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Unformatted text preview: 13.1 Valuing Stock Options:The BlackScholes Model Chapter 13 12.2 Reviewing Binomial Trees Valuing Options Using Binomial Trees We can replicate the bond We can replicate the option We can use risk neutral valuation How to build the tree? We can ignore the expected stock return We only need to construct u and d 12.3 Reviewing Binomial Trees The stock already has the required rate of return figured into the price We need the volatility to value the option The volatility tells us how much movement ( systematic and nonsystematic risk ) there is in stock prices This allows us to draw a tree to estimate the possible stock values at maturity CoxRossRubenstien use Use the tree to get the risk neutral probability t t e d e u  = = , 12.4 Estimating Volatility from Historical Data The real question in terms of constructing the binomial tree is how to estimate 1. Take observations S , S 1 , . . . , S n at intervals of years 2. Define the continuously compounded return, u , as: then taking natural logs u S S i i i =  ln 1 S S e i u i = 12.5 Estimating Volatility from Historical Data The real question in terms of constructing the binomial tree is how to estimate 1. Take observations S , S 1 , . . . , S n at intervals of years 2. Define the continuously compounded return, u , as: 3. Calculate the standard deviation, s , of the u i s over the time period years Because this is the volatility for years but we want it for one full year we adjust by u S S i i i =  ln 1 12.6 Estimating Volatility from Historical Data The real question in terms of constructing the binomial tree is how to estimate 1. Take observations S , S 1 , . . . , S n at intervals of years 2. Define the continuously compounded return, u , as: 3. Calculate the standard deviation, s , of the u i s over the time period years 4. The historical volatility estimate is: We do this b/c variance is proportional u S S i i i =  ln 1 s = 12.7 Nature of Volatility Example ( Excel ) How do we choose the data S , S 1 , . . . , S n ? 1) Daily, weekly, or monthly? 2) How far back should we go? Volatility is usually estimated using 1) Closing, daily returns 2) Less than one year of data If we use daily returns, how do we treat weekends and holidays? 12.8 Nature of Volatility Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed Time is usually measured in trading days not calendar days One year has 252 trading days A six month option is assumed to last for 252/2 = 126 days instead of 365/2 = 182.5 days 12.9 The Volatility What is volatility in the binomial tree?...
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This note was uploaded on 05/09/2008 for the course FIN 522 taught by Professor Roskelley during the Spring '08 term at University of Arizona Tucson.
 Spring '08
 ROSKELLEY
 Derivatives, Options, Valuation

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