The option-pricing model by Black and Scholes (1973) is based on the assumptions
that the underlying asset follows a geometric Brownian motion with a constant volatility :
Binomial trees, as a discretization of the Black-Scholes model,
can be constructed in several alternative ways.
Here we recall the classic Cox, Ross, and Rubinstein's (1979)
scheme (CRR), which has a constant logarithmic spacing between nodes
on the same level (this spacing represents the future price volatility).
A standard CRR tree is depicted in Figure 6.2. Starting at a
node , the price of an underlying asset can either
increase to
with probability
or
decrease to
with probability
:
A binomial tree corresponding to the risk-neutral underlying evaluation process is the same for all options on this asset, no matter what the strike price or time to expiration is. There are many extensions of the original Black-Scholes approach that try to capture the volatility variation and to price options consistently with the market prices (that is, to account for the volatility smile). Some extensions incorporate a stochastic volatility factor or discontinuous jumps in the underlying price; see for instance Franke, Härdle, and Hafner (2004) and Chapters 5 and 7. In the next section, we discuss an extension of the Black-Scholes model developed by Derman and Kani (1994) - the implied trees.