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.




One way to investigate the convergence properties of the BOPM is to compare the model results with the option value given by a closedform solution. For example, we can compare the BOPM values for a European option on a nondividend 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. 


Copyright 2013 Hedgebook Ltd.