The Black-ScholesModelChapter 12 1 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 2 The Lognormal Property(Equations and , page 235) It follows from this assumption that Since the logarithm of ST is normal, ST is lognormally distributed 3 The Lognormal Distribution pounded Return, h (Equations and ), page 236) 5 The Expected Return The expected value of the stock price is S0emT The expected return on the stock is m – s2/2 6 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 7 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: 8 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 9 The Derivation of the Black-Scholes Differential Equation 10
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