Page 243 - DCOM504_SECURITY_ANALYSIS_AND_PORTFOLIO_MANAGEMENT

Security Analysis and Portfolio Management

Notes The Binomial Model

The binomial model breaks down the time to expiration into potentially a very large number of
time intervals, or steps. A tree of stock prices is initially produced working forward from the
present to expiration. At each step it is assumed that the stock price will move up or down by an
amount calculated using volatility and time to expiration. This produces a binomial distribution,
or recombining tree, of underlying stock prices. The tree represents all the possible paths that
the stock price could take during the life of the option.
At the end of the tree – i.e. at expiration of the option – all the terminal option prices for each of
the final possible stock prices are known, as they simply equal their intrinsic values.
Next, the option prices at each step of the tree are calculated working back from expiration to the
present. The option prices at each step are used to derive the option prices at the next step of the
tree using risk neutral valuation based on the probabilities of the stock prices moving up or
down, the risk-free rate and the time interval of each step. Any adjustments to stock prices (at an
ex-dividend date) or option prices (as a result of early exercise of American options) are worked
into the calculations at the required point in time. At the top of the tree you are left with one
option price.
To get a feel for how the binomial model works you can use the on-line binomial tree calculators:
either using the original Cox, Ross and Rubinstein tree or the equal probabilities tree, which
produces equally accurate results while overcoming some of the limitations of the C-R-R
model. The calculators let you calculate European or American option prices and display
graphically the tree structure used in the calculation. Dividends can be specified as being discrete
or as an annual yield, and points at which early exercise is assumed for American options are
highlighted.

Advantages

The big advantage the binomial model has over the Black-Scholes model is that it can be used to
accurately price American options. This is because with the binomial model it is possible to
check at every point in an option’s life (i.e. at every step of the binomial tree) for the possibility
of early exercise (e.g. where, due to a dividend, or a put being deeply in the money, the option
price at that point is less than its intrinsic value).

Where an early exercise point is found it is assumed that the option holder would elect to
exercise, and the option price can be adjusted to equal the intrinsic value at that point. This then
flows into the calculations higher up the tree and so on.
The on-line binomial tree graphical option calculator highlights those points in the tree structure
where early exercise would have caused an American price to differ from a European price.

The binomial model basically solves the same equation, using a computational procedure that
the Black-Scholes model solves using an analytic approach and in doing so, provides opportunities
along the way to check for early exercise for American options.

Limitation

The main limitation of the binomial model is its relatively slow speed. It’s great for half a dozen
calculations at a time but even with today’s fastest PCs it’s not a practical solution for the
calculation of thousands of prices in a few seconds.

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