金融衍生品定价理论(期权定价)2.ppt

梁进
Financial Derivatives
Pricing for
Chapter 2
Arbitrage-Free Principle
Robert C. Merton
Financial Market
Two Kinds of Assets
Risk free asset
Bond
Risky asset
Stocks
Options
….
Portfolio – an investment strategy to hold different assets
Investment
At time 0, invest S
When t=T,
Payoff =
Return =
For a risky asset, the return is uncertain, ., S is a random variable
A Portfolio
a risk-free asset B
n risky assets
a portfolio

is called a investment strategy
on time t, wealth:
portion of the cor. Asset
Arbitrage Opportunity
Self-financing - during [0, T]
no add or withdraw fund
Arbitrage Opportunity - A self-financing investment,

and Probability

Prob
Arbitrage Free Theorem
Theorem
the market is arbitrage-free in time [0, T],
are any 2 portfolios satisfying
&
Proof of Theorem
Suppose false, .,
Denote
B is a risk-free bond satisfying
Construct a portfolio at
Proof of Theorem cont.
r – risk free interest rate, at t=T
Then
From the supposition
Proof of Theorem cont.
It follows
There is an Arbitrage Opportunity,
Contradiction!