Binomial trees are frequently used to approximate the movements in the price of a stock or other asset under the Black-Scholes-Merton model. There are several approaches to building the underlying binomial tree, such as Cox-Ross-Rubinstein, Jarrow-Rudd, and Tian. In all of the approaches above, the lattice is designed so as to minimize the discrepancy between the approximate (discrete) and target (continuous) distributions by matching, exactly or approximately, their first few moments. The rationale for this is that for any fixed number of time steps, a moment-matching lattice is believed to produce better option price estimates.
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If you denote by and the multiplicative constants for up and down movements in the tree, and by and the probabilities of the upward and the downward movements, then the stock price at the -th time step and the -th node is
for
and
You know that the three variables satisfy two equations, so there is some freedom to assign a value to one of the variables. This is the reason leading to the different versions of the binomial tree. In a risk-neutral binomial tree the transition probabilities can then be determined from the no-arbitrage condition
where is the known forward price of the stock. In a general constant volatility recombining binomial tree and have the form
and
for some reasonable value of .
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| (2.1) |
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| (2.2) |
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| (2.3) |
The corresponding transition probabilities are
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| (2.4) |
| (2.5) |
| (2.6) |
Consider the Black-Scholes process with the following parameters:
| (2.11) |
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Cox-Ross-Rubinstein Binomial Tree
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The binomial tree introduced by Cox, Ross and Rubinstein in 1979 (hereafter CRR) is one of the most important innovations to have appeared in the option pricing literature. Beyond its original use as a tool to approximate the prices of European and American options in the Black-Scholes (1973) framework, it is also widely used as a pedagogical device to introduce various key concepts in option pricing. CRR presented the fundamental economic principles of option pricing by arbitrage considerations in the most simplest manner. By application of a central limit theorem, they proved that their model merges into the Black and Scholes model when the time steps between successive trading instances approach zero. Additionally, the model was used to evaluate American type options and options on assets with continuous dividend payments. The Cox, Ross, and Rubinstein model makes the multiplication of up and down jumps equal
,
and
This corresponds to the case when
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Here is a logarithmic view of the same tree.
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| (2.1.2) |
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Jarrow-Rudd Binomial Tree
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There exist many extensions of the CRR model. Jarrow and Rudd (1983), JR, adjusted the CRR model to account for the local drift term
,
They constructed a binomial model where the first two moments of the discrete and continuous time-return processes match. As a consequence, a probability measure equal to one half results. This corresponds to the case when .
| (2.2.2) |
| (2.2.3) |
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