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01-Math(f) - Mathematical Preliminaries Primbs MS&E 345 1...

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Primbs, MS&E 345 1 Mathematical Preliminaries
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Primbs, MS&E 345 2 Math Preliminaries: Our first order of business is to develop mathematical models of asset prices and random factors. For most of this course, we will model prices as continuous time stochastic processes and stochastic differential equations.
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Primbs, MS&E 345 3 Math Preliminaries: Building Blocks: Stochastic Differential Equations Brownian Motion Poisson Processes Solutions to SDEs Ito’s Lemma
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Primbs, MS&E 345 4 Math Preliminaries: Building Blocks: Brownian Motion Poisson Processes Stochastic Differential Equations Solutions to SDEs Ito’s Lemma
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Primbs, MS&E 345 5 Gaussian (Normal) Random Variable An n-dimensional Gaussian (Normal) random variable is a random variable with density function: ( 29 ) ( ) ( exp ) 2 ( 1 ) ( ~ 1 2 1 2 μ μ π - Σ - - Σ = - x x x f X T X n where μ is the mean and Σ the covariance matrix. Notation: ) , ( ~ Σ μ N X ] [ X E = μ ] ) )( [( T x x E μ μ - - = Σ
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Primbs, MS&E 345 6 Brownian Motion: A real-valued stochastic process + t z t : is a Brownian motion if (1) 0 0 = z (2) s t s t N z z s t - - for ) , 0 ( ~ (3) 1 2 3 1 2 ,..., , - - - - n n t t t t t t z z z z z z are independent for n t t t ... 2 1
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Primbs, MS&E 345 7 Facts about Brownian Motion: (1) Sample paths of Brownian motion are (can be chosen to be) continuous with prob. 1 . (2) Brownian motion is nowhere differentiable. A good building block for prices that do not jump.
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Primbs, MS&E 345 8 Simulation of Brownian Motion 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time z t
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Primbs, MS&E 345 9 Math Preliminaries: Building Blocks: Brownian Motion Poisson Processes Stochastic Differential Equations Solutions to SDEs Ito’s Lemma
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Primbs, MS&E 345 10 Math Preliminaries: Building Blocks: Brownian Motion Poisson Processes Stochastic Differential Equations Solutions to SDEs Ito’s Lemma
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Primbs, MS&E 345 11 Poisson Random Variable: Remark: X is the number of events that occur in one time unit when the time between events is exponentially distributed with mean 1 /λ. A discrete random variable X is Possion with parameter λ >0 if λ λ - = = e k k X P k ! ) ( k=0,1,... λ = ] [ X E λ = ) ( X Var Statistics:
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Primbs, MS&E 345 12 Poisson Process: A Poisson Process with parameter (intensity) λ is a stochastic process π t that satisfies (1) 0 0 = π (2) s t π π - has a Poisson distribution with parameter λ (t-s) for s<t . (3) 1 2 3 1 2 ,..., , - - - - n n t t t t t t π π π π π π are independent for n t t t ... 2 1
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Primbs, MS&E 345 13 Simulation of a Poisson Process (λ= 1) 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 Time π τ
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Primbs, MS&E 345 14 Poisson processes Jump! Hence, they are a good building block for modeling: Market crashes or jumps. Bankruptcy. Other unexpected discontinuous price movements.
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Primbs, MS&E 345 15 Brownian Motion and the Poisson Process as limits: 1/2 1/2 1 -1 time = 1 time = 1/2 1/2 1/2 2 / 1 2 / 1 - 1/2 1/2 2 / 1 2 / 1 - time = 1/2 + time = 1/n 1/2 1/2 n / 1 n / 1 - +...+ time = 1/n 1/2 1/2 n / 1 n / 1 - Brownian Motion Chop time interval in half Chop finer and finer.
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Primbs, MS&E 345 16 Brownian Motion and the Poisson Process as limits: 1/2 1/2 1 -1 time = 1 Over dt Brownian motion looks like: dt ε ) 1 , 0 ( ~ N ε
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Primbs, MS&E 345 17
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