A Quasi-analytical Interpolation Method for Pricing American Options under General Multi-dimensional Diffusion Processes

We present a quasi-analytical method for pricing multi-dimensional American options based on interpolating two arbitrage bounds, along the lines of Johnson (1983). Our method allows for the close examination of the interpolation parameter on a rigorous theoretical footing instead of empirical regression. The method can be adapted to general diffusion processes as long as quick and accurate pricing methods exist for the corresponding European and perpetual American options. The American option price is shown to be approximately equal to an interpolation of two European option prices with the interpolation weight proportional to a perpetual American option. In the Black-Scholes model, our method achieves the same e±ciency as Barone-Adesi and Whaley's (1987) quadratic approximation with our method being generally more accurate for out-of-the-money and long-maturity options. When applied to Heston's stochastic volatility model, our method is shown to be extremely e±cient and fairly accurate.American option; Interpolation method; Quasi-analytical approximation; Critical bound- ary; Heston's Stochastic volatility model

Analytical Approximations for the Critical Stock Prices of American Options: A Performance Comparison

Many e±cient and accurate analytical methods for pricing American options now exist. However, while they can produce accurate option prices, they often do not give accurate critical stock prices. In this paper, we propose two new analytical approximations for American options based on the quadratic approximation. We compare our methods with existing analytical methods including the quadratic approximations in Barone-Adesi and Whaley (1987) and Barone-Adesi and Elliott (1991), the lower bound approximation in Broadie and Detemple (1996), the tangent approximation in Bunch and Johnson (2000), the Laplace inversion method in Zhu (2006b), and the interpolation method in Li (2008). Both of our methods give much more accurate critical stock prices than all the existing methods above.American option; Analytical approximation; Critical stock price

Distributions Implied by Exchange Traded Options: A Ghost’s Smile?

A new and easily applicable method for estimating risk neutral distributions (RND) implied by American futures options is proposed. It amounts to inverting the Barone-Adesi and Whaley method (1987) (BAW method) to get the BAW-implied volatility smile. Extensive empirical tests show that the BAW smile is equivalent to the volatility smile implied by corresponding European options. Therefore, the procedure leads to a legitimate RND estimation method. Further, the investigation of the currency options traded on the Chicago Mercantile Exchange and OTC markets in parallel provides us with insights on the structure and interaction of the two markets. Unequally distributed liquidity in the OTC market seems to lead to price distortions and an ensuing interesting `ghost- like' shape of the RND density implied by CME options. Finally, using the empirical results, we propose a parsimonious generalisation of the existing methods for estimating volatility smiles from OTC options. A single free parameter significantly improves the fit. Note:The revised version of this paper was published as: Cincibuch, M., 2004. Distributions Implied by American Currency Futures Options: A Ghosts' Smile ? Journal of Futures Markets, 2004, 24(2).

Valuing options in Heston's stochastic volatility model: Another analytical approach

We are concerned with the valuation of European options in Heston's stochastic volatility model with correlation. Based on Mellin transforms we present new closed-form solutions for the price of European options and hedging parameters. In contrast to Fourier-based approaches where the transformation variable is usually the log-stock price at maturity, our framework focuses on transforming the current stock price. Our solution has the nice feature that similar to the approach of Carr and Madan (1999) it requires only a single integration. We make numerical tests to compare our results to Heston's solution based on Fourier inversion and investigate the accuracy of the derived pricing formulae. --Stochastic volatility,European option,Mellin transform

Modelling FX smile : from stochastic volatility to skewness

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