probability for finance

Probability for Finance ContentsContents 1.3.3 Probability distributions and cumulative distributions 30 Stand out from the crowd Designed for graduates with less than one year of full-t

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Patrick Roger

Probability for Finance

Download free books at

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Probability for Finance

Patrick Roger Strasbourg University, EM Strasbourg Business School

May 2010

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Download free eBooks at bookboon.com

3

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Probability for Finance Contents

Contents

1.3.3 Probability distributions and cumulative distributions 30

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Contents

2.1.1 Expectations of discrete and continous random variables 39

2.1.3 Illustration: Jensen’s inequality and Saint-Peterburg paradox 43

2.4 Equivalent probabilities and Radon-Nikodym derivatives 63

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Probability for Finance Contents

4.1.4 Conditional expectation with respect to a random variable 98

4.1.5 Conditional expectation with respect to a substribe 100

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

your chance

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

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7

Contents

I was a

he s

Real work International opportunities

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al Internationa

or

�ree wo

I wanted real responsibili�

I joined MITAS because Maersk.com/Mitas

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supervisor in the North Sea advising and helping foremen solve problems

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I joined MITAS because

I was a

he s

al Internationa

or

�ree wo

I was a

he s

al Internationa

or

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www.discovermitas.com

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Probability for Finance Introduction

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Spend 3-4 weeks this summer at the highest ranked

technical university in Scandinavia

DTU’s English-taught Summer University is for dedicated

international BSc students of engineering or related

natural science programmes.

Application deadlines and programmes:

31 March Arctic Technology

15 March & 15 April Chemical/Biochemical Engineering

30 April Telecommunication

3 June Food Entrepreneurship

Visit us at www.dtu.dk

DTU Summer University

– for dedicated international students

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Probability for Finance Probability spaces and random variables

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Up to 25 % of the generating costs relate to nance 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

mainte-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 EngineeringBrain power

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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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MITIGATE RISK REDUCE COST ENABLE GROWTH

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

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       P (BBc) = P (B) + P (Bc)  B

 Bc     BBc = ,   P (B

Bc) = P () = 1

  P (Bc) = 1− P (B) 

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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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Challenge the way we run

1349906_A6_4+0.indd 1 22-08-2014 12:56:57

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Probability for Finance Moments of a random variable

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your career

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−10

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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

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Probability for Finance       Moments of a random variable

I was a

I joined MITAS because Maersk.com/Mitas

�e Graduate Programme for Engineers and Geoscientists

Month 16

I was a construction

supervisor in the North Sea advising and helping foremen

I was a

he

Real work

I was a

he

Real work

I was a

he

Real work

www.discovermitas.com

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Spend 3-4 weeks this summer at the highest ranked

technical university in Scandinavia

DTU’s English-taught Summer University is for dedicated

international BSc students of engineering or related

natural science programmes.

Application deadlines and programmes:

31 March Arctic Technology

15 March & 15 April Chemical/Biochemical Engineering

30 April Telecommunication

3 June Food Entrepreneurship

Visit us at www.dtu.dk

DTU Summer University

– for dedicated international students

Trang 56

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Up to 25 % of the generating costs relate to nance 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

mainte-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 EngineeringBrain power

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4((−1)2+ (−2)2+ (1)2+ (2)2) =√

2.5 = 1.58σ(Y ) =

1

4((1)

2+ (−1)2 + (−1)2+ (1)2) = 1

ρXY = 0.5

1.58 = 0.316

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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

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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



             

1    σ2(X) = σ2

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your career

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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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