Robust price bounds for the forward starting straddle

<Para ID="Par1">We consider the problem of giving a robust, model-independent, lower bound on the price of a forward starting straddle with payoff <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$|F_{T_{1}} - F_{T_{0}}|$</EquationSource> </InlineEquation>, where 0>T <Subscript>0</Subscript>>T <Subscript>1</Subscript>. Rather than assuming a model for the underlying forward price (F <Subscript> t </Subscript>)<Subscript> t≥0</Subscript>, we assume that call prices for maturities T <Subscript>0</Subscript>>T <Subscript>1</Subscript> are given and hence that the marginal laws of the underlying are known. The primal problem is to find the model that 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. <Para ID="Par2">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. Copyright Springer-Verlag Berlin Heidelberg 2015