Robust price bounds for the forward starting straddle

TL;DR

This paper derives explicit model-independent lower bounds for the price of a forward starting straddle using observed call prices, providing a robust hedging strategy without assuming specific underlying models.

Contribution

It introduces a novel approach to compute explicit bounds and semi-static subhedges for forward starting options based solely on marginal laws from call prices.

Findings

Explicit coupling minimizing the option price derived

Semi-static subhedge characterized explicitly

Bounds applicable under minimal assumptions on laws

Abstract

In this article we consider the problem of giving a robust, model-independent, lower bound on the price of a forward starting straddle with payoff $|F_{T_1} - F_{T_0}|$ where $0<T_0<T_1$. Rather than assuming a model for the underlying forward price $(F_t)_{t \geq 0}$, we assume that call prices for maturities $T_0<T_1$ are given and hence that the marginal laws of the underlying are known. The primal problem is to find the model which is consistent with the observed call prices, and for which the price of the forward starting straddle is minimised. The dual problem is to find the cheapest semi-static subhedge. Under an assumption on the supports of the marginal laws, but no assumption that the laws are atom-free or in any other way regular, we derive explicit expressions for the coupling which minimises the price of the option, and the form of the semi-static subhedge.