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

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

Assessing the performance of symmetric and asymmetric implied volatility functions

This study examines several alternative symmetric and asymmetric model specifications of regression-based deterministic volatility models to identify the one that best characterizes the implied volatility functions of S&P 500 Index options in the period 1996–2009. We find that estimating the models with nonlinear least squares, instead of ordinary least squares, always results in lower pricing errors in both in- and out-of-sample comparisons. In-sample, asymmetric models of the moneyness ratio estimated separately on calls and puts provide the overall best performance. However, separating calls from puts violates the put-call-parity and leads to severe model mis-specification problems. Out-of-sample, symmetric models that use the logarithmic transformation of the strike price are the overall best ones. The lowest out-of-sample pricing errors are observed when implied volatility models are estimated consistently to the put-call-parity using the joint data set of out-of-the-money options. The out-of-sample pricing performance of the overall best model is shown to be resilient to extreme market conditions and compares quite favorably with continuous-time option pricing models that admit stochastic volatility and random jump risk factors.     

International capital asset pricing: Evidence from options

We study the risk dynamics and pricing in international economies through a joint analysis of the time-series returns and option prices on three equity indexes underlying three economies: the S&P 500 Index of the United States, the FTSE 100 Index of the United Kingdom, and the Nikkei-225 Stock Average of Japan. We develop an international capital asset pricing model, under which the return on each equity index is decomposed into two orthogonal jump-diffusion components: a global component and a country-specific component. We apply separate stochastic time changes to the two components so that stochastic volatility can come from both global and country-specific risks. For each economy, we assign separate market prices for the two return risk components and the two volatility risk components. Under this specification, we obtain tractable option pricing solutions. Model estimation reveals several interesting insights. First, global and country-specific return and volatility risks show different dynamics. Global return movements contain a larger discontinuous component, and global return volatility is more persistent than the country-specific counterparts. Second, investors charge positive prices for global return risk and negative prices for volatility risk, suggesting that investors are willing to pay positive premiums to hedge against downside global return movements and upside volatility movements. Third, the three economies contain different risk profiles and also price risks differently. Japan contains the largest idiosyncratic risk component and smallest global risk component. Investors in the Japanese market also price more heavily against future volatility increases than against future market downfalls.     

The Impact of Jumps in Volatility and Returns

This paper examines continuous‐time stochastic volatility models incorporating jumps in returns and volatility. We develop a likelihood‐based estimation strategy and provide estimates of parameters, spot volatility, jump times, and jump sizes using S&P 500 and Nasdaq 100 index returns. Estimates of jump times, jump sizes, and volatility are particularly useful for identifying the effects of these factors during periods of market stress, such as those in 1987, 1997, and 1998. Using formal and informal diagnostics, we find strong evidence for jumps in volatility and jumps in returns. Finally, we study how these factors and estimation risk impact option pricing.     

An empirical examination of jump risk in asset pricing and volatility forecasting in China's equity and bond markets

Understanding jump risk is important in risk management and option pricing. This study examines the characteristics of jump risk and the volatility forecasting power of the jump component in a panel of high-frequency intraday stock returns and four index returns from Shanghai Stock Exchange. Across portfolio indexes, jump returns on average account for 45% to 64% of total returns when jumps occur. Market systematic jump risk is an important pricing factor for daily returns. The average jump beta is 62% of the average continuous beta for individual stocks. However, the contribution of jump risk to total risk is limited, indicating that statistically significant jumps in the stochastic process of asset price are rare events but have tremendous impacts on the prices of common stocks in China. We further document that accounting for jump components improves the performance of volatility forecasting for some equity and bond portfolios in China, which is confirmed by in-the-sample and out-of-sample forecasting performance analysis.     

Index Option Pricing Models with Stochastic Volatility and Stochastic Interest Rates

This paper specifies a multivariate stochasticvolatility (SV) model for the S & P500 index and spot interest rateprocesses. We first estimate the multivariate SV model via theefficient method of moments (EMM) technique based on observations ofunderlying state variables, and then investigate the respective effects of stochastic interest rates, stochastic volatility, and asymmetric S & P500 index returns on option prices. We compute option prices using both reprojected underlying historical volatilities and the implied risk premiumof stochastic volatility to gauge each model's performance through direct comparison with observed market option prices on the index. Our major empirical findings are summarized as follows. First, while allowing for stochastic volatility can reduce the pricing errors and allowing for asymmetric volatility or leverage effect does help to explain the skewness of the volatility smile, allowing for stochastic interest rates has minimal impact on option prices in our case. Second, similar to Melino and Turnbull (1990), our empirical findings strongly suggest the existence of a non-zero risk premium for stochastic volatility of asset returns. Based on the implied volatility risk premium, the SV models can largely reduce the option pricing errors, suggesting the importance of incorporating the information from the options market in pricing options. Finally, both the model diagnostics and option pricing errors in our study suggest that the Gaussian SV model is not sufficientin modeling short-term kurtosis of asset returns, an SV model withfatter-tailed noise or jump component may have better explanatory power.     

Empirical pricing kernels

This paper investigates the empirical characteristics of investor risk aversion over equity return states by estimating a time-varying pricing kernel, which we call the empirical pricing kernel (EPK). We estimate the EPK on a monthly basis from 1991 to 1995, using S&P 500 index option data and a stochastic volatility model for the S&P 500 return process. We find that the EPK exhibits counter cyclical risk aversion over S&P 500 return states. We also find that hedging performance is significantly improved when we use hedge ratios based the EPK rather than a time-invariant pricing kernel.     

A tale of two regimes: Theory and empirical evidence for a Markov-modulated jump diffusion model of equity returns and derivative pricing implications

We provide closed-form solutions for a continuous time, Markov-modulated jump diffusion model in a general equilibrium framework for options prices under a variety of jump diffusion specifications. We further demonstrate that the two-state model provides the leptokurtic return features, volatility smile, and volatility clustering observed empirically for the Dow Jones Industrial Average (DJIA) and its component stocks. Using 10 years of stock return data, we confirm the existence of jump intensity switching and clustering, illustrate transition probabilities, and verify superior empirical fit over competing Poisson-style models.     

The world price of jump and volatility risk

We study international integration of markets for jump and volatility risk, using index option data for the main global markets. To explain the cross-section of expected option returns we focus on return-based multi-factor models. For each market separately, we provide evidence that volatility and jump risk are priced risk factors. There is little evidence, however, of global unconditional pricing of these risks. We show that UK and US option markets have become increasingly interrelated, and using conditional pricing models generates some evidence of international pricing. Finally, the benefits of diversifying jump and volatility risk internationally are substantial, but declining.     

Realizing smiles: Options pricing with realized volatility

We develop a discrete-time stochastic volatility option pricing model exploiting the information contained in the Realized Volatility (RV), which is used as a proxy of the unobservable log-return volatility. We model the RV dynamics by a simple and effective long-memory process, whose parameters can be easily estimated using historical data. Assuming an exponentially affine stochastic discount factor, we obtain a fully analytic change of measure. An empirical analysis of Standard and Poor's 500 index options illustrates that our model outperforms competing time-varying and stochastic volatility option pricing models.     

Lévy processes driven by stochastic volatility

In this paper we extend option pricing under Lévy dynamics, by assuming that the volatility of the Lévy process is stochastic. We, therefore, develop the analog of the standard stochastic volatility models, when the underlying process is not a standard (unit variance) Brownian motion, but rather a standardized Lévy process. We present a methodology that allows one to compute option prices, under virtually any set of diffusive dynamics for the parameters of the volatility process. First, we use ‘local consistency’ arguments to approximate the volatility process with a finite, but sufficiently dense Markov chain; we then use this regime switching approximation to efficiently compute option prices using Fourier inversion. A detailed example, based on a generalization of the popular stochastic volatility model of Heston (Rev Financial Stud 6 (1993) 327), is used to illustrate the implementation of the algorithms. Computer code is available at www.theponytail.net/     

Empirical Performance of Alternative Option Pricing Models

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. We fill this gap by first deriving an option model that allows volatility, interest rates and jumps to be stochastic. Using S&P 500 options, we 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.     

The affine styled-facts price dynamics for the natural gas: evidence from daily returns and option prices

This study analyzes affine styled-facts price dynamics of Henry Hub natural gas price by incorporating the price features of jump risk, and seasonality within stochastic volatility framework. Affine styled-facts dynamics has the advantage of being able to incorporate mean reversion (MR), stochastic volatility (SV), seasonality trends (S), and jump diffusion (J) in a standardized inclusive framework. Our main finding is that models that incorporate jumps significantly improve overall out-of-sample option pricing performance. The combined MRSVJS model provides the best fit of both daily gas price returns and the related cross section of option prices. Incorporating seasonal effects tend to provide more stable pricing ability, especially for the long-term option contracts.     

News Arrival, Jump Dynamics, and Volatility Components for Individual Stock Returns

This paper models components of the return distribution, which are assumed to be directed by a latent news process. The conditional variance of returns is a combination of jumps and smoothly changing components. A heterogeneous Poisson process with a time‐varying conditional intensity parameter governs the likelihood of jumps. Unlike typical jump models with stochastic volatility, previous realizations of both jump and normal innovations can feed back asymmetrically into expected volatility. This model improves forecasts of volatility, particularly after large changes in stock returns. We provide empirical evidence of the impact and feedback effects of jump versus normal return innovations, leverage effects, and the time‐series dynamics of jump clustering.     

Estimation and pricing under long-memory stochastic volatility

We treat the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. We model the price process as a Geometric Brownian Motion with volatility evolving as a fractional Ornstein–Uhlenbeck process. We assume that the model has long-memory, thus the memory parameter H in the volatility is greater than 0.5. Although the price process evolves in continuous time, the reality is that observations can only be collected in discrete time. Using historical stock price information we adapt an interacting particle stochastic filtering algorithm to estimate the stochastic volatility empirical distribution. In order to deal with the pricing problem we construct a multinomial recombining tree using sampled values of the volatility from the stochastic volatility empirical measure. Moreover, we describe how to estimate the parameters of our model, including the long-memory parameter of the fractional Brownian motion that drives the volatility process using an implied method. Finally, we compute option prices on the S&P 500 index and we compare our estimated prices with the market option prices.     

Option pricing under stochastic volatility and tempered stable Lévy jumps

The purpose of this paper is to introduce a stochastic volatility model for option pricing that exhibits Lévy jump behavior. For this model, we derive the general formula for a European call option. A well known particular case of this class of models is the Bates model, for which the jumps are modeled by a compound Poisson process with normally distributed jumps. Alternatively, we turn our attention to infinite activity jumps produced by a tempered stable process. Then we empirically compare the estimated log-return probability density and the option prices produced from this model to both the Bates model and the Black–Scholes model. We find that the tempered stable jumps describe more precisely market prices than compound Poisson jumps assumed in the Bates model.     

Jump-diffusion volatility models for variance swaps: An empirical performance analysis

This paper studies a class of tractable jump-diffusion models, including stochastic volatility models with various specifications of jump intensity for stock returns and variance processes. We employ the Markov chain Monte Carlo (MCMC) method to implement model estimation, and investigate the performance of all models in capturing the term structure of variance swap rates and fitting the dynamics of stock returns. It is evident that the stochastic volatility models, equipped with self-exciting jumps in the spot variance and linearly-dependent jumps in the central-tendency variance, can produce consistent model estimates, aptly explain the stylized facts in variance swaps, and boost pricing performance. Moreover, our empirical results show that large self-exciting jumps in the spot variance, as an independent risk source, facilitate term structure modeling for variance swaps, whilst the central-tendency variance may jump with small sizes, but signaling substantial regime changes in the long run. Both types of jumps occur infrequently, and are more related to market turmoils over the period from 2008 to 2021.     

Expected returns, risk premia, and volatility surfaces implicit in option market prices

This article presents a pure exchange economy that extends Rubinstein (1976) to show how the jump-diffusion option pricing model of Merton (1976) is altered when jumps are correlated with diffusive risks. A non-zero correlation between jumps and diffusive risks is necessary in order to resolve the positively sloped implied volatility term structure inherent in traditional jump diffusion models. Our evidence is consistent with a negative covariance, producing a non-monotonic term structure. For the proposed market structure, we present a closed form asset pricing model that depends on the factors of the traditional jump-diffusion models, and on both the covariance of the diffusive pricing kernel with price jumps and the covariance of the jumps of the pricing kernel with the diffusive price. We present statistical evidence that these covariances are positive. For our model the expected stock return, jump and diffusive risk premiums are non-linear functions of time.     

基于分数维随机利率的分数跳-扩散外汇期权定价

假设利率为分数维随机利率,外汇汇率服从分数跳一扩散过程,并且波动率为常数,期望收益率为时间的非随机函数,本文利用保险精算方法,得出了看涨、看跌外汇欧武期权的一般定价公式,并建立了平价公式。     

The price of a smile: hedging and spanning in option markets

The volatility smile changed drastically around the crash of1987, and new option pricing models have been proposed to accommodatethat change. Deterministic volatility models allow for moreflexible volatility surfaces but refrain from introducing additionalrisk factors. Thus, options are still redundant securities.Alternatively, stochastic models introduce additional risk factors,and options are then needed for spanning of the pricing kernel.We develop a statistical test based on this difference in spanning.Using daily S&P 500 index options data from 1986-1995, ourtests suggest that both in- and out-of-the-money options areneeded for spanning. The findings are inconsistent with deterministicvolatility models but are consistent with stochastic modelsthat incorporate additional priced risk factors, such as stochasticvolatility, interest rates, or jumps.