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

Carraro, Laurent, Nicole El Karoui+1 more|arXiv (Cornell University)|Feb 8, 2009

Stochastic processes and financial applications25 references29 citations

TL;DR

This paper establishes a deep connection between Azéma-Yor processes, the Bachelier-drawdown equation, and optimal martingale representations. It proves that any process satisfying the drawdown constraint (staying above a function of its past maximum) is uniquely an Azéma-Yor process, and that solutions to the Bachelier SDE are precisely such drawdown-constrained processes. The key contribution is a characterization of Azéma-Yor martingales as optimal under concave order among martingales whose maximum stochastically dominates a given benchmark, unifying solutions to the Skorokhod embedding problem and portfolio insurance optimization.

ABSTRACT

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.

Motivation & Objective

• To characterize the class of processes satisfying the drawdown constraint, i.e., those that remain above a function of their past maximum.
• To establish the equivalence between the Bachelier stochastic differential equation and the drawdown equation, showing their solutions are precisely Azéma-Yor processes.
• To demonstrate that Azéma-Yor martingales are optimal under the concave order among uniformly integrable martingales whose maximum stochastically dominates a given benchmark distribution.
• To provide a new, unified interpretation of the Azéma-Yor solution to the Skorokhod embedding problem via terminal law construction.
• To apply the theory to portfolio insurance, showing that Azéma-Yor martingales are optimal for pathwise constraints in constrained portfolio optimization.

Proposed method

• Introduce a group structure on the set of Azéma-Yor processes indexed by increasing, absolutely continuous functions U, enabling a systematic analysis of their properties.
• Prove that any semimartingale with a continuous running maximum is an Azéma-Yor process by showing it satisfies the Bachelier-drawdown SDE.
• Establish the equivalence between the Bachelier SDE and the drawdown equation, where solutions are required to satisfy Y_t ≥ w(Y^*_t) for a given function w.
• Use the Hardy-Littlewood transform and its inverse to relate the average value at risk (AVaR) to the terminal law of the process.
• Construct Azéma-Yor martingales with a prescribed terminal law by solving for the appropriate function U, thereby recovering the Azéma-Yor solution to the Skorokhod embedding problem.
• Apply stochastic order theory to show that Azéma-Yor martingales dominate all other uniformly integrable martingales with the same maximum stochastic dominance in the concave order of terminal values.

Experimental results

Research questions

• RQ1 What is the precise mathematical relationship between the Bachelier stochastic differential equation and the drawdown constraint?
• RQ2 Can every process satisfying the drawdown property be represented as an Azéma-Yor process?
• RQ3 How can Azéma-Yor martingales be constructed to have a given terminal distribution?
• RQ4 What is the optimal martingale under the concave order among all uniformly integrable martingales whose maximum stochastically dominates a given benchmark?
• RQ5 How does the theory of Azéma-Yor processes unify solutions to the Skorokhod embedding problem and portfolio insurance problems?

Key findings

• The Bachelier SDE is equivalent to the drawdown equation, and its unique strong solutions are precisely the Azéma-Yor processes.
• Any process satisfying the drawdown constraint Y_t ≥ w(Y^*_t) for a given function w is necessarily an Azéma-Yor process MU(X) for some nonnegative semimartingale X.
• The Azéma-Yor martingale with terminal law ν is optimal in the concave order among all uniformly integrable martingales whose maximum stochastically dominates a given benchmark distribution.
• The construction of an Azéma-Yor martingale with a given terminal law ν is achieved by solving for the function U such that the terminal value is distributed as ν, which recovers the Azéma-Yor solution to the Skorokhod embedding problem.
• The optimal martingale for the concave order problem and the optimal martingale for the stochastic dominance problem (maximizing the maximum) are the same, both being Azéma-Yor martingales.
• For a floor process Ft = g(Nt), the Azéma-Yor martingale MU(N) is optimal under the concave order among all uniformly integrable martingales dominating Ft pathwise, and this optimality holds even when restricted to the larger class of martingales dominating Ft in distribution.