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Stochastic Volatility and Option Pricing in the Brazilian Stock MarkeAn Empirical Investigation
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The stochastic volatility model (SVPS) proposed by Fouque et al. (2000a) explores a rapid timescale fluctuation of the volatility process to end up with a parsimonious way of capturing the volatility smile implied by close to the money options. In this article we test the SVFPS model using options from a Brazilian telecommunications stock. First, we find evidence of fast mean reversion in the volatility process. In addition, to test the model's ability to price options not so close to the money, we extend its statistical estimators to consider, in the calibration process, a wider region for the options moneyness. As an illustration, we price an exotic option.
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A parsimonious generalization of the Heston model is proposed where the volatility-of-volatility is assumed to be stochastic. We follow the perturbation technique of Fouque et al (2011, CUP) to derive a first order approximation of the price of options on a stock and its volatility index. This approximation is given by Heston's quasi-closed formula and some of its Greeks. It can be very efficiently calculated since it requires to compute only Fourier integrals and the solution of simple ODE systems. We exemplify the calibration of the model with S&P 500 and VIX data.
This research paper examines one-day-ahead out-of-sample performance of the volatility smirk-based options pricing models, namely, Ad-Hoc-Black–Scholes (AHBS) models on the CNX Nifty index options of India. Further, we compare the performance of these models with that of a TSRV-based Black–Scholes (BS) model. For the purpose, the study uses tick-by-tick data. The results on the AHBS models are highly satisfactory and robust across all the subgroups considered in the study. Notably, a daily constant implied volatility based ad-hoc approach outperforms the TSRV-based BS model substantially. The performance of the ad-hoc approaches improves further when the smile/smirk effect is considered. For the estimation of the implied volatility smile, we apply three weighting schemes based on the Vega and liquidity of the options. All the schemes offer equally competing results. The major contribution of the study to the existing literature on options pricing is in terms of the ex-ante examination of the ad-hoc approaches to price the options by calibrating volatility smile/smirk on a daily basis.
In this paper we report an empirical analysis of the Ibovespa index of the São Paulo Stock Exchange and its respective option contracts. We compare the empirical data on the Ibovespa options with two option pricing models, namely the standard Black-Scholes model and an empirical model that assumes that the returns are exponentially distributed. It is found that at times near the option expiration date the exponential model performs better than the Black-Scholes model, in the sense that it fits the empirical data better than does the latter model.
Offering a unifying theoretical perspective not readily available in any other text, this innovative guide to econometrics uses simple geometrical arguments to develop students' intuitive understanding of basic and advanced topics, emphasizing throughout the practical applications of modern theory and nonlinear techniques of estimation. One theme of the text is the use of artificial regressions for estimation, reference, and specification testing of nonlinear models, including diagnostic tests for parameter constancy, serial correlation, heteroscedasticity, and other types of mis-specification. Explaining how estimates can be obtained and tests can be carried out, the authors go beyond a mere algebraic description to one that can be easily translated into the commands of a standard econometric software package. Covering an unprecedented range of problems with a consistent emphasis on those that arise in applied work, this accessible and coherent guide to the most vital topics in econometrics today is indispensable for advanced students of econometrics and students of statistics interested in regression and related topics. It will also suit practising econometricians who want to update their skills. Flexibly designed to accommodate a variety of course levels, it offers both complete coverage of the basic material and separate chapters on areas of specialized interest.
We consider maximum likelihood estimation for stochastic differential equations based on discrete observations when the likelihood function is unknown. A sequence of approximations to the likelihood function is derived, and convergence results for the sequence are proven. Estimation by means of the approximate likelihood functions is easy and very generally applicable. The performance of the suggested estimators is studied in two examples, and they are compared with other estimators.
Implied volatility "smiles" have been documented in a number of option markets worldwide. The volatilities implied by the Black-Scholes (1973) model tend to be systematically related to the option's exercise price and time to expiration. Derman and Kani (1994), Dupire (1994), and Rubinstein (1994) offer an explanation for this behavior, that is, the volatility of the return of the underlying asset is a deterministic function of the asset price level and time. Their option valuation methodology, dubbed the "implied binomial tree" approach, describes (perfectly) the observed structure of options prices and purportedly provides more accurate hedge ratios. We systematically evaluate the empirical properties of the implied binomial tree approach to option valuation using S&P 500 index options during the period June 1988 and December 1993.
Distinct from the market-index, most individual firms' risk-neutral return distributions share three characteristics: they are far more volatile, substantially less negatively-skewed, and more leptokurtotic. But, what are the implications of these properties for models of individual equity options? We present and empirically investigate a double-jump option-pricing model that allows for stochastic volatility, return-jumps, volatility-jumps, and admits many existing models as special cases. Using a sample of 100 most active firms on the CBOE, we find that (i) the double-jump process is the least misspecified and the least demanding in fitting the tail-size and tail-asymmetry of the individual return distributions; (ii) the double-jump model improves pricing performance beyond return-jumps absent volatility-jumps, and beyond volatility-jumps absent return-jumps; and (iii) between return-jumps and volatility-jumps, the former is empirically more relevant than the latter for pricing options. The inverse link between volatility-jumps and return-jumps is instrumental for reconciling the valuation of deep out-of-money options. Compared to risk-neutral skewness, the excess kurtosis is, by far, a more crucial determinant of the cross-section of pricing-errors, especially for puts.
Derman and Kani (1994), Dupire (1994), and Rubinstein (1994) hypothesize that asset return volatility is a deterministic function of asset price and time, and develop a deterministic volatility function (DVF) option valuation model that has the potential of fitting the observed cross section of option prices exactly. Using S&P 500 options from June 1988 through December 1993, we examine the predictive and hedging performance of the DVF option valuation model and find it is no better than an ad hoc procedure that merely smooths Black–Scholes (1973) implied volatilities across exercise prices and times to expiration.
This book addresses problems in financial mathematics of pricing and hedging derivative securities in an environment of uncertain and changing market volatility. These problems are important to investors from large trading institutions to pension funds. It presents mathematical and statistical tools that exploit the bursty nature of market volatility. The mathematics is introduced through examples and illustrated with simulations and the modeling approach that is described is validated and tested on market data. The material is suitable for a one semester course for graduate students who have had exposure to methods of stochastic modeling and arbitrage pricing theory in finance. It is easily accessible to derivatives practitioners in the financial engineering industry.