Without a doubt the Black Scholes formula dominates the competition when it comes to options pricing models. However, the binomial model has a loyal following and is often used in conjunction with the Black Scholes formula by option traders when analyzing option prices. In this post we take a closer look at the binomial model from its inception to its current place in the options market.
The Inception of the Binomial Model
Introduced in 1979 by John Cox, Stephen Ross, and Mark Rubinstein, the binomial model uses a “discrete time” model of the varying underlying stock price in the valuation of options. Because the binomial model is based on the description of the underlying stock over a period of time (binomial tree) rather than a single point (Black Scholes formula) the computation of option prices become very complex and time consuming. This was a big hindrance back in the day but with the advancements in computational power and speed over the last few decades this is no longer a real concern. The binomial model for options pricing yields a more accurate value for an option’s worth than the Black Scholes formula especially for longer-dated options on dividend paying stocks.
The Binomial Tree
The binomial model follows the evolution of the option’s key underlying variables in discrete time. So an option’s price is determined for each time increment (let’s use 1 day for simplicity sake) node. Option valuation using the binomial model is performed iteratively, starting at each of the final nodes (at expiration) and then working backwards until we reach the initial date. As you can imagine, the number of calculations needed to apply the binomial model can easily reach into the millions depending on the number of time increments. An example binomial tree is included below:
Binomial Model Assumptions
As in the case of the Black Scholes formula, there are assumptions underlying the binomial model that may or may not agree with practitioners in the options space. Two of the more prominent assumptions of the binomial model are:
- Discrete Time: The underlying stock is assumed to move in a discrete manner, which is more in line with reality (at least for now) given securities generally open and close every weekday and are typically closed on the weekend. The Black Scholes formula assumes a continuously traded market for the underlying security. One has to wonder how long it will be before worldwide exchanges work on a continuous time basis. Seems to be a likely evolution doesn’t it?
- Binomial Distribution: The binomial model assumes price movements follow a binomial distribution which when tested over many trials converges on the normal distribution assumed in the Black Scholes formula.
As we mentioned in the previous post “Why The Black Scholes Formula Dominates the Competition”, the Black Scholes formula maintains its dominance over other more accurate options pricing models like the binomial model primarily due to its simplicity. I’m sure the ultra advanced high-powered computers of 2010 have yielded the time difference in calculating the value of an option using the two methods virtually insignificant, but I guess there’s something to be said for good ‘ol comfort level.
For further information on the Binomial Model and other Options Trading Strategies visit OptionsUniversity.com.
