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Affine models
 In R. Cont (Ed.), Encyclopedia of Quantitative Finance
, 2009
"... Abstract. Affine term structure models have gained a lot of attention in the finance literature, which is due to their analytic tractability and statistical flexibility. The aim of this article is to present both, theoretical foundations and empirical aspects. Starting from the first short rate mode ..."
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Abstract. Affine term structure models have gained a lot of attention in the finance literature, which is due to their analytic tractability and statistical flexibility. The aim of this article is to present both, theoretical foundations and empirical aspects. Starting from the first short rate models, namely the Vasiček and the CoxIngersollRoss ones, we then give an overview of some properties of affine processes and explain their relation to affine term structure models. Pricing and estimation techniques are eventually mentioned, showing how the analytic tractability of affine models can be exploited for practical purposes.
The Structure of Risks in Equilibrium Affine Models of Bond Yields ∗
, 2013
"... Many equilibrium term structure models (ETSMs) in which the state of the economy follows an affine process imply that the variation in expected excess returns on bond portfolio positions is fully spanned by the conditional variances of the state variables. We show that these two assumptions alone – ..."
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Many equilibrium term structure models (ETSMs) in which the state of the economy follows an affine process imply that the variation in expected excess returns on bond portfolio positions is fully spanned by the conditional variances of the state variables. We show that these two assumptions alone – an affine state process with conditional variances that span expected excess returns – are sufficient to econometrically identify the factors determining risk premiums in these ETSMs from data on the term structure of bond yields. Using this result we derive maximum likelihood estimates of the conditional variances of the state – the “quantities of risk” – and evaluate the goodnessoffit of a large family of affine ETSMs. These assessments of fit are fully robust to the values of the parameters governing preferences and the evolution of the state, and to whether or not the economy is arbitrage free. Our findings suggest that, to be consistent with U.S. macroeconomic and Treasury yield data, affine ETSMs should have the features that: (i) inflation risk, and not longrun risks or variation in risk premiums arising from habitbased preferences, is a significant (and perhaps the dominant) risk underlying risk premiums in U.S. Treasury markets; and (ii) risks that are unspanned by bond yields have substantial explanatory power for risk premiums consistent with timevarying market prices of risks.
A New Linear Estimator for Gaussian Dynamic Term Structure Models
, 2013
"... Bank of Canada working papers are theoretical or empirical worksinprogress on subjects in economics and finance. The views expressed in this paper are those of the author. No responsibility for them should be attributed to the Bank of Canada. ..."
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Bank of Canada working papers are theoretical or empirical worksinprogress on subjects in economics and finance. The views expressed in this paper are those of the author. No responsibility for them should be attributed to the Bank of Canada.
Parametric Inference and Dynamic State Recovery from Option Panels.”Working Paper
, 2012
"... We develop a new parametric estimation procedure for option panels observed with error which relies on asymptotic approximations assuming an ever increasing set of observed option prices in the moneynessmaturity (crosssectional) dimension, but with a fixed time span. We develop consistent estimato ..."
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We develop a new parametric estimation procedure for option panels observed with error which relies on asymptotic approximations assuming an ever increasing set of observed option prices in the moneynessmaturity (crosssectional) dimension, but with a fixed time span. We develop consistent estimators of the parameter vector and the dynamic realization of the state vector that governs the option price dynamics. The estimators converge stably to a mixedGaussian law and we develop feasible estimators for the limiting variance. We provide semiparametric tests for the option price dynamics based on the distance between the spot volatility extracted from the options and the one obtained nonparametrically from highfrequency data on the underlying asset. We further construct new formal tests of the model fit for specific regions of the volatility surface and for the stability of the riskneutral dynamics over a given period of time. A largescale Monte Carlo study indicates that the inference procedures work well for empirically realistic model specifications and sample sizes. In an empirical application to S&P 500 index options we extend the popular doublejump stochastic volatility model to allow for timevarying risk premia of extreme events, i.e., jumps, as well as a more flexible relation between the risk
Mortgage Convexity ∗
, 2012
"... Most home mortgages in the United States are fixedrate loans with an embedded prepayment option. When longterm rates decline, the effective duration of mortgagebacked securities (MBS) falls due to heightened refinancing expectations. I show that these changes in aggregate MBS duration function as ..."
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Most home mortgages in the United States are fixedrate loans with an embedded prepayment option. When longterm rates decline, the effective duration of mortgagebacked securities (MBS) falls due to heightened refinancing expectations. I show that these changes in aggregate MBS duration function as largescale shocks to the quantity of interest rate risk that must be borne by professional bond investors. I develop a simple model in which the risk tolerance of bond investors is limited in the short run, so these fluctuations in MBS duration generate significant variation in bond risk premia. Specifically, bond risk premia are high when aggregate MBS duration is high. The model offers an explanation for why longterm rates may appear to be “excessively sensitive”to movements in short rates and explains how changes in MBS duration act as a positivefeedback mechanism that amplifies interest rate volatility. I find strong support for these predictions in the time series of US government bond returns. I am grateful to John Campbell, Robin Greenwood, Erik Stafford, Jeremy Stein, Larry Summers, and
Efficient RegressionBased Estimation of Dynamic Asset Pricing Models,” Federal Reserve Bank of New York Staff Reports
, 2013
"... This paper presents preliminary findings and is being distributed to economists and other interested readers solely to stimulate discussion and elicit comments. The views expressed in this paper are those of the authors and do not necessarily reflect the position of the Federal Reserve Bank of New Y ..."
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This paper presents preliminary findings and is being distributed to economists and other interested readers solely to stimulate discussion and elicit comments. The views expressed in this paper are those of the authors and do not necessarily reflect the position of the Federal Reserve Bank of New York or the Federal Reserve System. Any errors or omissions are the responsibility of the authors.
2010), “Why Do Term Structures in Different Currencies Comove?” mimeo, The University of North Carolina at Chapel
"... Yield curve fluctuations across different currencies are highly correlated. This paper investigates this phenomenon by exploring the channels through which macroeconomic shocks are transmitted across borders. Macroeconomic shocks affect current and expected future shortterm rates as central banks ..."
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Yield curve fluctuations across different currencies are highly correlated. This paper investigates this phenomenon by exploring the channels through which macroeconomic shocks are transmitted across borders. Macroeconomic shocks affect current and expected future shortterm rates as central banks react to changing economic environments. However, policy reactions are not the only channel through which the macroeconomy may affect bond yields. Investors may also respond to these shocks by altering their required compensation for risk. Macroeconomic shocks thus influence bond yields both through a “policy ” channel as well as through a “risk compensation” channel. In a noarbitrage vector autoregressive framework, we employ deviations from the expectations hypothesis to identify the two transmission channels, with particular attention to the degree to which each channel contributes to the covariation among term structures across the U.S., the U.K., and Germany. We find that a world inflation factor explains over 70 % of the covariance of medium to longmaturity yields between the U.S. and the U.K., as well as between the U.S. and Germany. Further, we find that the world inflation effect operates almost exclusively through the risk compensation channel for longterm bonds. ∗We are grateful for helpful comments from Levent Guntay, Craig Holden, and Bob Jennings. We are responsible for all remaining errors. 1
Core and “crust”: Consumer prices and the term structure of interest rates. Working Paper, Federal Reserve Bank of Chicago
, 2012
"... We estimate a model for nominal and real term structures of interest rates that includes dynamics for the three main components of total inflation: core, food, and energy. These dynamics combine together to produce a measure of expected total inflation that investors use to price nominal Treasuries. ..."
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We estimate a model for nominal and real term structures of interest rates that includes dynamics for the three main components of total inflation: core, food, and energy. These dynamics combine together to produce a measure of expected total inflation that investors use to price nominal Treasuries. This framework captures different frequencies in inflation fluctuations: shocks to core are more persistent and less volatile than shocks to food and, especially, energy (the ‘crust’). The model fits yields and inflation data well in sample, and produces inflation forecasts that outperform several benchmarks out of sample. A common structure of latent factors explains most of the variance of the forecasting error for core inflation and bond yields. This evidence suggests that interest rates contain useful predictive content for inflation. Moreover, we estimate real interest rates, as well as inflation and real rate risk premia, that are consistent with related marketbased measures. We are grateful to Larry Christiano, Charlie Evans, Spence Krane, Alejandro Justiniano, Michael McCracken,
Bond pricing and the macroeconomy
, 2012
"... This chapter reviews some of the academic literature that links nominal and real term structures with the macroeconomy. The main conclusion is that none of our models is consistent with basic properties of nominal yields. It is difficult to explain the average shape of the nominal yield curve, the v ..."
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This chapter reviews some of the academic literature that links nominal and real term structures with the macroeconomy. The main conclusion is that none of our models is consistent with basic properties of nominal yields. It is difficult to explain the average shape of the nominal yield curve, the variation of yields over time, and the predictability of excess bond returns. There are two overarching problems. First, much of the variation over time in economic activity is orthogonal to variation in nominal yields, and vice versa. Second, although mean excess returns to nominal Treasury bonds are positive, these returns do not appear to positively covary with risks that require compensation, at least according to standard assetpricing models.