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22
LinearityGenerating Processes: A Modelling Tool Yielding Closed Forms for Asset Prices
"... This methodological paper presents a class of stochastic processes with potentially appealing properties for theoretical and empirical work in finance and macroeconomics, the “linearitygenerating ” class. Its key property is that it yields simple exact closedform expressions for stocks and bonds, w ..."
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Cited by 34 (8 self)
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This methodological paper presents a class of stochastic processes with potentially appealing properties for theoretical and empirical work in finance and macroeconomics, the “linearitygenerating ” class. Its key property is that it yields simple exact closedform expressions for stocks and bonds, with an arbitrary number of factors. It operates in discrete and continuous time. Controlling for the covariance with the stochastic discount factor, the distribution of many disturbances does not affect stock or bond prices, which simplifies the modeller’s task. The paper presents a series of illustrative examples, including stocks with stochastic risk premia or stochastic dividend growth rates, macroeconomic environments with changing trend growth rates, and yield curve analysis.
Macroeconomics and the Term Structure
, 2010
"... This paper provides an overview of the analysis of the term structure of interest rates with a special emphasis on recent developments at the intersection of macroeconomics and finance. The topic is important to investors and also to policymakers, who wish to extract macroeconomic expectations from ..."
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Cited by 22 (1 self)
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This paper provides an overview of the analysis of the term structure of interest rates with a special emphasis on recent developments at the intersection of macroeconomics and finance. The topic is important to investors and also to policymakers, who wish to extract macroeconomic expectations from longerterm interest rates, and take actions to influence those rates. The simplest model of the term structure is the expectations hypothesis, which posits that longterm interest rates are expectations of future average shortterm rates. In this paper, we show that many features of the con…guration of interest rates are puzzling from the perspective of the expectations hypothesis. We review models that explain these anomalies using timevarying risk premia. Although the quest for the fundamental macroeconomic explanations of these risk premia is ongoing, in‡ation uncertainty seems to play a large role. Finally, while modern finance theory prices bonds and other assets in a single unified framework, we also consider an earlier approach based on segmented markets. Market segmentation seems important to understand the term structure of interest rates during the recent financial crisis.
Can Unspanned Stochastic Volatility Models Explain the Cross Section of Bond Volatilities? Working Paper
, 2006
"... In fixed income markets, volatility is unspanned if volatility risk cannot be hedged with bonds. We first show that all affine term structure models with state space RM+ ×RN−M can be drift normalized and show when the standard variance normalization can be obtained. Using this normalization, we find ..."
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Cited by 20 (5 self)
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In fixed income markets, volatility is unspanned if volatility risk cannot be hedged with bonds. We first show that all affine term structure models with state space RM+ ×RN−M can be drift normalized and show when the standard variance normalization can be obtained. Using this normalization, we find conditions for a wide class of affine term structure models to exhibit unspanned stochastic volatility (USV). We show that the USV conditions restrict both the mean reversions of risk factors and the cross section of conditional yield volatilities. The restrictions imply that previously studied affine USV models are unlikely to be able to generate the observed cross section of yield volatilities. However, more general USV models can match the cross section of bond volatilities. 1.
Nonparametric Estimation of StatePrice Densities Implicit in Interest Rate Cap Prices
 Review of Financial Studies
, 2009
"... Based on a multivariate extension of the constrained locally polynomial estimator of AtSahalia and Duarte (2003), we provide nonparametric estimates of the probability densities of LIBOR rates under forward martingale measures and the stateprice densities (SPDs) implicit in interest rate cap price ..."
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Cited by 11 (1 self)
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Based on a multivariate extension of the constrained locally polynomial estimator of AtSahalia and Duarte (2003), we provide nonparametric estimates of the probability densities of LIBOR rates under forward martingale measures and the stateprice densities (SPDs) implicit in interest rate cap prices conditional on the slope and volatility factors of LIBOR rates. Both the forward densities and the SPDs depend signicantly on the volatility of LIBOR rates, and there is a signicant impact of mortgage prepayment activities on the forward densities. The SPDs exhibit a pronounced Ushape as a function of future LIBOR rates, suggesting that the state prices are high at both extremely low and high interest rates, which tend to be associated with periods of economic recessions and high in
ations, respectively. Our results provide nonparametric evidence of unspanned stochastic volatility and suggest that the unspanned factors could be partly driven by renancing activities in the mortgage markets. Overthecounter interest rate derivatives, such as caps and swaptions, are among the most widely traded interest rate derivatives in the world. According to the Bank for International Settlements, in recent years, the notional value of caps and swaptions exceeds $ 10 trillion, which is many times
Modeling Yields at the Zero Lower Bound: Are Shadow Rates the Solution?,” Working Paper 201339, Federal Reserve Bank of San Francisco
, 2013
"... The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System. Working Paper 201339 ..."
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Cited by 8 (3 self)
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The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System. Working Paper 201339
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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Cited by 7 (1 self)
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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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Cited by 6 (4 self)
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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.
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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Cited by 3 (0 self)
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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,
Risk and Return Tradeoff in the U.S. Treasury Market∗
"... This paper characterizes the riskreturn tradeoff in the U.S. Treasury market through the lens of a discretetime noarbitrage term structure model, in which bond prices are solved in closed form and the conditional variances of bond yields feature a shortrun component and a longrun component. Us ..."
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Cited by 1 (0 self)
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This paper characterizes the riskreturn tradeoff in the U.S. Treasury market through the lens of a discretetime noarbitrage term structure model, in which bond prices are solved in closed form and the conditional variances of bond yields feature a shortrun component and a longrun component. Using Treasury yields data from January 1962 to August 2007, we find that for shortdated bonds, most of the variations in risk premiums are attributable to investors ’ changing attitudes toward risks. For longerdated bonds, risk premiums reflect both the amount of risks bond investors face as well as their tolerance for risks over time. Furthermore, we find that the shortrun volatility component of bond yields commands a positive risk premium whereas the longrun volatility component does not.
Can Spanned Term Structure Factors Drive Stochastic Volatility?
, 2010
"... The ability of the usual factors from empirical arbitragefree representations of the term structure—that is, spanned factors—to account for interest rate volatility dynamics has been much debated. We estimate new arbitragefree NelsonSiegel (AFNS) term structure specifications that allow for stoch ..."
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The ability of the usual factors from empirical arbitragefree representations of the term structure—that is, spanned factors—to account for interest rate volatility dynamics has been much debated. We estimate new arbitragefree NelsonSiegel (AFNS) term structure specifications that allow for stochastic volatility to be linked to one or more of the spanned AFNS yield curve factors. Our results with three separate daily data sets—U.S. Treasury yields, U.K. gilt yields, and U.S. dollar swap and LIBOR rates—suggest that much observed stochastic volatility cannot be associated with spanned term structure factors in terms of timeseries correlations at high frequency. However, some of the AFNS models with stochastic volatility do provide a close fit to our measure of realized yield volatility in addition to providing a good fit to the yield term structure.