Volatility Smiles

‣ Here there is a weighting assigned to each of the long-runvolatility estimate and the current estimate based on the last nreturns. This is called an ARCH model (for AutoregressiveConditional Heteroscedasticity).‣ Alpha is the weight that we give to our estimate of “meanvolatility” and the remainder is split between n instances ofrecent volatility measures.‣ If we combine the ARCH and Exponential Weighting, wearrive at a Generalized ARCH (or GARCH) model,‣ All notations maintain their identity, alpha arbitratingbetween our “mean volatility” and recorded data, and lambdadistributing the weights of recent measures in an exponentialfashion.14