新闻报道策划案例.ppt

A First Look at the Black-Scholes Equation
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Primbs, MS&E 345
Background:
Derivative Security:
Example: European Call Option.
The right, but not the obligation, to purchase a share of
stock at a specified price K (the strike price), at a specified
date T (the maturity date).
A derivative (or derivative security) is a financial instrument whose value depends on the values of other, more basic underlying variables. ([Hull, 1999]).
Arbitrage:
A riskless profit that involves no investment.
(A free lunch)
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Primbs, MS&E 345
Assumptions (to be used throughout most of the course)
There are no transaction costs (. markets are frictionless)
Trading may take place continuously
There is no prohibition on short selling
The risk free rate is the same for borrowing and lending
Assets are perfectly divisible.
These are the “standard assumptions”.
When I deviate from them, I will mention it specifically, otherwise assume that they are always in force.
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Primbs, MS&E 345
The Set-up:
Securities:
Bond:
Stock:
Bond:
-Deterministic
-Exponential Growth
-pounding
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Primbs, MS&E 345
The Set-up:
Securities:
Bond:
Stock:
Stock:
-Geometric Brownian Motion
-Log-Normal
-Always positive
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Primbs, MS&E 345
The Set-up:
Consider a derivative security whose price depends on St and t.
We will call it:
Securities:
Bond:
Stock:
By Ito’s lemma:
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Primbs, MS&E 345
Now we have 3 price processes:
Bond:
Stock:
Derivative:
es the Black-Scholes argument:
Let’s form a portfolio using two of the assets, so that it looks exactly like the third.
Then this portfolio must have the same price as the third.
We can choose any two assets for our portfolio. Let’s choose
the stock and derivative, and create a bond.
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Primbs, MS&E 345
Bond:
Stock:
Derivative:
Our portfolio will consist of D shares of the stock and b of the derivative.
Now we have 3 price processes:
To create a bond, we can dynamically choose D and b so that the portfolio is riskless (. dP