Maximizing functionals of the maximum in the Skorokhod embedding problem and an application to variance swaps

TL;DR

This paper extends the Azéma-Yor and Perkins solutions of the Skorokhod embedding problem, showing they optimize expected values of a broad class of functions involving the process and its maximum, with applications to variance swap pricing.

Contribution

It generalizes the optimality properties of Azéma-Yor and Perkins embeddings to a wider class of functions and explores convergence properties of associated stopping times.

Findings

Azéma-Yor maximizes and Perkins minimizes expected joint functions of process and maximum.

For monotonic functions g, Azéma-Yor minimizes and Perkins maximizes the integral of g over the stopped process.

Results provide model-independent bounds on variance swap prices.

Abstract

The Az\'{e}ma-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 $F(W_{\tau},S_{\tau})$ depending on the joint law of the stopped process and the maximum. Moreover, for monotonic functions $g$, they also maximize and minimize $\mathbb {E}[\int_0^{\tau}g(S_t)\,dt]$ amongst embeddings of $\mu$, although, perhaps surprisingly, we show that for increasing $g$ the Az\'{e}ma-Yor embedding minimizes this quantity, and the Perkins embedding maximizes it. For $g(s)=s^{-2}$ we show how these results are useful in calculating model independent bounds on the prices of variance swaps. Along the way we also consider whether $\mu_n$ converges weakly to $\mu$ is a sufficient condition for the associated Az\'{e}ma-Yor and Perkins stopping times to converge. In the case of the Az\'{e}ma-Yor embedding, if the potentials at zero also converge, then the stopping times converge almost surely, but for the Perkins embedding this need not be the case. However, under a further condition on the convergence of atoms at zero, the Perkins stopping times converge in probability (and hence converge almost surely down a subsequence).