15.4 Recommended Literature

The presentation of this chapter closely follows the work of Härdle and Hafner (2000). The standard ARCH model originated in Engle (1982), the development of EGARCH in Nelson (1991) and TGARCH in Zakoian (1994) (for the standard deviation) and Glosten et al. (1993) (for the variance). Non-parametric and semi-parametric variants of the ARCH model were suggested and studied by Carroll et al. (2002) and Hafner (1998). The classical option pricing model with stochastic volatility originated in Hull and White (1987). Hull and White implicitly assume that the market price of the risk of the volatility is zero, whereas in Melino and Turnbull (1990) it is different from zero, constant and exogenous. Empirical evidence for the valuation of risk of the volatility is given in Wiggins (1987). Renault and Touzi (1996) generalize the model from Hull and White (1987), in that they allow a market price of the risk for the volatility, which itself can vary over time. The concept of minimizing the quadratic loss of a hedge portfolio is given in Föllmer and Sondermann (1991) and Föllmer and Schweizer (1991). The practical procedure to implement ``15 minutes old'' implied volatility into the Black/Scholes formula, was successfully used in Bossaerts and Hillion (1993).