编者按:二叉树模型是金融衍生产品期权定价的离散模型.人们可以借助二叉树模型分别对欧式看涨看跌期权、美式看涨看跌期权进行期权金定价.抛开金融意义不谈,单从数学角度出发,这部分运用的数学知识仅是微积分的基本知识点.额外需要注意的是,在二叉树章节中反向归纳法(倒向归纳法)是特别重要的一种方法,其在涉及到有关期权问题的证明中显得尤为重要.之所以运用反向归纳法,是因为期权定价中我们已知未来某一时刻的期权状态,由此出发逐步倒向递推在时刻的价格.本系列是笔者学习二叉树模型所做的课堂笔记一部分,仅供参考!
[h1]Hedging Concept（套期保值概念）[/h1]Firstly,we should learn the definition of One-Period & Two-State.
Definition1.1(One-Period): Assets are traded at & only, hence the term one period.
Definition1.2(Two-State): At the risky asset has two possible values(states):& ,with their probabilities satisfying
Question:If risky asset and risk free asset ,known ,when two possibilities
,.(for strike price ,expired time .)
If known at ,how to find out when
Definition1.3(Hedging Definition):For a given option ,trade shares of the underlying asset in the opposite direction so that the portfolio
is risk-free.We can solve Meanwhile,we can getDefine a new Probability MeasureNotice that
[h1]期权价的期望表示和风险中性测度[/h1] Notice that denotes that the expectation of the random variable under the probability measure .
Let be a certain risky asset, and is a risk-free asset, then is called the discounted price(also known as the relative price) of the risky asset at time .
Theorem2.1:Under the probability measure ,an option's discounted price is its expectation on the expiration date.i.e
Remark:In order to examine the meaning of the probability measure ,consider is an underlying risky asset.It is easy to calculate
[h1]Risk-Neutral World(风险中性世界)[/h1]Definition3.1(Risk-Neutral World):Under the probability measure ,the expected return of a risky asset at is the same as the return of a risk-free bond.A financial market possessing this property is called a Risk-Neutral World.
Definition3.2(Risk-neutral measure):The probability measure defined by
is called by risk-neutral measure.
Definiton3.3(The risk-neutral price):The option price given under the risk-neutral measure is called the risk-neural price.
[h1]Replication(复制),等价性定理[/h1]In a market consisted of a risky asset and a risk-free asset ,if there exists a portfolio
such that the value of the portfolio is equal to the value of the option at ,then is called a replicating portfolio of the option ,then option price
Theorem4.1:In a market consisted of
- a risky asset ;
- a risk-free asset .
Then is true if and only if the market is arbitrage-free.
In fact, if the market is arbitrage-free, then there exists a risk-neutral measure defined by
such that
[h1]二叉树的构造[/h1]This means that if at the initial time the price of the underlying asset is , then at , will have possible values
Denote
未完待续......
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