When terminal facelift enforces Delta constraints

TL;DR

This paper explores conditions under which super-replication prices with convex constraints match the exact prices of facelift-transformed claims, providing insights into multi-dimensional models and practical constraints.

Contribution

It introduces a necessary and sufficient condition for the equivalence of super-replication and facelift prices, based on the dynamics and constraints in multi-dimensional models.

Findings

In 1D, the property holds for any local volatility model.

An analytical condition is established for multi-dimensional cases.

The study covers practical scenarios like Black-Scholes and short selling restrictions.

Abstract

This paper deals with the super-replication of non path-dependent European claims under additional convex constraints on the number of shares held in the portfolio. The corresponding super-replication price of a given claim has been widely studied in the literature and its terminal value, which dominates the claim of interest, is the so-called facelift transform of the claim. We investigate under which conditions the super-replication price and strategy of a large class of claims coincide with the exact replication price and strategy of the facelift transform of this claim. In one dimension, we observe that this property is satisfied for any local volatility model. In any dimension, we exhibit an analytical necessary and sufficient condition for this property, which combines the dynamics of the stock together with the characteristics of the closed convex set of constraints. To obtain this condition, we introduce the notion of first order viability property for linear parabolic PDEs. We investigate in details several practical cases of interest: multidimensional Black Scholes model, non-tradable assets or short selling restrictions.