Probability for Finance - eBooks and textbooks from bookboon.com

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 th[r]

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Probability for Finance

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Probability for Finance

Patrick Roger Strasbourg University, EM Strasbourg Business School

May 2010

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ISBN 978-87-7681-589-9

Download free eBooks at bookboon.com

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Probability for Finance Contents

Contents

1.1 Measurable spaces and probability measures 10

1.1.1 σ algebra (or tribe) on a set Ω 11

1.2 Conditional probability and Bayes theorem 18

1.2.1 Independant events and independant tribes 19

1.2.2 Conditional probability measures 21

1.3 Random variables and probability distributions 25

1.3.1 Random variables and generated tribes 25

1.3.2 Independant random variables 29

1.3.3 Probability distributions and cumulative distributions 30

1.3.4 Discrete and continuous random variables 34

1.3.5 Transformations of random variables 35

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Contents

2.1.1 Expectations of discrete and continous random variables 39 2.1.2 Expectation: the general case 40 2.1.3 Illustration: Jensen’s inequality and Saint-Peterburg paradox 43 2.2 Variance and higher moments 46

2.3 The vector space of random variables 50 2.3.1 Almost surely equal random variables 51 2.3.2 The space L 1 (Ω, A, P) 53

2.5.2 Application to portfolio choice 71

3 Usual probability distributions in financial models 73

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Probability for Finance Contents

3.3.2 The Student-t distribution 92

3.3.3 The Fisher-Snedecor distribution 93

4.1.2 Conditional distributions 96

4.1.3 Conditional expectation with respect to an event 97

4.1.4 Conditional expectation with respect to a random variable 98

4.1.5 Conditional expectation with respect to a substribe 100

4.2 Geometric interpretation in L 2 (Ω, A, P) 101

4.2.2 Conditional expectation as a projection in L 2 102

4.3 Properties of conditional expectations 104

4.3.1 The Gaussian vector case 105

4.4 The law of large numbers and the central limit theorem 108

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Contents

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Probability for Finance Introduction

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Probability for Finance Probability spaces and random variables

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  = {uu; ud; du; dd}         

      A′ = {∅; {uu; ud} ; {du; dd} ; } 

        P)   {du; dd} = {uu; ud}c   

       {uu; ud}{du; dd} =  ∈ A

              

        

1 րց

u րց

uu = u2

ud

d րց

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

×1

3;1 2

3− 0



×

1

2 − 13



= 118

  

P (B1|B2) =

1 18 1 6

= 1

3 = P (B1)

B1     B2

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3 ×(

)(

) =

63 120

3 ×(

)(

) =

21 120

1 120

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Probability for Finance Moments of a random variable

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−10

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4((−1)2+ (−2)2+ (1)2+ (2)2) =√

2.5 = 1.58σ(Y ) =

1

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+ θY

150110

+ θZ

11



=

00

+ 5

150110



− 350

11



=

00

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(eC− A)−1+(b− eA)−1

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Probability for Finance Usual probability distributions in financial models

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+ ∞



k=0

λkk! = λ exp(−λ) exp(λ) = λ

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+ ∞



k=0

λkk!

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= 12

(b2− a2)

b− a =

b + a2

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−



b + a2

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

−12



x− mσ

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2dx

=



−√σ2πexp

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

−1 2



ln(x)−m σ

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      

E(X) = 1

σ√2π

ln(x)− mσ

2dx

  y = ln(x),     

E(X) = 1

σ√2π



y− mσ

2dy

       

E(X) = 1

σ√2π

(y− (m + σ2))2

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

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Probability for Finance Conditional expectations and Limit theorems

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√2π

n

1

Det(X)exp

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(y2− m2)

X  |X  =x  = σ21− σ

2 12

σ2 2

exp

−1

2(x− m)′−1X (x− m)exp



−1 2

exp



−12

(x− m)′−1X (x− m) −

σ2

1σ2

2− σ2 12

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