The finite sample properties of the GARCH option pricing model

The authors explore the finite sample properties of the generalized autoregressive conditional heteroscedasticity (GARCH) option pricing model proposed by S. L. Heston and S. Nandi (2000). Simulation results show that the maximum likelihood estimators of the GARCH process may contain substantial estimation biases, even when samples as large as 3,000 observations are used. However, it was found that these biases cause significant mispricings only for short‐term, out‐of‐the‐money options. It is shown that, given an adequate estimation sample, this bias can be reduced considerably by employing the jackknife resampling method. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:599–615, 2007     

VIX option pricing

Substantial progress has been made in developing more realistic option pricing models for S&P 500 index (SPX) options. Empirically, however, it is not known whether and by how much each generalization of SPX price dynamics improves VIX option pricing. This article fills this gap by first deriving a VIX option model that reconciles the most general price processes of the SPX in the literature. The relative empirical performance of several models of distinct interest is examined. Our results show that state‐dependent price jumps and volatility jumps are important for pricing VIX options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:523–543, 2009     

Error analysis of finite difference and Markov chain approximations for option pricing

Mijatovi? and Pistorius proposed an efficient Markov chain approximation method for pricing European and barrier options in general one‐dimensional Markovian models. However, sharp convergence rates of this method for realistic financial payoffs, which are nonsmooth, are rarely available. In this paper, we solve this problem for general one‐dimensional diffusion models, which play a fundamental role in financial applications. For such models, the Markov chain approximation method is equivalent to the method of lines using the central difference. Our analysis is based on the spectral representation of the exact solution and the approximate solution. By establishing the convergence rate for the eigenvalues and the eigenfunctions, we obtain sharp convergence rates for the transition density and the price of options with nonsmooth payoffs. In particular, we show that for call‐/put‐type payoffs, convergence is second order, while for digital‐type payoffs, convergence is generally only first order. Furthermore, we provide theoretical justification for two well‐known smoothing techniques that can restore second‐order convergence for digital‐type payoffs and explain oscillations observed in the convergence for options with nonsmooth payoffs. As an extension, we also establish sharp convergence rates for European options for a rich class of Markovian jump models constructed from diffusions via subordination. The theoretical estimates are confirmed using numerical examples.     

Nonparametric American option pricing

A nonparametric method is introduced to accurately price American-style contingent claims. This method uses only historical stock price data, not option price data, to generate the American option price. The accuracy of this method is tested in a controlled experimental environment under both Black, F and Scholes, M (1973) and Heston, S (1993) assumptions, and an error-metric analysis is performed. These numerical experiments demonstrate that this method is an accurate and precise method of pricing American options under a variety of market conditions. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:717–748, 2008     

Pathwise moderate deviations for option pricing

We provide a unifying treatment of pathwise moderate deviations for models commonly used in financial applications, and for related integrated functionals. Suitable scaling enables us to transfer these results into small‐time, large‐time, and tail asymptotics for diffusions, as well as for option prices and realized variances. In passing, we highlight some intuitive relationships between moderate deviations rate functions and their large deviations counterparts; these turn out to be useful for numerical purposes, as large deviations rate functions are often difficult to compute.     

A flexible binomial option pricing model

This article develops a flexible binomial model with a “tilt” parameter that alters the shape and span of the binomial tree. A positive tilt parameter shifts the tree upward while a negative tilt parameter does exactly the opposite. This simple extension of the standard binomial model is shown to converge with any value of the tilt parameter. More importantly, the binomial tree can be recalibrated through the tilt parameter in order to position nodes relative to the strike price or barrier of an option. The rate of convergence is improved as a result. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 817–843, 1999     

Alternative tilts for nonparametric option pricing

This study generalizes the nonparametric approach to option pricing of Stutzer, M. (1996) by demonstrating that the canonical valuation methodology introduced therein is one member of the Cressie–Read family of divergence measures. Alhough the limiting distribution of the alternative measures is identical to the canonical measure, the finite sample properties are quite different. We assess the ability of the alternative divergence measures to price European call options by approximating the risk‐neutral, equivalent martingale measure from an empirical distribution of the underlying asset. A simulation study of the finite sample properties of the alternative measure changes reveals that the optimal divergence measure depends upon how accurately the empirical distribution of the underlying asset is estimated. In a simple Black–Scholes model, the optimal measure change is contingent upon the number of outliers observed, whereas the optimal measure change is a function of time to expiration in the stochastic volatility model of Heston, S. L. (1993). Our extension of Stutzer's technique preserves the clean analytic structure of imposing moment restrictions to price options, yet demonstrates that the nonparametric approach is even more general in pricing options than originally believed. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:983–1006, 2010     

The design and pricing of fixed‐ and moving‐window contracts: An application of Asian‐Basket option pricing methods to the hog‐finishing sector

Asian‐Basket‐type moving‐window contracts are an increasingly used risk‐management tool in the North American hog sector. The moving‐window contract is decomposed into a portfolio of a long Asian‐Basket put and a short Asian‐Basket call option. A projected break‐even price is used to determine the floor price, and then Monte Carlo simulation methods are used to price both a moving‐ and a fixed‐window contract. These methods provide unbiased pricing of fixed‐ and moving‐window hog‐finishing contracts of 1‐year duration. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:1047–1073, 2003     

Currency barrier option pricing with mean reversion

This article develops a barrier option pricing model in which the exchange rate follows a mean‐reverting lognormal process. The corresponding closed‐form solutions for the barrier options with time‐dependent barriers are derived. The numerical results show that barrier option values and the corresponding hedge parameters under the proposed model are different from those based on the Black‐Scholes model. For an up‐and‐out call, the mean‐reverting process keeps the exchange rate in a small range around the mean level. When the mean level is below the barrier but above the strike price, the risk of the call to be knocked out is reduced and its option value is enhanced compared with the value under the Black‐Scholes model. The parameters of the mean‐reverting lognormal process therefore have a material impact on the valuation of currency barrier options and their hedge parameters. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:939–958, 2006     

A new simple square root option pricing model

This study derives a simple square root option pricing model using a general equilibrium approach in an economy where the representative agent has a generalized logarithmic utility function. Our option pricing formulae, like the Black–Scholes model, do not depend on the preference parameters of the utility function of the representative agent. Although the Black–Scholes model introduces limited liability in asset prices by assuming that the logarithm of the stock price has a normal distribution, our basic square root option pricing model introduces limited liability by assuming that the square root of the stock price has a normal distribution. The empirical tests on the S&P 500 index options market show that our model has smaller fitting errors than the Black–Scholes model, and that it generates volatility skews with similar shapes to those observed in the marketplace. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark