金融衍生品定价理论第三章.doc

金融衍生品定价理论第三章
Chapter 3
Binomial Tree Methods
------ Discrete Models of Option Pricing
An Example
Question: When t=0, buying a call option of the stock at with strike price $40 and 1 month maturity1>. If the risk-free annual interest rate is 12% throughout the period [0, T], how much should the premium for the call option be?
Example
payoff =
Consider a portfolio
Example
When t=T,
has fixed value $35, no matter S is up or down

Example
If risk free interest r =12%, a bank
deposit of B=35/(1+) after 1 month
By arbitrage-free principle

Example
That is
Then
This is the investor should pay $ for this stock option.
Analysis of the Example
the idea of hedging: it is possible to construct an investment portfolio with S and c such that it is risk-free.
The option price thus determined (c_0=$) has nothing to do with any individual investor's expectation on the future stock price.
One-Period & Two-State
One-period: assets are traded at t=0 & t=T only, hence the term one period.
Two-state: at t=T the risky asset S has two possible values (states): , with their probabilities satisfying

One-Period & Two-State Model
The model is the simplest model.
Consider a market consisting of two assets: a risky S and a risk-free B
If: risky asset and risk free asset
known , when t=0,
t=T, 2 possibilities
Option Price at t=0?
(for strike price K, expired time T)
Analysis of the Model
- Stock Price, is a stochastic variable
Up, with probability p
Down, with probability 1-p
where is a stochastic variable.
Question & Analysis
If known at t=T,
how to find out when t=0?
Assume the risky asset to be a stock. Since the stock option price is a random variable, the seller of the option is faced with a risk in selling it. However, the seller can manage the risk by buying certain shares (denoted asΔ) of the stocks to hedge the risk in the option.
This is the idea!
Δ- Hedging