The Variance of Standard Options

The most commonly used theory of option pricing is the Black-Scholes PDE model, which is used to obtain the expectation (first moment) of an option's current value. We show that modified Black-Scholes PDEs can be used to obtain the n-th moment of an option's current value. We demonstrate how to find the second moment and the zeroth order moment for  different standard options (European options, barrier options, American options), and use this to find the variances of the option values, and the probabilities to  expire worthless.These latter two quantities give us a perspective of the option's risk, which is important in investment decisions and in pricing theories.