week 4a - Chapter 18 The Lognormal Distribution 1 The...

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1 Chapter 18 Chapter 18 The Lognormal Distribution The Lognormal Distribution
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2 The normal distribution Normal distribution (or density): Φ ( ; , ) x e x μ σ σ π μ σ - - 1 2 1 2 2
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3 The normal distribution Normal density is symmetric: If a random variable x is normally distributed with mean μ and standard deviation σ , z is a random variable distributed standard normal: The value of the cumulative normal [ P ( z < a )] distribution function N ( a ) or NormSDist(a) in Excel equals to the probability P of a number z drawn from the normal distribution to be less than a . Φ Φ ( ; , ) ( ; , ) x x μ σ μ σ = - x N ~ ( , ) μ σ 2 z N ~ ( , ) 0 1 N a e dx x a ( ) - -∞ 1 2 1 2 2 π
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4 The normal distribution The Normal Distribution allows us to make statements and inferences regarding future stock price levels. We are able to estimate probabilities of the terminal stock price being between two values, above a value, or below a value. Note: the probability of reaching an EXACT value is zero.
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5 The normal distribution (cont.)
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6 The normal distribution (cont.) The probability of a number drawn from the standard normal distribution of being between a and – a is : Prob ( z < a ) = N (– a ) Prob ( z < a ) = N ( a ) therefore Prob (– a < z < a ) = N ( a ) N (– a ) = N ( a ) [1 N ( a )] = N ( a ) 1 Example 18.1: Prob (–0.3 < z < 0.3) = 2·0.6179 1 = 0.2358
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7 The normal distribution (cont.) Converting a normal random variable to standard normal: If , then if And vice versa: If , then if Example 18.2: Suppose and , then and z N ~ ( , ) 0 1 x N ~ ( , ) μ σ 2 z x = - μ σ x N ~ ( , ) μ σ 2 z N ~ ( , ) 0 1 x z = + μ σ x N ~ ( , ) 3 5 z N ~ ( , ) 0 1 x N - 3 5 0 1 ~ ( , ) 3 5 3 25 + × z N ~ ( , )
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8 The normal distribution (cont.) Example : The number 7 is drawn from a Normal distribution of mean 4 and variance 9. What is the equivalent draw from a standard normal distribution? What is the probability of drawing a number larger than 7 ? Lower than 7 ? Exactly the number 7 ?
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9 Cumulative Normal Table d N(d) d N(d) d N(d) d N(d) -3 0.00135 -1.45 0.07353 0.05 0.51994 1.55 0.93943 -2.95 0.00159 -1.4 0.08076 0.1 0.53983 1.6 0.94520 -2.9 0.00187 -1.35 0.08851 0.15 0.55962 1.65 0.95053 -2.85 0.00219 -1.3 0.09680 0.2 0.57926
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week 4a - Chapter 18 The Lognormal Distribution 1 The...

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