L12a 专题： 期权定价的拓展

指数期权与货币期权

指数期权

中金所沪深300股指期权合约表

指数期权的用途：组合保险

  • Suppose the value of the index is and the strike price is

  • If a portfolio has a of , the portfolio insurance is obtained by buying put option contract on the index for each dollars held

  • If the is not , the portfolio manager buys put option for each dollars held

  • In both cases, is chosen to give the appropriate insurance level

组合保险：例一

  • Portfolio has a beta of 1.0
  • It is currently worth $500,000
  • The index currently stands at 1000
  • What trade is necessary to provide insurance against the portfolio value falling below $450,000?

组合保险：例二

  • Portfolio has a beta of 2.0
  • It is currently worth $500,000 and index stands at 1000
  • The risk-free rate is 12% per annum
  • The dividend yield on both the portfolio and the index is 4%
  • How many put option contracts should be purchased for portfolio insurance?

  • Solution

    • If index rises to 1040, it provides a 40/1000 or 4% return in 3 months
    • Total return (incl. dividends) = 5%
    • Excess return over risk-free rate = 2%
    • Excess return for portfolio = 4%
    • Increase in Portfolio Value = 4+3−1=6%
    • Portfolio value=$530,000
  • The Strike Price
Value of Index in 3 months Expected Portfolio Value in 3 months ($)
1,080 570,000
1,040 530,000
1,000 490,000
960 450,000
920 410,000
  • An option with a strike price of 960 will provide protection against a 10% decline in the portfolio value

外汇期权

  • 范围远期合约 (Range-Forward Contract)

    • Have the effect of ensuring that the exchange rate paid or received will lie within a certain range
    • When currency is to be paid it involves selling a put with strike and buying a call with strike (with )
    • When currency is to be received it involves buying a put with strike and selling a call with strike
    • Normally the price of the put equals the price of the call

期权性质：期权价格的下限

  • 由以下等价关系得到：

    • 股票期初价格为，股息收益率为
    • 股票期初价格为，无股息

  • 考虑投资组合A与投资组合B

    • 组合A： 一份欧式看涨期权+数量为的现金
    • 组合B： 股股票，股息被再投资于股票

  • 同理可得看跌期权价格下限：

期权性质：看跌期权-看涨期权平价关系

  • 考虑投资组合A与投资组合C

    • 组合A： 一份欧式看涨期权+数量为的现金
    • 组合C： 一份欧式看跌期权+股股票，股息被再投资于股票

  • 美式期权

定价

  • 偏微分方程

  • 定价公式

  • 风险中性定价

期货期权

概念

  • 即期期权（spot options / options on spot）vs期货期权（futures options / options on futures）

  • 期货期权

    • Referred to by the maturity month of the underlying futures
    • Usually they are American options
    • Expires on or a few days before the earliest delivery date of the underlying futures contract

期货期权的特性

  • When a call futures option is exercised the holder acquires

    • A long position in the futures
    • A cash amount equal to the excess of the futures price at the time of the most recent settlement over the strike price
    • Payoff:

  • When a put futures option is exercised the holder acquires

    • A short position in the futures
    • A cash amount equal to the excess of the strike price over the futures price at the time of the most recent settlement
    • Payoff:
  • Potential advantages of futures options over spot options
    • Futures contracts may be easier to trade and more liquid than the underlying asset
    • Exercise of option does not lead to delivery of underlying asset
    • Futures options and futures usually trade on same exchange
    • Futures options may entail lower transactions costs

期权性质：看跌期权-看涨期权平价关系

  • 欧式期货期权

    • European futures options and European spot options are equivalent when futures contract matures at the same time as the option
    • It is common to regard European spot options as European futures options when they are valued

  • 考虑投资组合A与投资组合B

    • 组合A： 一份欧式看涨期权+数量为的现金
    • 组合B： 一份欧式看跌期权+一份期货多头+数量为的现金

  • 美式期权

期权性质：期权价格的下限

  • 由看跌期权-看涨期权平价关系：

    • 看涨期权：
    • 看跌期权：

  • 美式期权

    • 看涨期权：
    • 看跌期权：

定价：二叉树

  • 与即期期权二叉树类似，但期初期货合约价格为0

  • Growth Rates For Futures Prices

    • A futures contract requires no initial investment
    • In a risk-neutral world the expected return should be zero
    • The expected growth rate of the futures price is therefore zero
    • The futures price can therefore be treated like a stock paying a dividend yield of

定价：Black's Model

  • 期货合约价值的漂移率

    • 风险中性：
    • 漂移率为0：
    • 服从的随机微分方程：

  • 偏微分方程

  • Black's Model

  • Application of Black's Model

    • Black's model is frequently used to value European options on the spot price of an asset
    • This avoids the need to estimate income on the asset