Black-Scholes Model 

? Maplesoft, a division of Waterloo Maple Inc., 2008 

Overview of the Model 

We consider the classical Black-Scholes model with single risky asset that follows a geometric Brownian motion 

 

 

 

where () is a standard Brownian motion, is the constant volatility, is the constant risk-free rate and is the initial asset price. Under these conditions, for any the stock price is given by the following formula. 

 

 

We consider a security with time to maturity and the payoff function.   

 

 

Payoff of the form corresponds to a digital call options with strike price, . 

We will consider several methods for computing the price of this security. 

Parameters 

Analytic Solution  

 

 

 

can be represented in the form 

 

(2.1)

 

where is a lognormal random variable with parameters and . 

 

The price of this option can be computed as the discounted expected payoff of the option 

 

 

 

 

(2.2)

 

 

 

(2.3)

 

Analytic Price 

 

We can use the analytic result to study the various market sensitivities. For example, we can symbolically compute the delta of our option. 

 

(2.4)

 

 

Here is a formula for the Gamma. 

 

(2.5)

 

 

We can also use the symbolic formula to plot the option price as a function of the parameters.  

Monte Carlo Simulation 

Alternatively, we can estimate the expectation using Monte Carlo simulation to compute the option price. The discrete-time version of the model is 

 

 

where , and is drawn from the lognormal distribution with parameters and . We can use this expression to generate a sample path for the price of our risky asset. 

 

Simulating Stock Prices 

*please be patient, it may take a few seconds to generate a sample
Number of Replications
Number of Updates

Note: We know the distribution of the final stock price. To compute the option price, we need only to simulate the final stock price, and not the whole stock path. 

 

We can verify the above analytic result using Monte Carlo simulation. 

 

Monte Carlo Simulation 

Number of Replications	Option Price
	Standard Error

Differential Equations 

 

Finally, we can use the Black-Scholes differential equations to compute the option price. 

 

 

(4.1)

 

The key boundary condition is: 

 

 

(4.2)

 

Another obvious condition: 

 

 

(4.3)

 

Finally, if for some , then it holds with a high probability that . Our option will thus be exercised and produce cash flow  

 

(4.4)

 

Numeric PDE Solver 

Space Step	Time Step
Option Price