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Unformatted text preview: Session 6. Black Scholes Model and Modeling the Stochastic Process (will not be included in final exam, but important for project) Model of the behavior of spot price Geometric Brownian motion continuous time, continuous stock prices Binomial discrete time, discrete stock prices approximation of geometric Brownian motion The Volatility The volatility is the standard deviation of the continuously compounded rate of return in 1 year The standard deviation of the return in time t is If a stock price is $50 and its volatility is 25% per year what is the standard deviation of the price change in one day? t Estimating Volatility from Historical Data 1. Take observations S , S 1 , . . . , S n at intervals of years 2. Define the continuously compounded return as: 3. Calculate the standard deviation, s , of the u i s 4. The historical volatility estimate is: u S S i i i =  ln 1 = s Nature of Volatility Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed For this reason time is usually measured in trading days not calendar days when options are valued Creation of synthetic option Geometric Brownian motion requires advanced calculus (Itos lemna) Binomial based on elementary algebra Options: the family tree Black Merton Scholes (1973) Analytical models Numerical models Analytical approximation models Term structure models B & S Merton Binomial Trinomial Finite difference Monte Carlo European Option European American Option American Option Options on Bonds & Interest Rates Analytical Numerical Modelling stock price behaviour Consider a small time interval t : S = S t + t S t 2 components of S: drift : E( S ) = S t [ = expected return (per year)] volatility : S / S = E( S/S) + random variable (rv) Expected value E(rv) = 0 Variance proportional to t Var(rv) = t Standard deviation = t rv = Normal (0, t) = Normal (0, t) = z z : Normal (0, t) = t : Normal(0,1) 2200 z independent of past values (Markov process) Geometric Brownian motion illustrated Geometric Brow nian motion100.0050.00 0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 184 192 200 208 216 224 232 240 248 256 Drift Random shocks Stock price Geometric Brownian motion model 2200 S/S = t + z 2200 S = S t + S z = S t + S t If t "small" (continuous model) dS = S dt + S dz Binomial representation of the geometric Brownian u , d and q are choosen to reproduce the drift and the volatility of the underlying process: Drift: Volatility: Cox, Ross, Rubinsteins solution: S uS dS q 1...
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This note was uploaded on 10/14/2010 for the course FINA 529 taught by Professor Jeromeyen during the Fall '10 term at HKUST.
 Fall '10
 JeromeYen
 Derivatives

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