【原】期权定价的二叉树模型学习笔记（II）

小朱的读书笔记 2021-11-25

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编者按:二叉树模型的第二部分学习笔记中涉及到欧式看涨看跌期权的定价公式和所谓的平价公式,从形式上来看,该公式还不算特别复杂的.由于欧式期权是在到期日时实施期权,因此它相比美式期权(在到期日之前皆可实施)来说还是较为简单的.关于欧式看涨和看跌期权的平价公式,其刻画了两个期权之间的等量关系,往后所要学习到的美式期权则没有类似的平价公式.因此可以说,平价公式是欧式期权所独有的,这也是欧式期权相比美式期权多的一个差异点.笔记后半部分涉及到的鞍和鞍测度等概念,严格来说其实涉及到测度论的知识,因此首先需要了解的是测度的基本概念.引进鞍的一大目的是为了阐述这样一个核心结论:在二叉树模型下,市场的无套利性质与鞍测度之间具有等价性(if and noly if).尽管我们假设市场是无套利的(动态的无套利),然而要想从数学这个视角精细地刻画这点就不得不寻找等价条件.毫无疑问的是,资产定价基本定理为我们揭示了鞍测度与市场无套利之间的微妙联系.

二叉树模型的期权价计算

Denote .,

We consider possible values of option at :

Question:If are given, how can we determine

In particular,

Answer:We can determine by us-ing backward induction in the one period and two-state model.

Notice that.

Meanwhile, we can calculate

Then we want to find

二叉树模型欧式期权定价公式

Define a risk-neural measure :

Then,we will get

So that for any ,

When ,=0.

折现价二叉树模型的平价公式

Denote Then the European call option valuation formula is

Especially,when ,,

For the binomial tree method,the call-put parity(in discrete form) becomes

鞅（Martingale）的概念

the bet at game,the next bet.If under the condition that complete information of all previous game are available,the expectation of equals the previous stake i.e.

then we say the gamble is fair.

In Mathematics, is called -algebra in stochastic theory.

Definition1(Martingale):The best sequence that satisfies condition

as a discrete random process,is called a Martingale.

Remark:Martingle is often used to refer to a fair gamble.

Then，we give mathematical definition of Martingale.

Definition1'(Martingale ):A sequence is a Martingale with respect to sequence if for all :

鞅测度

Under the risk-neutral measure ,the discount prices of an underlying asset ,as a discrete random process,satisfy the equation:

Remark:Hence the discount price sequence of an underlying asset is a martingale.

Definition2(Martingale measure):The risk-neutral measure is called the martingale measure.

概率测度等价定义

Definition3(Equivvalent measure):Probability measure and Probability measure are said to be equivalent if and only if for any probability event (set) there is

i.e. the Probability measure and have the same null set.

The European option valuation formula under the sense of equivalent Martingale measure ,can be written as

Especially,

鞅测度和无套利等价性;用倒向归纳法证明期权不等式

Theorem1(The fundamental theorem of asset pricing):If an underlying asset price moves as a binomial tree, there exists an equivalent Martingale measure if and only if the market is arbitrage-free.

Dividend-Paying(股息支付):

An underlying asset pays dividends in t-wo ways:

Pay dividends discretely at certain times in a year;
Pay dividends continuously at a certain rate.

We only consider the continuous model. For studying the continuous Model, there are two reasons.Meanwhile,we meet the example:A company needs to buy Euro at time to pay a German company. To avoid any loss if Euro goes up, the company buys a call option of Euro with Expiration date at rate .How much premium should the company pay?