Multilinear Superhedging of Lookback Options
Abstract
In a pathbreaking paper, Cover and Ordentlich (1998) solved a maxmin portfolio game between a trader (who picks an entire trading algorithm, $\theta(\cdot)$) and "nature," who picks the matrix $X$ of grossreturns of all stocks in all periods. Their (zerosum) 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 constantrebalanced portfolio (or fixedfraction 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) maxmin game. This obtains for performance benchmarks (read: derivatives) that are separately convex and homogeneous in each period's grossreturn vector. For completely arbitrary (even nonmeasurable) performance benchmarks, we show how the axiom of choice can be used to "find" an exact maximin strategy for the trader.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 arXiv:
 arXiv:1810.02447
 Bibcode:
 2018arXiv181002447G
 Keywords:

 Quantitative Finance  Pricing of Securities;
 Economics  Theoretical Economics;
 Quantitative Finance  Computational Finance;
 Quantitative Finance  General Finance;
 Quantitative Finance  Portfolio Management
 EPrint:
 41 pages, 3 figures