86 CHAPTER 2. BINOMIAL OPTION PRICING MODELS
7.7 Limitations of the Black-Scholes Model
The disadvantages of the binomial model are:
- Trading times are not really at discrete times, trading goes on contin-
uously. - Securities do not change value according to a Bernoulli (two-valued)
distribution on a single time step, or a binomial distribution on multiple
time periods, they change over a range of values with a continuous
distribution. - The calculations are tedious.
- Developing a more complete theory is going to take some detailed and
serious limit-taking considerations.
The advantages of the model are:- It clearly reveals the construction of the replicating portfolio.
- It clearly reveals that the probability distribution is not centrally in-
volved, since expectations of outcomes aren’t used to value the deriva-
tives. - It is simple to calculate, although it can get tedious.
- It reveals that we need more probability theory to get a complete un-
derstanding of path dependent probabilities of security prices.
It is possible, with considerable attention to detail, to make a limiting
argument and pass from the binomial tree model of Cox, Ross and Rubenstein
to the Black-Scholes pricing formula. However, this approach is not the most
instructive. Instead, we will back up from derivative pricing models, and
consider simpler models with only risk, that is, gambling, to get a more
complete understanding before returning to pricing derivatives.
Some caution is also needed when reading from other sources about the
Cox-Ross-Rubenstein or Binomial Option Pricing Model. Many other sources
derive the Binomial Option Pricing Model by discretizing the Black-Scholes
Option Pricing Model. The discretization is different from building the model