Black-Scholes Example

Suppose you want a function that implements the Black-Scholes pricing model. The Black-Scholes model, first published in 1973 by Fischer Black and Myron Scholes, is used to calculate the present value of a call option. Without going into the background theory here, we will simply present the resulting equations we wish to implement as a model.

The present value of a call option is:

P = the current price of the underlying security
E = the strike or exercise price in the options contract
t = the time to expiration of the option
v = the volatility of the underlying security's price
r = the risk-free interest rate

The function Cnorm is the cumulative normal distribution function; it is used frequently in statistics and management science. Cnorm is defined as:

The Cnorm function alone is a good candidate for a separate library model. In fact, to build the Black-Scholes model, it is best first to build a separate Cnorm model.

You will not find a closed form solution to the integral in the Cnorm definition because the integral cannot be solved. Instead, you are likely to find a table in the back of every statistics book that lists the value of this function for a range of input values. However, DecisionPro can handle the integration quite easily using the integral primitive, which performs numerical integration.

The Cnorm function can be implemented with two definitions taken straight from the equations above:

Once defined, this model should be saved in LBM format under the name Cnorm.lbm.

The complete Black-Scholes model can be implemented as follows:

Once created, this model should be saved in LBM format under the name Black.lbm. From then on, this model can be used in any other model simply by using the function Black(P,E,t,v,r) where the arguments P, E, t, v, and r are replaced with the appropriate input values.

Nesting Models

The Black-Scholes model has been split into two separate models, Black and Cnorm, where Black uses the model Cnorm. Library models can use other library models as components. In fact, there is no practical limit to how deeply you can nest library models.

The ability to nest models allows you to build libraries of models hierarchically. Each successive model can draw on previous models you have developed to perform increasingly more complex tasks. For example, once you have the Black model, you can build a model that uses Black to evaluate investment options.

When DecisionPro loads a library model, all component models are automatically loaded as well.