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Unformatted text preview: Lecture 7: Building the Tree This lecture shows that the model is a reasonably accurate approximation of more realistic dynamics of the underlying security. We also derive expressions for u , d , and r , and consequently q , for a binomial tree which matches the statistical properties of the underlying security. I. Why is the Binomial Model Realistic? A. LogNormal Model B. LogNormal Approximation II. Choosing Binomial Model Inputs A. Stock Price Parameters u , d , and q B. Number of Periods n III. Choosing the Number of Periods Using the Binomial Model I. Why is the Binomial Model Realistic? SI At first glance, the binomial model seems to do a terrible job describing reality. Stock dont either return u or d , They can take on any value. SI Were now going to derive a more realistic model for the stock price dynamics The lognormal model. SI Then well show that the binomial model can approximate the lognormal model arbitrarily well. Its just a matter of picking the periods short enough SI Well mention some of the problems with the log normal model. Bus 35100 Page 2 Robert NovyMarx Using the Binomial Model A. LogNormal Model Denote the stock prices at the end of each year by S t , for t D 0, 1, 2, 3, ... The simple annual returns on the stock is the gross return minus one: r t D R t NUL 1 where R t D DLE S t S t NUL 1 DC1 . The log return (or continuously compounded return) is given by ln R t D ln DC2 S t S t NUL 1 DC3 D ln S t NUL ln S t NUL 1 Bus 35100 Page 3 Robert NovyMarx Using the Binomial Model Example Stock price last year: $100; now: $10. SI What are the simple and log returns? The simple return is 10 NUL 100 100 D NUL 0.9 D NUL 90 % . The continuously compounded return is ln DC2 10 100 DC3 D NUL 2.30 D NUL 230 % . SI Note: with log returns were not limited to 100% losses. What is the log return when a company goes bankrupt? The log return over several years is the sum of the annual returns ln S T NUL ln S D ln R T C ... C ln R 1 D T X t D 1 ln R t . Bus 35100 Page 4 Robert NovyMarx Using the Binomial Model Lognormal Model Assumptions: These assumptions are approximately consistent with historical stock price data: 1. Log returns are independently distributed. SI I.e., ln R t is independent of ln R s , for t 6D s . 2. Log returns are identically distributed. SI I.e., the probability distribution of ln R t is the same as that of ln R s for all t and s . 3. The stock price evolves continuously. SI I.e., the stock price doesnt jump. Remark: Assumptions 1 and 2 jointly are often abbreviated by saying that logreturns are i.i.d. SI This stands for independently and identically distributed. Q: any problems with these assumptions?...
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This note was uploaded on 08/14/2009 for the course BUS 35100 taught by Professor Novymarx during the Winter '07 term at CHIC.
 Winter '07
 NovyMarx

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