154 Frequently Asked Questions In Quantitative Finance
variance, sometimes one uses high-low-open-close data
and not just closing prices, and sometimes one models
the logarithm of volatility. The latter seems to be quite
promising because there is evidence that actual volatil-
ity is lognormally distributed. Other work in this area
decomposes the volatility of a stock into components,
market volatility, industry volatility and firm-specific
volatility. This is similar toCAPMfor returns.
Deterministic models: The simple Black–Scholes formulæ
assume that volatility is constant or time dependent.
But market data suggests that implied volatility varies
with strike price. Such market behaviour cannot be con-
sistent with a volatility that is a deterministic function
of time. One way in which the Black–Scholes world can
be modified to accommodate strike-dependent implied
volatility is to assume that actual volatility is a func-
tion of both time and the price of the underlying. This
is thedeterministic volatility(surface) model. This is
the simplest extension to the Black–Scholes world that
can be made to be consistent with market prices. All it
requires is that we haveσ(S,t), and the Black–Scholes
partial differential equation is still valid. The interpre-
tation of an option’s value as the present value of the
expected payoff under a risk-neutral random walk also
carries over. Unfortunately the Black–Scholes closed-
form formulæ are no longer correct. This is a simple and
popular model, but it does not capture the dynamics of
implied volatility very well.
Stochastic volatility: Since volatility is difficult to measure,
and seems to be forever changing, it is natural to model
it as stochastic. The most popular model of this type is
due to Heston. Such models often have several param-
eters which can either be chosen to fit historical data
or, more commonly, chosen so that theoretical prices
calibrate to the market. Stochastic volatility models
are better at capturing the dynamics of traded option
