On peacocks and lyrebirds: Australian options, Brownian bridges, and the average of submartingales

Abstract

We introduce a class of stochastic processes, which we refer to as lyrebirds. These extend a class of stochastic processes, which have recently been coined peacocks, but are more commonly known as processes that are increasing in the convex order. We show how these processes arise naturally in the context of Asian and Australian options and consider further applications, such as the arithmetic average of a Brownian bridge and the average of submartingales, including the case of Asian and Australian options where the underlying features constant elasticity of variance or is of Merton jump diffusion type.

Supermartingale Brenier's Theorem with full-marginals constraint

“…Given a family of (integrable) probability measures µ = (µ t ) 0≤t≤1 , we denote by S ∞ (µ) ⊂ S ∞ the collection of all supermartingale measures on Ω such that X t ∼ P µ t for all t ∈ [0, 1]. In particular, from Ewald and Yor [33], we know that S ∞ (µ) is non-empty if and only if the family (µ t ) 0≤t≤1 is non-decreasing in convex-decreasing order and t → µ t is right-continuous.…”