Model Specification and Risk Premia: Evidence from Futures Options

This paper examines model specification issues and estimates diffusive and jump risk premia using S&P futures option prices from 1987 to 2003. We first develop a time series test to detect the presence of jumps in volatility, and find strong evidence in support of their presence. Next, using the cross section of option prices, we find strong evidence for jumps in prices and modest evidence for jumps in volatility based on model fit. The evidence points toward economically and statistically significant jump risk premia, which are important for understanding option returns.     

Out-of-sample density forecasts with affine jump diffusion models

We conduct out-of-sample density forecast evaluations of the affine jump diffusion models for the S&P 500 stock index and its options’ contracts. We also examine the time-series consistency between the model-implied spot volatilities using options & returns and only returns. In particular, we focus on the role of the time-varying jump risk premia. Particle filters are used to estimate the model-implied spot volatilities. We also propose the beta transformation approach for recursive parameter updating. Our empirical analysis shows that the inconsistencies between options & returns and only returns are resolved by the introduction of the time-varying jump risk premia. For density forecasts, the time-varying jump risk premia models dominate the other models in terms of likelihood criteria. We also find that for medium-term horizons, the beta transformation can weaken the systematic effect of misspecified AJD models using options & returns.     

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.     

Estimating Early Exercise Premiums on Gold and Copper Options Using a Multifactor Model and Density Matched Lattices

We use the standard geometric Brownian motion augmented by jumps to describe the spot underlying and mean regressive models of interest rates and convenience yields as state variables for gold and copper prices. Estimates of parameters of the diffusion processes are obtained by the Kalman filter. Using these estimates, jump parameters are estimated in the second stage by least squares. Early exercise premia on puts and calls are computed using a lattice with probabilities assigned by the density matching technique. We find that while deep in the money options have greater absolute early exercise premiums, the early exercise premium is roughly constant as a percent of option price. Our findings also confirm that gold behaves like an investment asset and copper behaves like a commodity.     

Sovereign default risk premia: Evidence from the default swap market

This study explores the risk premia embedded in sovereign default swaps using a term structure model. The risk premia remunerate investors for unexpected changes in the default intensity. A number of interesting results emerge from the analysis. First, the risk premia contribution to spreads decreases over the sample, 2003–07, and rebounds at the start of the ‘credit crunch.’ Second, daily risk premia co-move with US macro variables and corporate default risk. Third, global factors explain most of Latin American countries' premia, and local factors best explain European and Asian premia. The importance of global factors grows over time. Finally, conditioning on lagged local and global variables at a weekly frequency, sovereign risk premia are highly predictable.     

Tails,Fears, and Risk Premia

We show that the compensation for rare events accounts for a large fraction of the average equity and variance risk premia. Exploiting the special structure of the jump tails and the pricing thereof, we identify and estimate a new Investor Fears index. The index reveals large time‐varying compensation for fears of disasters. Our empirical investigations involve new extreme value theory approximations and high‐frequency intraday data for estimating the expected jump tails under the statistical probability measure, and short maturity out‐of‐the‐money options and new model‐free implied variation measures for estimating the corresponding risk‐neutral expectations.     

(Almost) Model‐Free Recovery

Under mild assumptions, we recover the model‐free conditional minimum variance projection of the pricing kernel on various tradeable realized moments of market returns. Recovered conditional moments predict future realizations and give insight into the cyclicality of equity premia, variance risk premia, and the highest attainable Sharpe ratios under the minimum variance probability. The pricing kernel projections are often U‐shaped and give rise to optimal conditional portfolio strategies with plausible market timing properties, moderate countercyclical exposures to higher realized moments, and favorable out‐of‐sample Sharpe ratios.     

A Nonlinear Factor Analysis of S&P 500 Index Option Returns

Growing evidence suggests that extraordinary average returns may be obtained by trading equity index options, and that at least part of this abnormal performance is attributable to volatility and jump risk premia. This paper asks whether such priced risk factors are alone sufficient to explain these average returns. To provide an answer in as general as possible a setting, I estimate a flexible class of nonlinear models using all S&P 500 Index futures options traded between 1986 and 2000. The results show that priced factors contribute to these expected returns but are insufficient to explain their magnitudes, particularly for short‐term out‐of‐the‐money puts.     

Option-implied volatility factors and the cross-section of market risk premia

The main goal of this paper is to study the cross-sectional pricing of market volatility. The paper proposes that the market return, diffusion volatility, and jump volatility are fundamental factors that change the investors’ investment opportunity set. Based on estimates of diffusion and jump volatility factors using an enriched dataset including S&P 500 index returns, index options, and VIX, the paper finds negative market prices for volatility factors in the cross-section of stock returns. The findings are consistent with risk-based interpretations of value and size premia and indicate that the value effect is mainly related to the persistent diffusion volatility factor, whereas the size effect is associated with both the diffusion volatility factor and the jump volatility factor. The paper also finds that the use of market index data alone may yield counter-intuitive results.     

Macroeconomic Sources of Risk in the Term Structure

We develop a new way of modeling time variation in term premia, based on the stochastic discount factor model of asset pricing. The joint distribution of excess U.S. bond returns of different maturity and the observable fundamental macroeconomic factors is modeled using multivariate GARCH with conditional covariances in the mean to capture the term premia. By testing the assumption of no arbitrage we derive a specification test of our model. We estimate the contribution made to the term premia at different maturities through real and nominal sources of risk. From the estimated term premia we recover the term structure of interest rates and examine how it varies through time. Finally, we examine whether the reported failures of the rational expectations hypothesis can be attributed to an omitted time-varying term premium.     

Jump risk premia across major international equity markets

We decompose the non-diversifiable market risk into continuous and discontinuous components and jump systematic risks into positive vs. negative and small vs. large components. We examine their association with equity risk premia across major equity markets. We show that developed markets jumps are more closely linked to the aggregate market index than emerging and frontier ones. The reward for bearing both the continuous and downside jump risks is positive during the pre-crisis period whereas the reward for bearing the upside and large jump risks is negative during the crisis and post-crisis periods. We also provide evidence of significant continuous and discontinuous leverage effects during the pre-crisis period, suggesting that both continuous and discontinuous price and volatility risks share compensations for common underlying risk factors.     

Bond risk premia and realized jump risk

We find that augmenting a regression of excess bond returns on the term structure of forward rates with an estimate of the mean realized jump size almost doubles the R² of the forecasting regression. The return predictability from augmenting with the jump mean easily dominates that offered by augmenting with options-implied volatility and realized volatility from high-frequency data. In out-of-sample forecasting exercises, inclusion of the jump mean can reduce the root mean square prediction error by up to 40%. The incremental return predictability captured by the realized jump mean largely accounts for the countercyclical movements in bond risk premia. This result is consistent with the setting of an incomplete market in which the conditional distribution of excess bond returns is affected by a jump risk factor that does not lie in the span of the term structure of yields.     

THE TERM PREMIA RELATIONSHIP IMPLICIT IN THE TERM STRUCTURE OF TREASURY BILLS

This paper considers a single coefficient representation of the term premia relationship that appears in treasury bill yield curves. Term premia are defined as positive or negative maturity-dependent differentials versus the instantaneous nominal spot rate. The term premia function is developed in the context of the Cox, Ingersoll, and Ross [ 3 ] Risk-Averse Preferred Habitat Model and proxies for the degree of risk aversion exhibited by the universe of treasury bill investors at a point in time. Empirical results indicate that term premia are influenced by a set of macroeconomic variables in the expected manner.     

The Valuation of Cash Flow Forecasts: An Empirical Analysis

This article compares the market value of highly leveraged transactions (HLTs) to the discounted value of their corresponding cash flow forecasts. For our sample of 51 HLTs completed between 1983 and 1989, the valuations of discounted cash flow forecasts are within 10 percent, on average, of the market values of the completed transactions. Our valuations perform at least as well as valuation methods using comparable companies and transactions. We also invert our analysis by estimating the risk premia implied by transaction values and forecast cash flows, and relating those risk premia to firm and industry betas, firm size, and firm book-to-market ratios.     

Cyclical variability of bond risk premia: A note

Using a newly developed model, this paper investigates the cyclical behavior of bond risk premia. It is shown that cyclical variability may result simply from the way risk premia are calculated and does not depend on changes over the business cycle in probability of default, investors' attitudes toward risk or institutional behavior. In addition, expectations concerning the likely time of default are shown to be a potential source of cyclical variability and to imply a specific cyclical pattern of risk premia.     

Stochastic Convenience Yield Implied from Commodity Futures and Interest Rates

We characterize a three‐factor model of commodity spot prices, convenience yields, and interest rates, which nests many existing specifications. The model allows convenience yields to depend on spot prices and interest rates. It also allows for time‐varying risk premia. Both may induce mean reversion in spot prices, albeit with very different economic implications. Empirical results show strong evidence for spot‐price level dependence in convenience yields for crude oil and copper, which implies mean reversion in prices under the risk‐neutral measure. Silver, gold, and copper exhibit time variation in risk premia that implies mean reversion of prices under the physical measure.     

International Interest Rates,Exchange Rates,and the Stochastic Structure of Supply

In a dual-currency, flexible exchange rate model, both nominal and real foreign exchange premia depend on investor risk attitudes, consumption parameters, and the stochastic structure of currency and commodity supplies. When supplies are random, their joint correlation structure determines the sign of the premia. If the money supplies are identically distributed, then all foreign exchange premia, regardless of the currency of denomination, are zero. A positive correlation between the value of a country's currency and its nominal interest rate need not indicate real interest rate movements. Relative bond prices can be negatively correlated with the terms of trade.     

Volatility components, leverage effects, and the return-volatility relations

This paper investigates the risk-return trade-off by taking into account the model specification problem. Market volatility is modeled to have two components, one due to the diffusion risk and the other due to the jump risk. The model implies Merton’s ICAPM in the absence of leverage effects, whereas the return-volatility relations are determined by interactions between risk premia and leverage effects in the presence of leverage effects. Empirically, I find a robust negative relationship between the expected excess return and the jump volatility and a robust negative relationship between the expected excess return and the unexpected diffusion volatility. The latter provides an indirect evidence of the positive relationship between the expected excess return and the diffusion volatility.     

Earnings announcement premia and the limits to arbitrage

We examine the factors underlying the presence of earnings announcement premia. We find that the premia persist beyond the sample period examined in prior studies (ending in 1988), although they decline in magnitude after 1988. Further, premia are lower on the expected than the actual earnings announcement dates. We document that increases in voluntary disclosures result in lower premia, despite the increase in return volatility over time. Finally, our evidence suggests that the premia are not completely eliminated because of the costs of arbitrage.     

Positive return premia in Japan

This paper examines Jensen's [J. Finance, 1968, 23, 389–416] alphas and the time-varying return premia unexplained by standard risk factors in Japan and presents several new findings. First, in contrast to the US experience, positive alphas remain after Fama and French's three factors are applied to excess stock returns in Japan. Second, positive alphas remain in Japan, even if the Fama–French three factors combined with momentum and reversal factors are applied to excess stock returns. Third, the positive return premia unexplained by these five factors bear little relation to the dynamics of the Japanese macroeconomy. Fourth, the time series evolution of the positive return premia indicates autonomous dynamics with at least three regimes. Fifth, we can predict or time the acquisition of the positive return premia for small-size portfolios in Japan by observing the direction and effect of the return premia of large-size portfolios and high-book equity to market equity (BE/ME) portfolios. Finally, application of the self-exciting threshold autoregressive (SETAR) model shows that the size effects are stronger than the BE/ME effects in Japan, given that the return premia from small-size portfolios in the SETAR model are bounded by positive thresholds, while the return premia from high-BE/ME portfolios are bounded by negative thresholds.