• 比较消费性资产与投资性资产的本质区别
• 探讨卖空机制 (Short Selling) 及其市场作用
• 解释正向现货-期货套利 (Cash-and-Carry) 与反向套利 (Reverse Cash-and-Carry)
• 运用无套利均衡原理推导以下基础资产的远期价格公式及存续期远期合约价值：
  • 无收益资产 (No Income)
  • 支付已知现金收益的资产 (Known Cash Income)
  • 提供已知红利收益率的资产 (Known Dividend Yield)
• 对比远期价格与期货价格的理论差异
• 推导不同大类期货资产的定价模型：
  • 外汇期货 • 股票指数期货 • 股指套利 • 大宗商品期货
• 界定并探讨持有成本 (Cost of Carry) 与便利收益 (Convenience Yield)
• 分析期货价格与未来即期价格期望值之间的关系

• Compare consumption and investment assets
• Discuss short selling
• Explain cash-and-carry arbitrage and reverse cash-and-carry arbitrage
• Employ arbitrage principles to derive forward price formulas and the value of a forward contract on a security that provides:
  • No income
  • A known cash income
  • A known dividend yield
• Compare forward and futures prices
• Derive formulas for futures prices:
  • Futures on currencies • Stock index futures • Index arbitrage • Futures on commodities
• Define and discuss cost of carry & convenience yield
• Compare futures prices and expected spot prices

• 投资性资产 (Investment Assets)： 由大量投资者完全出于投资目的而持有的资产。
  • 典型代表：股票、债券、黄金、白银。
• 消费性资产 (Consumption Assets)： 主要用于工业生产制造原料或日常消费而持有的商品。
  • 典型代表：农产品（玉米、小麦）、铜、原油等大宗工业品。
• 资产定价的核心区别：
  • 运用无套利证明可以推导确定投资性资产的精确远期与期货价格。
  • 套利机制仅能为消费性资产确定一个理论价格上限：
    • 持有实物商品本身带来收益，即便利收益 (Convenience Yield)；
    • 实物商品并非总能卖空；
    • 有些商品无法储存（如电力）。

• Investment assets are held by a significant number of investors purely for investment purposes.
  • Examples: stocks, bonds, gold, silver.
• Consumption assets are held primarily for consumption or manufacturing input.
  • Examples: agricultural goods (corn, wheat), copper, crude oil.
• Crucial Difference in Pricing:
  • Arbitrage arguments can be used to determine exact forward and futures prices of investment assets.
  • Arbitrage arguments provide only an upper bound for consumption assets:
    • Ownership of the physical commodity provides benefits – convenience yield.
    • Short selling of physical commodities is not always possible.
    • Some commodities cannot be stored easily or at all (e.g., electricity).

• 卖空是指投资者卖出自己当前并不拥有的证券资产：
  • 你的经纪人从其他客户账户中借入该证券，并在即期现货市场上正常抛售。
  • 在未来某一时间，你必须从市场上买回该证券（买入平仓 Short Covering），以归还给最初出借股票的客户账户。
  • 你必须向出借方足额补偿在借出期间所派发的一切股息红利及其他资本分红。
  • 你必须在经纪商处维持充足的保证金账户以防范违约风险。
• 为何进行卖空？
  • 投机策略： 预期基础资产价格未来将下跌，从中博取跌幅利润。
  • 套利交易： 当期货价格相对现货被严重低估时，执行反向现货-期货套利 (Reverse Cash-and-Carry)。

• Short selling involves selling securities you do not currently own:
  • Your broker borrows the securities from another client and sells them in the spot market in the usual way.
  • At some future stage, you must buy the securities back (short covering) so they can be replaced in the account of the original lender.
  • You must pay all dividends and other distributions that the owner of the borrowed securities would ordinarily have received.
  • You are required to maintain a margin account with the broker to protect against default risk.
• Why short sell?
  • Speculation: Profit from an anticipated decline in the asset price.
  • Arbitrage: Exploit relative mispricing when the forward contract is underpriced (reverse cash-and-carry).

• 交易情境设定：
  • 4月份： 某投资者以 $120 卖空 500 股 IBM 股票。
    → 投资者的经纪商借入 500 股并在即期现货市场上以 $120 售出。
  • 5月份： IBM 公司派发每股 $1 的现金红利。
  • 7月份： 投资者在市场上以每股 $100 买回股票并进行平仓。
    → 经纪商以 $100 价格买入 500 股并归还原出借人。
• 现金流最终核算：
  • 4月份卖空获得现金收入：$500\times \mathrm{\$}120=+\mathrm{\$}60,000$。
  • 5月份代为支付股利红利：$500\times \mathrm{\$}1=-\mathrm{\$}500$。
  • 7月份买回股票平仓支出：$500\times \mathrm{\$}100=-\mathrm{\$}50,000$。
  • 锁定投机净利润： $\mathrm{\$}60,000-\mathrm{\$}500-\mathrm{\$}50,000=+\mathrm{\$}9,500$。

• Setup:
  • In April: An investor shorts 500 IBM shares at $120.
    → The investor’s broker borrows 500 shares and sells them in the spot market for $120.
  • In May: IBM pays a dividend of $1 per share.
  • In July: The investor buys the shares back for $100 and closes out the short position.
    → The broker buys 500 shares at $100 to return to the original lender.
• The Outcome:
  • The investor receives: $500\times \mathrm{\$}120=+\mathrm{\$}60,000$ in April.
  • The investor must pay: $500\times \mathrm{\$}1=-\mathrm{\$}500$ in May (dividend reimbursement).
  • The investor pays: $500\times \mathrm{\$}100=-\mathrm{\$}50,000$ in July to close out.
  • Net Gain (ignoring time value of money): $$500\times (\mathrm{\$}120-\mathrm{\$}1-\mathrm{\$}100)=500\times \mathrm{\$}19=\mathrm{\$}9,500$$

• $S_0$：资产在时刻 $t=0$ 的即期现货价格。
• $S_t$：资产在时刻 $t$ 的即期现货价格。
• $F_0$：在时刻 $t=0$ 商定的远期价格：
  • 指在 $t=0$ 签订的、在到期日 $T$ 进行交割的远期合约中所约定的交割价格。
• $F_t$：在时刻 $t$ 市场上新签署的远期价格：
  • 指在时刻 $t$ 市场上达成的新远期合约中约定的到期 $T$ 的交割价格。
• $f_0$：远期合约在初始时刻 ($t=0$) 的合约价值。恒有 $f_0=0$。
• $f_t$：最初在 $t=0$ 签订、在 $T$ 交割的远期合约在存续期时刻 $t$ 的合约市场价值。
  • 核心区别： $F_t$ 是新合约的价格，而 $f_t$ 是存续期老合约本身的净现值！
• $r$：具有到期期限 $T$ 的年化无风险利率（采用连续复利形式）。

• $S_0$: spot price of the asset at time $t=0$.
• $S_t$: spot price of the asset at time $t$.
• $F_0$: forward price at time $t=0$:
  • The delivery price agreed in a contract negotiated at $t=0$ for delivery at maturity $T$.
• $F_t$: forward price at time $t$:
  • The delivery price in new forward contracts negotiated at time $t$ for delivery at $T$.
• $f_0$: value of the forward contract at initiation ($t=0$). Always $f_0=0$.
• $f_t$: value at time $t$ of a forward contract originally negotiated at $t=0$ for delivery at $T$.
  • Note: $F_t$ (price) and $f_t$ (value) are fundamentally different concepts!
• $r$: risk-free interest rate per annum with continuous compounding.

设市场实际远期报价为 $F^{\ast }$。与理论无套利远期价格对比：

• 若 $F^{\ast }>S_0{e}^{rT}$（远期价格被高估）：
  • 套利者执行正向现货-期货套利 (Cash-and-Carry Arbitrage，见第 1 讲第 25 页)： 按利率 $r$ 借入 $S_0$，以 $S_0$ 买入现货资产，并按 $F^{\ast }$ 签订远期卖出合约。 在到期时刻 $T$，交付资产收取 $F^{\ast }$，偿还贷款本息 $S_0{e}^{rT}$。 锁定确定性无风险净利润 = $F^{\ast }-S_0{e}^{rT}>0$。
• 若 $F^{\ast }<S_0{e}^{rT}$（远期价格被低估）：
  • 套利者执行反向现货-期货套利 (Reverse Cash-and-Carry Arbitrage，见第 1 讲第 26 页)： 卖空现货资产获得 $S_0$，将款项按 $r$ 贷出/存入银行，并按 $F^{\ast }$ 买入远期多头。 在到期日收取投资本息 $S_0{e}^{rT}$，支付 $F^{\ast }$ 接受远期交割并归还借股。 锁定确定性无风险净利润 = $S_0{e}^{rT}-F^{\ast }>0$。
• 理论无套利远期价格公式：
  $$F_0=S_0{e}^{rT}\text{在时刻 }t\text{ 满足：}{F}_t=S_t{e}^{r(T-t)}$$
  例如：不支付红利的股票、零息债券。

Let the market-quoted forward price be $F^{\ast }$. Compare with theoretical no-arbitrage price:

• If $F^{\ast }>S_0{e}^{rT}$ (Forward Overpriced):
  • Arbitrageur executes Cash-and-Carry Arbitrage (see slide 1.25): Borrow $S_0$ at rate $r$, buy the asset at $S_0$, and sell forward at $F^{\ast }$. At $T$, deliver the asset for $F^{\ast }$, repay loan $S_0{e}^{rT}$. Profit locked in = $F^{\ast }-S_0{e}^{rT}>0$.
• If $F^{\ast }<S_0{e}^{rT}$ (Forward Underpriced):
  • Arbitrageur executes Reverse Cash-and-Carry Arbitrage (see slide 1.26): Short sell the asset at $S_0$, invest proceeds in risk-free deposit at $r$, and buy forward at $F^{\ast }$. At $T$, deposit matures to $S_0{e}^{rT}$, buy asset via forward at $F^{\ast }$ to close short. Profit locked in = $S_0{e}^{rT}-F^{\ast }>0$.
• Theoretical Forward Price:
  $$F_0=S_0{e}^{rT}\text{or at time }t:F_t=S_t{e}^{r(T-t)}$$
  Examples: Non-dividend-paying stocks and zero-coupon bonds.

假定 $I$ 为合约存续期内将收到的所有已知现金收益的现值 (Present Value)，$I_t$ 为时刻 $t$ 时仍待收到的现金收益的现值：

• 若 $F^{\ast }>(S_0-I)e^{rT}$（远期价格被高估）：
  • 套利者借入资金 $S_0$ 买入现货资产，并做空卖出远期合约。
  • 存续期内收到的红利按利率 $r$ 滚动投资，到期用于冲抵偿还贷款本息。
• 若 $F^{\ast }<(S_0-I)e^{rT}$（远期价格被低估）：
  • 套利者以 $S_0$ 卖空现货资产，将 $(S_0-I)$ 的资金按 $r$ 贷出，并做多买入远期合约。
  • 借股期间需代为垫付的红利由留存备付资金 $I$ 及其利息精准支付。
• 理论无套利定价公式：
  $$F_0=(S_0-I)e^{rT}\text{在时刻 }t\text{ 满足：}{F}_t=(S_t-I_t)e^{r(T-t)}$$
  例如：支付已知红利的股票、附息债券。

Suppose $I$ is the present value of all known cash income received during the life of the contract, and $I_t$ is the present value at time $t$ of the income still to be received:

• If $F^{\ast }>(S_0-I)e^{rT}$ (Overpriced):
  • Arbitrageur borrows $S_0$ to buy the asset and enters a short forward contract.
  • The income received during the contract life is invested at rate $r$ and helps repay the loan at $T$.
• If $F^{\ast }<(S_0-I)e^{rT}$ (Underpriced):
  • Arbitrageur short sells the asset for $S_0$, deposits amount $(S_0-I)$ at $r$, and enters a long forward contract.
  • Income payments on the borrowed asset are paid out of the remaining short proceeds.
• Theoretical Forward Price:
  $$F_0=(S_0-I)e^{rT}\text{and at time }t:F_t=(S_t-I_t)e^{r(T-t)}$$
  Examples: Dividend-paying stocks with known dividend amounts and coupon-bearing bonds.

• 市场基础数据：
  • 股票即期现货价格：$S_0=\mathrm{\$}50$。
  • 10 个月远期市场报价：$F^{\ast }=\mathrm{\$}52$。
  • 连续复利年化无风险利率：所有期限均为 $r=8\mathrm{\%}$ p.a.。
  • 预期现金红利：在 3 个月、6 个月及 9 个月后各派发每股 $\mathrm{\$}0.75$。
• 是否存在无套利机会？
  • 步骤 1：计算红利现金流现值 ($I$)： $$I=0.75e^{-0.08\times 3/12}+0.75e^{-0.08\times 6/12}+0.75e^{-0.08\times 9/12}$$ $$I=0.75(0.98020+0.96079+0.94176)=0.75\times 2.88275=\mathrm{\$}2.162$$
  • 步骤 2：计算理论无套利远期价格 ($F_0$)： $$F_0=(50-2.162)e^{0.08\times 10/12}=47.838\times 1.06894=\mathrm{\$}51.14$$
  • 判定： 由于市场报价 $F^{\ast }=\mathrm{\$}52>\mathrm{\$}51.14$，远期价格被高估 $\mathrm{\$}(52-51.14)=\mathrm{\$}0.86$ → 执行正向现货-期货套利！

• Market Setup:
  • Spot price of the stock: $S_0=\mathrm{\$}50$.
  • Quoted 10-month forward price: $F^{\ast }=\mathrm{\$}52$.
  • Risk-free interest rate: $r=8\mathrm{\%}$ p.a. (continuously compounded) for all maturities.
  • Expected dividends: $\mathrm{\$}0.75$ per share in 3 months, 6 months, and 9 months.
• Is there an arbitrage opportunity?
  • Step 1: Calculate PV of dividends ($I$): $$I=0.75e^{-0.08\times 3/12}+0.75e^{-0.08\times 6/12}+0.75e^{-0.08\times 9/12}$$ $$I=0.75(0.98020+0.96079+0.94176)=0.75\times 2.88275=\mathrm{\$}2.162$$
  • Step 2: Calculate theoretical forward price: $$F_0=(S_0-I)e^{rT}=(50-2.162)e^{0.08\times 10/12}=47.838\times 1.06894=\mathrm{\$}51.14$$
  • Conclusion: Quoted price $F^{\ast }=\mathrm{\$}52>F_0=\mathrm{\$}51.14$.
    The forward contract is overpriced by $\mathrm{\$}52-\mathrm{\$}51.14=\mathrm{\$}0.86$.

正向现货-期货套利交易全流程 (Cash-and-Carry Arbitrage Walkthrough)

• 当前时刻 ($t=0$)：
  • 按 8% 年利率借入 $\mathrm{\$}50$ 购买 1 股现货股票。
  • 签订一份 10 个月期远期空头合约，锁定在 10 个月后以 $\mathrm{\$}52$ 卖出该股票。
• 在第 3、6、9 个月时刻：
  • 每次收到 $\mathrm{\$}0.75$ 股息红利，立即按 8% 年利率进行再投资，直至第 10 个月。
• 在第 10 个月到期时刻：
  • 偿还初始贷款本息：$\mathrm{\$}50e^{0.08\times 10/12}=\mathrm{\$}53.45$。
  • 交付股票履行远期合约，收取合同货款：$+\mathrm{\$}52.00$。
  • 收取 3 笔股息再投资的终值本利和： $$0.75e^{0.08\times 7/12}+0.75e^{0.08\times 4/12}+0.75e^{0.08\times 1/12}=0.7858+0.7703+0.7550=+\mathrm{\$}2.31$$
  • 锁定无风险净利润： $\mathrm{\$}52.00+\mathrm{\$}2.31-\mathrm{\$}53.45=+\mathrm{\$}0.86$（恰好等于定价偏差 $\mathrm{\$}52.00-\mathrm{\$}51.14=\mathrm{\$}0.86$）。

Cash-and-Carry Arbitrage Walkthrough

• Now ($t=0$):
  • Borrow $\mathrm{\$}50$ at $8\mathrm{\%}$ p.a. to buy one share in the spot market.
  • Enter into a short forward contract to sell the share at $\mathrm{\$}52$ in 10 months.
• In 3, 6, and 9 months:
  • Receive the $\mathrm{\$}0.75$ dividend each time and immediately invest at $8\mathrm{\%}$ until month 10.
• In 10 months (at maturity):
  • Repay the initial loan: $\mathrm{\$}50e^{0.08\times 10/12}=\mathrm{\$}53.45$.
  • Deliver the share under the forward contract for $+\mathrm{\$}52.00$.
  • Collect the compounded dividend investments: $$0.75e^{0.08\times 7/12}+0.75e^{0.08\times 4/12}+0.75e^{0.08\times 1/12}=\mathrm{\$}2.31$$
  • Net Arbitrage Profit: $$\text{Profit}=\mathrm{\$}52.00+\mathrm{\$}2.31-\mathrm{\$}53.45=\mathrm{\$}0.86\text{ per share}$$

假定 $q$ 为基础资产在合约存续期内的年化连续分红收益率 (Continuous Dividend Yield)：

• 典型应用 – 股票指数：
  • 股票指数代表由大量成分股构成的投资组合，全年持续派发股利红利。
  • 这部分密集派发的现金股利可极为精确地近似建模为连续红利率 $q$。
• 指数期货定价的核心前提条件：
  • 该公式有效成立的关键在于：该股票指数必须代表一项投资性资产。
  • 换言之，指数的涨跌变动必须完全能够通过在现货市场上实际复制并持有相应股票篮子资产来实现！
  • 反例：以美元数值衡量的日经指数 (Nikkei) 并不代表一项投资性资产。

Suppose $q$ is the average continuous dividend yield of the asset during the contract life:

• Example – Stock Index:
  • A stock index represents a portfolio paying dividends throughout the year.
  • Dividends can be modeled as a continuous yield $q$.
• Crucial Condition for Index Futures:
  • For this formula to hold, the index must represent an investment asset.
  • In other words, changes in the index must correspond to changes in the value of an actually tradable portfolio of securities.
  • Counter-example: The Nikkei index viewed as a dollar number does not represent an investment asset.

考虑一份在时刻 $0$ 订立、约定交割价格为 $F_0$、到期日为 $T$ 的远期合约：

• 在存续期时刻 $t$，多头远期合约的市场价值为： $$f_t=(F_t-F_0)e^{-r(T-t)}$$
• 在存续期时刻 $t$，空头远期合约的市场价值为： $$f_t=(F_0-F_t)e^{-r(T-t)}$$
• 其实质反映了在时刻 $t$ 提前对冲平仓该合约所能锁定的未来盈亏的贴现现值。
• 在合约成立初始时刻 ($t=0$)： $f_0=0$，因为初始交割价格恰好设定为理论远期价格 $F_0$。
• 经典计算算例： $S_0=\mathrm{\$}400$，$T=6$ 个月 (0.5年)，$r=5\mathrm{\%}$。
  • 在 $t=0$：$F_0=400e^{0.05\times 0.5}=\mathrm{\$}410.13$，且 $f_0=0$。
  • 2 个月后（$t=2/12$，剩余 $\tau =4/12$），股价升至 $S_t=\mathrm{\$}450$： $$F_t=450e^{0.05\times (4/12)}=\mathrm{\$}457.56$$ $$f_t=(457.56-410.13)e^{-0.05\times (4/12)}=47.43\times 0.98347=\mathrm{\$}46.65$$

Consider a forward contract negotiated at time $0$ with delivery price $F_0$ at maturity $T$:

• The value of a long position at time $t$ is: $$f_t=(F_t-F_0)e^{-r(T-t)}$$
• The value of a short position at time $t$ is: $$f_t=(F_0-F_t)e^{-r(T-t)}$$
• It reflects the present value of the gain or loss from closing out the position at time $t$.
• At initiation ($t=0$): $f_0=0$ because the delivery price is set to $F_0$.
• Worked Example: $S_0=\mathrm{\$}400$, $T=6$ months (0.5 yr), $r=5\mathrm{\%}$.
  • At $t=0$: $F_0=400e^{0.05\times 0.5}=\mathrm{\$}410.13$, and $f_0=0$.
  • After 2 months ($t=2/12$, remaining $\tau =4/12$), stock rises to $S_t=\mathrm{\$}450$: $$F_t=450e^{0.05\times (4/12)}=\mathrm{\$}457.56$$ $$f_t=(457.56-410.13)e^{-0.05\times (4/12)}=47.43\times 0.98347=\mathrm{\$}46.65$$

核心结论： 已经派发过的历史股息不进入后续估值；仅未来剩余尚未发放的股息红利现值才对新远期价格及存续期合约价值产生影响！

Core Insight: Past dividends paid out do not enter subsequent valuation; only the present value of future remaining dividends affects subsequent forward price and contract value!

• 当利率恒定或完全可预测时：
  • 若市场利率保持常数，或者是时间的确定性已知函数，远期价格与期货价格在理论上完全等价一致：$F_0={\text{期货}}_0$。
• 当利率随机不确定且存在每日逐日盯市结算时：
  • 基础资产价格与利率呈正相关： 当基础资产价格 $S$ 上涨时，期货多头持有者立即收到每日现金结算盈利，该现金流可按高于预期的利率再投资；而远期多头持有者不受这种不确定性影响。因此，期货价格 > 远期价格。
  • 基础资产价格与利率呈负相关： 远期价格 > 期货价格。
  • 但在实际绝大多数市场合约中，二者的理论差异微乎其微，在实际交易分析中通常视为等同。
• 远期合约的价值与期货合约的价值相同吗？ 不相同！
  • 期货：逐日盯市所实现的盈亏等于期货价格的变动额。
  • 远期：远期合约交易所实现的盈亏等于远期价格变动额的现值。

• Constant or Predictable Interest Rates:
  • When interest rates are constant or known functions of time, forward prices and futures prices are theoretically identical: $F_0={\text{Futures}}_0$.
• Uncertain Interest Rates & Daily Settlement:
  • Positive correlation between asset price and interest rates: As $S$ rises, long futures holders receive daily cash gains invested at higher interest rates. Holders of long forward positions are not affected by this uncertainty. Thus, futures price > forward price.
  • Negative correlation: Forward price > futures price.
  • In practice, for short/medium maturities, these differences are very small → assume $\text{Futures Price}=\text{Forward Price}$.
• Is the Value of a Forward Contract the Same as Futures? – NO!
  • Futures: Profit/loss is realized daily through marking to market. Value is reset to zero daily.
  • Forward: All profit/loss is locked until maturity $T$. Value at $t$ is the present value of the change: $f_t=(F_t-F_0)e^{-r(T-t)}$.

• 当 $F^{\ast }>S_0{e}^{(r-q)T}$（期货价格被高估）：
  • 套利者买入现货指数成分股一篮子股票，同时建立股指期货空头头寸（正向现货-期货套利）。
  • 通常由具有极低交易成本的专业做市机构或持有大量短期货币市场资金的企业执行。
• 当 $F^{\ast }<S_0{e}^{(r-q)T}$（期货价格被低估）：
  • 套利者建立股指期货多头头寸，同时在现货市场卖出（或借股卖空）指数成分股（反向现货-期货套利）。
  • 常由大型养老基金执行（直接售出现货股票换取现金存入银行，同时做多期货替代股票敞口）。
• 执行与市场现实：
  • 指数套利需要同时交易期货合约与大量成分股；通常由计算机生成交易指令。
  • 偶尔（如 1987 年 10 月 19 日“黑色星期一”，美国股市暴跌 30%）无法同时完成各项交易，此时 $F_t$ 与 $S_t$ 之间理论上的无套利关系不再成立。

• When $F^{\ast }>S_0{e}^{(r-q)T}$ (Futures Overpriced):
  • Arbitrageur buys the underlying index stocks and enters a short position in index futures contracts (Cash-and-carry).
  • Often done by institutions with low transaction costs or corporations holding short-term money market funds.
• When $F^{\ast }<S_0{e}^{(r-q)T}$ (Futures Underpriced):
  • Arbitrageur enters a long position in index futures contracts and sells (or short sells) the underlying stocks (Reverse cash-and-carry).
  • Frequently executed by large pension funds that can sell physical shares and invest in cash.
• Execution & Market Realities:
  • Index arbitrage requires simultaneous trades in hundreds of stocks and index futures; automated program trading is standard.
  • Black Monday (19 October 1987, when the US stock market dropped by 30%): Massive order imbalances caused computer trading delays. Cash and futures decoupled, breaking the theoretical parity relationship.

核心结论： 借股卖空现货并买入被低估的期货合约，在初始时刻无净资金流出的前提下锁定了确定性的到期无风险净套利收益！

Core Principle: The terminal index price $S_T$ cancels out completely! The locked-in profit at maturity is exactly the mispricing: $F_0-F^{\ast }=408.08-405=\mathrm{\$}3.08$ per contract unit (or $\mathrm{\$}3.08e^{-0.10\times 1/3}=\mathrm{\$}2.98$ in present value terms), with zero initial capital and zero market risk.

• 符号记法说明：
  • $S_0$：即期汇率 – 购买 1 单位外币所需的本币价格（直接标价法）。
  • $F_0$：远期汇率 – 在未来购买 1 单位外币预先锁定的本币价格。
  • $r$：本币本国无风险年化利率。
  • $r_f$：外币外国无风险年化利率。
• 外币作为支付连续收益的资产：
  • 持有外币可在外国存入银行并持续获得外币利息收入，利率为 $r_f$。
  • 因此，外币等价于提供连续分红收益率 $q=r_f$ 的投资性资产：
  $$F_0=S_0{e}^{(r-r_f)T}\text{在时刻 }t\text{ 满足：}{F}_t=S_t{e}^{(r-r_f)(T-t)}$$
• 该经典恒等式即为国际金融学著名的抛补利率平价定理 (Covered Interest Parity, CIP)！
• 报价惯例：
  • 上述公式所用的不是交易所实际的即期与远期汇率报价方式。
  • 交易所的汇率通常按每 1 美元兑多少外币报价，英镑、欧元、澳元和新西兰元除外。

• Notation:
  • $S_0$: spot exchange rate – price in domestic currency of 1 unit of foreign currency.
  • $F_0$: forward exchange rate – price in domestic currency of 1 unit of foreign currency.
  • $r$: risk-free interest rate in domestic currency.
  • $r_f$: risk-free interest rate in foreign currency.
• Foreign Currency as an Income-Yielding Asset:
  • Holding foreign currency earns interest at the foreign rate $r_f$.
  • Therefore, foreign currency is analogous to an investment asset paying a continuous dividend yield of $q=r_f$:
  $$F_0=S_0{e}^{(r-r_f)T}\text{and at time }t:F_t=S_t{e}^{(r-r_f)(T-t)}$$
• Quotation Conventions:
  • This is not the way spot and forward rates are quoted in exchanges.
  • Exchange rates are quoted as foreign currency per dollar, except for British pound, euro, Australian dollar and New Zealand dollar.

从时间 $0$ 到时间 $T$ 持有资金的两种等价投资路径：

S₀ 是即期汇率；F₀ 是远期汇率

Two equivalent strategies to hold money from time $0$ to time $T$:

S₀ is the spot exchange rate; F₀ is the forward exchange rate

By no-arbitrage, both riskless strategies must yield the identical dollar outcome:

• 若 $F^{\ast }<S_0{e}^{(r-r_f)T}$（远期汇率被低估）：
  • 按外国利率 $r_f$ 借入外币。
  • 在即期现货市场上按汇率 $S_0$ 将外币兑换为本币。
  • 将兑换得到的本币按本国利率 $r$ 存入银行进行投资。
  • 订立一份远期多头合约，约定在到期时按 $F^{\ast }$ 买入外币，以偿还外币借款本息。
  • 在时刻 $T$ 锁定确定性本币无风险套利净收益。
• 若 $F^{\ast }>S_0{e}^{(r-r_f)T}$（远期汇率被高估）：
  • 按本国利率 $r$ 借入本币。
  • 在即期现货市场上按汇率 $S_0$ 将本币兑换为外币。
  • 将外币按外国利率 $r_f$ 投资于外国货币市场。
  • 订立一份远期空头合约，约定在到期时按 $F^{\ast }$ 卖出外币（换回本币），以偿还本币贷款本息。
  • 在时刻 $T$ 同样锁定确定性无风险套利利润。

• If $F^{\ast }<S_0{e}^{(r-r_f)T}$ (Forward Exchange Rate Too Low):
  • Borrow foreign currency at rate $r_f$.
  • Convert borrowed foreign currency to domestic currency at spot rate $S_0$.
  • Invest domestic currency at domestic rate $r$.
  • Enter a long forward contract to buy foreign currency at rate $F^{\ast }$ at maturity $T$ and repay the foreign currency loan.
  • Guaranteed domestic currency profit locked in at $T$.
• If $F^{\ast }>S_0{e}^{(r-r_f)T}$ (Forward Exchange Rate Too High):
  • Borrow domestic currency at domestic rate $r$.
  • Convert to foreign currency at spot rate $S_0$.
  • Invest foreign currency at foreign rate $r_f$.
  • Enter a short forward contract to sell the foreign currency at rate $F^{\ast }$ (converting to domestic currency) at maturity $T$ and repay the domestic currency loan.
  • Guaranteed domestic currency profit locked in at $T$.

• 投资性商品 (如黄金、白银)：
  • 完全出于投资目的持有；其仓储保险成本可等价视为负收益 (Negative Income)。
  • 若仓储成本与商品价值成正比例（年化比率为 $u$）： $$F_0=S_0{e}^{(r+u)T}\text{在时刻 }t\text{ 满足：}{F}_t=S_t{e}^{(r+u)(T-t)}$$
  • 若仓储成本为固定金额，其贴现现值为 $U$（$U_t$ 为时刻 $t$ 时仍待支付的仓储成本的现值）： $$F_0=(S_0+U)e^{rT}\text{在时刻 }t\text{ 满足：}{F}_t=(S_t+U_t)e^{r(T-t)}$$
• 消费性商品 (如原油、铜、农产品)：
  • 若 $F^{\ast }>(S_0+U)e^{rT}$，正向现货-期货套利成立并发挥约束作用（借款、买入现货商品、储存并做空期货）。
  • 若 $F^{\ast }<(S_0+U)e^{rT}$，反向现货-期货套利受阻失效，因为：
    • 普通投资者无法借入并卖空实物大宗商品。
    • 现货库存持有者视其实物库存为生产平稳运行的必要保障（享受便利收益 Convenience Yield），不愿出清现货。
  • 因此，理论关系仅为价格上限： $$F_0\le S_0{e}^{(r+u)T}\text{或}{F}_0\le (S_0+U)e^{rT}$$ 在时刻 $t$：$F_t\le S_t{e}^{(r+u)(T-t)}$，$F_t\le (S_t+U_t)e^{r(T-t)}$。

• Investment Commodities (e.g., Gold, Silver):
  • Held for investment; storage costs can be treated as negative income.
  • If storage cost is proportional to price ($u$ per annum): $$F_0=S_0{e}^{(r+u)T}\text{or at time }t:F_t=S_t{e}^{(r+u)(T-t)}$$
  • If storage cost has present value $U$ ($U_t$ is the present value at time $t$ of the storage costs still to be paid): $$F_0=(S_0+U)e^{rT}\text{or at time }t:F_t=(S_t+U_t)e^{r(T-t)}$$
• Consumption Commodities (e.g., Oil, Copper, Agricultural Goods):
  • If $F^{\ast }>(S_0+U)e^{rT}$, cash-and-carry arbitrage works (borrow, buy commodity, store, sell futures).
  • If $F^{\ast }<(S_0+U)e^{rT}$, reverse cash-and-carry FAILS because:
    • Commercial inventory holders will not sell physical stock due to operational convenience.
    • Outside arbitrageurs cannot easily short-sell physical crude oil or grain.
  • Therefore, the theoretical relation is an upper bound: $$F_0\le S_0{e}^{(r+u)T}\text{or}{F}_0\le (S_0+U)e^{rT}$$ At time $t$: $F_t\le S_t{e}^{(r+u)(T-t)}$, $F_t\le (S_t+U_t)e^{r(T-t)}$.

• 市场基础数据：
  • 原油即期现货价格：$S_0=\mathrm{\$}70$ / 桶。
  • 1 年期原油期货市场报价：$F^{\ast }=\mathrm{\$}75.80$。
  • 连续复利年化无风险利率：所有期限均为 $r=5\mathrm{\%}$ p.a.。
  • 原油仓储成本：每桶每年 $\mathrm{\$}1.40$，按季度在季初预先支付（每季度初支付 $\mathrm{\$}0.35$，即在第 0、3、6、9 个月支付）。
• 是否存在无风险套利机会？
  • 步骤 1：计算仓储成本的现值 ($U$)： $$U=0.35+0.35e^{-0.05\times 3/12}+0.35e^{-0.05\times 6/12}+0.35e^{-0.05\times 9/12}$$ $$U=0.35\times (1+0.98758+0.97531+0.96319)=0.35\times 3.92608=\mathrm{\$}1.374$$
  • 步骤 2：计算包含仓储成本的理论期货价格上限： $$F_{\text{上限}}=(S_0+U)e^{rT}=(70+1.374)e^{0.05\times 1}=71.374\times 1.05127=\mathrm{\$}75.03$$
  • 判定结论： 市场实际期货价格 $F^{\ast }=\mathrm{\$}75.80>\mathrm{\$}75.03$，期货价格被高估 $\mathrm{\$}(75.80-75.03)=\mathrm{\$}0.77$！正向现货-期货套利完全可行！

• Setup:
  • Spot price of oil: $S_0=\mathrm{\$}70$ per barrel.
  • Quoted 1-year futures price: $F^{\ast }=\mathrm{\$}75.80$.
  • Interest rate: $r=5\mathrm{\%}$ p.a. with continuous compounding for all maturities.
  • Storage costs: $\mathrm{\$}1.40$ per barrel per annum, payable quarterly in advance ($\mathrm{\$}0.35$ per quarter at months 0, 3, 6, 9).
• Is there an arbitrage opportunity?
  • Step 1: Calculate PV of storage costs ($U$): $$U=0.35+0.35e^{-0.05\times 3/12}+0.35e^{-0.05\times 6/12}+0.35e^{-0.05\times 9/12}$$ $$U=0.35\times (1+0.98758+0.97531+0.96319)=0.35\times 3.92608=\mathrm{\$}1.374$$
  • Step 2: Calculate theoretical upper bound forward price: $$F_{\text{theoretical}}=(S_0+U)e^{rT}=(70+1.374)e^{0.05\times 1}=71.374\times 1.05127=\mathrm{\$}75.03$$
  • Conclusion: Quoted price $F^{\ast }=\mathrm{\$}75.80>\mathrm{\$}75.03$.
    The futures contract is overpriced by $\mathrm{\$}75.80-\mathrm{\$}75.03=\mathrm{\$}0.77$ per barrel.

正向现货-期货套利执行流程 (Cash-and-Carry Arbitrage Execution)

• 当前时刻 ($t=0$)：
  • 按 5% 年利率借入 $\mathrm{\$}(70+1.374)=\mathrm{\$}71.374$。
  • 以即期价格 $\mathrm{\$}70$ 买入 1 桶原油现货，并支付首季度仓储费 $\mathrm{\$}0.35$。
  • 签订一份 1 年期期货空头合约，锁定在 1 年后以 $\mathrm{\$}75.80$ 卖出该桶原油。
• 在第 3、6、9 个月时刻：
  • 支付仓储费用（将 $\mathrm{\$}1.374-\mathrm{\$}0.35=\mathrm{\$}1.024$ 现在分三笔投资，期限分别为 3、6、9 个月，到期用于支付各季度仓储费用）。
• 在 1 年后到期时刻：
  • 偿还贷款本息：$\mathrm{\$}71.374e^{0.05\times 1}=\mathrm{\$}75.03$。
  • 履行期货合约交付原油，收取期货货款：$+\mathrm{\$}75.80$。
  • 锁定无风险净套利利润： $\mathrm{\$}75.80-\mathrm{\$}75.03=+\mathrm{\$}0.77$ / 桶。
• 注意：对于原油这类消费性资产，反向现货-期货套利并非总是可行。

Cash-and-Carry Arbitrage Execution

• Now ($t=0$):
  • Borrow $\mathrm{\$}(70+1.374)=\mathrm{\$}71.374$ at $5\mathrm{\%}$ p.a.
  • Buy 1 barrel of oil at spot price $\mathrm{\$}70$ and pay first quarter storage of $\mathrm{\$}0.35$.
  • Enter into a short futures contract to sell oil at $\mathrm{\$}75.80$ in 1 year.
• In 3, 6, and 9 months:
  • Pay quarterly storage costs using the borrowed reserve funds (invest $\mathrm{\$}1.374-\mathrm{\$}0.35=\mathrm{\$}1.024$ in three instalments over 3, 6 and 9 months to finance the quarterly storage cost).
• In 1 year (at maturity):
  • Repay the loan: $\mathrm{\$}71.374e^{0.05\times 1}=\mathrm{\$}75.03$.
  • Deliver oil under the futures contract and receive $+\mathrm{\$}75.80$.
  • Arbitrage Profit: $$\text{Profit}=\mathrm{\$}75.80-\mathrm{\$}75.03=\mathrm{\$}0.77\text{ per barrel}$$
• Critical note: If futures were quoted below $\mathrm{\$}75.03$, reverse cash-and-carry is NOT always available for consumption assets like oil!

持有成本 (Cost of Carry) $c$ 综合概括了资产持有过程中的资金融资成本、仓储保险成本以及所产生的收益流：

• 对于无收益股票：$c=r$。
• 对于提供红利率 $q$ 的股票指数：$c=r-q$。
• 对于具有外国利率 $r_f$ 的外汇货币：$c=r-r_f$。
• 对于具有比例仓储成本 $u$ 的大宗商品：$c=r+u$。
• 对于纯投资性资产： 期货定价公式可统一表达为：
  $$F_0=S_0{e}^{cT}\text{在时刻 }t\text{ 满足：}{F}_t=S_t{e}^{c(T-t)}$$
  • 期货合约通常相对于基础资产的水平溢价交易，这一差额称为持有成本。
  • 由于基础资产可以买入并持有至未来交割，期货价格应反映将资产持有至期货到期日的持有成本。
  • 期货价格高于现货价格 – 升水市场 (Contango)（当 $c>0$ 时 $F_0>S_0$）。
• 对于消费性资产： $F_t\le S_t{e}^{c(T-t)}$
  • 期货价格通常低于现货价格 – 贴水市场 (Backwardation)。

The cost of carry $c$ summarizes financing costs, storage costs, and income earned:

• $c=r$ for a non-dividend paying stock.
• $c=r-q$ for a stock index providing dividend yield $q$.
• $c=r-r_f$ for a currency with foreign interest rate $r_f$.
• $c=r+u$ for a commodity with proportional storage cost $u$.
• For Investment Assets: $F_0=S_0{e}^{cT}$ (at time $t$: $F_t=S_t{e}^{c(T-t)}$).
  • The futures contract typically trades at a premium to the level of the underlying asset. This difference is referred to as the cost of carry.
  • As the underlying asset can be purchased and held for future delivery, the futures price should reflect the cost of carrying the asset to the maturity of the futures contract.
  • Futures prices trade higher than spot prices: $F_0>S_0$ whenever $c>0$ – Contango.
• For Consumption Assets: $F_0\le S_0{e}^{cT}$ (at time $t$: $F_t\le S_t{e}^{c(T-t)}$).
  • Due to convenience yield, futures often trade lower than spot – Backwardation.

• 定义： 便利收益率 $y$ 衡量了持有实体大宗商品现货相对于单纯持有纸面期货合约所带来的隐含经济利益与运营便利：
  $$F_0{e}^{yT}=S_0{e}^{cT}⟹F_0=S_0{e}^{(c-y)T}=S_0{e}^{(r+u-y)T}$$ $$\text{或采用金额仓储成本：}{F}_0=(S_0+U)e^{(r-y)T}$$ $$\text{在时刻 }t\text{ 满足：}{F}_t=S_t{e}^{(c-y)(T-t)},F_t=(S_t+U_t)e^{(r-y)(T-t)}$$
• 企业为何必须保有实体库存？
  • 在供应链遭遇意外中断或延误时，确保生产流水线不致停工。
  • 能够随时敏锐捕捉当地突发性的临时供给短缺溢价或应对客户爆发性需求。
• 市场预期与远期曲线走势形态：
  • 便利收益反映了市场对该商品未来可得性的预期：若预期出现短缺（过剩），便利收益率 $y$ 倾向于较高（较低）。
  • $y$ 较高（较低）时，远期曲线呈贴水 Backwardation（升水 Contango）形态，例如 $F_t<S_t$。

• Definition: The convenience yield $y$ measures the economic benefits of holding the physical commodity rather than holding a futures contract:
  $$F_0{e}^{yT}=S_0{e}^{cT}⟹F_0=S_0{e}^{(c-y)T}=S_0{e}^{(r+u-y)T}$$ $$\text{or with dollar storage: }{F}_0=(S_0+U)e^{(r-y)T}$$ $$\text{and at time }t:F_t=S_t{e}^{(c-y)(T-t)},F_t=(S_t+U_t)e^{(r-y)(T-t)}$$
• Why hold physical inventory?
  • Maintain production flow during supply chain disruptions.
  • Take advantage of local temporary shortages or unexpected customer demand.
• Market Expectations & Forward Curve Shape:
  • If shortage is expected → physical commodity is scarce → $y$ is high → $F_0<S_0$ (Backwardation).
  • If surplus is expected → warehouse inventories are high → $y$ is low → $F_0>S_0$ (Contango).

• 市场当下状态（今日观察到的远期曲线形态）：
  • 升水 (Contango)： 期货价格高于当前现货即期价格（$F_0>S_0$）。
  • 贴水 (Backwardation)： 期货价格低于当前现货即期价格（$F_0<S_0$）。
• 市场预期理论（期货价格与未来到期期望即期价格 $E[S_T]$ 的关系）：
  • 正常升水 / 正向市场 (Normal Contango，$F_0>E[S_T]$)：
    • 凯恩斯与希克斯 (Keynes-Hicks) 认为：套期保值者倾向于持有多头头寸（平均预期亏损），投机者倾向于持有空头头寸（平均预期盈利）。
    • 该资产的收益与市场负相关。
  • 正常贴水 / 现货溢价 (Normal Backwardation，$F_0<E[S_T]$)：
    • 凯恩斯与希克斯认为：套期保值者倾向于持有空头头寸（平均预期亏损），投机者倾向于持有多头头寸（平均预期盈利）。
    • 该资产的收益与市场正相关。

• Market Conditions (Forward Curve Shape Today):
  • Contango: Futures price > current spot price ($F_0>S_0$).
  • Backwardation: Futures price < current spot price ($F_0<S_0$).
• Market Expectations (Futures Price vs Expected Future Spot $E[S_T]$):
  • Normal Contango ($F_0>E[S_T]$):
    • Keynes and Hicks argument: Hedgers hold net long positions (purchasers seeking price caps; expect losses on average).
    • Speculators hold short positions and earn an expected positive return.
    • Asset return is negatively correlated with the broader market (systematic risk is negative).
  • Normal Backwardation ($F_0<E[S_T]$):
    • Keynes and Hicks argument: Hedgers hold net short positions (commodity producers seeking price floors; pay insurance premium).
    • Speculators hold long positions and expect to profit: $E[S_T]-F_0>0$.
    • Asset return is positively correlated with the broader market (positive systematic risk).

核心区别： 升水/贴水描述的是今日可直接观察到的现货与期货的静态价差形态；而正常升水/正常贴水描述的是期货价格相对于未来未知的期望即期价格的理论风险溢价关系！