回望期权的多重线性超套期保值
Multilinear Superhedging of Lookback Options
摘要：
在一篇开创性的论文中，Cover和Ordentlich（1998）解决了一个交易者（他选择了一个完整的交易算法，$θ(\cdot)$）和“自然”（他选择了所有股票在所有时期的总回报矩阵$X$）之间的最大-最小投资组合博弈。他们的（零和）博弈有一个收益核心$W_θ(X)/D(X)$，其中$W_θ(X)$是交易者的最终财富，$D(X)$是最终财富，它将累积到$\$1$存款中，存入事后确定的最佳恒定再平衡投资组合（或固定比例投注方案）。由此产生的“万能投资组合”以与事后看来的最佳再平衡规则相同的渐进速度将其货币复合，从而在极其一般的条件下渐进地击败市场。受这个结果的影响，本文解决了Cover和Ordentlich（1998）的最大-最小博弈的最一般的易处理版本。这对于在每个时期的总回报向量中分别是凸的和齐次的业绩基准（读作：衍生品）是可以得到的。对于完全任意的（甚至是不可测量的）性能基准，我们展示了如何选择公理可以用来“找到”一个确切的最大最小策略的交易者。
英文摘要：
In a pathbreaking paper, Cover and Ordentlich (1998) solved a max-min portfolio game between a trader (who picks an entire trading algorithm, $θ(\cdot)$) and "nature," who picks the matrix $X$ of gross-returns of all stocks in all periods. Their (zero-sum) game has the payoff kernel $W_θ(X)/D(X)$, where $W_θ(X)$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
