Robust price bounds for the forward starting straddle

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 T1 − F T0 | where 0 < T 0 < T 1 . Rather than assuming a model for the underlying forward price (F t ) t≥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.

Optimal Skorokhod Embedding Under Finitely Many Marginal Constraints

“…The main idea of his pioneering work is to exploit some solution of the SEP satisfying some optimality criteria, which yields the model-free hedging strategy and allows to solve together the model-free pricing and hedging problems. Since then, the optimal SEP has received substantial attention from the mathematical finance community and various extensions were achieved in the literature, such as Cox & Hobson [13], Hobson & Klimmek [35], Cox, Hobson & Ob lój [14], Cox & Ob lój [15] and Davis, Ob lój & Raval [16], Ob lój & Spoida [47], etc. A thorough literature is provided in Hobson's survey paper [34].…”

On the Monotonicity Principle of Optimal Skorokhod Embedding Problem

“…We also refer to the subsequent literature on martingale optimal transport by [6,7,8,14,15,19,20,21,23,25], etc.…”

Change of numeraire in the two-marginals martingale transport problem

“…Finally, Hobson and Klimmek [10] consider forward start straddles of type II, whose payoff is |Y − αX| when the strike is α, while we recall for later use that the payoff of a forward start straddle of type I is given by | Y X − α|. In the case α = 1, the authors construct another optimal transference plan giving the model-free subreplication price of a forward start straddle of type II, whose payoff does not satisfy the condition (1.1) above.…”

“…Our main results can be briefly stated as follows. Regarding the Hobson and Klimmek [10] optimal coupling measure, it turns out that the change of numeraire exchanges forward start straddles of type I and type II with strike 1. As consequence, this yields that the optimal transport plan in the subhedging problems is the same for both types of forward start straddles.…”