Book Review, Risk Magazine
Derivatives in Financial Markets with Stochastic Volatility, by
Jean-Pierre Fouque, George Papanicolaou and K. Ronnie Sircar,
Cambridge University Press 2000. xiv + 201pp. ISBN 0-521-79163-4
Reviewed by Mark Davis, Imperial College London
In an ideal Black-Scholes world, implied volatility would be constant across
all strikes and maturities. Needless to say, it isn’t. The ‘volatility surface’
of traded option implied volatilities exhibits the familiar ‘smile’ or ‘smirk’ be-
haviour with, typically, out-of-the-money calls and puts quoted at lower implied
vols. Further, a plot of daily implied volatility of at-the-money options with
fixed time to maturity looks quite ‘stochastic’. Of course, building these facts
into your model is an essential part of option trading and risk management.
The question is how to do it.
There are, roughly speaking, three ways:
• Make volatility a function of price level; thus no extra randomness is
introduced and vol only fluctuates because price does.
• Introduce a stochastic process model – generally mean-reverting – to rep-
resent the instantaneous vol in a price model that otherwise looks like
• Replace the Black-Scholes price model by something completely different,
for example a Lévy process model with jumps.
The first is the least radical departure from classic Black-Scholes. Its advantage
is that the market is still complete, so that pricing and hedging à la Black-
Scholes is still correct, and models can be built that fit the observed smile. On
the other hand, these models may be over-parametrized for hedging, and even
cursory econometric analysis tells you that price and volatility are not perfectly
correlated. The second alternative – emphasized for example by Hull and White
– has many attractions: it leads to quite simple formulas, and it gives a realistic
representation of the evolution of implied vol over time. The market is now
incomplete; it can be completed by including a traded option as an independent
asset, but in many dif