Seize the Moments: Approximating American Option Prices in the GARCH Framework

This paper proposes an efficient approach to compute the prices of American style options in the GARCH framework. Rubinstein's (1998) Edgeworth tree idea is combined with the analytical formulas for moments of the cumulative return under GARCH developed in Duan et al. (1999, 2002) to yield a simple recombining binomial tree for option valuation in the GARCH context. Since the resulting tree is univariate, the proposed approach represents a convenient approximation of the bivariate GARCH system. Numerical analyses are used to demonstrate the speed and accuracy of the proposed approximation.American Options; Edgeworth binomial tree; Garch process

Arbitrage-free prediction of the implied volatility smile

This paper gives an arbitrage-free prediction for future prices of an arbitrary co-terminal set of options with a given maturity, based on the observed time series of these option prices. The statistical analysis of such a multi-dimensional time series of option prices corresponding to $n$ strikes (with $n$ large, e.g. $n\geq 40$) and the same maturity, is a difficult task due to the fact that option prices at any moment in time satisfy non-linear and non-explicit no-arbitrage restrictions. Hence any $n$-dimensional time series model also has to satisfy these implicit restrictions at each time step, a condition that is impossible to meet since the model innovations can take arbitrary values. We solve this problem for any n\in\NN in the context of Foreign Exchange (FX) by first encoding the option prices at each time step in terms of the parameters of the corresponding risk-neutral measure and then performing the time series analysis in the parameter space. The option price predictions are obtained from the predicted risk-neutral measure by effectively integrating it against the corresponding option payoffs. The non-linear transformation between option prices and the risk-neutral parameters applied here is \textit{not} arbitrary: it is the standard mapping used by market makers in the FX option markets (the SABR parameterisation) and is given explicitly in closed form. Our method is not restricted to the FX asset class nor does it depend on the type of parameterisation used. Statistical analysis of FX market data illustrates that our arbitrage-free predictions outperform the naive random walk forecasts, suggesting a potential for building management strategies for portfolios of derivative products, akin to the ones widely used in the underlying equity and futures markets.Comment: 18 pages, 2 figures; a shorter version of this paper has appeared as a Technical Paper in Risk (30 April 2014) under the title "Smile transformation for price prediction

Pricing bivariate option under GARCH-GH model with dynamic copula: application for Chinese market

This paper develops the method for pricing bivariate contingent claims under General Autoregressive Conditionally Heteroskedastic (GARCH) process. In order to provide a general framework being able to accommodate skewness, leptokurtosis, fat tails as well as the time varying volatility that are often found in financial data, generalized hyperbolic (GH) distribution is used for innovations. As the association between the underlying assets may vary over time, the dynamic copula approach is considered. Therefore, the proposed method proves to play an important role in pricing bivariate option. The approach is illustrated for Chinese market with one type of better-of-two-markets claims : call option on the better performer of Shanghai Stock Composite Index and Shenzhen Stock Composite Index. Results show that the option prices obtained by the GARCH-GH model with time-varying copula differ substantially from the prices implied by the GARCH-Gaussian dynamic copula model. Moreover, the empirical work displays the advantage of the suggested method.Call-on-max option - GARCH process - generalized hyperbolic (GH) distribution - normal inverse Gaussian (NIG) distribution - copula - dynamic copula

The least squares method for option pricing revisited

It is shown that the the popular least squares method of option pricing converges even under very general assumptions. This substantially increases the freedom of creating different implementations of the method, with varying levels of computational complexity and flexible approach to regression. It is also argued that in many practical applications even modest non-linear extensions of standard regression may produce satisfactory results. This claim is illustrated with examples

Pricing bivariate option under GARCH-GH model with dynamic copula : application for Chinese market

This paper develops the method for pricing bivariate contingent claims under General Autoregressive Conditionally Heteroskedastic (GARCH) process. In order to provide a general framework being able to accommodate skewness, leptokurtosis, fat tails as well as the time varying volatility that are often found in financial data, generalized hyperbolic (GH) distribution is used for innovations. As the association between the underlying assets may vary over time, the dynamic copula approach is considered. Therefore, the proposed method proves to play an important role in pricing bivariate option. The approach is illustrated for Chinese market with one type of better-of-two-markets claims : call option on the better performer of Shanghai Stock Composite Index and Shenzhen Stock Composite Index. Results show that the option prices obtained by the GARCH-GH model with time-varying copula differ substantially from the prices implied by the GARCH-Gaussian dynamic copula model. Moreover, the empirical work displays the advantage of the suggested method.Call-on-max option, GARCH process, generalized hyperbolic (GH) distribution, normal inverse Gaussian (NIG) distribution, copula, dynamic copula.

Pricing bivariate option under GARCH processes with time-varying copula

This paper develops a method for pricing bivariate contingent claims under General Autoregressive Conditionally Heteroskedastic (GARCH) process. As the association between the underlying assets may vary over time, the dynamic copula with time-varying parameter offers a better alternative to any static model for dependence structure and even to the dynamic copula model determined by dynamic dependence measure. Therefore, the proposed method proves to play an important role in pricing bivariate options. The approach is illustrated with one type of better-of-two-markets claims : call option on the better performer of Shanghai and Shenzhen stock composite indexes. Results show that the option prices obtained by the time-varying copula model differ substantially from the prices implied by the static copula model and even the dynamic copula model derived from the dynamic dependence measure. Moreover, the empirical work displays the advantages of the suggested method.Call-on-max option, GARCH process, Kendall's tau, Copula, Dynamic Copula, time-varying parameter.

Option Pricing: The empirical tests of the Black-Scholes pricing formula and the feed-forward networks

In this article we evaluate the pricing performance of the rather simple but revolutionary Black-Scholes model and one of the more complex techniques (neural networks) on the European-style S&P Index call and put options over the period of 1.6.2006 till 8.6.2007. Our results on call options show that generally Black-Scholes model performs better than simple generalized feed-forward networks. On the other hand neural networks performance is improving as the option goes deep in the money and as days to expiration increase, compared to the worsening performance of the BS models. Neural networks seem to correct for the well-known Black-Scholes model moneyness and maturity biases.option pricing, neural networks

Pricing Weather Derivatives

This paper presents a general method for pricing weather derivatives. Specification tests find that a temperature series for Fresno, California follows a mean-reverting Brownian motion process with discrete jumps and ARCH errors. Based on this process, we define an equilibrium pricing model for cooling degree day weather options. Comparing option prices estimated with three methods: a traditional burn-rate approach, a Black-Scholes-Merton approximation, and an equilibrium Monte Carlo simulation reveals significant differences. Equilibrium prices are preferred on theoretical grounds, so are used to demonstrate the usefulness of weather derivatives as risk management tools for California specialty crop growers.derivative, jump-diffusion process, mean-reversion, volatility, weather, Demand and Price Analysis,

Bayesian Option Pricing Using Mixed Normal Heteroskedasticity Models

While stochastic volatility models improve on the option pricing error when compared to the Black-Scholes-Merton model, mispricings remain. This paper uses mixed normal heteroskedasticity models to price options. Our model allows for significant negative skewness and time varying higher order moments of the risk neutral distribution. Parameter inference using Gibbs sampling is explained and we detail how to compute risk neutral predictive densities taking into account parameter uncertainty. When forecasting out-of-sample options on the S&P 500 index, substantial improvements are found compared to a benchmark model in terms of dollar losses and the ability to explain the smirk in implied volatilities. Les modèles à volatilité stochastique apportent des améliorations en ce qui a trait à l’erreur d’établissement des prix des options comparativement au modèle de Black-Scholes-Merton. Toutefois, la fixation incorrecte des prix persiste. Le présent document a recours à des modèles mixtes avec hétéroscédasticité normale pour fixer les prix des options. Notre modèle permet de tenir compte de l’asymétrie négative importante et des moments d’ordre élevé variant dans le temps liés à la distribution du risque nul. Nous expliquons l’inférence des paramètres selon l’échantillonnage de Gibbs et détaillons la façon de traiter les densités prédictives de risque neutre en prenant en considération l’incertitude des paramètres. Dans le cas des prévisions concernant les options hors-échantillonnage sur l’indice S&P 500, nous constatons des améliorations importantes, par rapport à un modèle de référence, en termes de pertes exprimées en dollars et de capacité d’expliquer l’ironie des volatilités implicites.Bayesian inference, option pricing, finite mixture models, out-of-sample prediction, GARCH models, Inférence bayésienne, fixation du prix des options, modèles à mélanges finis, prédiction hors-échantillon, modèles GARCH.