Chapter 2: FAQs 155
prices better than deterministic models. However, differ-
ent markets behave differently. Part of this is because
of the way traders look at option prices. Equity traders
look at implied volatility versus strike, FX traders look
at implied volatility versus delta. It is therefore natu-
ral for implied volatility curves to behave differently in
these two markets. Because of this there have grown
up the sticky strike, sticky delta, etc., models, which
model how the implied volatility curve changes as the
underlying moves.
Poisson processes: There are times of low volatility and
times of high volatility. This can be modelled by volatil-
ity that jumps according to a Poisson process.
Uncertain volatility: An elegant solution to the problem of
modelling the unseen volatility is to treat it as uncertain,
meaning that it is allowed to lie in a specified range but
whereabouts in that range it actually is, or indeed the
probability of being at any value, are left unspecified.
With this type of model we no longer get a single option
price, but a range of prices, representing worst-case
scenario and best-case scenario.
References and Further Reading
Avellaneda, M, Levy, A & Paras, A 1995 Pricing and hedging ́
derivative securities in markets with uncertain volatilities.
Applied Mathematical Finance 2 73–88
Derman, E & Kani, I 1994 Riding on a smile.Risk magazine 7
(2) 32–39 (February)
Dupire, B 1994 Pricing with a smile.Risk magazine 7 (1) 18–20
(January)
Heston, S 1993 A closed-form solution for options with stochas-
tic volatility with application to bond and currency options.
Review of Financial Studies 6 327–343
