elle
(Elle)
#1
options (e.g., on a volatility index). The model we use is the one of Gruenbichler and
Longstaff (1996). They model the volatility process (e.g., the process of a volatility index)
in direct fashion by a square-root diffusion, provided in Equation 14-1. This process is
known to exhibit convenient features for volatility modeling, like positivity and mean
reversion.
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Equation 14-1. Square-root diffusion for volatility modeling
The variables and parameters in Equation 14-1 have the following meanings:
Vt
The time t value of the volatility index (for example, the VSTOXX)
θV
The long-run mean of the volatility index
κV
The rate at which Vt reverts to
ΣV
The volatility of the volatility (“vol-vol”)
θV, κV, and ΣV
Assumed to be constant and positive
Zt
A standard Brownian motion
Based on this model, Gruenbichler and Longstaff (1996) derive the formula provided in
Equation 14-2 for the value of a European call option. In the formula, D(T) is the
appropriate discount factor. The parameter denotes the expected premium for volatility
risk, while is the complementary noncentral
2
distribution.
Equation 14-2. Call option formula of Gruenbichler and Longstaff (1996)
