A closed-form GARCH option valuation model

This paper develops a closed-form option valuation formula fora spot asset whose variance follows a GARCH(p, q) process thatcan be correlated with the returns of the spot asset. It providesthe first readily computed option formula for a random volatilitymodel that can be estimated and implemented solely on the basisof observables. The single lag version of this model containsHeston's (1993) stochastic volatility model as a continuous-timelimit. Empirical analysis on S&P500 index options showsthat the out-of-sample valuation errors from the single lagversion of the GARCH model are substantially lower than thead hoc Black-Scholes model of Dumas, Fleming and Whaley (1998)that uses a separate implied volatility for each option to fitto the smirk/smile in implied volatilities. The GARCH modelremains superior even though the parameters of the GARCH modelare held constant and volatility is filtered from the historyof asset prices while the ad hoc Black-Scholes model is updatedevery period. The improvement is largely due to the abilityof the GARCH model to simultaneously capture the correlationof volatility, with spot returns and the path dependence involatility.     

Dynamic Volatility Trading Strategies in the Currency Option Market

The conditional volatility of foreign exchange rates can be predicted using GARCH models or implied volatility extracted from currency options. This paper investigates whether these predictions are economically meaningful in trading strategies that are designed only to trade volatility risk. First, this article provides new evidence on the issue of information content of implied volatility and GARCH volatility in forecasting future variance. In an artificial world without transaction costs both delta-neutral and straddle trading stratgies lead to significant positive profits, regardless of which volatility prediction method is used. Specifically, the agent using the Implied Stochastic Volatility Regression method (ISVR) earns larger profits than the agent using the GARCH method. Second, it suggests that the currency options market is informationally efficient. After accounting for transaction costs, which are assumed to equal one percent of option prices, observed profits are not significantly differentfrom zero in most trading strategies. Finally, these strategies offered returns have higher Sharpe ratio and lower correlation with several major asset classes. Consequently, hedge funds and institutional investors who are seeking alternative “marketneutral” investment methods can use volatility trading to improvethe risk-return profile of their portfolio through diversification. This revised version was published online in November 2006 with corrections to the Cover Date.     

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.     

Estimating and using GARCH models with VIX data for option valuation

This paper uses information on VIX to improve the empirical performance of GARCH models for pricing options on the S&P 500. In pricing multiple cross-sections of options, the models’ performance can clearly be improved by extracting daily spot volatilities from the series of VIX rather than by linking spot volatility with different dates by using the series of the underlying’s returns. Moreover, in contrast to traditional returns-based Maximum Likelihood Estimation (MLE), a joint MLE with returns and VIX improves option pricing performance, and for NGARCH, joint MLE can yield empirically almost the same out-of-sample option pricing performance as direct calibration does to in-sample options, but without costly computations. Finally, consistently with the existing research, this paper finds that non-affine models clearly outperform affine models.     

Option pricing based on hybrid GARCH-type models with improved ensemble empirical mode decomposition

The exploration of option pricing is of great significance to risk management and investments. One important challenge to existing research is how to describe the underlying asset price process and fluctuation features accurately. Considering the benefits of ensemble empirical mode decomposition (EEMD) in depicting the fluctuation features of financial time series, we construct an option pricing model based on the new hybrid generalized autoregressive conditional heteroskedastic (hybrid GARCH)-type functions with improved EEMD by decomposing the original return series into the high frequency, low frequency and trend terms. Using the locally risk-neutral valuation relationship (LRNVR), we obtain an equivalent martingale measure and option prices with different maturities based on Monte Carlo simulations. The empirical results indicate that this novel model can substantially capture volatility features and it performs much better than the M-GARCH and Black–Scholes models. In particular, the decomposition is consistently helpful in reducing option pricing errors, thereby proving the innovativeness and effectiveness of the hybrid GARCH option pricing model.     

Currency futures market in India: an empirical analysis of market efficiency and volatility

As the Indian currency futures market has been in existence for over 7 years, this paper analyses the effectiveness of the 1-month USD/INR currency futures rates in predicting the expected spot rate. The volatility of the USD/INR spot returns was also analysed. Modelling volatility of the USD/INR spot rate using a generalized autoregressive conditional heteroskedasticity (GARCH) and exponential generalized autoregressive conditional heteroskedasticity (EGARCH) model indicated the presence of volatility clustering. Using multivariate GARCH models such as the constant conditional correlation and dynamic conditional correlation, signs of a volatility spillover between the USD/INR spot and currency futures market were also observed.     

The effect of CBOE option listing on the volatility of NYSE traded stocks: a time-varying variance approach

This paper employs the standard General Auto-regressive Conditional Heteroskedasticity (GARCH(1,1)) process to examine the impact of option listing on volatility the underlying stocks. It takes into consideration the time variation in the individual stock's variance and explicitly tests whether option listing causes any permanent volatility change. It also investigates the impact of option listing on the speed at which information is incorporated into the stock price. The study uses clean samples to avoid sample selection biases and control samples to account for the change in the volatility and/or information flows that may be caused by factors other than option listing.     

The intraday behaviors and relationships with its underlying assets: evidence on option market in Taiwan

In this paper, we use daily data to investigate the information asymmetric effects and the relationships between the trading volume of options and their underlying spot trading volume. Our results reveal that options with higher liquidity are near-the-money and expiration periods with 2 to 4 weeks have higher trading activity. We classify them into two parts with the ARIMA model: the expected trading activity impact and the unexpected trading activity impact. Using the bivariate generalized autoregressive conditional heteroscedasticity (GARCH) model, we investigate the trading activity effect and information asymmetric effect. In conclusion, the trading volume volatility of the spot and options markets move together, and a greater expected and unexpected trading volume volatility of the spot (options) market is associated with greater volatility in the options (spot) market. However, both markets generate higher trading volume volatility when people expect such an impact rather than when they do not. We also find that there are feedback effects within these two markets. Furthermore, when the spot (options) market has negative innovations, it generates a greater impact on the options (spot) market than do positive innovations. Finally, the conditional correlation coefficient between the spot and the option markets changes over time based on the bivariate GARCH model.     

An empirical comparison of GARCH option pricing models

Recent empirical studies have shown that GARCH models can be successfully used to describe option prices. Pricing such contracts requires knowledge of the risk neutral cumulative return distribution. Since the analytical forms of these distributions are generally unknown, computationally intensive numerical schemes are required for pricing to proceed. Heston and Nandi (2000) consider a particular GARCH structure that permits analytical solutions for pricing European options and they provide empirical support for their model. The analytical tractability comes at a potential cost of realism in the underlying GARCH dynamics. In particular, their model falls in the affine family, whereas most GARCH models that have been examined fall in the non-affine family. This article takes a closer look at this model with the objective of establishing whether there is a cost to restricting focus to models in the affine family. We confirm Heston and Nandi's findings, namely that their model can explain a significant portion of the volatility smile. However, we show that a simple non affine NGARCH option model is superior in removing biases from pricing residuals for all moneyness and maturity categories especially for out-the-money contracts. The implications of this finding are examined. JEL Classification G13     

Intraday Return Volatility Process: Evidence from NASDAQ Stocks

This paper presents a comprehensive analysis of the distributional and time-series properties of intraday returns. The purpose is to determine whether a GARCH model that allows for time varying variance in a process can adequately represent intraday return volatility. Our primary data set consists of 5-minute returns, trading volumes, and bid-ask spreads during the period January 1, 1999 through March 31, 1999, for a subset of thirty stocks from the NASDAQ 100 Index. Our results indicate that the GARCH(1,1) model best describes the volatility of intraday returns. Current volatility can be explained by past volatility that tends to persist over time. These results are consistent with those of Akgiray (1989) who estimates volatility using the various ARCH and GARCH specifications and finds the GARCH(1,1) model performs the best. We add volume as an additional explanatory variable in the GARCH model to examine if volume can capture the GARCH effects. Consistent with results of Najand and Yung (1991) and Foster (1995) and contrary to those of Lamoureux and Lastrapes (1990), our results show that the persistence in volatility remains in intraday return series even after volume is included in the model as an explanatory variable. We then substitute bid-ask spread for volume in the conditional volatility equation to examine if the latter can capture the GARCH effects. The results show that the GARCH effects remain strongly significant for many of the securities after the introduction of bid-ask spread. Consistent with results of Antoniou, Homes and Priestley (1998), intraday returns also exhibit significant asymmetric responses of volatility to flow of information into the market.     

Uncovering a positive risk-return relation: the role of implied volatility index

We report empirical evidence suggesting a strong and positive risk-return relation for the daily S&P 100 market index if the implied volatility index is included as an exogenous variable in the conditional variance equation. This result holds for alternative GARCH specifications and conditional distributions. Monte Carlo evidence suggests that if implied volatility is not included, whilst is should be, the risk-return relation is more likely to be negative or weak.     

UK Housing Market: Time Series Processes with Independent and Identically Distributed Residuals

The paper examines whether a univariate data generating process can be identified which explains the data by having residuals that are independent and identically distributed, as verified by the BDS test. The stationary first differenced natural log quarterly house price index is regressed, initially with a constant variance and then with a conditional variance. The only regression function that produces independent and identically distributed standardised residuals is a mean process based on a pure random walk format with Exponential GARCH in mean for the conditional variance. There is an indication of an asymmetric volatility feedback effect but higher frequency data is required to confirm this. There could be scope for forecasting the index but this is tempered by the reduction in the power of the BDS test if there is a non-linear conditional variance process.     

Risk and the January effect

We use a time-series GARCH framework with the conditional variance/covariance as proxies for systematic risk to reexamine the proposition by Rozeff and Kinney (1976) and Rogalski and Tinic (1986) that the January effect may be a phenomenon of risk compensation in the month. We find no clear evidence that either conditional volatility or unconditional volatility in January is predominantly higher across the sampling years. Hence, against the proposition, the January effect is not due to risk per se. Rather, we find strong evidence that the January effect is due to higher compensation for risk in the month. This may be possible if investors have an increasing RRA utility function. Although many studies find that volatility tends to be higher in January, we find it to be period-specific and mostly in value-weighted return series, but not in equal-weighted return series. This is true both for the unconditional and conditional return volatility.     

A new approach for option pricing under stochastic volatility

We develop a new approach for pricing European-style contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time T variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a path-independent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying’s futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility. As a consequence of our SVRH, the partial differential equation (PDE) governing the arbitrage-free value of the contingent claim just depends on two state variables rather than the usual three. We then focus on the consistency of our SVRH with the standard assumption that the risk-neutral process for the instantaneous variance is a diffusion whose coefficients are independent of the variance swap maturity. We show that the combination of this maturity independent diffusion hypothesis (MIDH) and our SVRH implies a very special form of the risk-neutral diffusion process for the instantaneous variance. Fortunately, this process is tractable, well-behaved, and enjoys empirical support. Finally, we show that our model can also be used to robustly price and hedge volatility derivatives.     

Multivariate term structure models with level and heteroskedasticity effects

The paper introduces and estimates a multivariate level-GARCH model for the long rate and the term-structure spread where the conditional volatility is proportional to the γth power of the variable itself (level effects) and the conditional covariance matrix evolves according to a multivariate GARCH process (heteroskedasticity effects). The long-rate variance exhibits heteroskedasticity effects and level effects in accordance with the square-root model. The spread variance exhibits heteroskedasticity effects but no level effects. The level-GARCH model is preferred above the GARCH model and the level model. GARCH effects are more important than level effects. The results are robust to the maturity of the interest rates.     

Additional evidence on transparency and bank financial performance

This paper examines the relationship between volatility and the probability of occurrence of expected extreme returns in the Canadian market. Four measures of volatility are examined: implied volatility from firm option prices, conditional volatility calculated using an EGARCH model, idiosyncratic volatility, and expected shortfall. A significantly positive relationship is observed between a firm's idiosyncratic volatility and the probability of occurrence of an extreme return in the subsequent month for firms. A 10% increase in idiosyncratic volatility in a given month is associated with the probability of an extreme shock in the subsequent month (top or bottom 1.5% of the returns distribution) of 26.4%. Other firm characteristics, including firm age, price, volume and book‐to‐market ratio, are also shown to be significantly related to subsequent firm extreme returns. The effects of conditional and implied volatility are mixed. The E‐GARCH and expected shortfall measures of conditional volatility are consistent with mean reversion: high short term realizations of conditional volatility foreshadow a lower probability of extreme returns.     

Hedging with futures: Efficacy of GARCH correlation models to European electricity markets

European electricity markets have been subject to a broad deregulation process in the last few decades. We analyse hedging policies implemented through different hedge ratios estimation. More specifically we compare naïve, ordinary least squares, and GARCH conditional variance and correlations models to test if GARCH models lead to higher variance reduction in a context of high time varying volatility as the case of electricity markets. Our results show that the choice of the hedge ratio estimation model is central on determining the effectiveness of futures hedging to reduce the portfolio volatility.     

Central bank intervention and the volatility of foreign exchange rates: evidence from the options market

This paper tests the effects of central bank intervention on the ex ante volatility of $/DM and $/yen exchange rates between 1985 and 1991. In contrast to previous research which employed GARCH estimates of conditional volatility, we estimate ex ante volatility using the implied volatilities of currency option prices. We also control for the effects of other macroeconomic announcements. We find little support for the hypothesis that central bank intervention decreases expected exchange rate volatility. Instead, central bank intervention is generally associated with a positive change in ex ante exchange rate volatility, or with no change.     

An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates

This paper proposes an asymptotic expansion scheme of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas for the density functions of the underlying assets and for pricing currency options based on a third order asymptotic expansion scheme; we do not model a foreign exchange rate’s variance such as in Heston [(1993) The Review of Financial studies, 6, 327–343], but its volatility that follows a general time-inhomogeneous Markovian process. Further, the correlations among all the factors such as domestic and foreign interest rates, a spot foreign exchange rate and its volatility, are allowed. Finally, numerical examples are provided and the pricing formula are applied to the calibration of volatility surfaces in the JPY/USD option market.     

Optimal delta hedging for options

As has been pointed out by a number of researchers, the normally calculated delta does not minimize the variance of changes in the value of a trader's position. This is because there is a non-zero correlation between movements in the price of the underlying asset and movements in the asset's volatility. The minimum variance delta takes account of both price changes and the expected change in volatility conditional on a price change. This paper determines empirically a model for the minimum variance delta. We test the model using data on options on the S&P 500 and show that it is an improvement over stochastic volatility models, even when the latter are calibrated afresh each day for each option maturity. We also present results for options on the S&P 100, the Dow Jones, individual stocks, and commodity and interest-rate ETFs.