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2022 MCM  2022 年 MCM

Problem C: Trading Strategies
问题 C:交易策略

Background  背景

Market traders buy and sell volatile assets frequently, with a goal to maximize their total return. There is usually a commission for each purchase and sale. Two such assets are gold and bitcoin.
市场交易者经常买卖波动性资产,目的是最大限度地提高他们的总回报。每次购买和销售通常都会收取佣金。两种这样的资产是黄金和比特币。

Figure 1: Gold daily prices, U.S. dollars per troy ounce. Source: London Bullion Market Association, 9/11/2021
图 1:黄金每日价格,每金衡盎司美元。资料来源:伦敦金银市场协会,2021 年 9 月 11 日

Figure 2: Bitcoin daily prices, U.S. dollars per bitcoin. Source: NASDAQ, 9/11/2021
图 2:比特币每日价格,美元兑比特币。资料来源:纳斯达克,2021 年 9 月 11 日

Requirement  要求

You have been asked by a trader to develop a model that uses only the past stream of daily prices to date to determine each day if the trader should buy, hold, or sell their assets in their portfolio.
一位交易员要求您开发一个模型,该模型仅使用过去迄今为止的每日价格流来确定交易员每天是否应该购买、持有或出售其投资组合中的资产。
You will start with $ 1000 $ 1000 $1000\$ 1000 on 9 / 11 / 2016 9 / 11 / 2016 9//11//20169 / 11 / 2016. You will use the five-year trading period, from 9 / 11 / 2016 9 / 11 / 2016 9//11//20169 / 11 / 2016 to 9 / 10 / 2021 9 / 10 / 2021 9//10//20219 / 10 / 2021. On each trading day, the trader will have a portfolio consisting of cash, gold, and bitcoin [C, G, B] in U.S. dollars, troy ounces, and bitcoins, respectively. The initial state is [ 1000 , 0 , 0 ] [ 1000 , 0 , 0 ] [1000,0,0][1000,0,0]. The commission for each transaction (purchase or sale) costs α % α % alpha%\alpha \% of the amount traded. Assume α gold = 1 % α gold  = 1 % alpha_("gold ")=1%\alpha_{\text {gold }}=1 \% and α bitcoin = 2 % α bitcoin  = 2 % alpha_("bitcoin ")=2%\alpha_{\text {bitcoin }}=2 \%. There is no cost to hold an asset.
您将从 $ 1000 $ 1000 $1000\$ 1000 on 9 / 11 / 2016 9 / 11 / 2016 9//11//20169 / 11 / 2016 开始。您将使用五年交易期,从 9 / 11 / 2016 9 / 11 / 2016 9//11//20169 / 11 / 2016 9 / 10 / 2021 9 / 10 / 2021 9//10//20219 / 10 / 2021 。在每个交易日,交易者将拥有一个投资组合,包括现金、黄金和比特币 [C、G、B],分别是美元、金衡盎司和比特币。初始状态为 [ 1000 , 0 , 0 ] [ 1000 , 0 , 0 ] [1000,0,0][1000,0,0] 。每笔交易(购买或出售)的佣金成本 α % α % alpha%\alpha \% 为交易金额。假设 α gold = 1 % α gold  = 1 % alpha_("gold ")=1%\alpha_{\text {gold }}=1 \% α bitcoin = 2 % α bitcoin  = 2 % alpha_("bitcoin ")=2%\alpha_{\text {bitcoin }}=2 \% .持有资产无需任何成本。
Note that bitcoin can be traded every day, but gold is only traded on days the market is open, as reflected in the pricing data files LBMA-GOLD.csv and BCHAIN-MKPRU.csv. Your model should account for this trading schedule.
请注意,比特币可以每天交易,但黄金仅在市场开放的日子进行交易,如LBMA-GOLD.csv和BCHAIN-MKPRU.csv的定价数据文件所示。您的模型应考虑此交易时间表。
To develop your model, you may only use the data in the two spreadsheets provided:
要开发模型,您只能使用提供的两个电子表格中的数据:

LBMA-GOLD.csv and BCHAIN-MKPRU.csv.
LBMA-GOLD.csv 和 BCHAIN-MKPRU.csv。
  • Develop a model that gives the best daily trading strategy based only on price data up to that day. How much is the initial $ 1000 $ 1000 $1000\$ 1000 investment worth on 9 / 10 / 2021 9 / 10 / 2021 9//10//20219 / 10 / 2021 using your model and strategy?
    开发一个模型,仅根据截至当天的价格数据提供最佳每日交易策略。 9 / 10 / 2021 9 / 10 / 2021 9//10//20219 / 10 / 2021 使用您的模型和策略的初始 $ 1000 $ 1000 $1000\$ 1000 投资值多少钱?
  • Present evidence that your model provides the best strategy.
    提供证据证明您的模型提供了最佳策略。
  • Determine how sensitive the strategy is to transaction costs. How do transaction costs affect the strategy and results?
    确定策略对交易成本的敏感程度。交易成本如何影响策略和结果?
  • Communicate your strategy, model, and results to the trader in a memorandum of at most two pages.
    在最多两页的备忘录中将您的策略、模型和结果传达给交易者。
Your PDF solution of no more than 25 total pages should include:
总页数不超过 25 页的 PDF 解决方案应包括:
  • One-page Summary Sheet.  单页摘要表。
  • Table of Contents.  目录。
  • Your complete solution.  您的完整解决方案。
  • One- to two-page Memorandum.
    一到两页的备忘录。
  • Reference List.  参考列表。
Note: The MCM has a 25 -page limit. All aspects of your submission count toward the 25 -page limit (Summary Sheet, Table of Contents, Reference List, and any Appendices). You must cite the sources for your ideas, images, and any other materials used in your report.
注意: MCM 的页数限制为 25 页。您提交的所有方面都计入 25 页的限制(摘要表、目录、参考文献列表和任何附录)。您必须引用报告中使用的想法、图像和任何其他材料的来源。

Attachments
THE TWO DATA FILES PROVIDED CONTAIN THE ONLY DATA YOU SHOULD USE FOR THIS PROBLEM.
附件
提供的两个数据文件包含您应该用于此问题的唯一数据。

  1. LBMA-GOLD.csv
  2. BCHAIN-MKPRU.csv

Data Descriptions  数据说明

  1. LBMA-GOLD.csv
  • Date: The date in mm-dd-yyyy (month-day-year) format.
    日期:日期,格式为 mm-dd-yyyy(月-日-年)。
  • USD (PM): The closing price of a troy ounce of gold in U.S. dollars on the indicated date.
    美元 (PM):一金衡盎司黄金在指定日期的美元收盘价。
  1. BCHAIN-MKPRU.csv
  • Date: The date in mm-dd-yyyy (month-day-year) format.
    日期:日期,格式为 mm-dd-yyyy(月-日-年)。
  • Value: The price in U.S. dollars of a single bitcoin on the indicated date.
    价值:单个比特币在指定日期的美元价格。