Multilinear Superhedging of Lookback Options

TL;DR

This paper extends the universal portfolio framework to a broad class of path-dependent derivatives, providing a method to find optimal superhedging strategies for complex, convex, and homogeneous performance benchmarks.

Contribution

It generalizes the max-min portfolio game to include complex derivatives with convex and homogeneous payoffs, using the axiom of choice for arbitrary benchmarks.

Findings

Derived superhedging strategies for convex, homogeneous derivatives.

Extended the universal portfolio concept to path-dependent derivatives.

Provided a theoretical foundation for optimal superhedging in complex markets.

Abstract

In a pathbreaking paper, Cover and Ordentlich (1998) solved a max-min portfolio game between a trader (who picks an entire trading algorithm, $\theta(\cdot)$) and "nature," who picks the matrix $X$ of gross-returns of all stocks in all periods. Their (zero-sum) game has the payoff kernel $W_\theta(X)/D(X)$, where $W_\theta(X)$ is the trader's final wealth and $D(X)$ is the final wealth that would have accrued to a $\$1$ deposit into the best constant-rebalanced portfolio (or fixed-fraction betting scheme) determined in hindsight. The resulting "universal portfolio" compounds its money at the same asymptotic rate as the best rebalancing rule in hindsight, thereby beating the market asymptotically under extremely general conditions. Smitten with this (1998) result, the present paper solves the most general tractable version of Cover and Ordentlich's (1998) max-min game. This obtains for performance benchmarks (read: derivatives) that are separately convex and homogeneous in each period's gross-return vector. For completely arbitrary (even non-measurable) performance benchmarks, we show how the axiom of choice can be used to "find" an exact maximin strategy for the trader.