Chapter 15 The Black-Scholes-Merton model.docx - Lecture notes for Chapter 15 Options Futures and Other Derivatives Tenth Edition by John C Hull The

Chapter 15 The Black-Scholes-Merton model.docx - Lecture...

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Lecture notes for Chapter 15, Options Futures and Other Derivatives, Tenth Edition, by John C. Hull The Black-Scholes-Merton model Lognormal property of stock prices The assumption is that percentage changes in the stock price in a short period of time are normally distributed. |--------|---|----------------------| 0  t T Let S be the change in the stock price in the time interval t. Then the assumption is: S ~ (  t, t) Equation (15.1) S where S = stock price at the beginning of the time interval t = expected return on the stock per year = Volatility of the stock price per year = standard deviation of the continuously compounded return on the stock (m,v) represents the normal distribution with mean m and variance v The following graph illustrates a fitted normal distribution of the return on a stock with a mean of 0.5 and a standard deviation of 0.2. -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2 2 2 2 2 Cell A1 Comparison with Normal(0.5,0.2) Curve #1 Normal(0.5,0.2) xDelimiter xPDelimiter 1
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From the above assumption we obtain the following result: ln(S T ) ~ (ln S 0 +( - 2 /2)T, T) Equation (15.3) The natural logarithm of the stock price at time T has a normal distribution with a mean of ln S 0 +( - 2 /2)T and a standard deviation of  T, where S 0 is the stock price at the current time, time 0. This further implies that the stock price at time T has a lognormal probability distribution. The expected value of S T is given by: E(S T ) = S 0 e T Equation (15.4) The variance of S T is given by: V(S T ) = S 0 2 e 2 T (e 2 T - 1) Equation (15.5) The following illustrates a fitted lognormal distribution of a stock price with an expected value of $10 and a standard deviation of $5. -10 0 10 20 30 40 50 60 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.12 0.12 0.12 0.12 0.12 stockprice Comparison with Lognorm(10,5) stockprice Lognorm(10,5) xDelimiter xPDelimiter Comparing the lognormal distribution of the stock price with the normal distribution of the stock return, we note that the lognormal distribution is skewed, while the normal distribution is symmetric. Further the return on the stock can be less than zero, while the stock price can only take on values between 0 and infinity. 2
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Read problem 15.27 S 0 = 50 = 0.18 = 0.30 The probability distribution of the stock price in two years would be given by (applying equation (15.3)): ln(S T ) ~ (ln 50 +(0.18-0.3 2 /2)2, 0.3 2) ~ (4.1820,0.18) The mean of ln(S T ) is 4.1820. The standard deviation of ln(S T ) is 0.18= 0.4243. 95% confidence intervals for the stock price are calculated as follows: 4.1820 – 1.96 x 0.4243 ln(S T ) 4.1820 + 1.96 x 0.4243 3.3504 ln(S T ) 5.0136 e 3.3504 S T e 5.0136 28.51 S T 150.45 Mean and standard deviation of the stock price . E(S T ) = 50e 0.18x2 = $71.6665 V(S T ) = 50 2 e 2 x 0.18 x 2 (e 0.3x0.3x2 - 1) = 1012.9247 The standard deviation of the stock price = 1012.9247 1/2 = $31.8265 We can use this to answer the question as to what is the probability that the stock price in 2 years will be greater than $100.
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  • Winter '19
  • LOUIS CHARBONNEAU

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