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lec07B - Lecture 7 Building the Tree This lecture shows...

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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. Log-Normal Model B. Log-Normal 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
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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 don’t either return u or d , They can take on any value. SI We’re now going to derive a more realistic model for the stock price dynamics The log-normal model. SI Then we’ll show that the binomial model can approximate the log-normal model arbitrarily well. It’s just a matter of picking the periods short “enough” SI We’ll mention some of the problems with the log- normal model. Bus 35100 Page 2 Robert Novy-Marx
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Using the Binomial Model A. Log-Normal 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 Novy-Marx
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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 we’re 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 0 D ln R T C ... C ln R 1 D T X t D 1 ln R t . Bus 35100 Page 4 Robert Novy-Marx
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Using the Binomial Model Log-normal 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 doesn’t jump. Remark: Assumptions 1 and 2 jointly are often abbreviated by saying that “log-returns are i.i.d.” SI This stands for “independently and identically distributed.” Q: any problems with these assumptions? We’ll spend some time on these problems (and some “fixes”) later in the quarter Bus 35100 Page 5 Robert Novy-Marx
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Using the Binomial Model Mathematical Fact: these assumptions imply that log returns are normally distributed SI I.e., they follow a “Brownian motion”.
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