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Abstract
This thesis investigates implied volatility in general classes of stock price models.
To begin with, we take a very general view. We find that implied volatility is always, everywhere, and
for every expiry well-defined only if the stock price is a non-negative martingale. We also derive sufficient and close
to necessary conditions for an implied volatility surface to be free from static arbitrage. In this context, free from
static arbitrage means that the call price surface generated by the implied volatility surface is free from static
arbitrage.
We also investigate the small time to expiry behaviour of implied volatility. We do this in almost complete generality,
assuming only that the call price surface is non-decreasing and right continuous in time to expiry and that the call
surface satisfies the no-arbitrage bounds (S-K)+≤ C(K, τ)≤ S. We used S to denote the current stock price,
K to be a option strike price, τ denotes time to expiry, and C(K, τ) the price of the K strike option
expiring in τ time units. Under these weak assumptions, we obtain exact asymptotic formulae relating the call price
surface and the implied volatility surface close to expiry.
We apply our general asymptotic formulae to determining the small time to expiry behaviour of implied volatility in
a variety of models. We consider exponential Lévy models, obtaining new and somewhat surprising results. We
then investigate the behaviour close to expiry of stochastic volatility models in the at-the-money case. Our results
generalise what is already known and by a novel method of proof. In the not at-the-money case, we consider local
volatility models using classical results of Varadhan. In obtaining the asymptotics for local volatility
models, we use a representation of the European call as an integral over time to expiry. We devote an entire
chapter to representations of the European call option; a key role is played by local time and the argument of
Klebaner. A novel alternative that is especially useful in the local volatility case is also presented.