Automatic variance ratio test under conditional heteroskedasticity

An extensive Monte Carlo experiment is conducted to evaluate small sample properties of the automatic variance ratio test under conditional heteroskedasticity. It is found that the test shows serious size distortion in small samples. For improved small sample performance, this paper proposes the use of wild bootstrap. When wild bootstrapped, the automatic variance ratio test shows no size distortion, and it has power substantially higher than its competitors such as the Chen–Deo test and wild bootstrap Chow–Denning test.     

Option pricing under regime switching

Abstract

This paper develops a family of option pricing models when the underlying stock price dynamic is modelled by a regime switching process in which prices remain in one volatility regime for a random amount of time before switching over into a new regime. Our family includes the regime switching models of Hamilton (Hamilton J 1989 Econometrica 57 357–84), in which volatility influences returns. In addition, our models allow for feedback effects from returns to volatilities. Our family also includes GARCH option models as a special limiting case. Our models are more general than GARCH models in that our variance updating schemes do not only depend on levels of volatility and asset innovations, but also allow for a second factor that is orthogonal to asset innovations. The underlying processes in our family capture the asymmetric response of volatility to good and bad news and thus permit negative (or positive) correlation between returns and volatility. We provide the theory for pricing options under such processes, present an analytical solution for the special case where returns provide no feedback to volatility levels, and develop an efficient algorithm for the computation of American option prices for the general case.     

Option pricing under a stressed-beta model

Empirical studies have concluded that stochastic volatility is an important component of option prices. We introduce a regime-switching mechanism into a continuous-time Capital Asset Pricing Model which naturally induces stochastic volatility in the asset price. Under this Stressed-Beta model, the mechanism is relatively simple: the slope coefficient—which measures asset returns relative to market returns—switches between two values, depending on the market being above or below a given level. After specifying the model, we use it to price European options on the asset. Interestingly, these option prices are given explicitly as integrals with respect to known densities. We find that the model is able to produce a volatility skew, which is a prominent feature in option markets. This opens the possibility of forward-looking calibration of the slope coefficients, using option data, as illustrated in the paper.     

Option pricing and Esscher transform under regime switching

Summary We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).We would like to thank the referees for many helpful and insightful comments and suggestions.Correspondence to: R. J. Elliott     

Option pricing under fast-varying and rough stochastic volatility

Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatility asymptotics and their interrelation. The regimes discussed there include fast and slow volatility factors with strong or small volatility fluctuations and with the limits not commuting in general. The notion of a characteristic term structure exponent is introduced, this exponent governs the implied volatility term structure in the various asymptotic regimes.     

Option pricing under a Gamma-modulated diffusion process

We study a Gamma-modulated diffusion process as a long-memory generalization of the standard Black-Scholes model. This model introduces a time dependent volatility. The option pricing problem associated with this type of processes is computed.     

Option pricing under short-lived arbitrage: theory and tests

Models in financial economics derived from no-arbitrage assumptions have found great favour among theoreticians and practitioners. We develop a model of option prices where arbitrage is short lived. The arbitrage process is Ornstein–Uhlenbeck with zero mean and rapid adjustment of deviations. We find that arbitrage correlated with the underlying can have sizeable impact on option prices. We use data from five large capitalization firms to test implications of the model. Consistent with the existence of arbitrage, we find that idiosyncratic factors significantly effect arbitrage model parameters.     

Option pricing under non-normality: a comparative analysis

This paper carries out a comparative analysis of the calibration and performance of a variety of options pricing models. These include Black and Scholes (J Polit Econ 81:637–659, 1973), the Gram–Charlier (GC) approach of Backus et al. (1997), the stochastic volatility (HS) model of Heston (Rev Financ Stud 6:327–343, 1993), the closed-form GARCH process of Heston and Nandi (Rev Financ Stud 13:585–625, 2000) and a variety of Lévy processes including the Variance Gamma (VG), Normal Inverse Gaussian (NIG), and, CGMY and Kou (Manag Sci 48:1086–1101, 2002) jump-diffusion models. Unlike most studies of option pricing, we compare these models using a common point-in-time data which reflects the perspective of a new investor who wishes to choose between models using only the most minimal recent data set. For each of these models, we also examine the accuracy of delta and delta-gamma approximations to the valuation of both individual options and an illustrative option portfolio.     

Distributional dynamics under smoothly state-dependent pricing

Starting from the assumption that firms are more likely to adjust their prices when doing so is more valuable, this paper analyzes monetary policy shocks in a DSGE model with firm-level heterogeneity. The model is calibrated to retail price microdata, and inflation responses are decomposed into “intensive”, “extensive”, and “selection” margins. Money growth and Taylor rule shocks both have nontrivial real effects, because the low state dependence implied by the data rules out the strong selection effect associated with fixed menu costs. The response to sector-specific shocks is gradual, but inappropriate econometrics might make it appear immediate.     

Option pricing under time-varying risk-aversion with applications to risk forecasting

We present a two-factor option-pricing model, which parsimoniously captures the difference in volatility persistences under the historical and risk-neutral probabilities. The model generates an S-shaped pricing kernel that exhibits time-varying risk aversion. We apply our model for two purposes. First, we analyze the risk preference implied by S&P500 index options during 2001–2009 and find that risk-aversion level strongly increases during stressed market conditions. Second, we apply our model for Value-at-Risk (VaR) forecasts during the subprime crisis period and find that it outperforms several leading VaR models.     

Option pricing and the martingale restriction

In the absence of frictions, the value of the under-lying assetimplied by option prices must equal its actual market value.With frictions, however, this requirement need not hold. UsingS&P 100 index options data, I find that the implied costof the index is significantly higher in the options market thanin the stock market, and is directly related to measures oftransaction costs and liquidity. I show that the Black-Scholesmodel has strong bid-ask spread, trading volume, and open interestbiases. Option pricing models that relax the martingale restrictionperform significantly better.     

The cross-sectional relation between conditional heteroskedasticity,the implied volatility smile,and the variance risk premium

This paper estimates how the shape of the implied volatility smile and the size of the variance risk premium relate to parameters of GARCH-type time-series models measuring how conditional volatility responds to return shocks. Markets in which return shocks lead to large increases in conditional volatility tend to have larger variance risk premia than markets in which the impact on conditional volatility is slight. Markets in which negative (positive) return shocks lead to larger increases in future volatility than positive (negative) return shocks tend to have downward (upward) sloping implied volatility smiles. Also, differences in how volatility responds to return shocks as measured by GARCH-type models explain much, but not all, of the variations in excess kurtosis and multi-period skewness across different markets.     

The role of autoregressive conditional skewness and kurtosis in the estimation of conditional VaR

This paper investigates the role of high-order moments in the estimation of conditional value at risk (VaR). We use the skewed generalized t distribution (SGT) with time-varying parameters to provide an accurate characterization of the tails of the standardized return distribution. We allow the high-order moments of the SGT density to depend on the past information set, and hence relax the conventional assumption in conditional VaR calculation that the distribution of standardized returns is iid. The maximum likelihood estimates show that the time-varying conditional volatility, skewness, tail-thickness, and peakedness parameters of the SGT density are statistically significant. The in-sample and out-of-sample performance results indicate that the conditional SGT-GARCH approach with autoregressive conditional skewness and kurtosis provides very accurate and robust estimates of the actual VaR thresholds.     

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.     

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.     

On the conditional pricing effects of beta,size, and book-to-market equity in the Hong Kong market

Using Hong Kong equity stock data, this study examines empirically the pricing effects of beta, firm size, and book-to-market equity, but conditional on market situations, i.e. whether the market is up or down. Evidence supports the hypothesis that, if the risk variable is priced by the market, then there exists a systematic but conditional relation between the risk variable and average return, and this relation takes on opposite directions during up and down markets. However, the significance of the relations is often affected by the changing values of the risk variables as a result of changes in market conditions. Specifically, it is found that all three risk variables, namely beta, size, and book-to-market equity, exhibit conditional pricing effects. This is the first comprehensive study of its kind on Hong Kong market, which provides out-of-sample evidence relative to earlier tests on US data. The findings give important insights into capital market behaviour, which should prove useful in investment management and corporate financial decisions.     

Option pricing under stock market cycles with jump risks: evidence from the S&P 500 index

Review of Quantitative Finance and Accounting - This study incorporates the Markov switching model with return jumps to depict the behavior of stock returns. Based on the daily Standard &...