954 Continuous-Time Stochastic Volatility Models
mean reversion parameter has been found to alter dramatically at different data
frequencies, suggesting that volatility may be driven by multiple factors (for
example, Chacko and Viceira, 2003). Other studies (such as Bakshi, Ju and
Ou-Yang, 2006) suggest that volatility displays nonlinear mean reversion with
reversals at both high and low levels of the volatility spectrum. Empirical evi-
dence shows that volatility displays so-called level effects (for example, Jones,
2003) whereby periods of high volatility usually coincide with periods of volatile
volatility. Finally, recent studies suggest that volatility and asset returns display
correlated jumps during times of market stress (for example, Eraker, Johannes
and Polson, 2003).- Smiles, skews and implied volatility. It has long been documented that the Black–
Scholes (1973) model is not consistent with observed option prices. Given a
set of option prices, one can invert the Black–Scholes formula and backout
the implied volatility that sets the observed price equal to the Black–Scholes
price. If the Black–Scholes model was correct, then, as a function of strike prices,
implied volatility would be a flat line. However, it is well known that implied
volatility displays a U-shaped pattern (the implied volatility “smile”) in foreign
exchange derivatives markets and a downward sloping curve (the implied
volatility “skew”) in index option markets (see Bates, 1996a). Continuous-time
stochastic volatility models have been proposed as an alternative to Black–
Scholes in order to explain the empirical patterns of implied volatility curves
and smiles. Figure 19.2 depicts the evolution of daily S&P500 prices and the
1,8001,600S&P 500
VXO1,4001,2001,000800600400200060504030201002/01/199002/01/199302/01/199602/01/199902/01/2002 02/01/20050Figure 19.2 S&P500 prices (solid line) and VXO values (dotted line) over the time period
January 2, 1990, to December 31, 2007