文档介绍：The Black-Scholes Model Chapter 12
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The Stock Price Assumption
Consider a stock whose price is S
In a short period of time of length dt, the return on the stock is normally distributed:
where m is expected return and s is volatility
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The Lognormal Property (Equations and , page 235)
It follows from this assumption that
Since the logarithm of ST is normal, ST is lognormally distributed
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The Lognormal Distribution
pounded Return, h (Equations and ), page 236)
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The Expected Return
The expected value of the stock price is S0emT
The expected return on the stock is
m – s2/2
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The Volatility
The volatility of an asset is the standard deviation of the pounded rate of return in 1 year
As an approximation it is the standard deviation of the percentage change in the asset price in 1 year
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Estimating Volatility from Historical Data (page 239-41)
Take observations S0, S1, . . . , Sn at intervals of t years
Calculate the pounded return in each interval as:
Calculate the standard deviation, s , of the ui´s
The historical volatility estimate is:
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The Concepts Underlying Black-Scholes
The option price and the stock price depend on the same underlying source of uncertainty
We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty
The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate
This leads to the Black-Scholes differential equation
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The Derivation of the Black-Scholes Differential Equation
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