contmod - Plan 1. Introduction 2. Brownian motion 3....

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Unformatted text preview: Plan 1. Introduction 2. Brownian motion 3. Continuous-Time martingales 4. Ito integral / Ito formula 5. Black-Scholes Pricing/Hedging through PDE’s 6. Feynman-Kac’s Theorem 7. Risk-Neutral Pricing and Monte Carlo 8. Girsanov’s Theorem / Change of measure 1 (2) Brownian Motion Definition. A standard Brownian motion (BM) is a stochastic process W = { W t , t ≥ } satisfying (i) W = 0 (ii) for any 0 ≤ t < t 1 < ··· < t k , the rv’s W t k- W t k − 1 (increments) are independent, (iii) W t- W s ∼ N (0 , ( t- s )), where t > s . Note: Any linear transformation of the standard BM of the form ˜ W t = μt + σW t where μ ∈ R , σ > 0 are constants is called a Brownian motion with drift μ and diffusion coefficient (or volatility) σ . tim e in d e x (m u = 0 ) Brownian motion 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0-2-1 1 2 T ra je c to rie s o f a B ro w n ia n m o tio n T ra je c to rie s o f a B ro w n ia n m o tio n tim e in d e x (m u = 2 ) Brownian motion 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0-0.5 0.0 0.5 1.0 1.5 2.0 2 Properties of the BM : 1. The paths of a BM are almost surely continuous. 2. W is a Gaussian process. That is, for all 0 ≤ t 1 < t 2 < ... < t k , the vector ( W t 1 ,...,W t k ) has a multinormal distribution characterized by E( W t i ) = 0 and Cov( W t i ,W t j ) = min( t i ,t j ). 3. W is a Markovian process, and thus P ( W t n ≤ w n | W t i = w i , i = 1 ,...,n- 1) = P ( W t n ≤ w n | W t n − 1 = w n- 1 ) . 4. If W is a Brownian motion with zero drift and diffusion coefficient σ > 0, then- braceleftbig 1 σ W t ,t ≥ bracerightbig is a standard BM,- { μt + W t ,t ≥ } is a BM with drift μ and diff. coefficient σ ,- (stability) for each λ > 0 the process braceleftbigg ˜ W t = 1 √ λ W λt ,t ≥ bracerightbigg is a BM with drift μ = 0 and diffusion coefficient σ . 5. Before we state the two next properties, it will help to know that: (a) The BM will eventually hit any and every real value no matter how large or how negative. (b) No matter how far above or below the axis, the BM process will be back to zero at some later time with probability one. (c) Once the BM hits a value, it immediately hits it again infinitely often. (d) It doesn’t matter at what scale you examine the BM, it looks just the same. 6. W has almost surely sample paths that are nowhere differentiable. 7. For almost every w ∈ Ω a BM’s sample path has unbounded (total) variation on any interval [ a,b ], b > a . (For any real valued function h defined on an interval [ a,b ] the total variation of h on [ a,b ] is V ( h ) def = sup a = t <t 1 < ··· <t k = b k summationdisplay i =1 | h ( t i )- h ( t i- 1 ) | , where the supremum is taken over all partitions t < t 1 < ··· < t k of [ a,b ].) 3 Note that for functions h whose first derivative exists everywhere, we have that V ( h ) = integraldisplay b a | dh ( t ) | = integraldisplay b a | h ′ ( t ) | dt....
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This note was uploaded on 03/17/2012 for the course ACTSC 446 taught by Professor Adam during the Spring '09 term at Waterloo.

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contmod - Plan 1. Introduction 2. Brownian motion 3....

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