Probability for finance

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 2.1 Moments of a random variable Mathematical expectation 360° thinking 360° thinking 10 10 11 13 16 18 19 21 24 25 25 29 30 34 35 37 37 360° 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 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 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 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 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 Contents Probability for Finance 4.4.2 4.4.3 Law of large numbers Central limit theorem 109 112 Bibliography 114 With us you can shape the future Every single day For more information go to: www.eon-career.com Your energy shapes the future Download free eBooks at bookboon.com Click on the ad to read more Introduction Probability for Finance                                                                                                                                                                                                                                                                                                                                                                                                                    Download free eBooks at bookboon.com                                    Probability for Finance                                                                                                               Introduction                                                                                                                                                                                                   www.job.oticon.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 360° thinking                                                360° thinking 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 103 Discover the truth at www.deloitte.ca/careers Click on the ad to read more © Deloitte & Touche LLP and affiliated entities D 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                  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 15:13 107 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) ≤ 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 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                                                                                                                                                                                                                                                                      Download free eBooks at bookboon.com 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. .. 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