The Pricing Of American Options By The Canonical Implied Binomial Tree Method

In this paper a new approach to pricing American options is proposed and termed the canonical implied binomial (CIB) tree method. CIB takes advantage of both canonical valuation (Stutzer,1996) and the implied binomial tree method (Rubinstein,1994) and achieves its goal.Given a set of historical daily gross returns for the underlying asset, the daily canonical distribution can be easily estimated. From this canonical distribution, samples of daily gross returns are randomly drawn and multiplied together to obtain the risk-neutral gross terminal returns that match the days to maturity of an option to be priced. Using those terminal gross returns, all of which have equal probabilities, a risk-neutral implied binomial tree for the underlying asset can be constructed and utilized to price options, European or American.Using simulated returns from geometric Brownian motions (GBM), CIB produced very similar prices for calls and European puts as those of Black-Scholes (BS). Applied to a set of over15,000American-style S&P100Index puts, CIB outperformed BS with historic volatility in pricing out-of-the-money options. Furthermore, CIB suggests that the implied binomial tree can be constructed from the risk-neutral distribution simulated from regular GBM-based Monte Carlo and used to price American options. This would add another Monte Carlo method for American options pricing to the well-known least-squares Monte Carlo approach (Longstaff&Schwartz,2001).Compared with the earlier canonical valuation/least-squares approaches (Alcock&Carmichael,2008; Liu,2010), this new method is advantageous in several aspects. First, its tree directly provides the scheme necessary for the dynamic hedging of options; second, it makes readily available the optimal early exercise boundary needed for trading; third, it is easier to understand conceptually and to implement programmatically; and fourth, it is more computationally efficient. Finally, we may improve the pricing accuracy of CIB if additional information is used in deriving the canonical risk-neutral distribution. We could perhaps achieve this by using:traded option prices as constraints similar to those of Gray et al.(2007), the variance constraint of Liu (2011), or the generalized binomial trees of Jackwerth (1997).