Stochastic Processes for Finance Patrick Roger Download free books at Stochastic Processes for Finance Patrick Roger Strasbourg University, EM Strasbourg Business School June 2010 Download free eBooks at bookboon.com Stochastic Processes for Finance © 2010 Patrick Roger & Ventus Publishing ApS ISBN 978-87-7681-666-7 Download free eBooks at bookboon.com Contents Stochastic Processes for Finance Contents Introduction 1.1 1.2 1.3 1.3.1 1.3.2 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.4.5 1.5 1.5.1 1.5.2 1.5.3 1.5.4 Discrete-time stochastic processes Introduction The general framework Information revelation over time Filtration on a probability space Adapted and predictable processes Markov chains Introduction Definition and transition probabilities Chapman-Kolmogorov equations Classification of states Stationary distribution of a Markov chain Martingales Doob decomposition of an adapted process Martingales and self-financing strategies Investment strategies and stopping times Stopping times and American options 9 10 12 12 14 17 17 19 19 21 24 25 29 30 34 39 2.1 Continuous-time stochastic processes Introduction 43 43 Fast-track your career Masters in Management Stand out from the crowd Designed for graduates with less than one year of full-time postgraduate work experience, London Business School’s Masters in Management will expand your thinking and provide you with the foundations for a successful career in business The programme is developed in consultation with recruiters to provide you with the key skills that top employers demand Through 11 months of full-time study, you will gain the business knowledge and capabilities to increase your career choices and stand out from the crowd London Business School Regent’s Park London NW1 4SA United Kingdom Tel +44 (0)20 7000 7573 Email mim@london.edu Applications are now open for entry in September 2011 For more information visit www.london.edu/mim/ email mim@london.edu or call +44 (0)20 7000 7573 www.london.edu/mim/ Download free eBooks at bookboon.com Click on the ad to read more Contents Stochastic Processes for Finance 2.2 2.2.1 2.2.2 2.2.3 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6 2.3.7 General framework Filtrations, adapted and predictable processes Markov and diffusion processes Martingales The Brownian motion Intuitive presentation The assumptions Definition and general properties Usual transformations of the Wiener process The general Wiener process Stopping times Properties of the Brownian motion paths 44 48 51 53 55 55 57 61 64 67 69 71 3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 3.3.1 3.3.2 3.3.3 3.4 Stochastic integral and Itô’s lemma Introduction The stochastic integral An intuitive approach Counter-example Definition and properties of the stochastic integral Calculation rules Itô’s lemma Taylor’s formula, an intuitive approach to Itô’s lemma Itô’s lemma Applications The Girsanov theorem 73 73 75 75 78 80 83 85 86 88 88 91 Download free eBooks at bookboon.com Click on the ad to read more Contents Stochastic Processes for Finance 3.4.1 3.4.2 3.4.3 3.5 3.5.1 3.5.2 Preliminaries Girsanov theorem Application Stochastic differential equations Existence and unicity of solutions A specific case: linear equations 91 93 93 95 95 97 Bibliography 100 Index 103 your chance to change the world Here at Ericsson we have a deep rooted belief that the innovations we make on a daily basis can have a profound effect on making the world a better place for people, business and society Join us In Germany we are especially looking for graduates as Integration Engineers for • Radio Access and IP Networks • IMS and IPTV We are looking forward to getting your application! To apply and for all current job openings please visit our web page: www.ericsson.com/careers Download free eBooks at bookboon.com Click on the ad to read more Introduction Stochastic Processes for Finance 0, 1, , T [0; T ] Download free eBooks at bookboon.com Stochastic Processes for Finance Download free eBooks at bookboon.com Introduction Discrete-time stochastic processes Stochastic Processes for Finance T Download free eBooks at bookboon.com Discrete-time stochastic processes Stochastic Processes for Finance T = {0, 1, , T } T < +∞ T− = {0, 1, , T − 1} ; T − T T (, A, P ) A P A. X = (X0 , , XT ) (, A) R Xt n R BR Download free eBooks at bookboon.com 10 Stochastic Stochastic Processes for Finance √ integral and Itô’s lemma σ Xt Z dt dt, β) Z X t β α dX Xt dZ t = α(β − Xt )dt + σ t αβdt σ √ Xt Z dt dt, Z dXt = α(β − Xt )dt + σ Xt dZt αβdt β β Download free eBooks at bookboon.com 90 Click on the ad to read more Stochastic integral and Itô’s lemma Stochastic Processes for Finance W (, σ) , (, A, F , P ) F W Z, Wt = t + σZt Wt = ln(St ) St Wt − Ws [s; t] r Q W r P Q EP EQ P Q r = X ∼ N (0, 1) (, A, P ) Q α2 ∀A ∈ A, Q(A) = EP A exp αX − Q P X ∼ N (α, 1) Q = exp −α EP [exp (αX)] exp (αX) EP exp αX − α2 (0, α) , α EP [exp (αX)] = exp Q() = EP exp αX − α2 = P Q Download free eBooks at bookboon.com 91 Stochastic integral and Itô’s lemma Stochastic Processes for Finance EQ [X] = = = = α2 XdQ = X exp αX − dP +∞ α2 fX (x)dx x exp αx − −∞ 2 +∞ x α2 √ x exp αx − exp − dx 2 2π −∞ +∞ 1 √ x exp − (x − α) dx = α 2π −∞ fX X P X Q, α α exp αX − Q P, dQ dP Xt ∼ N (t, σ t) t = EP (Xt ) σ t = VP (Xt ) Xt rt = EQ (Xt ) σ t = VQ (Xt ) X √ Xt = t + σZt = t + σ tYt Z Yt ∼ N (0, 1) Yt N (α, 1) α Q √ EQ [Xt ] = t + σ tα = rt α √ (r − ) t α= σ √ √ 2 dQ (r − ) t (r − ) t −r t −r = exp Yt − = exp − Zt − dP σ σ σ σ Download free eBooks at bookboon.com 92 Stochastic integral and Itô’s lemma Stochastic Processes for Finance λ = (λt , t ∈ [0; T ]) F L = (Lt , t ∈ [0; T ]) t t Lt = exp − λs dZs − λ ds s λ T EP exp λ ds < +∞ s λ L P − Z ∗ t ∗ Zt = Zt + λs ds (, A, F, Q) Q dQ = LT dP λ L λ2 Lt = exp −λZt − t Download free eBooks at bookboon.com 93 Stochastic integral and Itô’s lemma Stochastic Processes for Finance dSt = St dt + σSt dZt St = S0 exp σ2 − t + σZt σ2 exp (−rt) St = S0 exp −r− t + σZt σ ∗ St = S0 exp − t + σZt Zt∗ = Zt + −r t σ Download free eBooks at bookboon.com 94 Click on the ad to read more Stochastic integral and Itô’s lemma Stochastic Processes for Finance σ F (, A, P ) Z X0 = c dXt = (Xt , t) dt + σ (Xt , t) dZt c c X [0; T ] X F σ T | (Xt , t)| dt < +∞ T σ (Xt , t) dt < +∞ X Xt = X0 + t (Xs , s) ds + t σ (Xs , s) dZs σ Download free eBooks at bookboon.com 95 Stochastic integral and Itô’s lemma Stochastic Processes for Finance P X T F, EP Xt2 dt < +∞ m > ∀t ∈ [0; T ] , ∀ (x, y) ∈ R2 max (|(x, t) − (y, t)| ; |σ(x, t) − σ(y, t)|) ≤ m |x − y| (x, t)2 + σ(x, t)2 ≤ m + x2 (x, t) X0 Ft t σ t σ X (1 + x2 ) (x, t) = exp(x) Z (X, Z) σ Xt c σ t, X σ Download free eBooks at bookboon.com 96 Stochastic integral and Itô’s lemma Stochastic Processes for Finance X0 = c dXt = aXt dt + σ t dZt a t Xt = c exp(at) + exp [a (t − s)] σ s dZs t Xt exp(−at) = c + exp (−as) σ s dZs Y, Y0 = c dYt = exp(−at)σ t dZt exp(at)Yt = f (Yt , t) , f ∂f = a exp(at)Yt ∂t ∂f = exp(at) ∂Yt ∂ 2f = ∂Yt2 df (Yt , t) = a exp(at)Yt dt + exp(at) exp(−at)σ t dZt = a exp(at)Yt dt + σ t dZt Yt exp(−at)Xt dXt = aXt dt + σ t dZt Download free eBooks at bookboon.com 97 Stochastic integral and Itô’s lemma Stochastic Processes for Finance X0 = c dXt = a(t)Xt dt + σ t dZt X t −1 γ s σ s dZs Xt = γ t c + γ ′ f (t) = a(t)f(t) Y Y0 = y0 dYt = α (β − Yt ) dt + σdZt Xt = (Yt − β) exp(αt) = f (Yt , t); f ∂f = α exp(αt) (Yt − β) ∂t ∂f = exp(αt) ∂Yt ∂2f = ∂Yt2 dXt = [α (Yt − β) exp(αt) + exp(αt)α (β − Yt )] dt + σ exp(αt)dZt = σ exp(αt)dZt X0 = y0 − β, Xt = y0 − β + σ t exp(αs)dZs Download free eBooks at bookboon.com 98 Stochastic integral and Itô’s lemma Stochastic Processes for Finance X Y Yt = β + Xt exp(−αt) Yt t = β + exp(−αt) y0 − β + σ exp(αs)dZs t exp(−α(t − s))dZs = β (1 − exp(−αt)) + y0 exp(−αt) + σ Yt EP [Yt |Y0 = y0 ] = β (1 − exp(−αt)) + y0 exp(−αt) t exp(−2α(t − s))ds VP [Yt |Y0 = y0 ] = σ σ2 = [1 − exp (−2αt)] 2α Download free eBooks at bookboon.com 99 Click on the ad to read more Bibliography Stochastic Processes for Finance Download free eBooks at bookboon.com 100 Stochastic Processes for Finance Download free eBooks at bookboon.com 101 Bibliography Bibliography Stochastic Processes for Finance Download free eBooks at bookboon.com 102 Index Stochastic Processes for Finance Index positively recurrent, 19 recurrent, 19 stationary distribution, 20 transient, 19 no memory process, 48 process, 15, 48 Brownian motion, 59 transition matrix, 15 Martingale, 21, 49 Brownian motion, 59 Doob, 24 submartingale, 21, 49 super-martingale, 21, 49 Modi cation stochastic process, 43 Brownian motion, 57 general, 63 geometric, 85 Markov process, 59 martingale, 59 path simulation, 65 stopping time, 67 transformation, 60 Doob decomposition, 25, 51 martingale, 24 Filtration, 9, 44 complete, 44 filtered probability space, 44 natural, 10, 44 right-continuous, 44 Girsanov theorem, 89 application, 89 Novikov condition, 89 Novikov condition, 89 Path, 40 càdlàg, 41 càglàd, 43 continuous, 40 Brownian motion, 58 LCRL, 43 nowhere differentiable, 59 RCLL, 41 Poisson distribution, 47 process, 47 Probability transition, 15 Itô lemma, 84 application, 84 Taylor series expansion, 82 process, 48 diffusion coefficient, 49 drift, 49 Landau notations, 55 Snell envelope, 36 Stochastic differential equation, 91 linear, 93 solution conditions, 92 de nition, 91 Markov, 92 Stochastic integral, 71 calculation rules, 79 definition, 76 properties, 78 stochastic differential, 78 Markov chain, 15 accessible, 17 aperiodic, 18 Chapman-Kolmogorov equations, 16 communicating class, 17 communication, 17 homogeneous, 15 irreducible, 18 periodicity, 18 Download free eBooks at bookboon.com 103 Index Stochastic Processes for Finance Stochastic process adapted, 10, 44 Brownian motion, 57 continuous-time, 40 diffusion, 48 diffusion coefficient, 49 discrete-time, drift, 49 increments independent, 45 stationary, 45 indistinguishable, 43 Itô, 48 Markov, 15, 48 modification, 43 Ornstein-Uhlenbeck, 85 path, 40 Poisson, 47 predictable, 12, 45 random walk, 22, 46, 51 square root, 86 stopped process, 33, 37 trajectory, 40 Wiener, 57 Stopping time, 32, 37, 65 American option, 35 optional stopping theorem, 32 Strategy doubling, 34 portfolio, 28 self- nancing, 28 Transition matrix, 15 probability, 15 Wiener process, 57 general, 63 104 .. .Stochastic Processes for Finance Patrick Roger Strasbourg University, EM Strasbourg Business School June 2010 Download free eBooks at bookboon.com Stochastic Processes for Finance ©... ssolve p Download free eBooks at bookboon.com 11 �e for Engin Click on the ad to read more Discrete-time stochastic processes Stochastic Processes for Finance ... Contents Stochastic Processes for Finance 2.2 2.2.1 2.2.2 2.2.3 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6 2.3.7 General framework Filtrations, adapted and predictable processes Markov and diffusion processes