13 - Volatility forecasting in a tick data model

Abstract

In the Black–Scholes paradigm, the variance of the change in log price during a time interval is proportional to the length of the time interval, but this appears not to hold in practice, as is evident by implied volatility smile effects. This chapter reviews how the variance depends on tin a tick data model. The approach of Rogers and Zane was to model the tick data itself. An event in the tick record of the trading of some asset consists of three numbers: the time at which the event happened, the price at which the asset traded, and an amount of the asset that changed hands. The natural model for tick data leads to the functional form for the “volatility” period. This appears to be consistent with the observed non-Black–Scholes behavior of share prices in various ways. First, implied volatility typically decreases with time to expiry, and the “volatility” in this model displays this feature. Second, log returns look more nearly Gaussian over longer time periods, and one may see this reflected here in that if one assumes the notional price is a Brownian motion with constant volatility and drift, then the log return is a sum of a Gaussian part and two noise terms with common variance.