Chapter 12 & 20 Chapter 21
The Black-Scholes Formula and Option Greeks
Adapted from Black & Scholes (1973), The Pricing of Options and Corporate Liabilities, The Journal of Political Economy, Vol. 81, No. 3., pp. 637-654.
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Black-Scholes Assumptions
• Assumptions about stock return distribution
Continuously compounded returns on the stock are normally distributed and there is no jumps in the stock price The volatility is a known constant Future dividends are known, either as discrete dollar amount or as a fixed dividend yield
• Assumptions about the economic environment
The risk-free rate is a known constant There are no transaction costs or taxes It is possible to short-sell costlessly and to borrow at the risk-free rate
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Black-Scholes Assumptions
• The original paper by Black and Scholes begins by assuming that the price of the underlying asset follows a process like the following
S(t) is the stock price dS(t) is the instantaneous change in the stock price is the continuously compounded expected return on the stock δ is the dividend yield on the stock is the continuously compounded standard deviation (volatility) Z(t) is the standard Brownian motion dZ(t) is the change in Z(t) over a short period of time
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Black-Scholes Assumptions
• There are 2 important implications of equation (20.1)
Suppose the stock price now is S(0). If the stock price follows equation (20.1), the distribution of S(T) is lognormal, i.e.
ln[ S (T )] ~ N (ln[ S (0)] [ 0.5 2 ]T , 2T )
Geometric Brownian motion allows us to describe the path the stock price takes in getting to a terminal point
Fischer Black, 1938 – 1995
Myron Scholes Robert Merton Nobel prize in economic sciences, 1997
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