American option pricing with imprecise risk-neutral probabilities

The aim of this paper is to price an American style option when there is uncertainty on the volatility of the underlying asset. An option contract can be either European or American style depending on whether the exercise is possible only at or also before the expiry date. A European option gives the holder the right to buy or sell the underlying asset only at the expiry date of the option. On the other hand, an American option gives the holder the right to buy or sell the underlying asset at any time up to the expiry date. Therefore, in American option pricing, the likelihood of the early exercise should be carefully taken into account. American option valuation is usually performed, under the risk-neutral valuation paradigm, by using numerical procedures such as the binomial option pricing model of Cox, Ross, Rubinstein (1979). A key input of the multiperiod binomial model is the volatility of the underlying asset, that is an unobservable parameter. The volatility parameter can be estimated either from historical data (historical volatility) or implied from the price of European options (implied volatility). In the first case, the lenght of the time series, the frequency and the estimation methodology may lead to different estimates. In the second case, as options differ in strike price, time to expiration and option type (call or put), which option class yields implied volatilities that are most representative of the markets’ volatility expectations, is still an open debate. Various papers have examined the predictive power of implied volatility extracted from different option classes. Christensen and Prabhala (1998) examine the relation between implied and realized volatility on SP&100 options. They found that at the money calls are good predictors of future realized volatility. Christensen and Strunk (2002) consider the relation between implied and realized volatility on the S&P100 options. They suggest to compute implied volatility as a weighted average of implied volatilities from both in the money and out of the money options and both puts and calls. Ederington and Guan (2005) examine how the information in implied volatility differs by strike price for options on S&P500 futures. They suggest to use implied volatilities obtained from high strike options (out of the money calls and in the money puts) since the information content in implied volatilities varies roughly in a mirror image of the implied volatility smile.