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

 can be represented in the form

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

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.

Here is a formula for the Gamma.

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

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

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

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

The key boundary condition is:

Another obvious condition:

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

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