Volatility Smiles

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‣ If there is a large one-day return it will increase the volatilityinstantaneously, but the estimate of volatility will stay raiseduntil N days later when that return drops out of the sample. Exponentially Weighted Moving Average‣ Now let us consider time-varying volatility. We don’t justhave one σ but must consider σ n , our estimate of the volatilityon the nth day, using data available up to that point.‣ Exponentially Weighted Moving Average estimates eliminatethe plateau-effect of the simpler moving average volatilityestimate.‣ The earlier models give every entry an equal weight whereasthis model weighs the most recent ones most and decays theweight exponentially as the returns become further.12
‣ The parameter λ must be greater than zero and less than one.The more recent the return, the more weight is attached. Thesum extends back to the beginning of time.‣ The coefficient of 1 − λ ensures that the weights all add toone. This expression can be simplified to, Mean Reversion and GARCH Models‣ If we believe that volatility tends to vary about a long-termmean σ, then we could use,13
Page 2 and 3: ‣ The differences between these t
Page 4 and 5: Empirical Results‣ We have just s
Page 6 and 7: ‣ The volatility used to price a
Page 8 and 9: Equity Options when Expecting a Lar
Page 10 and 11: ‣ The implied volatility of a Eur
Page 14: ‣ Here there is a weighting assig

Volatility Smiles

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