It is a numerical method based on the generation of a tree.

1
Lecture 6 Binomial trees

• It is a numerical method based on the generation
  of a tree.
• The tree represents the time evolution of the
  underlying equity, generated trough a lattice
  discretization of the stochastic process.
• At each step in the tree, the only allowed equity
  movements are an up or down moves.
• The lattice converges to the standard log normal
  model in the continuous limit.

2
Binomial trees

• Option valuation with binomial tree requires two
  main steps
• Generate the underlying equity price tree
  (according to CRR or Rubinstein method). I.e. at
  each node the underlying equity will move up or
  down by a multiplicative factor (u or d).
• Calculation of option value at each earlier node
  starting backward from the last point (option
  maturity). The value at the first node is the
  option price.

3
Binomial trees Rubinstein method (I)
The Rubinstein discrete formulas can be derived
starting from the basic integral equation
In order to transform a continuous problem
(continuous both in equity price as well as in
time) in a discrete problem (both in t and S), we
can simply transform w in a discrete random
variable
4
Binomial trees Rubinstein method (II)
As a result, over a discrete interval Dt, the
stock price can take only two possible values (up
or down)
In the limit Dt goes to 0 (infinite intervals) we
recover the standard log-normal process for stock
prices (central limit theorem).
5
Binomial trees Rubinstein method (III)

• It is an equal probabilities tree (50)
• The tree is not symmetric respect to the
  horizontal axis.
• I.e. ud ! 1.

6
Binomial trees CRR method

• The move up and down probabilities are not equal.
• The tree is symmetric respect to the horizontal
  axis (S0). I.e. ud1.

7
Option pricing calculation using binomial trees

• Generate the tree according to one of the two
  schemes CRR or Rubinstein
• Generate option prices backward along the tree,
  starting from the last point (corresponding to
  the option maturity) where the pay-off is known.

8
Binomial trees conclusions

• Simple to implement
• American optionality can be easily implemented
  within binomial scheme.

• It is restricted to low dimensional problems.
• Ad hoc implementation is required for each
  contract typology.
• For contracts with complicated features (e.g.
  asian option) the binomial tree becomes
  complicated and is not practical.