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We study short-maturity ("weekly") S&P 500 index options, which provide a direct way to analyze volatility and jump risks. Unlike longer-dated options, they are largely insensitive to the risk of intertemporal shifts in the economic environment. Adopting a novel seminonparametric approach, we uncover variation in the negative jump tail risk, which is not spanned by market volatility and helps predict future equity returns. As such, our approach allows for easy identification of periods of heightened concerns about negative tail events that are not always "signaled" by the level of market volatility and elude standard asset pricing models.

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... Cboe introduced weekly option expirations on the S&P 500 index in 2005 (see Andersen, Fusari, and Todorov (2017) 2 examine to what extent these risks are priced in the options market. ...

... Wright (2020) uses weekly options to estimate variance risk premium on the eve of FOMC and employment releases. More generally, Andersen et al. (2017) motivate the use of weekly expiration options to understand market participants' pricing of diffusive and jump risks. ...

... Options data are obtained from Optionmetrics. We apply an initial set of filters from Andersen et al. (2017). For each expiration, we require at least 10 distinct strike prices. ...

Using recently available daily SandP 500 index option expirations, we examine the ex ante pricing of uncertainty surrounding key economic releases and the determinants of risk premia associated with these releases. The cost of insurance against price, variance, and downside risk is higher for options that span U.S. CPI, FOMC, Nonfarm Payroll, and GDP releases compared to neighboring expirations. We calculate release-driven forward equity and variance risk premia and find that premia vary considerably across economic releases and increase with risk aversion as well as with monetary policy and real economic uncertainty. The empirical framework presented in this paper can be used to examine the ex ante pricing of a wide variety of events.

... Unlike the option-implied CCF (3), the CCF in (5) is fully parametric, that is, it requires parametric AJD model dynamics of the state vector X t . Although the AJD class is more restrictive than the general dynamics of F t in (1), it includes a myriad of popular option pricing models such as those in Heston (1993), Duffie et al. (2000), Pan (2002), Bates (2006), Broadie et al. (2007, Boswijk et al. (2015), and Andersen et al. (2017) among many others. ...

... The estimate of ρ being nearly equal to its boundary value might be due to the use of short-dated options that typically exhibit steep implied volatility slopes. Indeed, Andersen et al. (2017) also find this correlation to be close to −1 in their dataset dominated by option contracts with maturities of less than 2 months. ...

... See also, e.g.,Broadie, Chernov, and Johannes (2007), Aït-Sahalia, Cacho-Diaz, and Laeven(2015),Andersen, Fusari, and Todorov (2017) and the references therein. ...

We develop a novel filtering and estimation procedure for parametric option pricing models driven by general affine jump-diffusions. Our procedure is based on the comparison between an option-implied, model-free representation of the conditional log-characteristic function and the model-implied conditional log-characteristic function, which is functionally affine in the model's state vector. We formally derive an associated linear state space representation and establish the asymptotic properties of the corresponding measurement errors. The state space representation allows us to use a suitably modified Kalman filtering technique to learn about the latent state vector and a quasi-maximum likelihood estimator of the model parameters, which brings important computational advantages. We analyze the finite-sample behavior of our procedure in Monte Carlo simulations. The applicability of our procedure is illustrated in two case studies that analyze S&P 500 option prices and the impact of exogenous state variables capturing Covid-19 reproduction and economic policy uncertainty.

... While the underlying principles and potential applications of COS extend beyond financial mathematics, its primary use till now has been in pricing financial derivatives and insurance products. To name only few among numerous applications, see [8], [9], [29], [15], [16], [3], [1], [19], and [18]. Recently, COS has gained prominence as a computational tool for generating training data in deep learning studies in quantitative finance, owing to its exceptional speed and accuracy (see, e.g., [11] and [20]). ...

We provide a rigorous convergence proof demonstrating that the well-known semi-analytical Fourier cosine (COS) formula for the inverse Fourier transform of continuous probability distributions can be extended to discrete probability distributions, with the help of spectral filters. We establish general convergence rates for these filters and further show that several classical spectral filters achieve convergence rates one order faster than previously recognized in the literature on the Gibbs phenomenon. Our numerical experiments corroborate the theoretical convergence results. Additionally, we illustrate the computational speed and accuracy of the discrete COS method with applications in computational statistics and quantitative finance. The theoretical and numerical results highlight the method's potential for solving problems involving discrete distributions, particularly when the characteristic function is known, allowing the discrete Fourier transform (DFT) to be bypassed.

... Ultra-long-term options exhibit higher price uncertainty because of their extended time to expiration, resulting in increased volatility and unpredictability. Similarly, ultra-short-term options are difficult to forecast accurately due to their proximity to the exercise date, which leads to different pricing dynamics, as explored in Andersen, Fusari, and Todorov (2017), compared to other options. While transfer learning can effectively lessen the impact of volatility on price predictions by utilizing broad and diverse pricing data acquired during training in the source domain, deep learning is less adept at handling these challenges. ...

Structural models in economics can offer appealing insights but often suffer from a poor fit with the data. In contrast, machine learning models offer rich flexibility but tend to suffer from over-fitting. We propose a novel framework that incorporates useful economic restrictions from a structural model into a machine learning model through transfer learning. The core idea is to first construct a neural-network representation of the structural model, and then update the network using information from real data. In an example application to option pricing, our hybrid model significantly outperforms both the structural model and a conventional deep-learning model. The out-performance of the hybrid model is more significant when the sample size of real data is limited or under volatile market conditions.

... Much of the modern option pricing literature jointly considers the time-series of observable returns and option prices. See, for instance, Pan(2002),Eraker (2004),Bates (2006),Aït-Sahalia and Kimmel (2007),Hurn, Lindsay, and McClelland (2015), andAndersen, Fusari, and Todorov (2017). ...

The empirical option valuation literature specifies the pricing kernel through the price of risk, or defines it implicitly as the ratio of risk-neutral and physical probabilities. Instead, we extend the economically appealing Rubinstein-Brennan kernels to a dynamic framework that allows path- and volatility-dependence. Because of low statistical power, kernels with different economic properties can produce similar overall option fit, even when they imply cross-sectional pricing anomalies and implausible risk premiums. Imposing parsimonious economic restrictions such as monotonicity and path-independence (recovery theory) achieves good option fit and reasonable estimates of equity and variance risk premiums, while resolving pricing kernel anomalies.

... The risk-neutral distribution of jump sizes exceeding the truncation level exhibits strong evidence of heavy tails, as shown by the significantly positive shape parameter ± in most years. The evidence is consistent with the findings of Bollerslev and Todorov (2011b) and Andersen, Fusari and Todorov (2017) for the S&P 500 index. Our results suggest that the normally distributed risk-neutral jump size widely assumed in previous studies (Hilliard and Reis, 1999;Koekebakker and Lien, 2004) is inadequate to fully capture the impact of jumps on the pricing of commodity futures options. ...

The existence of a negative variance risk premium on agricultural futures contracts suggests that market participants pay to hedge unexpected increases in the volatility of these contracts. In this paper, we decompose the variance risk premium in corn and soybeans markets into jump and diffusive components using options and futures data from 2009 to 2021. We find that market participants on average only pay to hedge unexpected increases in jump volatility but not those in diffusive volatility. Furthermore, growing season uncertainty and the arrival of United States Department of Agriculture (USDA) announcements play important roles in driving the market’s fear of unexpectedly large price jumps.

... These prior studies were unable to study weekly options because their sample periods all ended prior to a significant number of firms having weekly options. Prior research on weekly options largely focuses on indices rather than individual stocks (e.g., Andersen et al. 2017;Oikonomou et al. 2019;Jain and Kotha 2022). The few studies that examine weekly options on individual stocks do not examine earnings announcements or straddle returns (e.g., Wen 2020; Bryzgalova et al. 2022). ...

This study examines the efficiency of weekly option prices around firms’ earnings announcements. With most of the largest firms now having options that expire on a weekly basis, option traders can hedge or speculate on earnings news using options that expire very close to a firm’s earnings announcement date. For earnings announcements near an options expiration date, one can estimate a firm’s expected stock price move in response to its earnings news (i.e., its option implied earnings announcement move) as the price of its at-the-money straddle as a proportion of its stock price. This study tests whether differences between historical earnings announcement moves and option implied earnings announcement moves predict straddle returns. Through the analysis of portfolio returns and Fama–MacBeth regressions, this study finds that straddle returns are significantly higher (lower) when the historical earnings announcement move is high (low) relative to the option implied earnings announcement move. In contrast to prior research, this study does not find an association between straddle returns and historical volatility, historical earnings announcement volatility, implied volatility, or the difference between historical volatility and implied volatility. Overall, this study suggests that weekly straddle prices around earnings announcements are not optimally efficient.

... Oikonomou et al. (2019) exploited the information content of weeklies and find that the implied variance of these options is a better predictor of the future weekly realized variance than not only monthly implied variance but also wellestablished time-series models of realized variance. Andersen et al. (2017) use the deep OTM weekly options as an indicator of a risk-neutral jump process and ATM options as a reflection of current spot volatility. The authors posit that market participants are better able to avoid diffusive and jump risks via weekly options as the expected volatility and jump intensity do not vary much over a short tenure. ...

This paper empirically examines the effect of weekly options introduction on the benchmark index of Indian stock market, NIFTY50. The paper evaluates the possible stabilizing or destabilizing nature of impact on underlying volatility focusing on the relation between information and volatility using GARCH framework. The results indicate that the onset of weekly index options has improved the information assimilation and reduced the persistence of old information on volatility. Further, similar changes are not evident on a control index, NIFTY NEXT50. Overall, the results indicate an increase in market efficiency with weekly index options trading.

... 7 A positive tempered stable distribution can be constructed from a one-sided α-stable law by exponential tilting, see Barndorff-Nielsen and Shephard (2001b, p.3) and Barndorff-Nielsen et al. (2002, p.14). It is a popular and basic tool in finance, see some recent applications in Carr et al. (2002), Bates (2012), Li and Linetsky (2013), Mendoza-Arriaga and Linetsky (2014), Todorov (2015) and Andersen et al. (2017). In particular, if α = 1 2 , it reduces to a very important distribution, the inverse Gaussian (IG) distribution (which can be interpreted as the distribution of the first passage time of a Brownian motion to an absorbing barrier). ...

We consider a parsimonious framework of jump-diffusion models for price dynamics with stochastic price volatilities and stochastic jump intensities in continuous time. They account for conditional heteroscedasticity and also incorporate key features appearing in financial time series of price volatilities and jump intensities, such as persistence of contemporaneous jumps (cojumps), mean reversion and feedback effects. More precisely, the stochastic variance and stochastic intensity are jointly modelled by a generalised bivariate shot-noise process sharing common jump arrivals with any non-negative jump-size distributions. This framework covers many classical and important models in the literature. The main contribution of this paper is that, we develop a very efficient scheme for its exact simulation based on perfect decomposition where neither numerical inversion nor acceptance/rejection scheme is required, which means that it is not only accurate but also the efficiency would not be sensitive to the parameter choice. Extensive numerical implementations and tests are reported to demonstrate the accuracy and effectiveness of this scheme. Our algorithm substantially outperforms the classical discretisation scheme. Moreover, we unbiasedly estimate the prices of discrete-barrier European options to show the applicability and flexibility of our algorithms.

High volatility in the stock market is often attributed to derivative expirations. The National Stock Exchange of India, largest derivatives exchange in the world by number of contracts traded, introduced weekly or short-dated derivatives in Feb 2019 to mitigate the expiration day effects. The study empirically examines the return and volatility data surrounding expiration days during the period before and after the introduction of weekly derivatives. First, for the period before the introduction of weekly contracts, the study gathers empirical evidence suggesting the presence of upward shift in price effects around expiration day and a transitory upward shift in volatility. Next, the study shows with a comprehensive time series analysis, these adverse expiration day effects disappear in the period after the introduction of weekly options. The paper concludes that the weekly, short-dated, contracts fill the gap in the market depth and are beneficial to the over all market welfare. As the weekly options are gaining popularity across the exchanges and investors, the study has important implications to the regulators and practitioners around the world.

We propose a method to identify the informativeness of a future scheduled announcement at the daily level, exploiting the discontinuity it creates in the term structure of option volatility. We implement the strategy in a panel data model to estimate the relation between prior signals and the future announcement. This method allows us to separate substitutes from complements, it can isolate multiple signals within the same quarter, and it can condition on the timing and signal characteristics. We find that analyst forecasts substitute earnings announcement information and that recommendations do not provide extra information on top of forecasts. Moreover, our evidence suggests that insiders sell to avoid uncertainty when the announcement is far away but pull forward earnings information when they trade one month before.
This paper was accepted by Eric So, accounting.
Funding: This publication is part of the project “Eliciting the properties of private signals” (with project number VI.Veni.201E.029 of the research program VENI SGW, which is (partly) financed by the Dutch Research Council (NWO).
Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.4970 .

We propose a novel measure of the market return tail risk premium based on minimum-distance state price densities recovered from high-frequency data. The tail risk premium extracted from intra-day S&P 500 returns predicts the market equity and variance risk premiums and expected excess returns on a cross section of characteristics-sorted portfolios. Additionally, we describe the differential role of the quantity of tail risk, and of the tail premium, in shaping the future distribution of index returns. Our results are robust to controlling for established measures of variance and tail risk, and of risk premiums, in the predictive models.

The cross section of options holds great promise for identifying return distributions and risk premia, but estimating dynamic option valuation models with latent state variables is challenging when using large option panels. We propose a particle Markov Chain Monte Carlo framework with a novel filtering approach and illustrate our method by estimating index option pricing models. Estimates of variance risk premiums, variance mean reversion, and higher moments differ from the literature. We show that these differences are due to the composition of the option sample. Restricting the option sample’s maturity dimension has the strongest impact on parameter inference and option fit in these models.

Market skewness risk is priced, but the components of its premium are not fully understood. We propose new trading strategies decomposing the skewness risk premium into jump and leverage effect components, and we analyze the skewness risk premia in the market for S&P 500 index options. We find that the skewness premium is higher when markets are closed than during trading hours, consistently with uncertainty resolution patterns by non-U.S investors; that it increases after left-tail market events; and that it is distinct from the variance premium. Moreover, during trading hours, the skewness premium is dominated by priced jump risk.
This paper was accepted by Kay Giesecke, finance.
Funding: P. Orłowski acknowledges financial support from the Doc.Mobility program of the Swiss National Science Foundation [Project P1TIP1_161875 “Option portfolio returns and dispersion”]. P. Schneider acknowledges financial support from the Swiss National Science Foundation [Projects 169582 “Model-free asset pricing” and 189086 “Scenarios”]. F. Trojani and P. Orłowski acknowledge financial support from the Swiss National Science Foundation [Project 150198 “Higher order robust resampling and multiple testing methods”] and the Swiss Finance Institute [Project “Term structures and cross-sections of asset risk premia”]. F. Trojani gratefully acknowledges support from the AXA Chair in Socioeconomic Risks of Financial Markets at the University of Turin.
Supplemental Material: The data files and online appendices are available at https://doi.org/10.1287/mnsc.2023.4734 .

This study used dummy variables to measure the influence of day-of-the-week effects and structural breaks on volatility. Considering day-of-the-week effects, structural breaks, or both, we propose three classes of HAR models to forecast electricity volatility based on existing HAR models. The estimation results of the models showed that day-of-the-week effects only improve the fitting ability of HAR models for electricity volatility forecasting at the daily horizon, whereas structural breaks can improve the in-sample performance of HAR models when forecasting electricity volatility at daily, weekly, and monthly horizons. The out-of-sample analysis indicated that both day-of-the-week effects and structural breaks contain additional ex ante information for predicting electricity volatility, and in most cases, dummy variables used to measure structural breaks contain more out-of-sample predictive information than those used to measure day-of-the-week effects. The out-of-sample results were robust across three different methods. More importantly, we argue that adding dummy variables to measure day-of-the-week effects and structural breaks can improve the performance of most other existing HAR models for volatility forecasting in the electricity market.

Among the first, we propose and implement a novel path-perturbation based closed-form expansion for approximating option prices under a general class of models featuring stochastic volatility and jumps in both asset return and volatility. The expansion naturally employs the formulas of, e.g., Merton (1976) or Kou (2002) for pricing options under jump-diffusions with constant volatility as the leading term and provides corrections up to an arbitrary order. It offers an efficient computational tool for empirical analysis on the models through, e.g., calibration or estimation based on option data, in particular for flexible yet analytically intractable cases.

In their paper, “Dark Matter in (Volatility and) Equity Option Risk Premiums,” Bakshi, Crosby, and Gao ask a provocative question: is there dark matter embedded in volatility and equity options? They consider a theoretical approach that allows them to introduce the constructs of risk premiums on jumps crossing the strike and on local time. The treatment of jumps crossing the strike and local time is integral to their theory because their absence would be counterfactual from an empirical standpoint. They label such abstract uncertainties—driven by unspanned risk components—“dark matter” as these uncertainties can be hard to identify, but their presence is implied in options data, and the workings of dark matter can be economically influential. Their empirical exercises are based on weekly equity index options (the “weeklys”) in addition to the farther-dated (index and futures) options up to 88 days maturity.

We analyze how informed investors trade in the options market ahead of corporate news when they receive private, but noisy, information about the timing and impact of these announcements on stock prices. We propose a framework that ranks options trading strategies (option type, maturity, and strike price) based on their maximum attainable leverage when price-taking investors face market frictions. We exploit the heterogeneity in announcement characteristics across twelve categories of corporate events to support that event-specific information signals are informative for announcement returns and that they impact the optimal choice of option moneyness and tenor.

This paper introduces new econometric tests to identify stochastic intensity jumps in high-frequency data. Our approach exploits the behavior of a time-varying stochastic intensity and allows us to assess how intensely stock market reacts to news. We describe the asymptotic properties of our test statistics, derive the associated central limit theorem and show in simulations that the tests have good size and reasonable power in finite-sample cases. Implementing our testing procedures on the S&P 500 exchange-traded fund data, we find strong evidence for the presence of intensity jumps surrounding the scheduled Federal Open Market Committee (FOMC) policy announcements. Intensity jumps occur very frequently, trigger sharp increases in realized volatility and arrive when differences in opinion among market participants are large at times of FOMC press releases. Unlike intensity jumps, volatility jumps fail to explain the variation in news-induced realized volatility.

We characterize jump dynamics in U.S. stock market returns using a novel series of intraday prices covering almost 90 years. Jump dynamics vary substantially over time. Trends in jump activity relate to secular shifts in the nature of news. Unscheduled news often involving major wars drives jump activity in early decades, whereas scheduled news and especially news pertaining to monetary policy drives jump activity in recent decades. Jump variation measures forecast excess stock market returns, consistent with theory. Results support models featuring a separate jump factor, such that risk premium dynamics are not fully captured by volatility state variables.

We develop a novel class of time-changed Lévy models, which are tractable and readily applicable, capture the leverage effect, and exhibit pure jump processes with finite or infinite activity. Our models feature four nested processes reflecting market, volatility and jump risks, and observation error of time changes. To operationalize the models, we use volume-based proxies of the unobservable time changes. To estimate risk premia, we derive the change of measure analytically. An extensive time series and option pricing analysis of sixteen time-changed Lévy models shows that infinite activity processes carry significant jump risk premia, and largely outperform many finite activity processes.

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 moneyness- maturity (cross-sectional) 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 mixed-Gaussian 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 high-frequency 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 risk-neutral dynamics over a given period of time. A large-scale Monte Carlo study indicates the inference procedures work well for empirically realistic specifications and sample sizes. In an empirical application to S&P 500 index options we extend the popular double-jump stochastic volatility model to allow for time-varying jump risk premia and a flexible relation between risk premia and the level of risk. Both extensions lead to an improved characterization of observed option prices.Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

The paper proposes an additive cascade model of volatility components defined over different time periods. This volatility
cascade leads to a simple AR-type model in the realized volatility with the feature of considering different volatility components
realized over different time horizons and thus termed Heterogeneous Autoregressive model of Realized Volatility (HAR-RV).
In spite of the simplicity of its structure and the absence of true long-memory properties, simulation results show that the
HAR-RV model successfully achieves the purpose of reproducing the main empirical features of financial returns (long memory,
fat tails, and self-similarity) in a very tractable and parsimonious way. Moreover, empirical results show remarkably good
forecasting performance.

The validity of the classic Black-Scholes option pricing formula depends on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying stock return dynamics can be described by a stochastic process with a continuous sample path. In this paper, an option pricing formula is derived for the more-general case when the underlying stock returns are generated by a mixture of both continuous and jump processes. The derived formula has most of the attractive features of the original Black-Scholes formula in that it does not depend on investor preferences or knowledge of the expected return on the underlying stock. Moreover, the same analysis applied to the options can be extended to the pricing of corporate liabilities.

We study the short-time asymptotics of conditional expectations of smooth and
non-smooth functions of a (discontinuous) Ito semimartingale; we compute the
leading term in the asymptotics in terms of the local characteristics of the
semimartingale. We derive in particular the asymptotic behavior of call options
with short maturity in a semimartingale model: whereas the behavior of
\textit{out-of-the-money} options is found to be linear in time, the short time
asymptotics of \textit{at-the-money} options is shown to depend on the fine
structure of the semimartingale.

We investigate the importance of diffusion and jumps in a new model for asset returns. In contrast to standard models, we allow for jump components displaying finite or infinite activity and variation. Empirical investigations of time series indicate that index dynamics are devoid of a diffusion component, which may be present in the dynamics of individual stocks. This leads to the conjecture, confirmed on options data, that the risk-neutral process should be free of a diffusion component. We conclude that the statistical and risk-neutral processes for equity prices are pure jump processes of infinite activity and finite variation.

We derive an asymptotic expansion formula for option implied volatility under a two-factor jump-diffusion stochastic volatility model when time-to-maturity is small. We further propose a simple calibration procedure of an arbitrary parametric model to short-term near-the-money implied volatilities. An important advantage of our approximation is that it is free of the unobserved spot volatility. Therefore, the model can be calibrated on option data pooled across different calendar dates to extract information from the dynamics of the implied volatility smile. An example of calibration to a sample of S&P 500 option prices is provided. (JEL G12) Copyright 2007, Oxford University Press.

This paper introduces a model in which the probability of a rare disaster varies over time. I show that the model can account for the high equity premium and high volatility in the aggregate stock market. At the same time, the model generates a low mean and volatility for the government bill rate, as well as economically significant excess stock return predictability. The model is set in continuous time, assumes recursive preferences and is solved in closed-form. It is shown that recursive preferences, as well as time-variation in the disaster probability, are key to the model's success.

: A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S&P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model ...

We study the convergence of at-the-money implied volatilities to the spot volatility in a general model with a Brownian component and a jump component of finite variation. This result is a consequence of the robustness of the Black-Scholes formula and of the central limit theorem for martingales.

Brownian motion and normal distribution have been widely used in the Black-Scholes option-pricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and an empirical phenomenon called "volatility smile" in option markets. To incorporate both of them and to strike a balance between reality and tractability, this paper proposes, for the purpose of option pricing, a double exponential jump-diffusion model. In particular, the model is simple enough to produce analytical solutions for a variety of option-pricing problems, including call and put options, interest rate derivatives, and path-dependent options. Equilibrium analysis and a psychological interpretation of the model are also presented.

The variance risk premium, defined as the difference between the actual and risk-neutral expectations of the forward aggregate market variation, helps predict future market returns. Relying on a new essentially model-free estimation procedure, we show that much of this predictability may be attributed to time variation in the part of the variance risk premium associated with the special compensation demanded by investors for bearing jump tail risk, consistent with the idea that market fears play an important role in understanding the return predictability.

We study the dynamic relation between market risks and risk premia using time series of index option surfaces. We find that priced left tail risk cannot be spanned by market volatility (and its components) and introduce a new tail factor. This tail factor has no incremental predictive power for future volatility and jump risks, beyond current and past volatility, but is critical in predicting future market equity and variance risk premia. Our findings suggest a wide wedge between the dynamics of market risks and their compensation, which typically displays a far more persistent reaction following market crises.

Part I Introduction and Preliminary Material.- 1.Introduction .- 2.Some Prerequisites.- Part II The Basic Results.- 3.Laws of Large Numbers: the Basic Results.- 4.Central Limit Theorems: Technical Tools.- 5.Central Limit Theorems: the Basic Results.- 6.Integrated Discretization Error.- Part III More Laws of Large Numbers.- 7.First Extension: Random Weights.- 8.Second Extension: Functions of Several Increments.- 9.Third Extension: Truncated Functionals.- Part IV Extensions of the Central Limit Theorems.- 10.The Central Limit Theorem for Random Weights.- 11.The Central Limit Theorem for Functions of a Finite Number of Increments.- 12.The Central Limit Theorem for Functions of an Increasing Number of Increments.- 13.The Central Limit Theorem for Truncated Functionals.- Part V Various Extensions.- 14.Irregular Discretization Schemes. 15.Higher Order Limit Theorems.- 16.Semimartingales Contaminated by Noise.- Appendix.- References.- Assumptions.- Index of Functionals.- Index.

We develop new methods for the estimation of time-varying risk-neutral jump tails in asset returns. In contrast to existing procedures based on tightly parameterized models, our approach imposes much fewer structural assumptions, relying on extreme-value theory approximations together with short-maturity options. The new estimation approach explicitly allows the parameters characterizing the shape of the right and the left tails to differ, and importantly for the tail shape parameters to change over time. On implementing the procedures with a panel of S&P 500 options, our estimates clearly suggest the existence of highly statistically significant temporal variation in both of the tails. We further relate this temporal variation in the shape and the magnitude of the jump tails to the underlying return variation through the formulation of simple time series models for the tail parameters.

In spite of the popularity of model calibration in finance, empirical researchers have put more emphasis on model estimation than on the equally important goodness-of-fit problem. This is due partly to the ignorance of modelers, and more to the ability of existing statistical tests to detect specification errors. In practice, models are often calibrated by minimizing the sum of squared difference between the modelled and actual observations. It is challenging to disentangle model error from estimation error in the residual series. To circumvent the difficulty, we study an alternative way of estimating the model by exact calibration. We argue that standard time series tests based on the exact approach can better reveal model misspecifications than the error minimizing approach. In the context of option pricing, we illustrate the usefulness of exact calibration in detecting model misspecification. Under heteroskedastic observation error structure, our simulation results shows that the Black-Scholes model calibrated by exact approach delivers more accurate hedging performance than that calibrated by error minimization.

This article provides the economic foundations for valuing derivative securities. In particular, it establishes how the characteristic function (of the future uncertainty) is basis augmenting and spans the payo! universe of most, if not all, derivative assets. From the characteristic function of the state-price density, it is possible to analytically price options on any arbitrary transformation of the underlying uncertainty. By di!erentiating (or translating) the characteristic function, limitless pricing and/or spanning opportunities can be designed. The strength and versatility of the methodology is inherent when valuing (1) average-interest options, (2) correlation options, and (3) discretely monitored knock-out options. ( 2000 Elsevier Science S.A. All rights reserved. JEL classixcation: G10; G12; G13

This paper applies the Bates (RFS, 2006) methodology to the problem of estimating and filtering time- changed Lévy processes, using daily data on U.S. stock market excess returns over 1926-2006. In contrast to density-based filtration approaches, the methodology recursively updates the associated conditional characteristic functions of the latent variables. The paper examines how well time-changed Lévy specifications capture stochastic volatility, the "leverage" effect, and the substantial outliers occasionally observed in stock market returns. The paper also finds that the autocorrelation of stock market excess returns varies substantially over time, necessitating an additional latent variable when analyzing historical data on stock market returns. The paper explores option pricing implications, and compares the results with observed prices of options on S&P 500 futures.

We build a new class of discrete time models where the distribution of daily returns is driven by two factors: dynamic volatility and dynamic jump intensity. Each factor has its own risk premium. The likelihood function for the models is available using analytical filtering, which makes them much easier to implement than most existing models. Estimating the models on S&P500 returns, we find that they significantly outperform standard models without jumps. We find very strong empirical support for time-varying jump intensities, and thus for flexible skewness and kurtosis dynamics. Compared to the risk premium on dynamic volatility, the risk premium on the dynamic jump intensity has a much larger impact on option prices. We confirm these findings using joint estimation on returns and large option samples, which is feasible in our class of models.

We study the convergence of at-the-money implied volatilities to the spot volatility in a general model with a Brownian component and a jump component of finite variation. This result is a consequence of the robustness of the Black-Scholes formula and of the central limit theorem for martingales.

Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein-Uhlenbeck equations driven by one-sided discontinuous Lévy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.

This paper examines the empirical performance of jump diffusion models of stock price dynamics from joint options and stock markets data. The paper introduces a model with discontinuous correlated jumps in stock prices and stock price volatility, and with state-dependent arrival intensity. We discuss how to perform likelihood-based inference based upon joint options/returns data and present estimates of risk premiums for jump and volatility risks. The paper finds that while complex jump specifications add little explanatory power in fitting options data, these models fare better in fitting options and returns data simultaneously.

Uncertainty plays a key role in economics, finance, and decision sciences. Financial markets, in particular derivative markets,
provide fertile ground for understanding how perceptions of economic uncertainty and cash-flow risk manifest themselves in
asset prices. We demonstrate that the variance premium, defined as the difference between the squared VIX index and expected
realized variance, captures attitudes toward uncertainty. We show conditions under which the variance premium displays significant
time variation and return predictability. A calibrated, generalized long-run risks model generates a variance premium with
time variation and return predictability that is consistent with the data, while simultaneously matching the levels and volatilities
of the market return and risk-free rate. Our evidence indicates an important role for transient non-Gaussian shocks to fundamentals
that affect agents' views of economic uncertainty and prices.

We study optimal investment strategies given investor access not only to bond and stock markets but also to the derivatives market. The problem is solved in closed form. Derivatives extend the risk and return tradeoffs associated with stochastic volatility and price jumps. As a means of exposure to volatility risk, derivatives enable non-myopic investors to exploit the time-varying opportunity set; and as a means of exposure to jump risk, they enable investors to disentangle the simultaneous exposure to diffusive and jump risks in the stock market. Calibrating to the S&P 500 index and options markets, we find sizable portfolio improvement from derivatives investing.

This paper examines the joint time series of the S&P 500 index and near-the-money short-dated option prices with an arbitrage-free model, capturing both stochastic volatility and jumps. Jump-risk premia uncovered from the joint data respond quickly to market volatility, becoming more prominent during volatile markets. This form of jump-risk premia is important not only in reconciling the dynamics implied by the joint data, but also in explaining the volatility “smirks” of cross-sectional options data.

Post-crash distributions inferred from S&P 500 future option prices have been strongly negatively skewed. This article examines two alternate explanations: stochastic volatility and jumps. The two option pricing models are nested, and are fitted to S&P 500 futures options data over 1988–1993. The stochastic volatility model requires extreme parameters (e.g., high volatility of volatility) that are implausible given the time series properties of option prices. The stochastic volatility/jump-diffusion model fits option prices better, and generates more plausible volatility process parameters. However, its implicit distributions are inconsistent with the absence of large stock index moves over 1988–93.

Here we develop an option pricing method for European options based on the Fourier-cosine series, and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fourier-cosine expansion of the density function. In most cases, the convergence rate of the COS method is exponential and the computational complexity is linear. Its range of application covers different underlying dynamics, including L\'evy processes and Heston stochastic volatility model, and various types of option contracts. We will present the method and its applications in two separate parts. The first one is this paper, where we deal with European options in particular. In a follow-up paper we will present its application to options with early-exercise features.

We propose a new and flexible non-parametric framework for estimating the jump tails of It� semimartingale processes. The approach is based on a relatively simple-to-implement set of estimating equations associated with the compensator for the jump measure, or its "intensity", that only utilizes the weak assumption of regular variation in the jump tails, along with in-fill asymptotic arguments for uniquely identifying the "large" jumps from the data. The estimation allows for very general dynamic dependencies in the jump tails, and does not restrict the continuous part of the process and the temporal variation in the stochastic volatility. On implementing the new estimation procedure with actual high-frequency data for the S&P 500 aggregate market portfolio, we find strong evidence for richer and more complex dynamic dependencies in the jump tails than hitherto entertained in the literature.

This paper implements the time-state preference model in a multi-period economy, deriving the prices of primitive securities from the prices of call options on aggregate consumption. These prices permit an equilibrium valuation of assets with uncertain payoffs at many future dates. Furthermore, for any given portfolio, the price of a $1.00 claim received at a future date, if the portfolio's value is between two given levels at that time, is derived explicitly from a second partial derivative of its call-option pricing function. An intertemporal capital asset pricing model is derived for payoffs that are jointly lognormally distributed with aggregate consumption. It is shown that using the Black-Scholes equation for options on aggregate consumption implies that individuals' preferences aggregate to isoelastic utility.

This paper argues that in an uncertain world options written on existing assets can improve efficiency by permitting an expansion
of the contingencies that are covered by the market. The two major results obtained are, first, that complex contracts can
be “built up” as portfolios of simple options and, second, that there exists a single portfolio of the assets, the efficient
fund, on which all options can be written with no loss of efficiency.

The role of ordinary options in facilitating the completion of securities markets is examined in the context of a model of
contigent claims sufficiently general to accommodate the continuous distributions of asset pricing theory and option pricing
theory. In this context, it is shown that call options written on a single security approximately span all contingent claims
written on this security and that call options written on portfolios of call options on individual primitive securities approximately
span all contingent claims that can be written on these primitive securities. In the case of simple options, explicit formulas
are given for the approximating options and portfolios of options. These results are applied to the pricing of contingent
claims by arbitrage and to irrelevance propositions in corporate finance.

I modify the uniform-price auction rules in allowing the seller to ration bidders. This allows me to provide a strategic foundation for underpricing when the seller has an interest in ownership dispersion. Moreover, many of the so-called "collusive-seeming" equilibria disappear.

We find that the difference between implied and realized variances, or the variance risk premium, is able to explain more than fifteen percent of the ex-post time series variation in quarterly excess returns on the market portfolio over the 1990 to 2005 sample period, with high (low) premia predicting high (low) future returns. The magnitude of the return predictability of the variance risk premium easily dominates that afforded by standard predictor variables like the P/E ratio, the dividend yield, the default spread, and the consumption-wealth ratio (CAY). Moreover, combining the variance risk premium with the P/E ratio results in an R^2 for the quarterly returns of more than twenty-five percent. The results depend crucially on the use of "model-free", as opposed to standard Black-Scholes, implied variances, and realized variances constructed from high-frequency intraday, as opposed to daily, data. Our findings suggest that temporal variation in risk and risk-aversion both play an important role in determining stock market returns.

We introduce adaptive learning behavior into a general-equilibrium life-cycle economy with capital accumulation. Agents form forecasts of the rate of return to capital assets using least-squares autoregressions on past data. We show that, in contrast to the perfect-foresight dynamics, the dynamical system under learning possesses equilibria that are characterized by persistent excess volatility in returns to capital. We explore a quantitative case for theselearning equilibria. We use an evolutionary search algorithm to calibrate a version of the system under learning and show that this system can generate data that matches some features of the time-series data for U.S. stock returns and per-capita consumption. We argue that this finding provides support for the hypothesis that the observed excess volatility of asset returns can be explained by changes in investor expectations against a background of relatively small changes in fundamental factors.

This paper extends the class of stochastic volatility diffusions for asset returns to encompass Poisson jumps of time-varying intensity. We find that any reasonably descriptive continuous-time model for equity-index returns must allow for discrete jumps as well as stochastic volatility with a pronounced negative relationship between return and volatility innovations. We also find that the dominant empirical characteristics of the return process appear to be priced by the option market. Our analysis indicates a general correspondence between the evidence extracted from daily equity-index returns and the stylized features of the corresponding options market prices. Copyright The American Finance Association 2002.

Substantial progress has been made in developing more realistic option pricing models. Empirically, however, it is not known whether and by how much each generalization improves option pricing and hedging. The authors fill this gap by first deriving an option model that allows volatility, interest rates, and jumps to be stochastic. Using S&P 500 options, they examine several alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant time-series data, (2) out-of-sample pricing, and (3) hedging. Overall, incorporating stochastic volatility and jumps is important for pricing and internal consistency. But for hedging, modeling stochastic volatility alone yields the best performance. Copyright 1997 by American Finance Association.

In the setting of affine' jump-diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applications include fixed-income pricing models, with a role for intensityy-based models of default, as well as a wide range of option-pricing applications. An illustrative example examines the implications of stochastic volatility and jumps for option valuation. This example highlights the impact on option 'smirks' of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both amplitude as well as jump timing.

This paper studies the empirical performance of jump-diffusion models that allow for stochastic volatility and correlated jumps affecting both prices and volatility. The results show that the models in question provide reasonable fit to both option prices and returns data in the in-sample estimation period. This contrasts with previous findings where stochastic volatility paths are found to be too smooth relative to the option implied dynamics. While the models perform well during the high volatility estimation period, they tend to overprice long dated contracts out-of-sample. This evidence points towards a too simplistic specification of the mean dynamics of volatility.

Building on Duffie and Kan (1996) , we propose a new representation of affine models in which the state vector comprises infinitesimal maturity yields and their quadratic covariations. Because these variables possess unambiguous economic interpretations, they generate a representation that is "globally identifiable". Further, this representation has more identifiable parameters than the "maximal" model of Dai and Singleton (2000) . We implement this new representation for select three-factor models and find that model-independent estimates for the state vector can be estimated directly from yield curve data, which present advantages for the estimation and interpretation of multifactor models. Copyright 2008 by The American Finance Association.

We develop a simple robust method to distinguish the presence of continuous and discontinuous components in the price of an asset underlying options. Our method examines the prices of at-the-money and out-of-the-money options as the option's time-to-maturity approaches zero. We show that these prices converge to zero at speeds that depend upon whether the underlying asset price process is purely continuous, purely discontinuous, or a combination of both. We apply the method to S&P 500 index options and find the existence of both a continuous component and a jump component in the index. Copyright 2003 by the American Finance Association.

Transactions prices of S&P 500 futures options over 1985-87 are examined for evidence of expectations prior to October 1987 of an impending stock market crash. First, it is shown that out-of-the-money puts became unusually expensive during the year preceding the crash. Second, a model is derived for pricing American options on jump-diffusion processes with systematic jump risk. The jump-diffusion parameters implicit in options prices indicate that a crash was expected and that implicit distributions were negatively skewed during October 1986 to August 1987. Both approaches indicate no strong crash fears during the 2 months immediately preceding the crash. Copyright 1991 by American Finance Association.

Three processes reecting persistence of volatility are formulated by evaluating three Levy processes at a time change given by the integral of a square root process. A positive stock price process is then obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating the processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. Our empirical results on index options and single name options suggest advantages to employing higher dimensional Levy systems for index options and lower dimensional structures for single names. In general, mean corrected exponentiation performs better than employing the stochastic exponential. Martingale laws for the mean corrected exponential are also studied and two new concepts termed Levy and martingale marginals are introduced.

This paper proposes a methodology for the valuation of contingent securities. In particular, it establishes how the characteristic function (of the future uncertainty) is basis augmenting and spans the payoff universe of most, if not all, derivative assets. In one specific application, from the characteristic function of the state-price density, it is possible to analytically price options on any arbitrary transformation of the underlying uncertainty. By differentiating (or translating) the characteristic function, limitless pricing and/or spanning opportunities can be designed. As made lucid via example contingent claims, by exploiting the unifying spanning concept, the valuation approach affords substantial analytical tractability. The strength and versatility of the methodology is inherent when valuing (1) Average-interest options; (2) Correlation options; and (3) Discretely-monitored knock-out options. For each option-like security, the characteristic function is strikingly simple (although the corresponding density is unmanageable/indeterminate). This article provides the economic foundations for valuing derivative securities.

This paper studies the empirical performance of jump-di#usion models that allow for stochastic volatility and correlated jumps a#ecting both prices and volatility. The results show that the models in question provide reasonable fit to both option prices and returns data in the in-sample estimation period. This contrasts previous findings where stochastic volatility paths are found to be too smooth relative to the option implied dynamics. While the models perform well during the high volatility estimation period, they tend to overprice long dated contracts out-of-sample. This evidence points towards a too simplistic specification of the mean dynamics of volatility. Keywords: Market crashes, jump-di#usion, Stochastic volatility, jump in volatility, a#ne models, Markov Chain Monte Carlo JEL Class: C11, C15, G12 # I thank Fredrico Bandi, Lars Hansen, Nick Polson, Pietro Veronesi, and Mike Johannes, as well as seminar participants at the University of Chicago, Duke University, McGill University, University de Montreal, University of Toronto, London Business School, Norwegian School of Economics, and Northwestern University for helpful comments and discussions. 1 1

Specification and Risk Premiums: The Information in S&P 500 Futures Options

- M Broadie
- M Chernov
- M Johannes

Broadie, M., M. Chernov, and M. Johannes (2007). Specification and Risk Premiums: The Information in S&P 500 Futures Options. Journal of Finance 62, 1453-1490.

- D S Bates

Bates, D. S. (2012). U.S. Stock Market Crash Risk, 1926 -2010. Journal of Financial Economics 105, 229-259.

- D Duffie
- J Pan
- K Singleton

Duffie, D., J. Pan, and K. Singleton (2000). Transform Analysis and Asset Pricing for Affine
Jump-Diffusions. Econometrica 68, 1343-1376.