Functional It^ o calculus and Applications

Field: Mathematics (Physical Sciences)

This thesis studies extensions of the Itô calculus to a functional setting, using analytical and probabilistic methods, and applications to the pricing and hedging of derivative securities. The first chapter develops a non-anticipative pathwise calculus for functionals of two cadlag paths, with a predictable dependence in the second one. This calculus is a functional generalization of Follmer's analytical approach to Itô calculus. An Itô-type change of variable formula is obtained for non-anticipative functionals on the space of right-continuous paths with left limits, using purely analytical methods. The main tool is the Dupire derivative, a Gateaux derivative for non-anticipative functionals on the space of right-continuous paths with left limits. Our framework implies as a special case a pathwise functional Itô calculus for cadlag semimartingales and Dirichlet processes. It is shown how this analytical HO' formula implies a probabilistic Itô formula for general cadlag semimartingales. In the second chapter, a functional extension of the Itô formula is derived using stochastic analysis tools and used to obtain a constructive martingale representation theorem for a class of continuous martingales verifying a regularity property. By contrast with the Clark-Haussmann-Ocone formula, this representation involves non-anticipative quantities which can be computed pathwise. These results are used to construct a weak derivative acting on square-integrable martingales, which is shown to be the inverse of the Itô integral, and derive an integration by parts formula for Itô stochastic integrals. We show that this weak derivative may be viewed as a non-anticipative "lifting" of the Malliavin derivative. Regular functionals of an Itô martingale which have the local martingale property are characterized as solutions of a functional differential equation, for which a uniqueness result is given. It is also shown how a simple verification theorem based on a functional version of the Hamilton-Jacobi-Bellman equation can be stated for a class of path-dependent stochastic control problems. In the third chapter, a generalization of the martingale representation theorem is given for functionals satisfying the regularity assumptions for the functional Itô formula only in a local sense, and a sufficient condition taking the form of a functional differential equation is given for such locally regular functional to have the local martingale property. Examples are given to illustrate that the notion of local regularity is necessary to handle processes arising as the prices of financial derivatives in computational finance. In the final chapter, functional Itô calculus for locally regular functionals is applied to the sensitivity analysis of path-dependent derivative securities, following an idea of Dupire. A general valuation functional differential equation is given, and many examples show that all usual options in local volatility model are actually priced by this equation. A definition is given for the usual sensitivities of a derivative, and a rigorous expression of the concept of Γ– Θ tradeoff is given. This expression is used together with a perturbation result for stochastic differential equations to give an expression for the Vega bucket exposure of a path-dependent derivative, as well as its of Black-Scholes Delta and Delta at a given skew stickiness ratio. An efficient numerical algorithm is proposed to compute these sensitivities in a local volatility model.