Chapter 5: Models and Equations 279
For other contracts replace the maximum function with
the relevant, even path-dependent, payoff function.
Hull & White (1987) Hull & White considered both
general and specific volatility models. They showed that
when the stock and the volatility are uncorrelated and
the risk-neutral dynamics of the volatility are unaffected
by the stock (i.e.p−λqandqare independent ofS)
then the fair value of an option is the average of the
Black–Scholes values for the option, with the average
taken over the distribution ofσ^2.
Square-root model/Heston (1993) In Heston’s model
dv=(a−bv)dt+c√
vdX 2 ,wherev=σ^2. This has arbitrary correlation between
the underlying and its volatility. This is popular because
there are closed-form solutions for European options.
3/2 model
dv=(av−bv^2 )dt+cv^3 /^2 dX 2 ,wherev=σ^2. Again, this has closed-form solutions.
GARCH-diffusion In stochastic differential equation form
the GARCH(1,1) model is
dv=(a−bv)dt+cv dX 2.Herev=σ^2.
Ornstein–Uhlenbeck process Withy=logv,v=σ^2 ,
dy=(a−by)dt+cdX 2.This model matches real, as opposed to risk-neutral,
data well.