances, and is based on the duration concept and the second one is an application of ideas by Samuelson and Vasicek.     

A preference free partial differential equation for the term structure of interest rates

The objectives of this paper are two-fold: the first is the reconciliation of the differences between the Vasicek and the Heath-Jarrow-Morton approaches to the modelling of term structure of interest rates. We demonstrate that under certain (not empirically unreasonable) assumptions prices of interest-rate sensitive claims within the Heath-Jarrow-Morton framework can be expressed as a partial differential equation which both is preference-free and matches the currently observed yield curve. This partial differential equation is shown to be equivalent to the extended Vasicek model of Hull and White. The second is the pricing of interest rate claims in this framework. The preference free partial differential equation that we obtain has the added advantage that it allows us to bring to bear on the problem of evaluating American style contingent claims in a stochastic interest rate environment the various numerical techniques for solving free boundary value problems which have been developed in recent years such as the method of lines.     

Cross‐sectional Restrictions on the Spot and Forward Term Structures of Interest Rates and Panel Unit Root Tests

In this paper we examine the stationarity of all the rates comprising the USD, GBP, DM and JPY spot and forward term structures. Instead of focussing on short maturity interest rates, as most other papers do, we perform a detailed analysis of the whole range of spot and forward interest rates of the 4 main currencies. We investigate the issue of stationarity within the framework of an equilibrium interest rate model such as Vasicek (1977), that defines the cross-sectional and time series properties that interest rates of various maturities must satisfy. We show that within a one-factor interest rate model, such as Vasicek, all interest rates are restricted to exhibit the same mean reverting behaviour. This restriction allows us to apply more powerful panel unit root tests. This methodology increases considerably the number of observations available and as a result the power of the unit root tests. The higher power of these tests allows us to demonstrate that there does exist mean reversion on the spot and forward US interest rates and the forward DM and GBP interest rates.     

Covariance of discounted compound renewal sums with a stochastic interest rate

Formulas have been obtained for the moments of the discounted aggregate claims process, for a constant instantaneous interest rate, and for a claims number process that is an ordinary or a delayed renewal process. In this paper, we present explicit formulas on the first two moments and the joint moment of this risk process, for a non-trivial extension to a stochastic instantaneous interest rate. Examples are given for Erlang claims number processes, and for the Ho–Lee–Merton and the Vasicek interest rate models.     

Nonparametric density estimation and tests of continuous time interest rate models

I study the finite sample distribution of one of Ait-Sahalia's(1996c) nonparametric tests of continuous-time models of theshort-term riskless rate. The test rejects true models too oftenbecause interest rate data are highly persistent but the asymptoticdistribution of the test (and of the kernel density estimatoron which the test is based) treats the data as if it were independentlyand identically distributed. To attain the accuracy of the kerneldensity estimator implied by its asymptotic distribution with22 years of data generated from the Vasicek model in fact requires2755 years of data.     

Interest Rate Term Structure Estimation with Exponential Splines: A Note

Vasicek and Fong 11 developed exponential spline functions as models of the interest rate term structure and claim such models are superior to polynomial spline models. It is found empirically that i) exponential spline term structure estimates are no more stable than estimates from a polynomial spline model, ii) data transformations implicit in the exponential spline model frequently condition the data so that it is difficult to obtain approximations in which one can place confidence, and iii) the asymptotic properties of the exponential spline model frequently are unrealistic. Estimation with exponential splines is no more convenient than estimation with polynomial splines and gives substantially identical estimates of the interest rate term structure as well.     

Continuous time and nonparametric modelling of U.S. interest rate models

In this paper we compare the forecasting performance of different models of interest rates using parametric and nonparametric estimation methods. In particular, we use three popular nonparametric methods, namely, artificial neural networks (ANN), k-nearest neighbour (k-NN), and local linear regression (LL). These are compared with forecasts obtained from two-factor continuous time interest rate models, namely, Chan, Karolyi, Longstaff, and Sanders [CKLS, J. Finance 47 (1992) 1209]; Cos, Ingersoll, and Ross [CIR, Econometrica 53 (1985) 385]; Brennan and Schwartz [BR-SC, J. Financ. Quant. Anal. 15 (1980) 907]; and Vasicek [J. Financ. Econ. 5 (1977) 177]. We find that while the parametric continuous time method, specifically Vasicek, produces the most successful forecasts, the nonparametric k-NN performed well.     

An Exact Bond Option Formula

This paper derives a closed-form solution for European options on pure discount bonds, assuming a mean-reverting Gaussian interest rate model as in Vasicek [8]. The formula is extended to European options on discount bond portfolios.     

A model of the term structure of interest rates for an economically dependent country

This paper presents a closed form model of the term structure of interest rates for an economically dependent country. Using monthly Euroyen rates and Eurodollar rates in the London Market of the period January 1981 to December 1992, we conduct empirical tests and show that our model is consistent with the term structure of the Euroyen rates. Furthermore, comparing the predictive power of our model with that of Vasicek model, our model is shown to perform better.     

Pricing Participating Inflation Retirement Funds Through Option Modeling and Copulas

Abstract

Pension plans and life insurances offering minimum performance guarantees are very common worldwide. In the Brazilian market, the customers of a common type of defined contribution plan have the right to receive, over their savings, the positive difference between the return of a specified investment fund, usually a fixed income fund, and the minimum guaranteed rate, commonly defined as the composition of a fixed interest rate and a floating inflation rate. This instrument can be characterized as an option to exchange one asset, the minimum guaranteed rate, for another, the return of the specified investment fund. In this paper we provide a closed formula to evaluate this liability that depends on two stochastic rates assuming bivariate normality. We also explore the use of copulas for the modeling of the dependence structure and price the options using Monte Carlo simulation to compare the effects of the copula specification in their values. An application with real data is provided. The model makes use of a one-factor Vasicek framework for the term structures of interest rate and inflation rate.