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oBIN( ) Convergence Properties

The option values derived from the BOPM are a function of the number of steps that are used in the tree construction. The option value is more accurate for larger number of timesteps, but the number of steps that are required to produce reasonable results is dependent on some features of the option under consideration such as time to maturity, the degree of moneyness, and the underlying volatility.

In general, the BOPM values converge to the 'true' value as shown in the following graph.

 

oBIN( ) Convergence 1

 

One way to investigate the convergence properties of the BOPM is to compare the model results with the option value given by a closed-form solution. For example, we can compare the BOPM values for a European option on a non-dividend paying stock with the value given by the Black Scholes model: the Black Scholes value gives the 'true' value in this case.

The following table shows the (absolute) percentage pricing errors given by the BOPM for a range of timesteps. The pricing errors are computed relative to the Black Scholes model and across a variety of different European put option deals.

 

Option Parameters

Option 1

Option 2

Option 3

Option 4

Option 5

.

Spot

50

50

50

50

50

Strike

50

48

52

50

50

Time to Maturity

180 days

180 days

180 days

270 days

180 days

Volatility

20%

20%

20%

20%

40%

Riskless Rate

5%

5%

5%

5%

5%

 

 

 

 

 

 

Black Scholes Value

$2.2121

$1.4245

$3.2189

$2.5537

$4.9441

 

 

 

 

 

 

Pricing Errors

 

 

 

 

 

.

20 Timesteps

1.5377%

1.8336%

0.8969%

1.6117%

1.3803%

75 Timesteps

0.4081%

0.2789%

0.1720%

0.4256%

0.3665%

100 Timesteps

0.3090%

0.4160%

0.1846%

0.3239%

0.2776%

200 Timesteps

0.1545%

0.1268%

0.0939%

0.1619%

0.1389%

350 Timesteps

0.0882%

0.0762%

0.0521%

0.0925%

0.0794%

500 Timesteps

0.0617%

0.0412%

0.0090%

0.0647%

0.0556%

 

 

 

 

 

 

We suggest that for most options, a reasonable level of accuracy can be achieved using between 100 and 200 timesteps.

 

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