Estimating affine multifactor term structure models using closed-form likelihood expansions

We develop and implement a technique for closed-form maximum likelihood estimation (MLE) of multifactor affine yield models. We derive closed-form approximations to likelihoods for nine Dai and Singleton (2000) affine models. Simulations show our technique very accurately approximates true (but infeasible) MLE. Using US Treasury data, we estimate nine affine yield models with different market price of risk specifications. MLE allows non-nested model comparison using likelihood ratio tests; the preferred model depends on the market price of risk. Estimation with simulated and real data suggests our technique is much closer to true MLE than Euler and quasi-maximum likelihood (QML) methods.     

Identifying Term Structure Volatility from the LIBOR-Swap Curve

This paper proposes a new family of specification tests andapplies them to affine term structure models of the London InterbankOffered Rate (LIBOR)-swap curve. Contrary to Dai and Singleton(2000), the tests show that when standard estimation techniquesare used, affine models do a poor job of forecasting volatilityat the short end of the term structure. Improving the volatilityforecast does not require different models; rather, it requiresa different estimation technique. The paper distinguishes betweentwo econometric procedures for identifying volatility. The "cross-sectional"approach backs out volatility from a cross section of bond yields,and the "time-series" approach imputes volatility from time-seriesvariation in yields. For an affine model, the volatility impliedby the time-series procedure passes the specification tests,while the cross-sectionally identified volatility does not.This is surprising, since under correct specification, the "cross-sectional"approach is maximum likelihood. One explanation is that affinemodels are slightly misspecified; another is that bond yieldsdo not span volatility, as in Collin-Dufresne and Goldstein(2002).     

Gaussian models for Euro high grade government yields

This paper tests affine, quadratic and Black-type Gaussian models on Euro area triple A Government bond yields for maturities up to 30 years. Quadratic Gaussian models beat affine Gaussian models both in-sample and out-of-sample. A Black-type model best fits the shortest maturities and the extremely low yields since 2013, but worst fits the longest maturities. Even for quadratic models we can infer the latent factors from some yields observed without errors, which makes quasi-maximum likelihood (QML) estimation feasible. New specifications of quadratic models fit the longest maturities better than does the ‘classic’ specification of Ahn et al. [2002. ‘Quadratic Term Structure Models: Theory and Evidence.’ The Review of Financial Studies 15 (1): 243–288], but the opposite is true for the shortest maturities. These new specifications are more suitable to QML estimation. Overall quadratic models seem preferable to affine Gaussian models, because of superior empirical performance, and to Black-type models, because of superior tractability. This paper also proposes the vertical method of lines (MOL) to solve numerically partial differential equations (PDEs) for pricing bonds under multiple non-independent stochastic factors. ‘Splitting’ the PDE drastically reduces computations. Vertical MOL can be considerably faster and more accurate than finite difference methods.     

Term Premia and Interest Rate Forecasts in Affine Models

The standard class of affine models produces poor forecasts of future Treasury yields. Better forecasts are generated by assuming that yields follow random walks. The failure of these models is driven by one of their key features: Compensation for risk is a multiple of the variance of the risk. Thus risk compensation cannot vary independently of interest rate volatility. I also describe a broader class of models. These "essentially affine" models retain the tractability of standard models, but allow compensation for interest rate risk to vary independently of interest rate volatility. This additional flexibility proves useful in forecasting future yields.     

Diagnosing affine models of options pricing: Evidence from VIX

Affine jump-diffusion models have been the mainstream in options pricing because of their analytical tractability. Popular affine jump-diffusion models, however, are still unsatisfactory in describing the options data and the problem is often attributed to the diffusion term of the unobserved state variables. Using prices of variance-swaps (i.e., squared VIX) implied from options prices, we provide fresh evidence regarding the misspecification of affine jump-diffusion models, as variance-swap prices are affine functions of the state variables in a broader class of models that do not restrict the diffusion term of the state variables. We apply the nonparametric methodology used by Aït-Sahalia (1996b), supplemented with bootstrap tests and other parametric tests, to the S&P 500 index options data from January 1996 to September 2008. We find that, while the affine diffusion term of the state variables may contribute to the misspecification as the literature has suggested, the affine drift of the state variables, jump intensities, and risk premiums are also sources of misspecification.     

Properties of foreign exchange risk premiums

We study the properties of foreign exchange risk premiums that can explain the forward bias puzzle, defined as the tendency of high-interest rate currencies to appreciate rather than depreciate. These risk premiums arise endogenously from the no-arbitrage condition relating the term structure of interest rates and exchange rates. Estimating affine (multi-currency) term structure models reveals a noticeable tradeoff between matching depreciation rates and accuracy in pricing bonds. Risk premiums implied by our global affine model generate unbiased predictions for currency excess returns and are closely related to global risk aversion, the business cycle, and traditional exchange rate fundamentals.     

Specification Analysis of Affine Term Structure Models

This paper explores the structural differences and relative goodness-of-fits of affine term structure models (ATSMs). Within the family of ATSMs there is a trade-off between flexibility in modeling the conditional correlations and volatilities of the risk factors. This trade-off is formalized by our classification of N -factor affine family into N + 1 non-nested subfamilies of models. Specializing to three-factor ATSMs, our analysis suggests, based on theoretical considerations and empirical evidence, that some subfamilies of ATSMs are better suited than others to explaining historical interest rate behavior.     

Market price of risk specifications for affine models: Theory and evidence

We extend the standard specification of the market price of risk for affine yield models, and apply it to U.S. Treasury data. Our specification often provides better fit, sometimes with very high statistical significance. The improved fit comes from the time-series rather than cross-sectional features of the yield curve. We derive conditions under which our specification does not admit arbitrage opportunities. The extension has extremely strong statistical significance for affine yield models with multiple square-root type variables. Although we focus on affine yield models, our specification can be used with other asset pricing models as well.     

Credit risk modeling with affine processes

This article combines an orientation to credit risk modeling with an introduction to affine Markov processes, which are particularly useful for financial modeling. We emphasize corporate credit risk and the pricing of credit derivatives. Applications of affine processes that are mentioned include survival analysis, dynamic term-structure models, and option pricing with stochastic volatility and jumps. The default-risk applications include default correlation, particularly in first-to-default settings. The reader is assumed to have some background in financial modeling and stochastic calculus.     

Affine forward variance models

We introduce the class of affine forward variance (AFV) models of which both the conventional Heston model and the rough Heston model are special cases. We show that AFV models can be characterised by the affine form of their cumulant-generating function, which can be obtained as solution of a convolution Riccati equation. We further introduce the class of affine forward order flow intensity (AFI) models, which are structurally similar to AFV models, but driven by jump processes, and which include Hawkes-type models. We show that the cumulant-generating function of an AFI model satisfies a generalised convolution Riccati equation and that a high-frequency limit of AFI models converges in distribution to an AFV model.

     

Are interest rate options important for the assessment of interest rate risk?

Fixed income options contain substantial information on the price of interest rate volatility risk. In this paper, we ask if those options will also provide information related to other moments of the objective distribution of interest rates. Based on dynamic term structure models within the class of affine models, we find that interest rate options are useful for the identification of interest rate quantiles. Two three-factor models are adopted and their adequacy to estimate Value at Risk of zero-coupon bonds is tested. We find significant difference on the quantitative assessment of risk when options are (or not) included in the estimation process of each of these dynamic models. Statistical backtests indicate that bond estimated risk is clearly more adequate when options are adopted, although not yet completely satisfactory.     

An empirical comparison of GARCH option pricing models

Recent empirical studies have shown that GARCH models can be successfully used to describe option prices. Pricing such contracts requires knowledge of the risk neutral cumulative return distribution. Since the analytical forms of these distributions are generally unknown, computationally intensive numerical schemes are required for pricing to proceed. Heston and Nandi (2000) consider a particular GARCH structure that permits analytical solutions for pricing European options and they provide empirical support for their model. The analytical tractability comes at a potential cost of realism in the underlying GARCH dynamics. In particular, their model falls in the affine family, whereas most GARCH models that have been examined fall in the non-affine family. This article takes a closer look at this model with the objective of establishing whether there is a cost to restricting focus to models in the affine family. We confirm Heston and Nandi's findings, namely that their model can explain a significant portion of the volatility smile. However, we show that a simple non affine NGARCH option model is superior in removing biases from pricing residuals for all moneyness and maturity categories especially for out-the-money contracts. The implications of this finding are examined. JEL Classification G13     

Estimation and Evaluation of Conditional Asset Pricing Models

We find that several recently proposed consumption‐based models of stock returns, when evaluated using an optimal set of managed portfolios and the associated model‐implied conditional moment restrictions, fail to capture key features of risk premiums in equity markets. To arrive at these conclusions, we construct an optimal Generalized Method of Moments (GMM) estimator for models in which the stochastic discount factor (SDF) is a conditionally affine function of a set of priced risk factors, and we show that there is an optimal choice of managed portfolios to use in testing a null model against a proposed alternative generalized SDF.     

The role of time-varying jump risk premia in pricing stock index options

This paper examines out-of-sample option pricing performances for the affine jump diffusion (AJD) models by using the S&P 500 stock index and its associated option contracts. In particular, we investigate the role of time-varying jump risk premia in the AJD specifications. Our empirical analysis shows strong evidence in favor of time-varying jump risk premia in pricing cross-sectional options. We also find that, during a period of low volatility, the role of jump risk premia becomes less pronounced, making the differences across pricing performances of the AJD models not as substantial as during a period of high volatility. This finding can possibly explain poor pricing perfomances of the sophisticated AJD models in some previous studies whose sample periods can be characterized by low volatility.     

A General Affine Earnings Valuation Model

We introduce a methodology, with two applications, that incorporates stochastic interest rates, heteroskedasticity and risk aversion into the residual income model. In the first application, goodwill is an affine (constant plus linear term) function where the constant and linear coefficients are time-varying. Homoskedastic risk gives rise to a constant risk premium, while heteroskedastic risk gives rise to linear state-dependent risk premiums. In the second application, we present a class of models where a non-linear function for the price-to-book ratio can be derived. We show how interest rates, risk, profitability and growth affect the price-to-book ratio.     

Do Bonds Span Volatility Risk in the U.S. Treasury Market? A Specification Test for Affine Term Structure Models

We propose using model‐free yield quadratic variation measures computed from intraday data as a tool for specification testing and selection of dynamic term structure models. We find that the yield curve fails to span realized yield volatility in the U.S. Treasury market, as the systematic volatility factors are largely unrelated to the cross‐section of yields. We conclude that a broad class of affine diffusive, quadratic Gaussian, and affine jump‐diffusive models cannot accommodate the observed yield volatility dynamics. Hence, the Treasury market per se is incomplete, as yield volatility risk cannot be hedged solely through Treasury securities.     

Unconditional Tests of Linear Asset Pricing Models with Time‐Varying Betas

In conditional affine factor models, estimated risk prices should satisfy certain unconditional constraints. Specifically, a cross‐sectional estimate of the unconditional slope associated with a risk factor should equal the average price of risk of the factor. The estimated slope associated with the product of a risk factor and an instrument should be equal to the covariance of the factor risk premium with the instrument. We show that the constraints only apply to the conditional models with time‐varying betas. We identify an unconditional constraint on unconditional betas for time‐varying beta models and incorporate it into model tests. We show that imposing this unconditional constraint changes estimates of unconditional betas and risk prices significantly.     

Affine Models for Credit Risk Analysis

Continuous-time affine models have been recently introducedin the theoretical financial literature on credit risk. Theyprovide a coherent modeling, rather easy to implement, but havenot yet encountered the expected success among practitionersand regulators. This is likely due to a lack of flexibilityof these models, which often implied poor fit, especially comparedto more ad hoc approaches proposed by the industry. The aimof this article is to explain that this lack of flexibilityis mainly due to the continuous-time assumption. We developa discrete-time affine analysis of credit risk, explain howdifferent types of factors can be introduced to capture separatelythe term structure of default correlation, default heterogeneity,correlation between default, and loss-given-default; we alsoexplain why the factor dynamics are less constrained in discretetime and are able to reproduce complicated cycle effects. Thesemodels are finally used to derive a credit-VaR and various decompositionsof the spreads for corporate bonds or first-to-default basket.     

Inflation risk premia and the expectations hypothesis

We study the properties of the nominal and real risk premia of the term structure of interest rates. We develop and solve the bond pricing implications of a structural monetary version of a real business cycle model, with taxes and endogenous monetary policy. We show the relation of this model with the class of essentially affine models that incorporate an endogenous state-dependent market price of risk. We characterize and estimate the inflation risk premium and find that over the last 40 years the ten-year inflation risk premium has been has averaged 70 basis points. It is time-varying, ranging from 20 to 140 basis points over the business cycle and its term structure is sharply upward sloping. The inflation risk premium explains 23% (42%) of the time variation in the five (ten)-year forward risk premium and it plays an important role in help explain deviations from the expectations hypothesis of interest rates.     

Modeling Foreign Exchange Risk Premium in Armenia

This paper applies stochastic discount factor methodology to modeling the foreign exchange risk premium in Armenia. We use weekly data on foreign and domestic currency deposits, which coexist in the Armenian banking system. This coexistence implies elimination of the cross-country risks and transaction costs, leaving the pure foreign exchange risk. It is shown that there exists a systematic time-varying risk premium that increases with maturity. Using two-currency affine term structure and generalized autoregressive conditional heteroskedasticity (GARCH)-in-mean models, we find that the central bank's foreign exchange market interventions and ratio-of-deposit volumes significantly affect public expectations about foreign exchange fluctuations. We also find that the foreign exchange risk premium accounts for the largest part of the interest differential. When accounting for economic and institutional differences, our results can be extended to other countries.