productivity differences across employers the roles of employer size age and human capital recent research using longitudinal microeconomic databases on businesses and individuals has uncovered tremendous heterogeneity across businesses even within narrowly defined industrial sectors we observe large differences in the levels and growth rates of inputs outputs and productivity across workers even controlling for demographic characteristics wage and employment outcomes vary a great deal and there are sizable worker flows across jobs in the economy these worker flows are closely connected to firm outcomes reflecting in large part the ongoing shift in resources from less productive to more productive employers the primary message is that the continual process of matching and sorting as businesses and workers seek their niches is heterogeneous with experimentation learning and selection all playing important roles understanding the connection between the matching and sorting of businesses and workers is at the very heart of the overlap between industrial organization and labor economics however due to the paucity of data linking businesses and workers at the micro level we know relatively little about these interactions although a large empirical literature suggests that worker outcomes are associated with firm characteristics notable examples are firm size wage effects compensating wage differentials and interindustry wage differentials very little is known about the converse the process by which business outcomes are associated with the characteristics of their employees there are a number of questions associated with this process do employers choose different modes of production resulting in skill segregation as suggested by the sorting and assignment hypotheses in michael kremer 1993 and michael sattinger 1993 or do all firms in a given industry select roughly the same workforce composition how do employers change their workforce composition either to accommodate demand shocks or to experiment with new modes of production in turn what is the connection between these different decisions regarding workforce composition and observable business outcomes such as survival growth and productivity in this short paper our empirical objectives are relatively modest compared to these broad research questions our focus is to provide an initial investigation into the connection between a key measure of firm performance labor productivity and the composition of the firm s workforce as measured by observable worker characteristics we exploit longitudinal matched employeremployee micro data developed under a recent pilot project at the u s census bureau our data set contains longitudinal information on key outcomes of the businesses as well as many of the basic demographic characteristics for the universe of workers at these businesses although longitudinal matched employer employee micro data have been developed in a variety of countries over the last decade or so see john abowd and francis kramarz 1999 ours is the first large scale micro level data set in the united states that allows the investigation haltiwanger department of economics university of maryland college park md 20742 u s bureau of the census and nber lane department of economics american university washington dc 20016 and u s bureau of the census spletzer bureau of labor statistics 2 massachusetts ave n e washington dc 20212 we thank adela luque and javier miranda for superb research assistance and charles brown and marilyn manser for comments on an earlier draft this research was conducted at the center for economic studies at the u s bureau of the census while the second author was an american statistical association census bureau fellow the analysis and results are attributable to the authors and do not necessarily reflect concurrence of the institutions they represent

appropriate business strategy for leaders and laggards we develop a series of hypotheses that predict that the appropriateness of different business strategies is conditional on the firm s relative performance or distance to the industry frontier we use data on three 2 digit high tech manufacturing industries in the united states over the period 1972 1999 and apply semi parametric quantile regressions to investigate the contribution of firm behavior to market value at various points of the conditional distribution of tobin s q among our results we observe that innovative activity measured in terms of r d expenditure or patents has a strong positive association with market value at the upper quantiles corresponding to the leader firms whereas the innovative efforts of laggard firms are valued significantly less laggard firms we suggest should instead achieve productivity growth through efficient exploitation of existing technologies and imitation of industry leaders employment growth in leader firms is encouraged whereas growth of backward firms is not as well received on the stock market

micro data heterogeneity and the evaluation of public policy nobel lecture this paper summarizes the contributions of microeconometrics to economic knowledge four main themes are developed 1 microeconometricians developed new tools to respond to econometric problems raised by the analysis of the new sources of micro data produced after the second world war 2 microeconometrics improved on aggregate time series methods by building models that linked economic models for individuals to data on individual behavior 3 an important empirical regularity detected by the field is the diversity and heterogeneity of behavior this heterogeneity has profound consequences for economic theory and for econometric practice 4 microeconometrics has contributed substantially to the scientific evaluation of public policy

an analysis of the true notional bond system applied to the cbot t bond futures abstract the conversion factor system cfs is used in the determination of the invoice price of the chicago board of trade treasury bond futures as an alternative to the cfs oviedo oviedo r a 2006 improving the design of treasury bond futures contracts the journal of business 79 1293 1315 proposed the true notional bond system tnbs and showed that it outperforms the cfs when interest rates are deterministic the main purpose of this paper is to compare the effectiveness of the two systems in a stochastic environment in order to do so we price the cbot t bond futures as well as all its embedded delivery options under both the cfs and the tnbs our pricing procedure is an adaptation of the dynamic programming algorithm described in ben abdallah et al ben abdallah r ben ameur h breton m 2007 pricing cbot treasury bond futures les cahiers du gerad g 2006 77 numerical illustrations show that in a stochastic framework tnbs does not always outperform the cfs however as the long term mean moves away from the level of the notional rate the tnbs performs increasingly better than the cfs

box cox quantile regression and the distribution of firm sizes using the box cox quantile regression model we analyse the size distribution of firms in portuguese manufacturing during the 1980s specifically we estimate the effect of selected industry attributes on the location scale skewness and kurtosis of the conditional size distributions of firms we find that industry attributes affect the size of firms in the same direction across the distribution but the effects of these variables are typically much greater at the largest quantiles over time the distribution shifted towards smaller firms due mainly to the way the economy responds to industry characteristics rather than to changes of the level of these characteristics the prediction of lognormality implied by gibrat s law is soundly rejected by the observed distribution of firm sizes however we found that at least in 1983 lognormality is a reasonable description of the conditional size distribution copyright 2000 john wiley sons ltd

counterfactual decomposition of changes in wage distributions using quantile regression we propose a method to decompose the changes in the wage distribution over a period of time in several factors contributing to those changes the method is based on the estimation of marginal wage distributions consistent with a conditional distribution estimated by quantile regression as well as with any hypothesized distribution for the covariates comparing the marginal distributions implied by different distributions for the covariates one is then able to perform counterfactual exercises the proposed methodology enables the identification of the sources of the increased wage inequality observed in most countries specifically it decomposes the changes in the wage distribution over a period of time into several factors contributing to those changes namely by discriminating between changes in the characteristics of the working population and changes in the returns to these characteristics we apply this methodology to portuguese data for the period 1986 1995 and find that the observed increase in educational levels contributed decisively towards greater wage inequality copyright 2005 john wiley sons ltd

persistence survival and growth a closer look at 20 years of fast growing firms in austria for survival persistence and growth hifs show a much higher persistence than hgfs the average growth rate after the high growth episode is quite modest for both hifs and hgfs policy implications of the ndings are discussed

a dynamic programming approach for pricing options embedded in bonds abstract we propose a dynamic programming dp approach for pricing options embedded in bonds the focus being on call and put options with advance notice an efficient procedure is developed for the cases where the interest rate process follows the vasicek cox ingersoll ross cir or generalized vasicek models our dp methodology uses the exact joint distribution of the interest rate and integrated interest rate at a future date conditional on the current value of the interest rate we provide numerical illustrations for the vasicek and cir models comparing our dp method with finite difference methods our procedure compares quite favorably in terms of both efficiency and accuracy an important advantage of the our dp approach is that it can be applied to more general models calibrated to capture the term structure of interest rates e g the generalized vasicek model

changes in the u s wage structure 1963 1987 application of quantile regression the 1980 s witnessed rapid changes in the structure of wages in the united states in this study recently developed techniques for quantile regression are applied to every march current population survey since 1964 and changes in the return to schooling and experience at different points of the wage distribution are examined the quantile regression technique parsimoniously describes the entire conditional wage distribution this conditional distribution is used to examine changes in within group wage inequality as measured by the difference between conditional quantiles two linear models are considered a simple one group model and a 16 group model the results suggest that the returns to schooling and experience differ across quantiles of the wage distribution but their patterns of change are similar significant differences in wage inequality are also found across the various skill groups the paper also presents a new imputation method for cps data

pricing stock and bond options when the default free rate is stochastic we derive formulas for the valuation of call options on stocks and bonds when the defaultfree rate is stochastic the formulas highlight the role of the correlation between the unanticipated returns on the underlying security and the changes in the short term rate in determining the options value our numerical analysis indicates that option prices predicted by the proposed formula differ from those predicted by the black and scholes formula when this correlation is relatively large and the short term rate s instantaneous variance is relatively large as well moreover the proposed formula predicts higher lower stock option prices than those predicted by the black and scholes formula for correlation values that are lower higher than some positive critical value

hysteresis effects under cir interest rates most decision making research in real options focuses on revenue uncertainty assuming discount rates remain constant however for many decisions revenue or cost streams are relatively static and investment is driven by interest rate uncertainty for example the decision to invest in durable machinery and equipment using interest rate models from cox et al 1985b we generalize the work of ingersoll and ross 1992 in two ways firstly we include real options on perpetuities in addition to zero coupon cash flows secondly we incorporate abandonment or disinvestment as well as investment options and thus model interest rate hysteresis parallel to revenue uncertainty in dixit 1989a under stochastic interest rates economic hysteresis is found to be significant even for small sunk costs

product market regulation innovation and distance to frontier this article contributes to the literature on competition and innovation it tests the impact of market regulation on innovation conditional to the closeness to the technological frontier with a panel of 15 industries for 17 oecd countries over the period 1979 2003 one of the main conclusions of this literature is that of a negative impact of regulation growing in intensity with the proximity to the frontier a simple model of innovation and growth shows that one should not necessarily expect this result empirical tests on a variety of pecifications show that the impact of regulation can be positive when industries are close to the technological frontier we argue that this result is in fact line with previous evidence

an intertemporal general equilibrium model of asset prices this paper develops a continuous time general equilibrium model of a simple but complete economy and uses it to examine the behavior of asset prices in this model asset prices and their stochastic properties are determined endogenously one principal result is a partial differential equation which asset prices must satisfy the solution of this equation gives the equilibrium price of any asset in terms of the underlying real variables in the economy in this paper we develop a general equilibrium asset pricing model for use in applied research an important feature of the model is its integration of real and financial markets among other things the model endogenously determines the stochastic process followed by the equilibrium price of any financial asset and shows how this process depends on the underlying real variables the model is fully consistent with rational expectations and maximizing behavior on the part of all agents our framework is general enough to include many of the fundamental forces affecting asset markets yet it is tractable enough to be specialized easily to produce specific testable results furthermore the model can be extended in a number of straightforward ways consequently it is well suited to a wide variety of applications for example in a companion paper cox ingersoll and ross 7 we use the model to develop a theory of the term structure of interest rates many studies have been concerned with various aspects of asset pricing under uncertainty the most relevant to our work are the important papers on intertemporal asset pricing by merton 19 and lucas 16 working in a continuous time framework merton derives a relationship among the equilibrium expected rates of return on assets he shows that when investment opportunities are changing randomly over time this relationship will include effects which have no analogue in a static one period model lucas considers an economy with homogeneous individuals and a single consumption good which is produced by a number of processes the random output of these processes is exogenously determined and perishable assets are defined as claims to all or a part of the output of a process and the equilibrium determines the asset prices our theory draws on some elements of both of these papers like merton we formulate our model in continuous time and make full use of the analytical tractability that this affords the economic structure of our model is somewhat similar to that of lucas however we include both endogenous production and

education returns across quantiles of the wage function alternative explanations for returns to education by race in south america private wage returns to schooling in south africa in 1993 are twice as high for nonwhites as for whites and substantially higher at advanced levels of schooling for both races to explore how these returns might be expected to change as the proportion of africans with secondary and higher education increases quantile regressions for the wage function are estimated for african males and white males if the residuals based on a standard wage function specification are interpreted as omitted ability and it is assumed that ability increases decreases returns to schooling then we expect quantile returns to increase decrease at higher quantiles this is not observed for africans at the higher and secondary school level but is evident among whites at higher educational levels the opposite pattern of differential returns favoring the least able i e lowest deciles is reported for the lower tail of the educational distribution that is for africans at the level of primary school and for whites at secondary school authors

schooling data technological diffusion and the neoclassical model this research was supported by the spanish ministry of science and technology through cicyt grants sec99 1189 and sec99 0820 and by the spanish ministry of finance and fundacion caixa galicia

quantile estimates of the impact of r d intensity on firm performance this paper investigates the relationship between initial research and development r d intensity and firm growth using a unique data set for firms with r d activities in austria during the period 1995 2006 results based on the least absolute deviation lad estimator show that initial r d intensity has a positive and significant impact on both employment and sales growth in the subsequent 2 years quantile regressions for each cross section reveal that the impact of r d intensity is significant from 0 3 to the highest quantile of the conditional distribution of employment growth furthermore the elasticity of employment growth with respect to r d intensity is highest for firms at or slightly below the median of the distribution of firm growth finally we find that the impact of r d decreases significantly over time

high growth entrepreneurial firms in africa a quantile regression approach this article analyses the growth performance of a large set of entrepreneurial firms in ten manufacturing sectors of 11 sub saharan african countries the focus of the article is on identifying those entrepreneurs attributes and firm characteristics that tend to generate a significant number of high growth firms in these countries to this end we use a quantile regression which provides a more complete estimation of the growth distribution of firms conditional on different attributes the results indicate that firms that engage in product innovation have their own transport means and are connected to the internet through their own website are especially characterized by higher growth rates and also display a distribution of growth rates skewed to the right hosting a higher number of high growth firms the effect of the last two variables which relate to distance bridging modes of infrastructure points to the self reinforcing growth effects they generate in creating wider input and output markets education raises growth opportunities by affecting the lower quantiles but it does not appear to influence the upper quantiles the estimated conditional growth distributions for the technology intensive machinery and electronics sectors show more extreme tails and a lower mean in comparison to the traditional industries indicating the more risky nature of doing business in these industries

innovation and firm growth does firm age play a role this paper explores the relationship between innovation and firm growth for firms of different ages we hypothesize that young firms undertake riskier innovation activities which may have greater performance benefits if successful or greater losses if unsuccessful using an extensive spanish community innovation survey sample for the period 2004 2012 we apply panel quantile regressions to study the effect of r d activities on firm growth i e sales growth productivity growth and employment growth our results show that young firms face larger performance benefits from r d at the upper quantiles of the growth rate distribution but face larger decline at the lower quantiles r d investment by young firms therefore appears to significantly riskier than r d investment by more mature firms which suggests some policy implications

a letter to the editor on wage bargaining abstract the rubinstein perfect information alternating offers bargaining model is problematic when applied to wage negotiations a strike or any other industrial action is not an automatic consequence of a delay in reaching an agreement because production can continue normally also when negotiations take place this paper extends the rubinstein model to incorporate the choice of calling a strike and it is shown that in this model there is no longer a unique subgame perfect equilibrium and that strikes with a length in real time can occur in equilibrium

the pricing of options on default free bonds recent developments in the finance literature dealing with the valuation of contingent claims have been triggered by black and scholes 2 with their valuation of european options on corporate stocks the type of valuation model derived by black and scholes is attractive because it is independent of preferences this independence is possible due to the fact that the contingent claims are usually assumed to be contingent on traded assets or traded state variables for example a european call option is a claim contingent on the value of the corresponding stock to maintain the independence of the claim from preferences however some variables have to be discarded and considered as constants for example the rate of interest which is not a traded state variable is considered a constant in most of the literature see black and scholes 2 merton 11 and ingersoll 9 though this approximation might not cause important discrepancies in the case of corporate liabilities and options on corporate liabilities the effect of such an approximation on the value of default free bonds and options on default free bonds is more important since these liabilities depend only on the rate of interest

evaluating callable and putable bonds an eigenfunction expansion approach abstract we propose an efficient method to evaluate callable and putable bonds under a wide class of interest rate models including the popular short rate diffusion models as well as their time changed versions with jumps the method is based on the eigenfunction expansion of the pricing operator given the set of call and put dates the callable and putable bond pricing function is the value function of a stochastic game with stopping times under some technical conditions it is shown to have an eigenfunction expansion in eigenfunctions of the pricing operator with the expansion coefficients determined through a backward recursion for popular short rate diffusion models such as cir vasicek 3 2 the method is orders of magnitude faster than the alternative approaches in the literature in contrast to the alternative approaches in the literature that have so far been limited to diffusions the method is equally applicable to short rate jump diffusion and pure jump models constructed from diffusion models by bochner s subordination with a levy subordinator

one factor interest rate models and the valuation of interest rate derivative securities this paper compares different approaches to developing arbitrage free models of the term structure it presents a numerical procedure that can be used to construct a wide range of one factor models of the short rate that are both markov and consistent with the initial term structure of interest rates

risky innovation the impact of internal and external r d strategies upon the distribution of returns a b s t r a c t external innovation increases the profits of the median firm but also increases dispersion and the kur tosis of the distribution of profits this means that external strategies are risky and may require a very large number of attempts before average returns are obtained this puts smaller firms into a position of disproportionately high risk despite the earlier evidence that the rewards from innovation are positively skewed we find no effect of innovation strategies upon the skewness of the distribution of firms profits

allocation of human capital and innovation at the frontier firm level evidence on germany and the netherlands this paper examines how productivity eects of human capital and innovation vary at dierent points of the conditional productivity distribution our analysis draws upon two large unbalanced panels of 6 634 enterprises in germany and 14 586 enterprises in the netherlands over the period 2000 2008 considering 5 manufacturing and services industries that dier in the level of technological intensity industries in the netherlands are characterized by a larger average proportion of high skilled employees and indus tries in germany by a more unequal distribution of human capital intensity except for low technology manufacturing average innovation performance is higher in all industries in germany and the innova tion performance distributions are more dispersed in the netherlands in both countries we observe non linearities in the productivity eects of investing in product innovation in the majority of indus tries frontier rms enjoy the highest returns to product innovation whereas the most negative returns to process innovation are observed in the best performing enterprises of most industries in both countries we nd that the returns to human capital increase with proximity to the technological frontier in industries with a low level of technological intensity strikingly a negative complementarity eect between human capital and proximity to the technological frontier is observed in knowledge intensive services which is most pronounced for the netherlands suggestive evidence for the latter points to a winner takes all interpretation of this nding

does unmeasured ability explain the wage premium associated with technological change quantile regression analysis by using the quantile regressions of earnings equation we find that the educational wage premium is higher in industries with rapid technological change than in industries with slower technological change at every decile in the distribution of wage residuals the wage premium associated with the technological change is mostly explained by the returns to workers unobserved heterogeneities which are correlated with education rather than the rents of high tech industries

the returns to education microeconomics in this paper we focus on education as a private decision to invest in human capital and the estimation of the rate of return to that private investment while the literature is replete with studies that estimate the rate of return using regression methods where the estimated return is obtained as the coefficient on a years of education variable in a log wage equation that contains controls for work experience and other individual characteristics the issue is surrounded with difficulties we outline the theoretical arguments underpinning the empirical developments and show that the evidence on private returns to the individual is compelling despite some of these issues surrounding the estimation of the return to schooling our evidence based on estimates from a variety of datasets and specifications is that there is an unambiguously positive effect on the earnings of an individual from participation in education moreover the size of the effect seems large relative to the returns on other investments

does education raise productivity or just reflect it education has an important effect on wages but it not clear whether this is because education raises productivity or because education is simply a signal of ability we implement a number of existing tests for discriminating between these two explanations and find that they do not support the signalling hypothesis however we have severe reservations about these results and we propose an alternative test based on changes in education incentives caused by changes in the minimum school leaving age in the 1970s using this idea we find that uk data appear to strongly support the human capital explanation an important issue in the economics of education is that an individual s education has an effect on wages paid in the labour market for example blundell et al 2003 uses detailed education and subsequent earnings information for a cohort of male individuals born in 1958 to estimate that the returns to a uk degree typically of 3 year duration relative to graduating from high school at 18 with 2 a level qualifications a necessary but not sufficient condition for admission to university is a 24 wage premium human capital explanations pioneered by becker 1962 and schultz 1963 suggest that the correlation between education and wages is due to education enhancing productivity however earnings may rise in response to education not because of any effect on productivity but simply because education may act as a signal of productivity employers believing that education is correlated with productivity will screen workers for their education and pay higher wages to the more educated the employers beliefs will be confirmed by their experience if it is the case that high productivity individuals signal this by choosing high levels of education it will be optimal for individuals to behave in this way if the cost of acquiring education is less for high productivity individuals than it is for low productivity individuals thus under reasonable conditions the market will be characterised by a separating equilibrium where higher productivity individuals choose higher levels of education than lower productivity workers and earn correspondingly higher wages the theory is largely due to spence 1973 1979 and the subsequent empirical literature was reviewed by groot and oosterbeek 1994 riley 2001 provides an excellent modern survey chevalier harmon and walker are fellows of iza in bonn and are affiliated with the institute for the study of social change issc in dublin this paper is part of the issc policy evaluation programme chevalier and walker are also affiliated with the centre for the economics of education at the london school of economics lse the support of all of these bodies for facilitating our collaboration is acknowledged harmon is grateful to university college london for their hospitality during his nuffield foundation new career development fellowship in the social sciences comments from steve machin and three anonymous referees greatly improved the exposition of this paper we are also grateful to richard blundell and to seminar participants at the lse and the tinbergen institute in amsterdam for their comments and to vincent o sullivan for excellent research assistance the usual disclaimer applies

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

the returns to education macroeconomics we offer an extensive summary and a critical discussion of the empirical literature on the impact of human capital on macro economic performance with a particular focus on uk policy we also highlight methodological issues and make recommendations for future research priorities taking the studies as a whole the evidence that human capital increases productivity is compelling though still largely divided on whether the stock of education affects the long run level or growth rate of gdp a one year increase in average education is found to raise the level of output per capita by between three and six percent according to augmented neo classical specifications while leading to an over one percentage point faster growth according to estimates from the new growth theories still over the short run planning horizon four years the empirical estimates of the change in gdp are of similar orders of magnitude in the two approaches the impact of increases at different levels of education appear to depend on the level of a country s development with tertiary education being the most important for growth in oecd countries education is found to yield additional indirect benefits to growth more preliminary evidence seems to indicate that type quality and efficiency of education matter for growth too

is there a return risk link in education abstract using results for 16 countries the positive relationship between return to education and the risk involved in this investment is studied it seems that most of the countries fit the pattern well higher risk higher return and the tradeoff is rather large

stochastic processes for interest rates and equilibrium bond prices default free bonds represent in essence money sold forward the investor gives dollars say to the government now in exchange for a contract which promises to pay with probability one a given number of dollars at maturity and possibly a stream of intermediate coupon payments prior to maturity only recently have financial economists begun to examine the pricing of those bonds in a portfolio context in that framework nominal liquidity premiums or risk premiums on long term default free bonds are attributable to uncertainty about taste technological and price level changes which may occur prior to maturity of the bonds and prior to each coupon payment if any to the extent that the investor is unable to diversify away that uncertainty there is a systematic risk attached to the marginal utility which he or she can derive from the use in consumption or investment of the dollar payoffs on the bonds in some term structure models the instantaneous or short term rate of interest denoted here as a function of time t by r t in nominal terms and r t in real terms can be taken as a sufficient statistic1 for the source of systematic uncertainty about the marginal utility of the proceeds to be derived from defaultfree bonds maturing beyond the next instant e g merton 1974 vasicek 1977 cox ingersoll and ross 1977 specific assumptions about tastes technology and utility functions lead to specific equilibrium stochastic processes for the instantaneous short term riskless rate and longer term bond prices which are modelled as functionally dependent upon maturity and that short term riskless rate if a closed form solution exists for the long term bond price as a function of maturity and the short term interest rate it will be conditional inter alia on the parameters of the stochastic process for that short term rate marsh 1981 proposed that the closed form solutions could be tested in terms of the restrictions implied cross sectionally on the parameters of the time series process for the returns or prices of longer term bonds each with a given term to maturity for an analogy outside the term structure literature think of the closed form solution given by black and scholes 1972 for the price of an option on a nondividend paying stock with lognormal price dynamics given the model the cross section of time series for the changes in prices for each of the several options written on the same stock are not unrestricted their dispersion is functionally linked to the same volatility parameter for the stock price thus the option model including

numerical valuation of high dimensional multivariate american securities we consider the problem of pricing an american contingent claim whose payoff depends on several sources of uncertainty several efficient numerical lattice based techniques exist for pricing american securities depending on one or few up to three risk sources however these methods cannot be used for high dimensional problems since their memory requirement is exponential in the number of risk sources we present an efficient numerical technique that combines monte carlo simulation with a particular partitioning method of the underlying assets space which we call stratified state aggregation ssa using this technique we can compute accurate approximations of prices of american securities with an arbitrary number of underlying assets our numerical experiments show that the method is practical for pricing american claims depending on up to 400 risk sources

pricing interest rate derivative securities this article shows that the one state variableinterest rate models of vasicek 1977 and cox ingersoll and ross 1985b can be extended sothat they are consistent with both the current termstructure of interest rates and either the currentvolatilities of all spot interest rates or the currentvolatilities of all forward interest rates theextended vasicek model is shown to be very tracta ble analytically the article compares option pricesobtained using the extended vasicek model withthose obtained using a number of other models