Volatility and jump risk in option returns

We examine the importance of volatility and jump risk in the time-series prediction of S&P 500 index option returns. The empirical analysis provides a different result between call and put option returns. Both volatility and jump risk are important predictors of put option returns. In contrast, only volatility risk is consistently significant in the prediction of call option returns over the sample period. The empirical results support the theory that there is option risk premium associated with volatility and jump risk, and reflect the asymmetry property of S&P 500 index distribution.     

Jump variance risk: Evidence from option valuation and stock returns

We study jump variance risk by jointly examining both stock and option markets. We develop a GARCH option pricing model with jump variance dynamics and a nonmonotonic pricing kernel featuring jump variance risk premium. The model yields a closed-form option pricing formula and improves in fitting index options from 1996 to 2015. The model-implied jump variance risk premium has predictive power for future market returns. In the cross-section, heterogeneity in exposures to jump variance risk leads to a 6% difference in risk-adjusted returns annually.     

MCMC ESTIMATION OF LÉVY JUMP MODELS USING STOCK AND OPTION PRICES

We examine the performances of several popular Lévy jump models and some of the most sophisticated affine jump‐diffusion models in capturing the joint dynamics of stock and option prices. We develop efficient Markov chain Monte Carlo methods for estimating parameters and latent volatility/jump variables of the Lévy jump models using stock and option prices. We show that models with infinite‐activity Lévy jumps in returns significantly outperform affine jump‐diffusion models with compound Poisson jumps in returns and volatility in capturing both the physical and risk‐neutral dynamics of the S&P 500 index. We also find that the variance gamma model of Madan, Carr, and Chang with stochastic volatility has the best performance among all the models we consider.     

EQUILIBRIUM ASSET AND OPTION PRICING UNDER JUMP DIFFUSION

This paper develops an equilibrium asset and option pricing model in a production economy under jump diffusion. The model provides analytical formulas for an equity premium and a more general pricing kernel that links the physical and risk‐neutral densities. The model explains the two empirical phenomena of the negative variance risk premium and implied volatility smirk if market crashes are expected. Model estimation with the S&P 500 index from 1985 to 2005 shows that jump size is indeed negative and the risk aversion coefficient has a reasonable value when taking the jump into account.     

Oil jump risk

The risk premium associated with large upside jumps in oil market is a significant driver of the cross-section of stock returns from 1986 to 2014. In contrast to previous research, variance risk is priced only when we do not control for jumps. Upward jumps are priced in tight supply-demand conditions but not in more abundant supply periods. There is some evidence that downward jumps are priced in abundant supply conditions but not in tight conditions. Innovations in risk neutral jumps have predictive power for important economic indicators, including notably consumption growth. This helps explain the pricing of jump risks.     

Incorporating time-varying jump intensities in the mean-variance portfolio decisions

This paper examines the role of time-varying jump intensities in forming mean-variance portfolios. We find that compared with the no-jump or constant-jump models, the model which incorporates time-varying jump intensities better fits the dynamics of the assets returns, and yields mean-variance portfolios with higher Sharpe ratios. Our research suggests that using a better econometric model that captures non-normal features in the data has benefits for portfolio allocation even for a mean-variance investor.     

A generalization of the Barone‐Adesi and Whaley approach for the analytic approximation of American options

This article introduces a general quadratic approximation scheme for pricing American options based on stochastic volatility and double jump processes. This quadratic approximation scheme is a generalization of the Barone‐Adesi and Whaley approach and nests several option models. Numerical results show that this quadratic approximation scheme is efficient and useful in pricing American options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:478–493, 2009     

Market Excess Returns,Variance and the Third Cumulant

In this paper, we develop an equilibrium asset pricing model for market excess returns, variance and the third cumulant by using a jump‐diffusion process with stochastic variance and jump intensity in Cox et al. (1985) production economy. Empirical evidence with the S&P 500 index and options from January, 1996 to December, 2005 strongly supports our model prediction that the lower the third cumulant, the higher the market excess returns. Consistent with existing literature, the theoretical mean–variance relation is supported only by regressions on risk‐neutral variance. We further demonstrate empirically that the third cumulant explains significantly the variance risk premium.     

Closed‐form option pricing formulas with extreme events

This paper explores the effect of extreme events or big jumps downwards and upwards on the jump‐diffusion option pricing model of Merton (1976). It starts by obtaining a special case of the jump‐diffusion model where there is a positive probability of a big jump downwards. Then, it obtains a new limiting case where there is an asymptotically big jump upwards. The paper extends these models to allow both types of jumps. In some cases these formulas nest Samuelson's (1965) formulas. This simple analysis leads to several closed‐form solutions for calls and puts, which are able to generate smiles, and skews with similar shapes to those observed in the marketplace. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:213–230, 2008     

A NEW LOOK AT SHORT‐TERM IMPLIED VOLATILITY IN ASSET PRICE MODELS WITH JUMPS

We analyze the behavior of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalization of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal–Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short‐end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model‐independent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on short‐maturity option prices.     

Pricing American exchange options in a jump‐diffusion model

A way to estimate the value of an American exchange option when the underlying assets follow jump‐diffusion processes is presented. The estimate is based on combining a European exchange option and a Bermudan exchange option with two exercise dates by using Richardson extrapolation as proposed by R. Geske and H. Johnson (1984). Closed‐form solutions for the values of European and Bermudan exchange options are derived. Several numerical examples are presented, illustrating that the early exercise feature may have a significant economic value. The results presented should have potential for pricing over‐the‐counter options and in particular for pricing real options. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:257–273, 2007     

Pricing Defaultable Bonds Using a Lévy Jump‐Diffusion Model

This paper uses a reduced‐form approach to derive a closed‐form pricing formula for defaultable bonds. The authors specify the default hazard rate as an affine function of multiple variables which follow the Lévy jump‐diffusion processes. Because such specification allows greater flexibility in the generation of a valid probability of default, their pricing model should be more accurate than the valuation models in traditional studies, which ignore the jump effects. This paper also proposes a new method for estimating the parameters in a Lévy Jump‐diffusion process. The real data from the Taiwanese bond market are used to illustrate how their model can be applied in practical situations. The authors compare the pricing results for the influential variables with no jump effects, with jump magnitudes following the normal distribution, and with jump magnitudes following the gamma distribution. The results reveal that the predictive ability is the best for the model with the jump components. The valuation model shown in this paper should help portfolio managers more accurately price defaultable bonds and more effectively hedge their portfolio holdings.     

The valuation of European options when asset returns are autocorrelated

This article derives the closed‐form formula for a European option on an asset with returns following a continuous‐time type of first‐order moving average process, which is called an MA(1)‐type option. The pricing formula of these options is similar to that of Black and Scholes, except for the total volatility input. Specifically, the total volatility input of MA(1)‐type options is the conditional standard deviation of continuous‐compounded returns over the option's remaining life, whereas the total volatility input of Black and Scholes is indeed the diffusion coefficient of a geometric Brownian motion times the square root of an option's time to maturity. Based on the result of numerical analyses, the impact of autocorrelation induced by the MA(1)‐type process is significant to option values even when the autocorrelation between asset returns is weak. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:85–102, 2006     

APPROXIMATING GARCH-JUMP MODELS, JUMP-DIFFUSION PROCESSES, AND OPTION PRICING

This paper considers the pricing of options when there are jumps in the pricing kernel and correlated jumps in asset prices and volatilities. We extend theory developed by Nelson (1990) and Duan (1997) by considering the limiting models for our approximating GARCH Jump process. Limiting cases of our processes consist of models where both asset price and local volatility follow jump diffusion processes with correlated jump sizes. Convergence of a few GARCH models to their continuous time limits is evaluated and the benefits of the models explored.     

The Pricing Kernel Puzzle: A Real Phenomenon or a Statistical Artifact?

A large literature finds evidence that the pricing kernels estimated from option prices and historical returns are not monotonically decreasing in market returns. We show that the inevitable coalescence of contingencies, especially in the left‐tail, associated with estimating a distribution function may give rise to a nonmonotonic empirical pricing kernel even if the actual pricing kernel is monotonic. Hence, the observed nonmonotonicity of pricing kernels may be a statistical artifact rather than a real phenomenon. We argue that empirical work should explicitly correct for this effect by widening the option pricing bounds associated with monotonic pricing kernels.     

Option pricing with a non‐zero lower bound on stock price

Black, F. and Scholes, M. (1973) assume a geometric Brownian motion for stock prices and therefore a normal distribution for stock returns. In this article a simple alternative model to Black and Scholes (1973) is presented by assuming a non‐zero lower bound on stock prices. The proposed stock price dynamics simultaneously accommodate skewness and excess kurtosis in stock returns. The feasibility of the proposed model is assessed by simulation and maximum likelihood estimation of the return probability density. The proposed model is easily applicable to existing option pricing models and may provide improved precision in option pricing. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:775–794, 2005     

Time‐varying jump risk premia in stock index futures returns

This study tests the presence of time‐varying risk premia associated with extreme news events or jumps in stock index futures return. The model allows for a dynamic jump component with autoregressive jump intensity, long‐range dependence in volatility dynamics, and a volatility in mean structure separately for the normal and extreme news events. The results show significant jump risk premia in four stock market index futures returns including the DAX, FTSE, Nikkei, and S&P500 indices. Our results are robust to various specifications of conditional variance including the plain GARCH, component GARCH, and Fractionally Integrated GARCH models. We also find the time‐varying risk premium associated with normal news events is not significant across all indices. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 32:639–659, 2012     

CONTINUOUSLY MONITORED BARRIER OPTIONS UNDER MARKOV PROCESSES

In this paper, we present an algorithm for pricing barrier options in one‐dimensional Markov models. The approach rests on the construction of an approximating continuous‐time Markov chain that closely follows the dynamics of the given Markov model. We illustrate the method by implementing it for a range of models, including a local Lévy process and a local volatility jump‐diffusion. We also provide a convergence proof and error estimates for this algorithm.     

Volatility options: Hedging effectiveness,pricing, and model error

Motivated by the growing literature on volatility options and their imminent introduction in major exchanges, this article addresses two issues. First, the question of whether volatility options are superior to standard options in terms of hedging volatility risk is examined. Second, the comparative pricing and hedging performance of various volatility option pricing models in the presence of model error is investigated. Monte Carlo simulations within a stochastic volatility setup are employed to address these questions. Alternative dynamic hedging schemes are compared, and various option‐pricing models are considered. It is found that volatility options are not better hedging instruments than plain‐vanilla options. Furthermore, the most naïve volatility option‐pricing model can be reliably used for pricing and hedging purposes. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:1–31, 2006     

EQUITY CORRELATIONS IMPLIED BY INDEX OPTIONS: ESTIMATION AND MODEL UNCERTAINTY ANALYSIS

We propose a method for constructing an arbitrage‐free multiasset pricing model which is consistent with a set of observed single‐ and multiasset derivative prices. The pricing model is constructed as a random mixture of N reference models, where the distribution of mixture weights is obtained by solving a well‐posed convex optimization problem. Application of this method to equity and index options shows that, whereas multivariate diffusion models with constant correlation fail to match the prices of index and component options simultaneously, a jump‐diffusion model with a common jump component affecting all stocks enables to do so. Furthermore, we show that even within a parametric model class, there is a wide range of correlation patterns compatible with observed prices of index options. Our method allows, as a by product, to quantify this model uncertainty with no further computational effort and propose static hedging strategies for reducing the exposure of multiasset derivatives to model uncertainty.