Further extensions

The model can be further generalized by investigating other price processes. The returns of financial assets are usually not normally distributed as assumed in the BSM model, but their tails are fatter than predicted by the Gauss curve. This phenomenon can be described by the GARCH model (General Autoregressive Conditional Heteroscedasticity), where the variance is autocorrelated, which causes a clustering of volatility. Another way of catching the higher probability of extreme returns can be building random jumps into the process. Applying these processes in the model will make the hedging of the derivative even more expensive, thereby increasing the expected value and also the variance of the cost distribution.

We can see that changing the spot price causes the change of the delta that can be measured by the gamma, which is the second derivative of the option price with respect to the spot price. A gamma-neutral portfolio cannot be achieved by exclusively holding the...