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.

A variance swap is a derivative with a path-dependent payoff which allows investors to take positions on the future variability of an asset. In the idealised setting of a continuously monitored variance swap written on an asset with continuous paths it is well known that the variance swap payoff can be replicated exactly using a portfolio of puts and calls and a dynamic position in the asset. This fact forms the basis of the VIX contract.But what if we are in the more realistic setting where the contract is based on discrete monitoring, and the underlying asset may have jumps? We show that it is possible to derive model-independent, no-arbitrage bounds on the price of the variance swap, and corresponding sub-and super-replicating strategies. Further, we characterise the optimal bounds. The form of the hedges depends crucially on the kernel used to define the variance swap.

The Azéma-Yor solution (resp., the Perkins solution) of the Skorokhod embedding problem has the property that it maximizes (resp., minimizes) the law of the maximum of the stopped process. We show that these constructions have a wider property in that they also maximize (and minimize) expected values for a more general class of bivariate functions F (Wτ , Sτ ) depending on the joint law of the stopped process and the maximum. Moreover, for monotonic functions g, they also maximize and minimize E[ 1. Introduction. Let W = (W t ) t≥0 be Brownian motion, null at 0, and µ a centered probability measure. Then the Skorokhod embedding problem (SEP) (Skorokhod [21]) is to find a stopping time τ such that the stopped process satisfies W τ ∼ µ. There are many classical solutions to this problem (for a survey, see Ob lój [16]), and further solutions continue to appear in the literature, including most recently Hirsch et al. [9]. Further impetus to the investigation of old and new solutions is derived from the connections between solutions of the SEP and model independent bounds for the prices of options; for a survey, see Hobson [10].Given the multiplicity of solutions to the SEP, it is natural to search for embeddings with additional optimality properties. In particular, if Ψ is a functional of the stopped Brownian path (W t ) 0≤t≤τ , then these constructions aim to maximize Ψ over (a suitable subclass of) embeddings of µ. For example, if F is an increasing function, and S t = sup s≤t W s , then the Azéma-Yor solution [2] maximizes E[F (S τ )] over uniformly integrable embeddings, and the Perkins embedding [17] minimizes the same quantity.Our goal in this paper is to extend this result to functions F = F (W τ , S τ ). Then, subject to regularity conditions, our first result (Theorem 5.3) is that:Suppose Fs(w, s)/(s − w) is monotonic decreasing in w. Then E[F (Wτ , Sτ )] is minimized (resp., maximized) over uniformly integrable embeddings τ of µ by the Azéma-Yor (resp., Perkins) embedding.This result is a tool in the derivation of our second result, Theorem 7.1, which, again subject to regularity conditions is as follows:Suppose g is increasing. Then E[ τ 0 g(Su) du] is minimized (resp., maximized) over uniformly integrable embeddings τ of µ by the Azéma-Yor (resp., Perkins) embedding.

Consider a set of discounted optimal stopping problems for a one-parameter family of objective functions and a fixed diffusion process, started at a fixed point. A standard problem in stochastic control/optimal stopping is to solve for the problem value in this setting.In this article we consider an inverse problem; given the set of problem values for a family of objective functions, we aim to recover the diffusion. Under a natural assumption on the family of objective functions we can characterise existence and uniqueness of a diffusion for which the optimal stopping problems have the specified values. The solution of the problem relies on techniques from generalised convexity theory.