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The aim of this article is to nd bounds on the prices of exotic derivatives , and in particular the lookback option, in terms of the (market) prices of call options. This is achieved without making explicit assumptions about the dynamics of the price process of the underlying asset, but rather by inferring information about the potential distribution of(More)
The paper proposes an original class of models for the continuous time price process of a nancial security with non-constant volatility. The idea is to deene instantaneous volatility in terms of exponentially-weighted moments of historic log-price. The instantaneous volatility is therefore driven by the same stochastic factors as the price process, so that(More)
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(More)
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(More)
A variance swap is a derivative with a path-dependent payoff which allows investors to take positions on the future variability of an asset. In the idealised setting of a continuously monitored variance swap written on an asset with continuous paths it is well known that the variance swap payoff can be replicated exactly using a portfolio of puts and calls(More)
We develop a class of pathwise inequalities of the form H(Bt)≥ Mt + F (Lt), where Bt is Brownian motion, Lt its local time at zero and Mt a local martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive constructions and optimality results of Vallois’(More)