衍生5-determination of forward and future price

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5-determination of forward and futures price

• asset category & short selling (价格由市场决定，而不是由交易双方)
  • sort
    • consumption asset: primarily for consumption eg.copper,oil
    • investment asset: held purely for investment purpose eg.gold,silver
  • short selling(卖空)
    • 概念： sell security you don't own (your broker borrows the security from another client and sell them in the market)
      • short selling in China(融券交易) 投资者以资金或证券作为抵押，向证券公司借入证券卖出，在约定的期限内，买入相同数量和品种的证券归还券商并支付相应的融券费用
    • 注意事项
      • 承担相应的义务： must pay dividend & benefits to the onwer of security
      • there's a small fee for borrowing
      • there's a margin account (like margin call for future contract)
    • 计算
      • • short sell准确来说指的是借了钱卖出去那个时候
• arbitrage (套利)
  • 概念： lock in a riskless profit by simultaneously(同时) enter into two or more market
    • markets like: spot/money(for short-time funding)/derivative
    • necessary condition: pricing inequality
  • find the arbitrage opportunity (key is that,sell the contract with high price and buy the contract with low price)
    • forward > spot price
      • borrow money at money market
      • purchase the stock at spot market
      • short a forward contract at forward market(卖方)
      • payoff
        • PF - Ps*E(Rf)
          • 
    • spot > forward price
      • short sell at spot,then invest at riskfree rate
      • long a forward contract(买方)
      • at the expiration date, purchase the stock(结束远期合同) and close short-selling position
      • payoff
        • Ps*E(Rf) - PF
  • the adjust of forward price
    • 原理： in an efficient market,arbitrage will force market to evaluate the equilibrium forward price
    • 公式： S-spot price F-forward price T-交付的时间 r-0.25-year risk-free rate
      • if F0 > S*E, short the forward contract
      • if F0 < S*E, long the forward contract
      • if F0 = S*E, there's no arbitrage chance
        • F是理论远期价格，F0是现实的价格，要将两个价格作比较
• forward price： known income/yield
  • 相关信息： stock paying known dividend & coupon-bearing bond these provide predictable cash income and the cash income should be adjusted when calculating forward price
    • forward YS future price when maturity & asset price are the same,forward = future price however,when interest rate different,they are slightly different positive correlation between i&p implies:future price is slightly higher than forward price
      • explanation: 因为期货是daily settled a gian(increase of future price) could be invested into higher i,loss could be financed by lower i 当标的资产价格上升，期货价格通常也会随之升高，期货多头将因每日结算制而立即获利，并可按高于平均利率的水平将所获利润进行再投资。当标的资产价格下降，期货多头将立即亏损，这是他可以按低于平均利率的水平从市场上融资以补充资金。相比之下，远期合约的多头将不会因利率变动而受到上述影响。
  • sort
    • provide a known income
      • 公式： I：PV of the income F0:这里是均衡状态的forward price 另外，使用的r应该不是一样的
      • 原理： 不管是哪种套利策略，都要减去PV of PMT,再算与forward的差价
        • spot price-900,t2=9 month,r-4% payment-40,t1=4 month,r-3% (1) 若forward price=910，采取borrow to buy bond & short forward的策略 PV of coupon payment:39.6(r=3%) the remaining 860.4 is borrowed for 9 month(r=4%), PV=886.6 为什么要borrow？ 因为套利默认是不需要自己出本金的 而39.6相当于一种赔偿，可以在总借款中省一笔钱 payoff: 910-886.6=23.4 (2) 若forward price=870，采取short the bond & long forward的策略 the remaining 860.4 is invested,PV=886.6 因为从券商那里借债券过来是要付债权人息票的，减去付的钱，剩下的才能拿去投资 payoff:886.6-870=16.6
    • provide a known yield
      • 相关信息： income of asset could express as a percentage of the asset's price at the time the income is paid
      • 衡量 yield： e(q*n): principal + yield
        • eg.provide a yield of 5% per annum, if income is paid semiannually,each 6 month the income is paid 2.5%
      • 套利公式： q: the average yield during the life of the contract
      • market value/price 公式
        • total market value: St1:one unit of the asset worth St1*E(qt1)-St1:a cash proceeds worth
        • is the market price of E(qt1)-1 units of the asset sold/purchased at t1
• value a forward contract
  • 原理： a forward contract is worth zero when it's first negotiated later it may have positive/negative value 之前说的价格都是在说标的物，只有这个是在说合同本身
  • 基本信息
    • K(交割价格)：delivery price of a forward contract
    • F0(远期价格):the applicable forward price if we negotiate a forward contract with the same expiration date today
      • 在交易的0时点，远期价格与交割价格是一样的，在0时点到交割日t之间，远期价格会被即期价格，到期时间，和市场利率所影响，所以远期价格在这个时刻是一直在变化的
    • f(合同价值): the value of forward contract today
      • 在0时点远期价值为0，但是之后就不一定的，比如签订合约约定的玉米交割价格是50，但是后来玉米涨到了60，这个时候远期多头就赚了，此时对多头来说远期价值就是10。
    • 关系
      • for forward contract negotiated today: K = F0 f = 0
      • for a forward contract negotiated some times ago: K ≠ F0, or K = F0 f could be zero,positive,negative
  • 计算价值
    • 帮助理解： at T-1,long a forward contract(ContractⅠ),the forward price is K,maturity is T1 at T0,take another forward contract(Contract Ⅱ),the forward price is F0,maturity is T1 the spot price at T1 is S then,the payoff of ContractⅠ is S - K,the payoff of Contract Ⅱ is F0 - K so the total payoff is (S - K)+(F0 - K) = F0 - K 也就是说，收益被固定了
    • value of a long forward contract:
    • value of a short forward contract:
  • variation 升级版（增加附加条件）
    • asset with no-income:
    • asset with known income:
    • asset with known yield:
• futures on commodity
  • storage cost(存储费用)
    • 原理： for consumption goods,sell them will generate extra loss, due to the lack of raw material
      • 商家认为持有商品比持有期货更好 一个工厂不会将持有原油库存和持有原油期货等同来看，因为库存能够生产，还能从暂时短缺中盈利 而期货做不了
    • 表示符号
      • u: the storage cost per unit as a percent of the asset value
      • U: PV of the storage cost
    • 策略选择
      • arbitrage work
      • arbitrage doesn't work (onwer unwilling to sell the asset)
    • the convenience yield (便利收益率)
      • 概念： 将持有商品所带来的好处称为商品所具有的便利收益率 它衡量了持有期货小于持有商品的程度 它反映了市场对将来能够购买商品的可能性的期望， 商品短缺的可能越大，便利收益率越高
      • 公式：
• future price of stock indice
  • 相关信息： as an investment asset paying a dividend yield index should represent an investment asset (Nikkei就不行，index of yen while contract of dollar)
  • 公式： q: the average dividend yield
    • buy the stocks & short the future
      • occasionally simultaneous trades are not possible
    • short/sell the stock & long the future
      • short or sell 一个是借钱卖出，一个是直接卖出
• futures and forwards on currencies