Probability for Finance Patrick Roger Download free books at Probability for Finance Patrick Roger Strasbourg University, EM Strasbourg Business School May 2010 Download free eBooks at bookboon.com Probability for Finance © 2010 Patrick Roger & Ventus Publishing ApS ISBN 978-87-7681-589-9 Download free eBooks at bookboon.com Contents Probability for Finance Contents Introduction 1.1 1.1.1 1.1.2 1.1.3 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 Probability spaces and random variables Measurable spaces and probability measures σ algebra (or tribe) on a set Ω Sub-tribes of A Probability measures Conditional probability and Bayes theorem Independant events and independant tribes Conditional probability measures Bayes theorem Random variables and probability distributions Random variables and generated tribes Independant random variables Probability distributions and cumulative distributions Discrete and continuous random variables Transformations of random variables 10 10 11 13 16 18 19 21 24 25 25 29 30 34 35 2.1 Moments of a random variable Mathematical expectation 37 37 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 Probability for Finance 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.4 2.4.1 2.4.2 2.5 2.5.1 2.5.2 Expectations of discrete and continous random variables Expectation: the general case Illustration: Jensen’s inequality and Saint-Peterburg paradox Variance and higher moments Second-order moments Skewness and kurtosis The vector space of random variables Almost surely equal random variables The space L1 (Ω, A, P) The space L2 (Ω, A, P) Covariance and correlation Equivalent probabilities and Radon-Nikodym derivatives Intuition Radon Nikodym derivatives Random vectors Definitions Application to portfolio choice 39 40 43 46 46 48 50 51 53 54 59 63 63 67 69 69 71 3.1 3.1.1 3.1.2 3.1.3 Usual probability distributions in financial models Discrete distributions Bernoulli distribution Binomial distribution Poisson distribution 73 73 73 76 78 Download free eBooks at bookboon.com Click on the ad to read more Contents Probability for Finance 3.2 3.2.1 3.2.2 3.2.3 3.3 3.3.1 3.3.2 3.3.3 Continuous distributions Uniform distribution Gaussian (normal) distribution Log-normal distribution Some other useful distributions The X distribution The Student-t distribution The Fisher-Snedecor distribution 81 81 82 86 91 91 92 93 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.2 4.2.1 4.2.2 4.3 4.3.1 4.4 4.4.1 Conditional expectations and Limit theorems Conditional expectations Introductive example Conditional distributions Conditional expectation with respect to an event Conditional expectation with respect to a random variable Conditional expectation with respect to a substribe Geometric interpretation in L2 (Ω, A, P) Introductive example Conditional expectation as a projection in L2 Properties of conditional expectations The Gaussian vector case The law of large numbers and the central limit theorem Stochastic Covergences 94 94 94 96 97 98 100 101 101 102 104 105 108 108 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 Contents Probability for Finance 4.4.2 4.4.3 Law of large numbers Central limit theorem 109 112 Bibliography 114 I joined MITAS because I wanted real responsibili� I joined MITAS because I wanted real responsibili� Real work International Internationa al opportunities �ree wo work or placements �e Graduate Programme for Engineers and Geoscientists Maersk.com/Mitas www.discovermitas.com M Month 16 I was a construction M supervisor ina cons I was the North Sea super advising and the No he helping foremen advis ssolve problems Real work he helping f International Internationa al opportunities �ree wo work or placements ssolve p Download free eBooks at bookboon.com �e for Engin Click on the ad to read more Introduction Probability for Finance Download free eBooks at bookboon.com Probability for Finance Introduction DTU Summer University – for dedicated international students Spend 3-4 weeks this summer at the highest ranked Application deadlines and programmes: technical university in Scandinavia 31 15 30 DTU’s English-taught Summer University is for dedicated international BSc students of engineering or related natural science programmes March Arctic Technology March & 15 April Chemical/Biochemical Engineering April Telecommunication June Food Entrepreneurship Visit us at www.dtu.dk Download free eBooks at bookboon.com Click on the ad to read more Probability spaces and random variables Probability for Finance t = T = T P P Download free eBooks at bookboon.com 10 Conditional expectations and Limit theorems Probability for Finance Z (z1 ; z2 ; z3 ; z4 ) B1 B2 Z B B1 B2 z1 = z2 z3 = z4 [p1 x1 + p2 x2 ] = E (X |B1 ) p1 + p2 = z4 = [p3 x3 + p4 x4 ] = E (X |B2 ) p3 + p4 z1 = z2 = z3 B1 (B2 ) X B1 (B2 ) X B, E (X |B ) X L2 , A, P ) L2 (, A, P ) R2 , d(x, y) = (x1 − y1 )2 + (x2 − y2 )2 x′ = (x1 , x2 ) y ′ = (y1 , y2 ) x ∈ R2 , z = (z1 , z1 ) x minz (x1 − z1 )2 + (x2 − z1 )2 z1 = z2 Download free eBooks at bookboon.com 101 Probability for Finance L2 (, A, P ) Conditional expectations and Limit theorems z1 = x +x z x ∈ R z − x z < z − x, z >= (z1 − x1 )z1 + (z1 − x2 )z1 x2 − x1 x1 − x2 = z1 + z1 = 2 R2 ∗ d (x, y) = p(x1 − y1 )2 + q (x2 − y2 )2 p + q = 1, p > 0, q > z1 = px1 + qx2 z1 x L2 X L2 (, A, P ) , E(X |B ) B B L2 (, B, P ) L2 (, A, P ) R4 L2 (, B, P ) R2 E(X |B ) X L2 (, B, P ) E (X |B ) minZ∈L ,B,P ) E (X − Z)2 = minZ∈L ,B,P ) d(X, Z)2 = E (X − E (X |B ))2 E (X |B ) B z1 = z2 z3 = z4 P PB B L , B, P ) P B Download free eBooks at bookboon.com 102 Conditional expectations and Limit theorems Probability for Finance E (X − Z)2 = p1 (x1 −z1 )2 +p2 (x2 −z1 )2 +p3 (x3 −z3 )2 +p4 (x4 −z3 )2 z1 z3 ∂E (X − Z)2 = −2 [p1 (x1 − z1 ) + p2 (x2 − z1 )] = ∂z1 ∂E (X − Z)2 = −2 [p3 (x3 − z3 ) + p4 (x4 − z3 )] = ∂z3 (p1 x1 + p2 x2 ) = E (X |B ) (ω ) = E (X |B ) (ω2 ) p1 + p2 = z4 = (p3 x3 + p4 x4 ) = E (X |B ) (ω ) = E (X |B ) (ω4 ) p3 + p4 z1 = z2 = z3 DTU Summer University – for dedicated international students Application deadlines and programmes: Spend 3-4 weeks this summer at the highest ranked technical university in Scandinavia 31 15 30 DTU’s English-taught Summer University is for dedicated international BSc students of engineering or related natural science programmes March Arctic Technology March & 15 April Chemical/Biochemical Engineering April Telecommunication June Food Entrepreneurship Visit us at www.dtu.dk Download free eBooks at bookboon.com 103 Click on the ad to read more Conditional expectations and Limit theorems Probability for Finance (X, Y ) L2 (, A, P ) B, B ′ A B ⊂ B′ X c ∈ R, E (X |B ) = c ∀(a, b) ∈ R2 , E (aX + bY |B ) = aE (X |B ) + bE (Y |B ) X ≤ Y, E (X |B ) ≤ E (Y |B ) E (E (X |B′ ) |B ) = E (X |B ) X B E (XY |B ) = X E (Y |B ) X B, E (X |B ) = E(X) c c B c L2 (, B, P ) L2 (, B, P ) L2 (, A, P ) , E (X |B′ ) X L2 (, B′ , P ) E (E (X |B′ ) |B ) L2 (, B, P ) E (X |B ′ ) L2 (, B ′ , P ) L (, B, P ) L2 (, B, P ) B = {∅, } E (X |B ) = E(X) E (E (X |B ′ )) = E (X) B′ E (X − E(X) |B ) = E(X) X − E(X) Y L2 (, B, P ) E((X − E(X)) Y ) = E (X − E(X)) E(Y ) = Download free eBooks at bookboon.com 104 Conditional expectations and Limit theorems Probability for Finance X −E(X) Y X = (X1 , , Xn ) n Xi i=1 m′ = (E(X1 ), , E(Xn )) X X fX X n 1 n ′ −1 ∀x ∈ R , f(x) = √ exp − (x − m) X (x − m) 2π Det(X ) Det(X ) X = (X1 , , Xn ) m X ; p < n Y1 = (X1 , , Xp ) Y2 = (Xp+1 , , Xn ) X Σ11 Σ12 X = Σ21 Σ22 Σii Yi Σij Yi Yj i, j = 1, 2, i = j Y1 Y2 = y2 ∈ Rn−p E (Y1 |Y2 = y2 ) = E(Y1 ) + Σ12 Σ−1 22 (y2 − E(Y2 )) Y |Y =y = Σ11 − Σ12 Σ−1 22 Σ21 Download free eBooks at bookboon.com 105 Conditional expectations and Limit theorems Probability for Finance p = n = p = n = σ 12 (y2 − m2 ) σ 22 σ2 = σ 21 − 12 σ 22 E (X1 |X2 = x2 ) = m1 + X |X =x ρ12 X |X =x = σ 21 (1 − ρ212 ) x′ = (x1 , x2 )) −1 √1 exp − 12 (x − m)′ X (x − m) (2π) |Det(X )| fX (x1 , x2 ) fX |X (x1 |x2 ) = = 2 fX (x2 ) 1 x −m √ exp − σ σ 2π −1 exp − 12 (x − m)′ X (x − m) σ2 = √ 2 2 2π σ σ − σ 212 x −m exp − σ 2 σ2 − m x 2 −1 = √ 2 exp − (x − m)′ X (x − m) − 2 σ 2π σ σ − σ 12 −1 X = 2 σ σ − σ 212 −σ 12 σ 22 −σ 12 σ 21 −1 (x − m), A = (x − m)′ X A = σ 22 x21 − 2σ 22 x1 m1 − 2x1 σ 12 x2 + 2x1 σ 12 m2 + σ 21 σ 22 − σ 212 σ 22 m21 + 2m1 σ 12 x2 − 2m1 σ 12 m2 + σ 21 x22 − 2σ 21 x2 m2 + σ 21 m22 σ 21 σ 22 − σ 212 fX (x1 , x2 ) σ2 (−σ 22 x1 + σ 22 m1 + σ 12 x2 − σ 12 m2 ) =√ 2 exp − fX (x2 ) σ 22 (σ 21 σ 22 − σ 212 ) 2π (σ σ − σ 212 ) Download free eBooks at bookboon.com 106 Probability for Finance expectations and Limit theorems Conditional σ 12 (x2 − m2 ) σ 22 σ 212 = σ1 − σ2 E (X1 |X2 = x2 ) = m1 + X |X =x g 2 σ x − m − (x − m ) 1 2 1 σ g(x1 ) = exp − √ σ 2 σ σ − σ σ − 2π σ (−σ 22 x1 + σ 22 m1 + σ 12 x2 − σ 12 m2 ) σ2 exp − = √ 2 σ 22 (σ 21 σ 22 − σ 212 ) 2π (σ σ − σ 212 ) g(x1 ) = fX |X (x1 |x2 ) X |X =x = σ 21 (1 − ρ212 ) X2 = x2 X1 X1 ρ12 Brain power By 2020, wind could provide one-tenth of our planet’s electricity needs Already today, SKF’s innovative knowhow is crucial to running a large proportion of the world’s wind turbines Up to 25 % of the generating costs relate to maintenance These can be reduced dramatically thanks to our systems for on-line condition monitoring and automatic lubrication We help make it more economical to create cleaner, cheaper energy out of thin air By sharing our experience, expertise, and creativity, industries can boost performance beyond expectations Therefore we need the best employees who can meet this challenge! The Power of Knowledge Engineering Plug into The Power of Knowledge Engineering Visit us at www.skf.com/knowledge Download free eBooks at bookboon.com 107 Click on the ad to read more Conditional expectations and Limit theorems Probability for Finance β L1 L2 (Xn , n ∈ N) X (, A, P ) ; P (Xn , n ∈ N) X Xn → X ε > 0 lim P (|Xn − X| > ε) = n→+∞ Download free eBooks at bookboon.com 108 Conditional expectations and Limit theorems Probability for Finance a.s (Xn , n ∈ N) X Xn → X 0 ⊂ P (0 ) = ∀ω ∈ 0 , lim Xn (ω) = X(ω) n→+∞ PXn PX Xn X (Xn , n ∈ N) L X Xn → X) f f(x).dPXn (x) = f(x).dPX (x) lim n→+∞ R R X E(X) = A > P (X ≥ A) ≤ A A > 1 X X Download free eBooks at bookboon.com 109 Conditional expectations and Limit theorems Probability for Finance X ∈ L2 (, A, P ) E(X) = m V (X) = σ ; B > σ2 P (|X − | ≥ B) ≤ B P (|X − | ≥ Aσ) ≤ A2 A X 2A2 A = 2×0.01 = 7.0711 A = 2.32, P (X − −Aσ) ≤ The financial industry needs a strong software platform That’s why we need you SimCorp is a leading provider of software solutions for the financial industry We work together to reach a common goal: to help our clients succeed by providing a strong, scalable IT platform that enables growth, while mitigating risk and reducing cost At SimCorp, we value commitment and enable you to make the most of your ambitions and potential Are you among the best qualified in finance, economics, IT or mathematics? Find your next challenge at www.simcorp.com/careers www.simcorp.com MITIGATE RISK REDUCE COST ENABLE GROWTH Download free eBooks at bookboon.com 110 Click on the ad to read more Conditional expectations and Limit theorems Probability for Finance (Xn , n ∈ N) σ), Zn = n ni=1 Xi (Zn , n ∈ N) ε > 0 P (|Zn − | ≥ ε) ≤ σ2 nε2 (Xn , n ∈ N) Xn X X L2 ) limn→+∞ E(Xn ) = E(X) limn→+∞ V (Xn − X) = (Xn , n ∈ N) Zn = n1 ni=1 Xi (Zn , n ∈ N) E(|Xn |) = +∞, Zn K ri = E(ri ) + β ik Fk + εi k=1 Download free eBooks at bookboon.com 111 Conditional expectations and Limit theorems Probability for Finance ri i, F1 , , FK β ik i k εi i Cov(Fk , Fj ) = j = k) Cov(Fk , εi ) = 0) Cov(εi , εm ) = i = m) N N N N N K ri = E(ri ) + β ik Fk + εi N i=1 N i=1 N i=1 k=1 N i=1 N K N N 1 = E(ri ) + β Fk + εi N i=1 N i=1 ik N i=1 k=1 N N εi i=1 (Xn , n ∈ N) p; Tn n Xi − np Tn = i=1 np(1 − p) p Download free eBooks at bookboon.com 112 Conditional expectations and Limit theorems Probability for Finance n u d up n , n ≥ Y = Y1n , , Yk(n) k(n) n n, sn = V Y i=1 Yi n ,n ≥ ε > 0, U = U1n , , Uk(n) Uin = Yin |Yin | ≤ εsn = V lim n→+∞ k(n) i=1 s2n Yin =1 n Y = Y1n , , Yk(n) , n ≥ n n n n Y1 − E (Y1 ) , , Yk(n) − E Yk(n) , n ≥ k(n) n n ≥ 1, Zn = i=1 Yi E (Zn ) → V (Zn ) → σ = 0 Zn Z u d u d Download free eBooks at bookboon.com 113 Bibliography Probability for Finance ◦ Download free eBooks at bookboon.com 114 Bibliography Probability for Finance 115 .. .Probability for Finance Patrick Roger Strasbourg University, EM Strasbourg Business School May 2010 Download free eBooks at bookboon.com Probability for Finance © 2010 Patrick... Contents Probability for Finance Contents Introduction 1.1 1.1.1 1.1.2 1.1.3 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 Probability spaces and random variables Measurable spaces and probability. .. Sub-tribes of A Probability measures Conditional probability and Bayes theorem Independant events and independant tribes Conditional probability measures Bayes theorem Random variables and probability