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Cont_IR_Models_One

Cont_IR_Models_One - BrownianMo*onandCon*nuous (PartOne...

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Brownian Mo*on and Con*nuous Time Interest Rate Models (Part One)
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Outline Brownian Mo*on Defini*on Important Proper*es Brief introduc*on to stochas*c calculus Con*nuous Time Interest Rate Models
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Brownian Mo*on
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Stochas*c Process A con$nuous $me stochas$c process is a collec*on of random variables, one for each *me t ≥ 0. Common examples (building blocks of most financial models): Poisson process, Brownian Mo*on Typically, we assume that the value of X t is known (revealed) at *me t . X = { X t | t [0, )}
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Brownian Mo*on A con*nuous *me stochas*c process W is a Brownian mo*on (Wiener process) if: W 0 = 0 For any 0≤ t 0 < t 1 < … < t N the random variables (increments): are independent For any t > s: W t n W t n 1 , n = 1,..., N W t W s ~ N (0, t s )
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Simula*ng a Brownian mo*on Set W 0 = 0, define a step‐size h and number of desired *me steps N. For n=1 to N, set: Where ξ n , n=1,…,N is a sequence of standard normal random variables. W nh = W ( n 1) h + h ξ n
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Simula*ons of Brownian Mo*on Note the jagged paths.
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Proper*es of Brownian Mo*on A given realiza*on (sample) of the stochas*c process W t , t≥0 is called a sample path . The sample paths of a Brownian mo*on are con*nuous func*on of t (with probability 1). Any linear transforma*on of a Brownian mo*on of the form: is called Brownian mo*on with dri6 μ and vola$lity (diffusion) σ. X t = μ t + σ W t
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Proper*es of Brownian Mo*on A Brownian Mo*on W is a Markov process . For any 0≤ t 0 < t 1 < … < t N : If W is Brownian mo*on with zero dri[ and vola*lity σ>0: σ ‐1 W t is a standard Brownian mo*on. μt+W t is a Brownian mo*on with dri[ μ and vola*lity σ. (Brownian scaling). For λ > 0, the following process is also Brownian mo*on with zero dri[ and vola*lity σ. P W t N xW t n = w t n , n = 1,..., N 1 ( ) = P W t N xW t N 1 = w t N 1 ( ) ˜ W t = 1 λ W λ t
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Proper*es of Brownian Mo*on Brownian mo*on will eventually hit (with probability one) every real value, no ma‘er how large or how nega*ve. No ma‘er how far above or below the axis, the Brownian mo*on process will be back to zero at some later *me with probability one. Once BM hits a value, it hits it again infinitely o[en. Brownian mo*on is a fractal; it doesn’t ma‘er what scale you examine it on, it looks the same.
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Proper*es of Brownian Mo*on Calculus with Brownian mo*on is more complex because of the jagged nature of the paths of Brownian mo*on.
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