The Price of Correlation Risk: Evidence from Equity Options

We study whether exposure to marketwide correlation shocks affects expected option returns, using data on S&P100 index options, options on all components, and stock returns. We find evidence of priced correlation risk based on prices of index and individual variance risk. A trading strategy exploiting priced correlation risk generates a high alpha and is attractive for CRRA investors without frictions. Correlation risk exposure explains the cross-section of index and individual option returns well. The correlation risk premium cannot be exploited with realistic trading frictions, providing a limits-to-arbitrage interpretation of our finding of a high price of correlation risk.     

Cross-sectional tests of deterministic volatility functions

We study the cross-sectional performance of option pricing models in which the volatility of the underlying stock is a deterministic function of the stock price and time. For each date in our sample of FTSE 100 index option prices, we fit an implied binomial tree to the panel of all European style options with different strike prices and maturities and then examine how well this model prices a corresponding panel of American style options. We find that the implied binomial tree model performs no better than an ad-hoc procedure of smoothing Black–Scholes implied volatilities across strike prices and maturities. Our cross-sectional results complement the time-series findings of Dumas et al. [J. Finance 53 (1998) 2059].     

A dynamic equilibrium model for U-shaped pricing kernels

This paper proposes a dynamic equilibrium model that can provide a unified explanation for the stylized facts observed in stock index markets such as the fat tails of the risk-neutral return distribution relative to the physical distribution, negative expected returns on deep OTM call options and negative realized variance risk premiums. In particular, we focus on the U-shaped pricing kernel against the stock index return, which is closely related to the negative call returns. We assume that the stock index return follows a time-changed Lévy process and that a representative investor has power utility over the aggregate consumption that forms a linear regression of the stock index return and its stochastic activity rate. This model offers a macroeconomic interpretation of the stylized facts from the perspective of the sensitivity of the activity rate and stock index return on aggregate consumption as well as the investor’s risk aversion.     

Implied volatility skews and stock return skewness and kurtosis implied by stock option prices

The Black-Scholes* option pricing model is commonly applied to value a wide range of option contracts. However, the model often inconsistently prices deep in-the-money and deep out-of-the-money options. Options professionals refer to this well-known phenomenon as a volatility ‘skew’ or ‘smile’. In this paper, we examine an extension of the Black-Scholes model developed by Corrado and Su that suggests skewness and kurtosis in the option-implied distributions of stock returns as the source of volatility skews. Adapting their methodology, we estimate option-implied coefficients of skewness and kurtosis for four actively traded stock options. We find significantly nonnormal skewness and kurtosis in the option-implied distributions of stock returns.     

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 information content of implied skewness and kurtosis changes prior to earnings announcements for stock and option returns

We use option prices to examine whether changes in stock return skewness and kurtosis preceding earnings announcements provide information about subsequent stock and option returns. We demonstrate that changes in jump risk premiums can lead to changes in implied skewness and kurtosis and are also associated with the mean and variability of the stock price response to the earnings announcement. We find that changes in both moments have strong predictive power for future stock returns, even after controlling for implied volatility. Additionally, changes in both moments predict call returns, while put return predictability is primarily linked to changes in skewness.     

Evaluation of Black-Scholes and GARCH Models Using Currency Call Options Data

This paper empirically examines the performance of Black-Scholes and Garch-M call option pricing models using call options data for British Pounds, Swiss Francs and Japanese Yen. The daily exchange rates exhibit an overwhelming presence of volatility clustering, suggesting that a richer model with ARCH/GARCH effects might have a better fit with actual prices. We perform dominant tests and calculate average percent mean squared errors of model prices. Our findings indicate that the Black-Scholes model outperforms the GARCH models. An implication of this result is that participants in the currency call options market do not seem to price volatility clusters in the underlying process.     

A lattice model for option pricing under GARCH-jump processes

This study extends the GARCH pricing tree in Ritchken and Trevor (J Financ 54:366–402, 1999) by incorporating an additional jump process to develop a lattice model to value options. The GARCH-jump model can capture the behavior of asset prices more appropriately given its consistency with abundant empirical findings that discontinuities in the sample path of financial asset prices still being found even allowing for autoregressive conditional heteroskedasticity. With our lattice model, it shows that both the GARCH and jump effects in the GARCH-jump model are negative for near-the-money options, while positive for in-the-money and out-of-the-money options. In addition, even when the GARCH model is considered, the jump process impedes the early exercise and thus reduces the percentage of the early exercise premium of American options, particularly for shorter-term horizons. Moreover, the interaction between the GARCH and jump processes can raise the percentage proportions of the early exercise premiums for shorter-term horizons, whereas this effect weakens when the time to maturity increases.     

Stochastic risk premiums,stochastic skewness in currency options,and stochastic discount factors in international economies

We develop models of stochastic discount factors in international economies that produce stochastic risk premiums and stochastic skewness in currency options. We estimate the models using time-series returns and option prices on three currency pairs that form a triangular relation. Estimation shows that the average risk premium in Japan is larger than that in the US or the UK, the global risk premium is more persistent and volatile than the country-specific risk premiums, and investors respond differently to different shocks. We also identify high-frequency jumps in each economy but find that only downside jumps are priced. Finally, our analysis shows that the risk premiums are economically compatible with movements in stock and bond market fundamentals.     

The Effect of Executive Stock Option Plans on Stockholders and Bondholders

Executive stock option plans have asymmetric payoffs that could induce managers to take on more risk. Evidence from traded call options and stock return data supports this notion. Implicit share price variance, computed from the Black-Scholes option pricing model, and stock return variance increase after the approval of an executive stock option plan. The event is accompanied by a significant positive stock and a negative bond market reaction. This evidence is consistent with the notion that executive stock options may induce a wealth transfer from bondholders to stockholders.     

Pricing Reinsurance Contracts on FDIC Losses

This paper proposes a pricing model for the FDIC's reinsurance risk. We derive a closed‐form Weibull call option pricing model to price a call‐spread a reinsurer might sell to the FDIC. To obtain the risk‐neutral loss‐density necessary to price this call spread we risk‐neutralize a Weibull distributed FDIC annual losses by a tilting coefficient estimated from the traded call options on the BKX index. An application of the proposed approach yield reasonable reinsurance prices.     

Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices

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.     

Valuation of American call options on dividend-paying stocks: Empirical tests

This paper examines the pricing performance of the valuation equation for American call options on stocks with known dividends and compares it with two suggested approximation methods. The approximation obtained by substituting the stock price net of the present value of the escrowed dividends into the Black-Scholes model is shown to induce spurious correlation between prediction error and (1) the standard deviation of stock return, (2) the degree to which the option is in-the-money or out-of-the-money, (3) the probability of early exercise, (4) the time to expiration of the option, and (5) the dividend yield of the stock. A new method of examining option market efficiency is developed and tested.     

GARCH vs. stochastic volatility: Option pricing and risk management

In this paper we compare the out-of-sample performance of two common extensions of the Black–Scholes option pricing model, namely GARCH and stochastic volatility (SV). We calibrate the three models to intraday FTSE 100 option prices and apply two sets of performance criteria, namely out-of-sample valuation errors and Value-at-Risk (VaR) oriented measures. When we analyze the fit to observed prices, GARCH clearly dominates both SV and the benchmark Black–Scholes model. However, the predictions of the market risk from hypothetical derivative positions show sizable errors. The fit to the realized profits and losses is poor and there are no notable differences between the models. Overall, we therefore observe that the more complex option pricing models can improve on the Black–Scholes methodology only for the purpose of pricing, but not for the VaR forecasts.     

Asset pricing theory for two price economies

We show that nonlinearly discounted nonlinear martingales are related to no arbitrage in two price economies as linearly discounted martingales were related to no arbitrage in economies satisfying the law of one price. Furthermore, assuming risk acceptability requires a positive physical expectation, we demonstrate that expected rates of return on ask prices should be dominated by expected rates of return on bid prices. A preliminary investigation conducted here, supports this hypothesis. In general we observe that asset pricing theory in two price economies leads to asset pricing inequalities. A model incorporating both nonlinear discounting and nonlinear martingales is developed for the valuation of contingent claims in two price economies. Examples illustrate the interactions present between the severity of measure changes and their associated discount rates. As a consequence arbitrage free two price economies can involve unique discount curves and measure changes that are however specific to both the product being priced and the trade direction. Furthermore the developed valuation operators call into question the current practice of Debt Valuation Adjustments.     

American option pricing with discrete and continuous time models: An empirical comparison

This paper considers discrete time GARCH and continuous time SV models and uses these for American option pricing. We first of all show that with a particular choice of framework the parameters of the SV models can be estimated using simple maximum likelihood techniques. We then perform a Monte Carlo study to examine their differences in terms of option pricing, and we study the convergence of the discrete time option prices to their implied continuous time values. Finally, a large scale empirical analysis using individual stock options and options on an index is performed comparing the estimated prices from discrete time models to the corresponding continuous time model prices. The results show that, while the overall differences in performance are small, for the in the money put options on individual stocks the continuous time SV models do generally perform better than the discrete time GARCH specifications.     

The multivariate Variance Gamma model: basket option pricing and calibration

In this paper, we propose a methodology for pricing basket options in the multivariate Variance Gamma model introduced in Luciano and Schoutens [Quant. Finance 6(5), 385–402]. The stock prices composing the basket are modelled by time-changed geometric Brownian motions with a common Gamma subordinator. Using the additivity property of comonotonic stop-loss premiums together with Gauss-Laguerre polynomials, we express the basket option price as a linear combination of Black & Scholes prices. Furthermore, our new basket option pricing formula enables us to calibrate the multivariate VG model in a fast way. As an illustration, we show that even in the constrained situation where the pairwise correlations between the Brownian motions are assumed to be equal, the multivariate VG model can closely match the observed Dow Jones index options.     

Disasters Implied by Equity Index Options

We use equity index options to quantify the distribution of consumption growth disasters. The challenge lies in connecting the risk‐neutral distribution of equity returns implied by options to the true distribution of consumption growth. First, we compare pricing kernels constructed from macro‐finance and option‐pricing models. Second, we compare option prices derived from a macro‐finance model to those we observe. Third, we compare the distribution of consumption growth derived from option prices using a macro‐finance model to estimates based on macroeconomic data. All three perspectives suggest that options imply smaller probabilities of extreme outcomes than have been estimated from macroeconomic data.     

Nonparametric,conditional pricing of higher order multivariate contingent claims

This paper describes and applies a nonparametric model for pricing multivariate contingent claims. Multivariate contingent claims are contracts whose payoffs depend on the future prices of more than one underlying variable. The pricing however of these kinds of contracts represents a challenge. All known models are adaptations of earlier ones that have been introduced to price plain vanilla calls and puts. They are imposing strong assumptions on the distributional properties of the underlying variables. In contrast, this study adopts a methodology that relaxes such restrictions. Following [Barone-Adesi, G., Bourgoin, F., Giannopoulos, K., 1998. Don’t Look Back, Risk 11 (August), 100–104; Barone-Adesi, G., Engle, R., Mancini, L., 2004. GARCH Options in Incomplete Markets, mimeo, University of Applied Sciences of Southern Switzerland; Long, X., 2004. Semiparametric Multivariate GARCH Model, mimeo, University of California, Riverside], multivariate pathways for a set of underlying variables are constructed before the option payoffs are computed. This enables the covariances, in addition to the means and variances, to be modelled in a dynamic and nonparametric manner. The model is particular suitable for options whose payoffs depend on variables that are characterised by high nonlinearities and extremes and on higher order multivariate options whose underlying variables are more unlikely to conform to a common theoretical distribution.