远期价格与期货价格的确定

作者: pdnbplus | 发布时间: 2024/06/20 | 阅读量: 597

远期价格与期货价格的确定--潘登同学的期权、期货及其他衍生品学习笔记

  • 远期价格
  • 股票指数

    • 对期货和远期合约估值的假设

    • 市场参与者进行交易时没有手续费(There are no commissions for market participants to trade)
    • 市场参与者对所有交易净利润都使用同一税率(Market participants are taxed at the same rate on all trading net profits)
    • 市场参与者能够以同样的无风险利率借入和借出资金(Market participants are able to borrow and lend funds at the same risk-free rate)
    • 当套利机会出现时，市场参与者会马上利用套利机会(When arbitrage opportunities arise, market participants immediately take advantage of them)
    符号 定义 英文
    $S_0$ 期货或远期合约的资产当前现货价格 The current Spot Price of the asset in the futures or forward contract
    $F_0$ 期货或远期的当前价格 Futures or Forward Price
    $T$ 到交货日期的时间 Time until Delivery
    $r$ 到T 年无风险利率 (Risk-free Interest Rate)，连续复利 Risk-free Interest Rate until T years, continuously compounded

    远期价格

    • 如果不提供收益的投资资产的现货价格为$S_0$，r 是到T 年无风险利率，T 年内可交割合约的远期/期货价格为$F_0$(If the spot price of an investment asset that does not provide income is $S_0$, r is the risk-free interest rate until T years, the forward/futures price of the deliverable contract within T years is $F_0$): $$ F_0 = S_0e^{rT} $$
    • 当投资资产的收入已知 (Income Known) 时,I 是远期合约有效期内的收入现值(When the income of the investment asset is known,I is the present value of income over the life of the forward contract) $$ F_0 = (S_0-I)e^{rT} $$
    • 当投资资产收益率已知 (Yield Known) 时,q 是合约有效期内的平均收益率 (连续复利)(When the yield on investment assets is known,q is the average yield over the life of the contract (continuously compounded)) $$ F_0 = S_0e^{(r-q)T} $$

    远期合约的定价

    • 远期合约在首次谈判时价值为零 (买卖价差效应除外),稍后其价值或正或负(A forward contract is worth zero when it is first negotiated (except for bid-ask spread effects),Later its value will be positive or negative)
      • 假设K 是交割价格(Suppose K is the delivery price)
      • $F_0$是当前合约 (今天谈判时) 的远期价格($F_0$ is the forward price of the current contract (when negotiating today))
      • 根据交割价格为K 的合同与交割价格为$F_0$的合同之间的差异(According to the difference between the contract with delivery price K and the contract with delivery price $F_0$)
      • 远期多头合约的价值为(The value of the forward long contract is) $$ f = (F_0-K)e^{-rt} $$
      • 远期空头合约的价值是(The value of the forward short contract is) $$ f = (K-F_0)e^{-rt} $$

    将远期价格与远期合约的计算结合可得(多头合约)

    • 不提供收益的投资资产的现货价格为$S_0$ $$ f = (F_0-K)e^{-rt} = S_0 - Ke^{-rt} $$
    • 当投资资产的收入已知 (Income Known) 时,I 是远期合约有效期内的收入现值 $$ f = (F_0-K)e^{-rt} = S_0 -I- Ke^{-rt} $$
    • 当投资资产收益率已知 (Yield Known) 时,q 是合约有效期内的平均收益率 (连续复利) $$ f = (F_0-K)e^{-rt} = S_0e^{-qt} - Ke^{-rt} $$

    远期价格与期货价格的关系

    • 当期限和资产价格相同时,通常假设远期价格和期货价格相等(When Maturity and Asset Price are the same,It is usually assumed that Forward Prices and Futures Prices are equal)
      • 理论上，当利率不确定时，二者略有不同(In theory, when the interest rate is uncertain, the two are slightly different)
      • 利率与资产价格之间的强正相关 (系数) 意味期货价格略高于远期价格(A strong positive correlation (coefficient) between interest rates and asset prices means that futures prices are slightly higher than forward prices)
        • 通过考虑标的资产价格 S 与利率高度正相关的情形。当S上涨时，期货多头的持有者因为期货的每日结算会马上获利。期货价格与利率的正相关性说明利率也很可能上涨，这时对所获利润的投资收益很可能会高于平均利率。同样当S下跌时，投资者马上会遭受损失，这时亏损的融资费用很可能会低于平均利率。而持有远期多头(而不是期货多头) 的投资者不会因为利率的这种上下变动而受到影响。因此，在其他条件相同的情况下，期货多头比远期多头更具吸引力。
        • When S increases, r increases => higher interest from recognized profit
        • When S decreases, r decreases => lower interest from financing
      • 强负相关 (系数) 意味反转(Strong negative correlation (coefficient) means inversion)

    股票指数

    • 股票指数(Stock Index)

      • 支付股息收益率的投资资产(Investment assets that pay a dividend yield)
      • 期货价格与现货价格的关系为,q 是指数所代表的投资组合在合约有效期内的平均股息收益率(Dividend Yield) $$ F_0 = S_0e^{(r-q)T} $$
      • 指数代表投资资产(Indices represent investment assets)
      • 指数的变化必须对应可交易投资组合价值的变化(Changes in the index must correspond to changes in the value of the tradable portfolio)

    • 指数套利（Index Arbitrage）

      • 当$F_0 > S_0e^{(r-q)T}$时(When $F_0 > S_0e^{(r-q)T}$)
        • 可以通过买入指数标的股票并卖出指数期货套利(You can arbitrage by buying the underlying stocks of the index and selling the index futures)

      • 当$F_0 < S_0e^{(r-q)T}$时(When $F_0 < S_0e^{(r-q)T}$)

        • 可以通过买入指数期货并做空或卖出指数相关的股票套利(You can arbitrage by buying index futures and shorting or selling index-related stocks)

      • 指数套利包含期货和许多不同股票的同时交易(Index Arbitrage involves simultaneous trading of futures and many different stocks)

      • 经常通过计算机生成交易(Often computer generated transactions)
      • 有时不可能同时进行交易，且𝐹0和𝑆0之间的理论上的无套利关系不成立(Sometimes it is impossible to trade at the same time, and the theoretical no-arbitrage relationship between $𝐹_0$ and $𝑆_0$ does not hold)

    货币的期货和远期

    • 货币与可提供收益的证券相似(Currencies are similar to securities that can provide yield)
    • 收益率为外币对应国家的无风险利率(The rate of return is the risk-free interest rate of the country corresponding to the foreign currency)
    • 假设$𝑟_𝑓$是外币对应国家的无风险利率(Suppose $𝑟_𝑓$ is the risk-free interest rate of the country corresponding to the foreign currency) $$ F_0 = S_0e^{(r-𝑟_𝑓)T} $$

    消费资产与投资资产

    消费资产（Consumption Assets）

    存储带来负收入,u 是单位时间的存储成本占资产价值的百分比（Storage Brings Negative Revenue,u is the storage cost per unit time as a percentage of the asset value） $$ F_0 = S_0e^{(r+u)T} $$

    或者U 是存储成本的现值（U is the present value of the storage cost） $$ F_0 = (S_0+U)e^{rT} $$

    持有成本(Cost of Carry),加成本(u)减收益(q) $$ c = r-q+u $$

    对于消费资产(For consumption assets),当个人和公司持有商品的目的是其消费价值而不是其投资价值时，他们不愿意出售商品并买人期货合约，因为期货合约并不能用于加工或其他形式的消费。 $$ F_0\leq S_0e^{cT} $$

    因为商品持有者可能会认为持有商品比持有期货能提供更多的便利，由于持有商品而带来的好处有时称为商品所具有的便利收益率(convenience vield)。

    对于消费资产(For consumption assets),期货价格为 $$ F_0 = S_0e^{(c-y)T} $$

    投资资产

    对于投资资产(For investment assets),期货价格为 $$ F_0 = S_0e^{cT} $$

    期货价格与预期的未来现价

    假设 k 是投资者对一项资产要求的预期回报,我们以无风险利率投资 $𝐹_0𝑒^{-rT}$并签订一份多头期货合约，这样在到期时可以创造 $𝑆_𝑇$ 的现金流入(Assume k is the expected return required by investors on an asset, We invest $𝐹_0𝑒^{-rT}$ at the risk-free rate and enter into a long futures contract, which creates a cash inflow of $𝑆_𝑇$ at maturity)

    这表明(This indicates) $$ 𝐹_0𝑒^{-rT}𝑒^{kT}=E(S_T) $$

    则$k=r \Rightarrow F_0=E(S_T)$

    这一关系式说明:在标的资产的收益与股票市场无关时($k=r$,k就是无风险收益率)，期货价格是对未来即期价格期望值的无偏估计。

    当一项资产价格与股票市场有正相关性时，由$k>r$和上式得出$F_0

    标的资产(Underlying Assets) 预期资产收益率与无风险利率之间的关系(Association between the expected return on assets and the risk-free rate) 期货价格与预期未来即期价格之间的关系(Association between futures prices and expected future spot prices)
    无系统风险(No system risk) 𝑘 = 𝑟 $F_0=E(S_T)$
    系统风险为正(Positive system risk) 𝑘 > 𝑟 $F_0
    系统风险为负(Negative system risk) 𝑘 < 𝑟 $F_0>E(S_T)$
    • 系统风险为正/Positive system risk：股票指数/Stock Index
    • 系统风险为负/Negative system risk：黄金/Gold（某些时期/Specific periods）