Article

# Short-Term Market Risks Implied by Weekly Options

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## Abstract

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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... 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. ...
Preprint
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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.
... Indeed, the standardized strikeK is an imperfect substitute for moneyness, as moneyness is best represented as a function of strike price and time-to-maturity. Following Andersen, Fusari, and Todorov (2017), we define a moneyness indicator m as ...
... To do so, they argue that the illiquidity makes these options prices unreliable (see, e.g. Andersen et al., 2015Andersen et al., , 2017Kadan and Tang, 2020). However, the relatively good performance of our RF shows that there is some reliable pattern in the implied volatility smile. ...
... 21 Table 6 in appendix D shows an extended version of this table. Andersen et al. (2017) who highlight that weekly options are very sensitive to jump risks. Second, we see that the upward trends in replication errors after the 2008 crisis exists across different time to maturity subsample and can therefore not be explained by the change in market composition highlighted in figure 4(a). ...
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We propose a novel structural estimation framework in which we train a surrogate of an economic model with deep neural networks. Our methodology alleviates the curse of dimensionality and speeds up the evaluation and parameter estimation by orders of magnitudes, which significantly enhances one's ability to conduct analyses that require frequent parameter re-estimation. As an empirical application, we compare two popular option pricing models (the Heston and the Bates model with double-exponential jumps) against a non-parametric random forest model. We document that: a) the Bates model produces better out-of-sample pricing on average, but both structural models fail to outperform random forest for large areas of the volatility surface; b) random forest is more competitive at short horizons (e.g., 1-day), for short-dated options (with less than 7 days to maturity), and on days with poor liquidity; c) both structural models outperform random forest in out-of-sample delta hedging; d) the Heston model's relative performance has deteriorated significantly after the 2008 financial crisis.
... Using USO weeklies has two main advantages. First, as Andersen et al. (2017) state, the pricing of short-dated ATM options depends mainly on its diffusive component, whereas deep OTM options mainly reflect the characteristics of the risk-neutral jump process. The use of weekly ATM options can be an effective way to control for the effects of jump risk and to focus primarily on diffusive risks. ...
... Even after controlling for jump risk, we find that the coefficient of vov is still negative and significant; however, the coefficient of vol is no longer significant. Consistent with Andersen et al. (2017), where the effect of jump risk is lower for weekly ATM options, the predictive power of jump risk proxies is weaker than that associated with one-month deltahedged option returns. Specifically, the coefficients of risk-neutral skewness (skew), riskneutral kurtosis (kurt), and downside realized jump risk (J − ) are not significant, whereas those associated with realized jumps (J) and upside realized jumps (J + ) are statistically significant. ...
... For example, according toAndersen et al. (2017), the trading share of S&P 500 weeklies has reached 40% of the total trading volume of S&P 500 options. ...
Preprint
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Under the stochastic volatility-of-volatility framework, we show that oil volatility-of-volatility risk is a significant pricing factor for cross-sectional delta-hedged gains constructed from 1-month United States Oil Fund (USO) options, and is negatively priced. Moreover, oil volatility-of-volatility risk can significantly and negatively predict one-period ahead delta-hedged option gains. The findings are robust after implementing several tests such as controlling for jump risk measures, another measure of oil volatility-of-volatility and delta-hedged gains constructed from 1-week USO options. The information content of oil volatility-of-volatility is also distinctive from its equity counterpart, which can contribute to predicting the future real personal consumption expenditure.
... Many no-arbitrage and simultaneous equilibrium 5 See Constantinides et al. (2020, figure 1). models impose PCP to obtain parameter estimates from the midpoint of the bid-ask spread of OTM options, which are then used to calibrate the model and to estimate option prices, as in, for instance, Andersen, Fusari, and Todorov (2017), Bondarenko (2014), and Pan (2002). For a frictionless equilibrium option pricing model to be consistent with the data, the OTM option cross-sections, separately for calls and puts, must lie within the SD bounds; otherwise, the model would be inconsistent with the P-distribution data. ...
... At the very least, the SD bounds should serve to verify the equilibrium model results, given their generality in terms of assumptions. This is particularly important for the highly liquid short-term options, with maturities of one month or less, for which volatility can be taken as constant and only jumps are important; see Andersen et al. (2017). ...
Article
We compare the equilibrium jump diffusion option prices with the endogenously determined stochastic dominance (SD) option bounds. We use model parameters from earlier studies and find that most of the equilibrium model prices consistent with the SD bounds yield economically meaningless results. Further, the SD bounds' implied distributions exhibit tail risk comparable to that of the underlying return data, thus shedding light on the dark matter of the inconsistency of physical and risk‐neutral tail probabilities. Since the SD bounds' assumptions are weaker, we conclude that these bounds should either replace, or be used to verify, the equilibrium models' results. This article is protected by copyright. All rights reserved.
... 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. ...
Article
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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). ...
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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.
... The list of references that adopted models similar to (3.4) and (3.5) is very large and only partially reported here. For continuous time SVJ models it includes, in addition to those already mentioned, models of S&P 500 options in Andersen et al. (2017), Bates (1996Bates ( , 2003Bates ( , 2006, Bondarenko (2003Bondarenko ( , 2014, Broadie et al. (2007Broadie et al. ( , 2009 (2002), Ziegler (2007). In the GARCH class of models there are empirical studies by Christoffersen and Jacobs (2004), and theoretical results by Christoffersen et al. (2010) and Duan et al. (2006) that adopt different risk neutralization approaches than the representative investor of Duan (1995), and also GARCH with jumps models by Christoffersen et al. (2012), Orthanalai (2014). ...
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This paper surveys several of the most important applications of the continuous time finance paradigm in portfolio selection and derivatives pricing. While it recognizes the powerful insights that the paradigm offered to researchers and practitioners, it finds that several methodological approaches that it introduced have themselves hardened into paradigms and become dysfunctional. They have downgraded and neglected significant real-world problems because of their inability to model them, or adopted simplifications that had little relevance to the problems that they were supposed to solve. The paper then offers in all cases an alternative methodology that can reach the desired solution via rigorous economic and mathematical reasoning, by replacing mathematical elegance with numerical estimations and approximations.
... The single-factor models are forced to compromise between slow and fast mean reversion, leading to a deterioration in fit in some parts of the sample. This confirms existing findings by, among others, Bates (2000), Christoffersen, Heston, and Jacobs (2009), and Andersen, Fusari, and Todorov (2015a, 2015b. Figures 2 and 3 provide additional perspective on the differences between the GARCH(1,1) and component models. ...
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We nest multiple volatility components, fat tails, and a U-shaped pricing kernel in a single option model and compare their contribution in describing returns and option data. All three features lead to statistically significant model improvements. A U-shaped pricing kernel is economically most important and improves option fit by 17%, on average, and more so for two-factor models. A second volatility component improves the option fit by 9%, on average. Fat tails improve option fit by just over 4%, on average, but more so when a U-shaped pricing kernel is applied. Overall, these three model features are complements rather than substitutes: the importance of one feature increases in conjunction with the others. Received date September 27, 2016; Accepted date July 09, 2017 By Editor Raman Uppal
... Assumption 2.1 rules out processes with discontinuous sample paths from our analysis. Hence, our results are not expected to provide good approximations for applications involving high-frequency data for which jumps are likely to be important; e.g., Andersen et al. (2016), Bandi and Renò (2016) and Hounyo et al. (2020) for financial-oriented applications and Li and Xiu (2016) for a GMM setup; for a textbook account, see Aït-Sahalia and Jacod (2014). Another important difference from the high-frequency statistics centers on the "mean effect" which is captured here by the drift process. ...
Article
Building upon the continuous record asymptotic framework recently introduced by Casini and Perron (2020a) for inference in structural change models, we propose a Laplace-based (Quasi-Bayes) procedure for the construction of the estimate and confidence set for the date of a structural change. It is defined by an integration rather than an optimization-based method. A transformation of the least-squares criterion function is evaluated in order to derive a proper distribution, referred to as the Quasi-posterior. For a given choice of a loss function, the Laplace-type estimator is the minimizer of the expected risk with the expectation taken under the Quasi-posterior. Besides providing an alternative estimate that is more precise—lower mean absolute error (MAE) and lower root-mean squared error (RMSE)—than the usual least-squares one, the Quasi-posterior distribution can be used to construct asymptotically valid inference using the concept of Highest Density Region. The resulting Laplace-based inferential procedure is shown to have lower MAE and RMSE, and the confidence sets strike a better balance between empirical coverage rates and average lengths of the confidence sets relative to traditional long-span methods, whether the break size is small or large.
... Assumption 2.1 rules out processes with discontinuous sample paths from our analysis. Hence, our results are not expected to provide good approximations for applications involving highfrequency data for which jumps are likely to be important; e.g., Andersen, Fusari, and Todorov (2016), Bandi and Renò (2016) and Hounyo, Liu, and Varneskov (2020) for financial-oriented applications and Li and Xiu (2016) for a GMM setup; for a textbook account, see Aït-Sahalia and Jacod (2014). Another important difference from the high-frequency statistics centers on the "mean effect" which is captured here by the drift process. ...
Preprint
Full-text available
Building upon the continuous record asymptotic framework recently introduced by Casini and Perron (2020a) for inference in structural change models, we propose a Laplace-based (Quasi-Bayes) procedure for the construction of the estimate and confidence set for the date of a structural change. It is defined by an integration rather than an optimization-based method. A transformation of the least-squares criterion function is evaluated in order to derive a proper distribution, referred to as the Quasi-posterior. For a given choice of a loss function, the Laplace-type estimator is the minimizer of the expected risk with the expectation taken under the Quasi-posterior. Besides providing an alternative estimate that is more precise-lower mean absolute error (MAE) and lower root-mean squared error (RMSE)-than the usual least-squares one, the Quasi-posterior distribution can be used to construct asymptotically valid inference using the concept of Highest Density Region. The resulting Laplace-based inferential procedure is shown to have lower MAE and RMSE, and the confidence sets strike a better balance between empirical coverage rates and average lengths of the confidence sets relative to traditional long-span methods, whether the break size is small or large. JEL Classification: C12, C13, C22
... having an expiration date within one week (Andersen, Fusari, and Todorov, 2017). The introduction of weekly options represents a semi-natural experiment that enables us to estimate the causal impact of the introduction of options on the underlying equity market. ...
Article
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... Data Source: [24] Here we see the implied volatility is quite high for options expiring in the near future. This is expected an expected equilibrium in options markets [25]. The general case here is the implied volatility for before and after the Halving is approximately the same at about 80%. ...
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... Inthe-money option quotes are excluded because of small transaction volume (Bliss and Panigirtzoglou, 2005). Scholars usually do not analyze option contracts with time to maturity of less than 7 days (Andersen et al., 2017). However, as these options are getting popular recently, e.g., weekly index options, we here analyze option contracts with a short time to maturity and only exclude the contracts with maturity of less than 2 days. ...
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... Inthe-money option quotes are excluded because of small transaction volume (Bliss and Panigirtzoglou, 2005). Scholars usually do not analyze option contracts with time to maturity of less than 7 days (Andersen et al., 2017). However, as these options are getting popular recently, e.g., weekly index options, we here analyze option contracts with a short time to maturity and only exclude the contracts with maturity of less than 2 days. ...
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We model the S&P500 index options dynamics using the CGMY distribution, with independent "up" and "down" return jumps, and diffusive jump intensities. Allowing the up and down parts to be separately parameterised accounts for the dynamic smirk effect, without correlation between returns and intensities. We filter the up and down intensity factors and associated options risk premia. Both factors are informative for dynamic risk premia of equity and corporate debt. We argue that the former is a liquidity effect, and the latter an equilibrium effect. The magnitudes of the estimated, dynamic equity premia are consistent with the "equity premium puzzle". JEL Classifications: G10, G12, G13
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We exploit weekly options on the S&P 500 index to compute the weekly implied variance. We show that the weekly implied variance is a strong predictor of the weekly realized variance. In an encompassing regression test, it crowds out the information content of the monthly implied variance. Further tests reveal that the weekly implied variance outperforms not only the monthly implied variance but also well-established time series models of realized variance. This result holds both in- and out-of-sample and the forecast accuracy gains are significant.
Chapter
There is much research whose efforts have been devoted to discovering the distributional defects in the Black–Scholes model, which are known to cause severe biases. However, with a free specification for the distribution, one can only find upper and lower bounds for option prices. In this chapter, we derive a new nonparametric lower bound and provide an alternative interpretation of Ritchken’s (J Finance 40:1219–1233, 1985) upper bound to the price of European option. In a series of numerical examples, our new lower bound is substantially tighter than previous lower bounds.
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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.
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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.
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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.
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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.
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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.
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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.
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: 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 ...
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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.
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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.
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Book
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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
Article
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.
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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.
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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.
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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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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