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 option and the underlying asset, as the gamma of the latest is zero, but we have to buy options for the same underlying asset with any maturity or strike price.

Furthermore, if we disregard the assumption of constant volatility, the value of the derivative will be affected not only by the change of the underlying asset's spot price and the change of the remaining time to maturity, but also the change of the underlying asset's volatility. The effect of the changing volatility can be measured by the vega, the first derivative of the option price according to the volatility. A high value of vega causes a notable effect of the volatility on the option price (Hull, 2009). This can cause a situation where the price of the underlying asset is increasing, so the value of a call option should increase while the implied volatility has decreased, and the price of the option may decrease as well. In order to offset the effect of vega, either other options for the same underlying asset are to be bought, or we can hedge volatility with an index called the VIX index, which is a traded index that contains the implied volatilities of options.

This chapter was dedicated to analyzing delta hedging; detailing gamma and vega neutralization is beyond our focus.