Silverio Foresi

Affine Term Structure Models and the Forward Premium Anomaly

One of the most puzzling features of currency prices is the forward premium anomaly: the tendency for high interest rate currencies to appreciate. We characterize the anomaly in the context of affine models of the term structure of interest rates. In affine models, the anomaly requires either that state variables have asymmetric effects on state prices in different currencies or that nominal interest rates take on negative values with positive probability. We find the quantitative properties of either alternative to have important shortcomings. PERHAPS THE MOST PUZZLING FEATURE of currency prices is the tendency for high interest rate currencies to appreciate when one might guess, instead, that investors would demand higher interest rates on currencies expected to fall in value. This departure from uncovered interest parity, which we term the forward premium anomaly, has been documented in dozens—and possibly hundreds—of studies, and has spawned a second generation of papers attempting to account for it. One of the most inf luential of these is Fama ~1984!, who attributes the behavior of forward and spot exchange rates to a time-varying risk premium. Fama shows that the implied risk premium on a currency must ~1! be negatively correlated with its expected rate of depreciation and ~2! have greater variance. We refer to this feature of the data as an anomaly because asset pricing theory to date has been notably unsuccessful in producing a risk premium with the requisite properties. Attempts include applications of the capital asset pricing model to currency prices ~Frankel and Engel ~1984!, Mark ~1988!!, statistical models relating risk premiums to changing second moments ~Hansen and Hodrick ~1983!, Domowitz and Hakkio ~1985!, Cumby ~1988!!, and consumption-based asset pricing theories, including departures from time

A Simple Approach to Three-Factor Affine Term Structure Models

RANGARAJAN SUNDARAM is assistant professor at New York University. his article develops a simple estimation approach for three-factor models of the term structure of interest rates, exploiting the exponential-affine structure of these models. The three factors in the model are the short-term rate of interest, the long-run mean of the short-term rate, and the volatility of the short-term rate. The objective of term structure modeling from a practitioner’s viewpoint is to develop a parsimonious representation of the yield curve matching the time series and cross-sectional variation of bond yields. The more factors, the richer the time series and cross-sectional properties of bond returns the model can accommodate. Of course, complexity is often traded off against parsimony for practical reasons of implementation. So far, practicality has kept researchers in the field from going beyond three-factor models. The simple estimation method in this article should make the implementation of three-factor models easier. Implementation of the model on data over a thirty-year period indicates that it captures known theoretical features and aspects of the term structure well. Duffle and Kan [1996] show that a wide range of choices of stochastic processes for interest rate factors yield bond pricing solutions of a form now widely called exponential-a

Accounting for Biases in Black-Scholes

ne models. It is the affine form that we use in this article to develop our implementation methodology. Even though there are three factors, the affine class of solutions enables the use of a single equation maximum-likelihood model to approximate the multivariate estimation of a three-factor model. The model fits the data well, and also enables the filtering of the

Exact solutions for bond and option prices with systematic jump risk

Prices of currency options commonly differ from the Black-Scholes formula along two dimensions: implied volatilities vary by strike price (volatility smiles) and maturity (implied volatility of at-the-money options increases, on average, with maturity). We account for both using Gram-Charlier expansions to approximate the conditional distribution of the logarithm of the price of the underlying security. In this setting, volatility is approximately a quadratic function of moneyness, a result we use to infer skewness and kurtosis from volatility smiles. Evidence suggests that both kurtosis in currency prices and biases in Black-Scholes option prices decline with maturity.

Predictable changes in yields and forward rates

A variety of realistic economic considerations make jump-diffusion models of interest rate dynamics an appealing modeling choice to price interest-rate contingent claims. However, exact closed-form solutions for bond prices when interest rates follow a mixed jump-diffusion process have proved very hard to derive. This paper puts forward two new models of interest-rate dynamics that combine infrequent, discrete changes in the interest-rate level, modeled as a jump process, with short-lived, mean reverting shocks, modeled as a diffusion process. The two models differ in the way jumps affect the central tendency of interest rates; in one case shocks are temporary, in the other shocks are permanent. We derive exact closed-form solutions for the price of a discount bond and computationally tractable schemes to price bond options.

Interest Rate Targeting and the Dynamics of Short-Term Rates

We consider the patterns in the predictability of interest rates expectations hypothesis (EH), and attempt to account for them with affine models. We make the following points: (i) Discrepancies in the data from the EH take a particularly simple form with forward rates: as theory suggests, the largest discrepancies are at short maturities. (ii) Reasonable estimates of one-factor Cox-Ingersoll-Ross models imply regressions on the opposite side of the EH than we see in the data: regression slopes are greater than one (iii) Multifactore affine models can nevertheless approximate both departures from the EH and other properties of interest rates.

The Conditional Distribution of Excess Returns: An Empirical Analysis

We find that in 1989-1996, when U.S. monetary policy tightly targeted overnight fed funds rates, the volatility and persistence of spreads between target and term fed funds levels were larger for longer-maturity loans. We show that such patterns are consistent with an expectational model where target revisions are infrequent and predictable. In our model, the (autoco-) variance of the spreads of term fed funds rates from the target increases with maturity because longer-term rates are more heavily influenced by persistent expectations of future target changes.

Crash–O–Phobia

Abstract In this article we describe the cumulative distribution function of excess returns conditional on a broad set of predictors that summarize the state of the economy. We do so by estimating a sequence of conditional logit models over a grid of values of the response variable. Our method uncovers higher-order multidimensional structure that cannot be found by modeling only the first two moments of the distribution. We compare two approaches to modeling: one based on a conventional linear logit model and the other based on an additive logit. The second approach avoids the “curse of dimensionality” problem of fully nonparametric methods while retaining both interpretability and the ability to let the data determine the shape of the relationship between the response variable and the predictors. We find that the additive logit fits better and reveals aspects of the data that remain undetected by the linear logit. The additive model retains its superiority even in out-of-sample prediction and portfolio s...

Interpreting the Forward Premium Anomaly

From a large options data set on major equity indexes across the world, the authors find that worldwide, implied volatilities of options on equity indexes exhibit strikingly similar behaviors. When plotted against moneyness, implied volatilities show a heavily skewed smirk pattern, implying that out–of–the–money put options are more expensive than the corresponding out–of–the–money call options and that the risk–neutral return distribution for these indexes is heavily negatively skewed. Furthermore, as the option maturity increases from one month to five years, the implied volatility smirk does not flatten out but steepens, indicating that the risk–neutral distribution of equity index returns becomes even more negatively skewed at longer horizons. The average term structure of the implied volatility level is relatively flat, and the standard deviation of the implied volatility declines as maturity increases. Although fairly persistent, the implied volatility series are stationary. Finally, principal component analysis reveals that equity index volatility movements across the world share one global component.

Price Barriers and the Dynamics of Asset Prices in Equilibrium

One of the central issues in international finance concerns the forward premium anomaly: changes in spot exchange rates are inversely related to the premium of forward rates over spot rates. The authors construct a numerical example of a theoretical economy with this property and discuss its potential as an explanation of the anomaly.