摘要翻译：
这张纸定价并复制金融衍生品，其以$T美元的回报是在事后确定的最佳连续再平衡投资组合（或固定部分投注方案）中积累的1美元存款。对于单一股票的Black-Scholes市场，Ordentlich and Cover(1998)只在时间-0时对该衍生品定价，给出$C_0=1+\sigma\sqrt{t/(2\pi)}$.当然，一般的time-$t$price不等于$1+\sigma\sqrt{(T-t)/(2\pi)}$。我完成了Ordentlich-Cover(1998)的分析，得出了任何时候的价格$T$。相比之下，我还研究了事后看来最好的杠杆再平衡规则的更自然的情况。这将产生$C(S，t)=\sqrt{t/t}\cdot\,\exp\{rt+\sigma^2b(S，t)^2\cdot t/2\}$,其中$B(S，t)$是经过观察的历史$[0,t]$后知后觉的最佳再平衡规则。我证明了复制策略相当于在$[t，t+dt]区间内将财富的分数$b(S，t)$押注于股票。$这一事实适用于具有$n$相关股票的几何布朗运动的一般市场：我们得到$c(S，t)=(t/t)^{n/2}\exp(Rt+b'\sigma b\cdot t/2)$,其中$\sigma是单位时间内瞬时收益的协方差。这一结果与Cover在他的“通用投资组合理论”(1986，1991，1996，1998)中导出的$\Mathcal{O}(T^{n/2}）$“普遍性成本”相吻合，后者在离散时间内超复制了相同的导数。事后看来，复制策略以与最佳杠杆再平衡规则相同的渐进速度复合资金，从而渐进地击败市场。很自然地，我们发现，美国风格的Cover的导数从来没有在均衡的早期使用过。
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英文标题：
《Exact Replication of the Best Rebalancing Rule in Hindsight》
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作者：
Alex Garivaltis
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最新提交年份：
2019
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Pricing of Securities        证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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一级分类：Economics        经济学
二级分类：General Economics        一般经济学
分类描述：General methodological, applied, and empirical contributions to economics.
对经济学的一般方法、应用和经验贡献。
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一级分类：Economics        经济学
二级分类：Theoretical Economics        理论经济学
分类描述：Includes theoretical contributions to Contract Theory, Decision Theory, Game Theory, General Equilibrium, Growth, Learning and Evolution, Macroeconomics, Market and Mechanism Design, and Social Choice.
包括对契约理论、决策理论、博弈论、一般均衡、增长、学习与进化、宏观经济学、市场与机制设计、社会选择的理论贡献。
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一级分类：Quantitative Finance        数量金融学
二级分类：Economics        经济学
分类描述：q-fin.EC is an alias for econ.GN. Economics, including micro and macro economics, international economics, theory of the firm, labor economics, and other economic topics outside finance
q-fin.ec是econ.gn的别名。经济学，包括微观和宏观经济学、国际经济学、企业理论、劳动经济学和其他金融以外的经济专题
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一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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一级分类：Quantitative Finance        数量金融学
二级分类：Portfolio Management        项目组合管理
分类描述：Security selection and optimization, capital allocation, investment strategies and performance measurement
证券选择与优化、资本配置、投资策略与绩效评价
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英文摘要：
  This paper prices and replicates the financial derivative whose payoff at $T$ is the wealth that would have accrued to a $\$1$ deposit into the best continuously-rebalanced portfolio (or fixed-fraction betting scheme) determined in hindsight. For the single-stock Black-Scholes market, Ordentlich and Cover (1998) only priced this derivative at time-0, giving $C_0=1+\sigma\sqrt{T/(2\pi)}$. Of course, the general time-$t$ price is not equal to $1+\sigma\sqrt{(T-t)/(2\pi)}$. I complete the Ordentlich-Cover (1998) analysis by deriving the price at any time $t$. By contrast, I also study the more natural case of the best levered rebalancing rule in hindsight. This yields $C(S,t)=\sqrt{T/t}\cdot\,\exp\{rt+\sigma^2b(S,t)^2\cdot t/2\}$, where $b(S,t)$ is the best rebalancing rule in hindsight over the observed history $[0,t]$. I show that the replicating strategy amounts to betting the fraction $b(S,t)$ of wealth on the stock over the interval $[t,t+dt].$ This fact holds for the general market with $n$ correlated stocks in geometric Brownian motion: we get $C(S,t)=(T/t)^{n/2}\exp(rt+b'\Sigma b\cdot t/2)$, where $\Sigma$ is the covariance of instantaneous returns per unit time. This result matches the $\mathcal{O}(T^{n/2})$ "cost of universality" derived by Cover in his "universal portfolio theory" (1986, 1991, 1996, 1998), which super-replicates the same derivative in discrete-time. The replicating strategy compounds its money at the same asymptotic rate as the best levered rebalancing rule in hindsight, thereby beating the market asymptotically. Naturally enough, we find that the American-style version of Cover's Derivative is never exercised early in equilibrium.
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PDF链接：
https://arxiv.org/pdf/1810.02485