Preliminaries

The inputs

Simulations

Black-Scholes formula

The input vector for all formulas is

The outputs

The stock price scenario for the above parameters is graphed below.

The option price is highly correlated with the underlying asset's price. It might not be obvious graphically, but the volatility of the option price approaches zero as we reach the maturity and the option reaches its deterministic intrinsic payoffs.

The delta of the option follows more or less the patterns of the option price or the price of the underlying asset. One very important difference is that the delta will approach either one (if the option ends up in the money) or zero (if the option ends up out of the money). This can be verified graphically below.

All rebalances of the delta hedged portfolio are shown below. Theory says that as the rebalancing interval goes to zero, the value of the delta insurance will be equal to the price of the option that provides this insurance.

The value of the insurance as the derived by the delta hedging strategy is comparable to the value of the option.