No-arbitrage bounds for the forward smile given marginals

TL;DR

This paper develops a computational framework to determine no-arbitrage bounds for forward-start options using market data, reconciling semi-infinite programming and dual martingale measure approaches in different models.

Contribution

It introduces a discretisation scheme for the semi-infinite linear programming problem and formulates the dual problem as a finite-dimensional linear program, enabling numerical reconciliation.

Findings

Discretisation reduces problem complexity.

Dual approach can be implemented as a finite linear program.

Reconciliation achieved in Black-Scholes and Heston models.

Abstract

We explore the robust replication of forward-start straddles given quoted (Call and Put options) market data. One approach to this problem classically follows semi-infinite linear programming arguments, and we propose a discretisation scheme to reduce its dimensionality and hence its complexity. Alternatively, one can consider the dual problem, consisting in finding optimal martingale measures under which the upper and the lower bounds are attained. Semi-analytical solutions to this dual problem were proposed by Hobson and Klimmek (2013) and by Hobson and Neuberger (2008). We recast this dual approach as a finite dimensional linear programme, and reconcile numerically, in the Black-Scholes and in the Heston model, the two approaches.