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.
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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.
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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:
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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).
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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.
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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. ud=1.
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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.
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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.
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