No abstract

In this article we are interested in option pricing in markets with bubbles. A bubble is defined to be a price process which, when discounted, is a local martingale under the risk-neutral measure but not a martingale. We give examples of bubbles both where volatility increases with the price level, and where the bubble is the result of a feedback mechanism. In a market with a bubble many standard results from the folklore become false. Put-call parity fails, the price of an American call exceeds that of a European call and call prices are no longer increasing in maturity (for a fixed strike). We show how these results must be modified in the presence of a bubble. It turns out that the option value depends critically on the definition of admissible strategy, and that the standard mathematical definition may not be consistent with the definitions used for trading. Copyright Springer-Verlag Berlin/Heidelberg 2005Bubbles, feedback, local martingales, derivative pricing, put-call parity,

Suppose we are given a set of prices of European call options over a finite range of strike prices and exercise times, written on a financial asset with deterministic dividends which is traded in a frictionless market with no interest rate volatility. We ask: when is there an arbitrage opportunity? We give conditions for the prices to be consistent with an arbitrage-free model (in which case the model can be realized on a finite probability space). We also give conditions for there to exist an arbitrage opportunity which can be locked in at time zero. There is also a third boundary case in which prices are recognizably misspecified, but the ability to take advantage of an arbitrage opportunity depends upon knowledge of the null sets of the model.

This set of lecture notes is concerned with the following pair of ideas and concepts:1) The Skorokhod Embedding problem (SEP) is, given a stochastic process X = (Xt) t≥0 and a measure µ on the state space of X, to find a stopping time τ such that the stopped process Xτ has law µ. Most often we take the process X to be Brownian motion, and µ to be a centred probability measure.2) The standard approach for the pricing of financial options is to postulate a model and then to calculate the price of a contingent claim as the suitably discounted, risk-neutral expectation of the payoff under that model. In practice we can observe traded option prices, but know little or nothing about the model. Hence the question arises, if we know vanilla option prices, what can we infer about the underlying model?If we know a single call price, then we can calibrate the volatility of the Black-Scholes model (but if we know the prices of more than one call then together they will typically be inconsistent with the Black-Scholes model). At the other extreme, if we know the prices of call options for all strikes and maturities, then we can find a unique martingale diffusion consistent with those prices.If we know call prices of all strikes for a single maturity, then we know the marginal distribution of the asset price, but there may be many martingales with the same marginal at a single fixed time. Any martingale with the given marginal is a candidate price process. On the other hand, after a time change it becomes a Brownian motion with a given distribution at a random time. Hence there is a 1-1 correspondence between candidate price processes which are consistent with observed prices, and solutions of the Skorokhod embedding problem.These notes are about this correspondence, and the idea that extremal solutions of the Skorokhod embedding problem lead to robust, model independent prices and hedges for exotic options.