We will consider a stochastic variable, which follows the standard Brownian motion with drift 0.055 and diffusion 0.3.

Here ae sample paths for .

You can compute the expected value of any expression involving .

Consider another stochastic process.

So  define the same stochastic process.

Note that the previous value is the expected payoff of a European call option with strike price 1 maturing in 3 years. In order to compute the current option price you have to discount this expected value at the risk-free rate (which is the drift parameter of ).

Compare this with the analytic price obtained using the Black-Scholes formula.

Try to compute some market sensitivities of the option price.

So is the standard Wiener process. Using tools from the Malliavin Calculus you can show that for any payoff function  

Here are multiple stocks.

This is the correlation structure.