Bowmain Consulting

Binomial Trees

Part of Project Denning

Denning contains a number of implementations of a binomial tree as part of this package.

Convergence

The convergence of binomial trees is a function of the number of steps taken. The diagram below shows the relationship between the calculated price and the number of steps for a European Call. There is a clear pattern. The calculated price oscillates around true price.

Three obvious strategies suggest themselves:

Use a large value for N, say 1000.
Use Richardson Interpolation - Calculate values using N steps and N+1 steps. Use the average of the 2 numbers.
If we only ever look at even values for N, the calculated prices show a upward trend with sudden "drops". Calculate the price just before the "drop" and afterwards, and use the average of the 2 numbers.

There is a problem with strategy 1. It takes 2¹⁰ (= 1024) longer to use 1000 steps than use 100 steps. Even on a fast computer, strategy 1 can become unfeasible.

The second strategy is widely used. The average of the 2 prices is consistently better than a single price (see below), and as a result convergence is more rapid. The strategy works over a wide of strikes.

The third strategy gives extremely rapid convergence. It can be more accurate to use strategy 3 with 50 steps than to use 1000 steps. The problems with strategy 3 are

It is necessary to calculate where the "drops" occur.
The strategy does not work if the spot is near the strike.
For out-of-the-money options, the first value of suitable value of N may be very large.

Calculating Critical values for N

How do we know what values to choose for N? Trees work best when the strike value corresponds to a node at Expiry. It is relatively simple to produce a formula for this.

Options near strike

Why does the strategy not work near the strike? The answer can be seen simply by looking at the formula for calculating critical step values.

Out-of-the-money Options

Why does the strategy not work near heavily out-of-the-money options?

Consequences

Analytic formulas exist for European Calls, however the same approach can be applied to other options with a single strike (and sometimes with more than one strike).

Download spreadsheet and option models. Source code is available here.