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(2) Stochastic Processes for Finance Patrick Roger Strasbourg University, EM Strasbourg Business School June 2010. Download free ebooks at bookboon.com 2.
(3) Stochastic Processes for Finance © 2010 Patrick Roger & Ventus Publishing ApS ISBN 978-87-7681-666-7. Download free ebooks at bookboon.com 3.
(4) Contents. Stochastic Processes for Finance. Contents Introduction. 7. 1 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 9 10 12 12 14 17 17 19 19 21 24 25 29 30 34 39. 2 2.1. Continuous-time stochastic processes Introduction. 43 43. Please click the advert. 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 [email protected]. Applications are now open for entry in September 2011.. For more information visit www.london.edu/mim/ email [email protected] or call +44 (0)20 7000 7573. www.london.edu/mim/. Download free ebooks at bookboon.com 4.
(5) Please click the advert. 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 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. You’re full of energy and ideas. And that’s just what we are looking for.. © UBS 2010. All rights reserved.. Contents. Stochastic Processes for Finance. Looking for a career where your ideas could really make a difference? UBS’s Graduate Programme and internships are a chance for you to experience for yourself what it’s like to be part of a global team that rewards your input and believes in succeeding together. Wherever you are in your academic career, make your future a part of ours by visiting www.ubs.com/graduates.. www.ubs.com/graduates. Download free ebooks at bookboon.com 5.
(6) Contents. Stochastic Processes for Finance. 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. Please click the advert. 3.4.1 3.4.2 3.4.3 3.5 3.5.1 3.5.2. Download free ebooks at bookboon.com 6.
(7) Introduction. Stochastic Processes for Finance. . . 0, 1, .., T. [0; T ] . . . Download free ebooks at bookboon.com 7.
(8) Stochastic Processes for Finance. . . Introduction. . . . Download free ebooks at bookboon.com 8.
(9) Discrete-time stochastic processes. Stochastic Processes for Finance. . . T . . . Download free ebooks at bookboon.com 9.
(10) Discrete-time stochastic processes. Stochastic Processes for Finance. . . . . . T = {0, 1, ..., T } T < +∞. T− = {0, 1, ..., T − 1} ; T − 1 T. T (, A, P ) A P A. X = (X0 , ..., XT ) (, A) R. Xt n R BR . Download free ebooks at bookboon.com 10.
(11) Discrete-time stochastic processes. Stochastic Processes for Finance. . . Xt X Xt = Xt−1 × Yt Yt u d p 1 − p. X0 X Xt Xt R+ . your chance Please click the advert. 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 11.
(12) Discrete-time stochastic processes. Stochastic Processes for Finance. . ω1 ω2 ω3 ω4. X1 (ω) . X2 (ω) . X1 X2. . . Xt X1 X2 X1 X2 . = {ω i , i = 1, ..., 4} A = P() X1 X2 X1 X1 = 100 {ω 1 , ω 2 } X1 = 105 {ω 3 , ω 4 } . X2 X1 {ω 1 , ω 2 } X1 T = 2 X2 X1 X2 . . Download free ebooks at bookboon.com 12.
(13) Discrete-time stochastic processes. Stochastic Processes for Finance. . . X1 X2 σ X1 σ (X1 , X2 ) . t t s t (, A, P ) F = {F0 , F1 , ..., FT } A. (, A, P, F) Ft Ft−1 ⊆ Ft t ≥ 1. T, T. FT = A. F0 = {∅ } F0 = {∅, } F1 = {∅, {ω 1 , ω 2 } , {ω 3 , ω 4 } , } F2 = P(). Download free ebooks at bookboon.com 13.
(14) Discrete-time stochastic processes. Stochastic Processes for Finance. . . t = 1 {ω 1 , ω 2 } {ω 3 , ω 4 } X1 . . . X Xt t. t, Xt Xt t + 1 Xt+1 Xt t t. F = {F0 , F1 , ..., FT } Ft t {Xt ≤ 100} Xt Xt Ft t). X = (X0 , ..., XT ) F t, Xt Ft X F X A FtX Xs , s ≤ t. Xt = Xt−1 Yt Yt u d p 1 − p. T = 2, X t = 0 T = 2. 4 = {uu, ud, du, dd} X1 up down . {uu, ud} {du, dd} . . Download free ebooks at bookboon.com 14.
(15) Discrete-time stochastic processes. Stochastic Processes for Finance. . . F1 = {∅, {uu, ud} , {du, dd} , } . X1 X1 (uu) = X1 (ud) = uX0 X1 (du) = X1 (dd) = dX0 F1 F = F X . . Please click the advert. what‘s missing in this equation?. You could be one of our future talents. MAERSK INTERNATIONAL TECHNOLOGY & SCIENCE PROGRAMME Are you about to graduate as an engineer or geoscientist? Or have you already graduated? If so, there may be an exciting future for you with A.P. Moller - Maersk.. www.maersk.com/mitas Download free ebooks at bookboon.com 15.
(16) Discrete-time stochastic processes. Stochastic Processes for Finance. . ω1 ω2 ω3 ω4. X0 1 1 1 1. X1 (ω) u u d d. X2 (ω) u2 ud du d2. X Card() = n n L2 (, A, P ) Rn . L2 (, Ft , P ) , Ft L2 (, A, P ) t. L2 (, F1 , P ) L2 (, F1 , P ) = (x, y, z, t) ∈ R4 x = y z = t F1 X0 = 1 . K X k , k = 1, ..., K. t t + 1. t t + 1 1 ′ K θ′t = θ1t , ...θK = θ , ...θ θ t t+1 t+1 t+1 t t + 1. θt+1 t X = (X1 ..., XT ) t ≥ 1, Xt Ft−1 t = 1. Download free ebooks at bookboon.com 16.
(17) Discrete-time stochastic processes. Stochastic Processes for Finance. . . X1 F0 θ θ0 θ0 θ1 . rt [t; t + 1] Bt t rt t r F , B t−1 (1 + rs ) Bt = s=0. Bt t − 1 Bt Ft−1 . . . p, p + 1. p p + 1. . Download free ebooks at bookboon.com 17.
(18) Discrete-time stochastic processes. Stochastic Processes for Finance. . . p p + 1. . Turning a challenge into a learning curve. Just another day at the office for a high performer. Please click the advert. 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 18.
(19) Discrete-time stochastic processes. Stochastic Processes for Finance. . . . . X (, A, P, F ) (B1 , ..., Bn ) ∈ BRn (t1 , ..., tn ) ∈ T n t1 < .... < tn P Xt ∈ Bn Xt ∈ Bj , j = 1, .., n − 1 = P Xt ∈ Bn Xt− ∈ Bn−1. tn−1 tn−1 tn . tn−1 . (Xp , p ∈ N) (x1 , ..., xn ) (Xp , p ∈ N) (x1 , ..., xn ) ; π(xi , p − 1, p, xj ) = P (Xp = xj |Xp−1 = xi ). . Πp−1 = (π(xi , p − 1, p, xj ), i, j = 1, ..., n) p − 1. p, π(xi , p − 1, p, xj ) = π ij π(xi , p − 1, p, xj ) X xj p xi p − 1. . . . m . Download free ebooks at bookboon.com 19.
(20) Discrete-time stochastic processes. Stochastic Processes for Finance. . . m X = (Xp , p ∈ N) N π = (π ij , i, j = 1, ..., N ) . (m+n) = P (Xm+n = xj |X0 = xi ) m + n π ij i j. n (m+n) (m) (n) π ij = π ik π kj k=1. (2). m = n = 1 N = 3. π 12 x2 x1 . π . x1. π . π. x2. π . ր. x1 →. ց. π . x3. ց → x2 ր. π . (2). . π 12 = π 11 π 12 + π 12 π 22 + π 13 π 32. xi xj m + n (m+n) π ij i m (m) π ij , i, j = 1, ..., n j n (n) π ij , i, j = 1, ..., n . . X = (Xp , p ∈ N) N π = (π ij , i, j = 1, ..., N ) . (m) πij = P (Xm = xj |X0 = xi ) m i j π (m) m (m) π ij . π (m) = π m π m m π.. Download free ebooks at bookboon.com 20.
(21) Discrete-time stochastic processes. Stochastic Processes for Finance. . . . . xj X xi k > 0 (k) π ij > 0. . xi xj k k ′ (k). (k′ ). . π ij > 0 πji > 0 π. 0.6 0.4 0 0 0.3 0.7 0 0 π= 0 0 0.5 0.5 0 0 0.7 0.3. . . x1 x2 , x3 x4 x1 x2 x3 x4 x1 x2 {x1 , x2 } {x3 , x4 } . X = (Xp , p ∈ N) N π = (π ij , i, j = 1, ..., N) . (C1 , C2 , ...CM ) N Ck . R, x R x x x . Download free ebooks at bookboon.com 21.
(22) Discrete-time stochastic processes. Stochastic Processes for Finance. . . S t + 1 uSt p St+1 = dSt 1 − p d = 1/u. S0 X = (Xp , p ∈ N) N π = (π ij , i, j = 1, ..., N) . xi , t(i) m π (m) (i, i) > 0. t(i) = 0 m, π (m) (i, i) = 0. X t(i) = 1 i. X = (Xp , p ∈ N) π.. Download free ebooks at bookboon.com 22.
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