Convex order for path-dependent derivatives: a dynamic programming approach

We investigate the (functional) convex order of for various continuous martingale processes, either with respect to their diffusions coefficients for L\'evy-driven SDEs or their integrands for stochastic integrals. Main results are bordered by counterexamples. Various upper and lower bounds can be derived for path wise European option prices in local volatility models. In view of numerical applications, we adopt a systematic (and symmetric) methodology: (a) propagate the convexity in a {\em simulatable} dominating/dominated discrete time model through a backward induction (or linear dynamical principle); (b) Apply functional weak convergence results to numerical schemes/time discretizations of the continuous time martingale satisfying (a) in order to transfer the convex order properties. Various bounds are derived for European options written on convex pathwise dependent payoffs. We retrieve and extend former results obtains by several authors since the seminal 1985 paper by Hajek . In a second part, we extend this approach to Optimal Stopping problems using a that the Snell envelope satisfies (a') a Backward Dynamical Programming Principle to propagate convexity in discrete time; (b') satisfies abstract convergence results under non-degeneracy assumption on filtrations. Applications to the comparison of American option prices on convex pathwise payoff processes are given obtained by a purely probabilistic arguments.Comment: 48

Max-Plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance

We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any supermartingale of class $(\mathcal{D})$ as a conditional expectation of some running supremum process. As an application, we show how the Max-Plus supermartingale decomposition allows, in particular, to solve the American optimal stopping problem without having to compute the option price. Some illustrative examples based on one-dimensional diffusion processes are then provided. Another interesting application concerns the portfolio insurance. Hence, based on the ``Max-Plus martingale,'' we solve in the paper an optimization problem whose aim is to find the best martingale dominating a given floor process (on every intermediate date), w.r.t. the convex order on terminal values.Comment: Published in at http://dx.doi.org/10.1214/009117907000000222 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Az\'ema-Yor processes, their optimal properties and the Bachelier-drawdown equation

We study the class of Az\'ema-Yor processes defined from a general semimartingale with a continuous running maximum process. We show that they arise as unique strong solutions of the Bachelier stochastic differential equation which we prove is equivalent to the drawdown equation. Solutions of the latter have the drawdown property: they always stay above a given function of their past maximum. We then show that any process which satisfies the drawdown property is in fact an Az\'ema-Yor process. The proofs exploit group structure of the set of Az\'ema-Yor processes, indexed by functions, which we introduce. We investigate in detail Az\'ema-Yor martingales defined from a nonnegative local martingale converging to zero at infinity. We establish relations between average value at risk, drawdown function, Hardy-Littlewood transform and its inverse. In particular, we construct Az\'ema-Yor martingales with a given terminal law and this allows us to rediscover the Az\'ema-Yor solution to the Skorokhod embedding problem. Finally, we characterize Az\'ema-Yor martingales showing they are optimal relative to the concave ordering of terminal variables among martingales whose maximum dominates stochastically a given benchmark.Comment: Published in at http://dx.doi.org/10.1214/10-AOP614 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal dual martingales, their analysis and application to new algorithms for Bermudan products

In this paper we introduce and study the concept of optimal and surely optimal dual martingales in the context of dual valuation of Bermudan options, and outline the development of new algorithms in this context. We provide a characterization theorem, a theorem which gives conditions for a martingale to be surely optimal, and a stability theorem concerning martingales which are near to be surely optimal in a sense. Guided by these results we develop a framework of backward algorithms for constructing such a martingale. In turn this martingale may then be utilized for computing an upper bound of the Bermudan product. The methodology is pure dual in the sense that it doesn't require certain input approximations to the Snell envelope. In an It\^o-L\'evy environment we outline a particular regression based backward algorithm which allows for computing dual upper bounds without nested Monte Carlo simulation. Moreover, as a by-product this algorithm also provides approximations to the continuation values of the product, which in turn determine a stopping policy. Hence, we may obtain lower bounds at the same time. In a first numerical study we demonstrate the backward dual regression algorithm in a Wiener environment at well known benchmark examples. It turns out that the method is at least comparable to the one in Belomestny et. al. (2009) regarding accuracy, but regarding computational robustness there are even several advantages.Comment: This paper is an extended version of Schoenmakers and Huang, "Optimal dual martingales and their stability; fast evaluation of Bermudan products via dual backward regression", WIAS Preprint 157