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 2.1 Continuous-time stochastic processes Introduction 360° thinking 360° thinking 9 10 12 12 14 17 17 19 19 21 24 25 29 30 34 39 43 43 360° thinking Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities Discover the truth at www.deloitte.ca/careers Download free eBooks at bookboon.com © Deloitte & Touche LLP and affiliated entities © Deloitte & Touche LLP and affiliated entities Discover the truth at www.deloitte.ca/careers Click on the ad to read more © Deloitte & Touche LLP and affiliated entities D 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 Increase your impact with MSM Executive Education For almost 60 years Maastricht School of Management has been enhancing the management capacity of professionals and organizations around the world through state-of-the-art management education Our broad range of Open Enrollment Executive Programs offers you a unique interactive, stimulating and multicultural learning experience Be prepared for tomorrow’s management challenges and apply today For more information, visit www.msm.nl or contact us at +31 43 38 70 808 or via admissions@msm.nl For more information, visit www.msm.nl or contact us at +31 43 38 70 808 the globally networked management school or via admissions@msm.nl Executive Education-170x115-B2.indd Download free eBooks at bookboon.com 18-08-11 15:13 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 GOT-THE-ENERGY-TO-LEAD.COM We believe that energy suppliers should be renewable, too We are therefore looking for enthusiastic new colleagues with plenty of ideas who want to join RWE in changing the world Visit us online to find out what we are offering and how we are working together to ensure the energy of the future 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 Turning a challenge into a learning curve Just another day at the office for a high performer Accenture Boot Camp – your toughest test yet Choose Accenture for a career where the variety of opportunities and challenges allows you to make a difference every day A place where you can develop your potential and grow professionally, working alongside talented colleagues The only place where you can learn from our unrivalled experience, while helping our global clients achieve high performance If this is your idea of a typical working day, then Accenture is the place to be It all starts at Boot Camp It’s 48 hours that will stimulate your mind and enhance your career prospects You’ll spend time with other students, top Accenture Consultants and special guests An inspirational two days packed with intellectual challenges and activities designed to let you discover what it really means to be a high performer in business We can’t tell you everything about Boot Camp, but expect a fast-paced, exhilarating and intense learning experience It could be your toughest test yet, which is exactly what will make it your biggest opportunity Find out more and apply online Visit accenture.com/bootcamp 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 Download free eBooks at bookboon.com t exp(αs)dZs 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 The Wake the only emission we want to leave behind QYURGGF 'PIKPGU /GFKWOURGGF 'PIKPGU 6WTDQEJCTIGTU 2TQRGNNGTU 2TQRWNUKQP 2CEMCIGU 2TKOG5GTX 6JG FGUKIP QH GEQHTKGPFN[ OCTKPG RQYGT CPF RTQRWNUKQP UQNWVKQPU KU ETWEKCN HQT /#0 &KGUGN 6WTDQ 2QYGT EQORGVGPEKGU CTG QHHGTGF YKVJ VJG YQTNFoU NCTIGUV GPIKPG RTQITCOOG s JCXKPI QWVRWVU URCPPKPI HTQO VQ M9 RGT GPIKPG )GV WR HTQPV (KPF QWV OQTG CV YYYOCPFKGUGNVWTDQEQO 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 Download free eBooks at bookboon.com 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 ©... 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... 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