The next time your dealer quotes the implied volatility of an American option, be sure to ask "is that trinomial, BAW, or something else?" Implied volatilities are obtained by matching the market price of an option to its corresponding model price, and finding the value of volatility that equates both. Implied volatilities are therefore dependent on the model used. For European options the industry-standard implied volatility model is the Black-Scholes model. Implied volatilities from European options should be denoted "Black-Scholes implied volatilities" but since there is no ambiguity about the model used to extract them they are simply known as "implied volatilities." Consequently, given the correct inputs such as the strike price and underlying spot price an operator should, in principle, be able to replicate the same implied volatility as that which has been quoted, up to rounding-off error.

For American options (example) the situation is different as there is no industry-standard model for pricing American options, even under Black-Scholes assumptions. The available models include binomial trees and trinomial trees of various sorts, the Barone-Adesi and Whaley approximation (BAW) and its variants, the Longstaff and Schwartz algorithm, and many others. Hence, without the choice of model being used to extract an implied volatility, a trader cannot obtain the same quoted implied volatility. Even given the choice of model, approximation error (such as the number of time steps used in the tree, or simulation noise, for example) would preclude an exact replication of the quoted implied volatility.

One very promising method to valuing American puts has been proposed by Alexey Medvedev and Olivier Scaillet in their 2010 paper, published in the Journal of Financial Economics (link). Their method takes the form of an infinite expansion with analytic terms, truncated in practice to usually five terms or less. Being of closed form, prices are produced very fast. An implied volatility extracted with this model could be replicated, provided the number of expansion terms is specified.

Fabrice Rouah is the author of the book "The Heston Model in Matlab and C#" to be published in early 2013 by John Wiley & Sons.


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