stochastic processes for finance

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Stochastic Processes for Finance

Patrick Roger Strasbourg University, EM Strasbourg Business School

June 2010

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Download free eBooks at bookboon.com

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Stochastic Processes for Finance Contents

Contents

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

Masters in Management

Fast-track

your career

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Click on the ad to read more

5

Contents

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Stochastic Processes for Finance Contents

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

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Stochastic Processes for Finance Introduction

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Stochastic Processes for Finance Discrete-time stochastic processes

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I was a

he s

Real work International opportunities

�ree work placements

al Internationa

or

�ree wo

I wanted real responsibili�

�e Graduate Programme for Engineers and Geoscientists

Month 16

I was a construction

supervisor in the North Sea advising and helping foremen solve problems

I was a

he s

al Internationa

or

�ree wo

I joined MITAS because

I was a

he s

al Internationa

or

�ree wo

I was a

he s

al Internationa

or

�ree wo

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

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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 Engineering

Brain power

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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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

 lim

0.5 0.50.5 0.5





           

             

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2s

s

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Stochastic Processes for Finance Continuous-time stochastic processes

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Stochastic Processes for Finance     Continuous-time stochastic processes

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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

Masters in Management

Fast-track

your career

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

•           

         

0  t     

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N n=1δn   δn   

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

= E

Zs+ (Zt− Zs)

FZ s



= E

Zs

FZ s

+ E

Zt− Zs



FZ s

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

= E

Zt2

FZ s



− t

  

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+ 2E

ZtZs



FZ s



− EZs2

FZ s

+ 2E

ZtZs



FZ s

+ 2ZsE

Zt



FZ s

+ 2Zs2− Z2

s − t

E(Zt− Zs)2

+ Zs2 − t(t− s) + Zs2− t



Eexp (γZt)

FZ s



Eexp (γ(Zt− Zs)) exp(γZs)

FZ s



γx− γ

2t2



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= √12πt

= exp



−γ

2(t + s)2

1

√2πt

2t2

1

√2πt

= exp



−γ

2s2



= g(0, s)

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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

our web page: www.ericsson.com/careers

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�e Graduate Programme for Engineers and Geoscientists

Month 16

I was a construction

supervisor in

I was a

www.discovermitas.com

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Stochastic Processes for Finance Stochastic integral and Itô’s lemma

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N n=1n−1

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σsdZs

           

     

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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ZsdZs= Z

2

T − Z2 0

ZT22

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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 Engineering

Brain power

Plug into The Power of Knowledge Engineering

Visit us at www.skf.com/knowledge

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sds +

 t 0

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dWs = Wt=

 t 0

ds +

 t 0

σdZs

= 

 t 0

ds + σ

 t 0

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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

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∂2f

∂x2(x0, t0)(dx)2 

+12

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2

∂2f

∂x2 [(Xt, t)dt + σ(Xt, t)dZt]2+1

∂2f

∂x2σ2(Xt, t)

dt+σ(Xt, t)∂f

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∂2g

∂x2σ2(Yt, t) = ∂g

∂xYt+

12

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Stochastic Processes for Finance       Stochastic integral and Itô’s lemma

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   Q() = EP

exp

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= √12π

2

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λsdZs− 1

2

 t 0

12

 T 0

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σ2(Xt, t) dt < +∞ 

 X 

Xt= X0+

 t 0

 (Xs, s) ds +

 t 0

σ (Xs, s) dZs

        σ   

     

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exp [a (t− s)] σsdZs 

       

Xtexp(−at) = c +

 t 0

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exp(αs)dZs



= β (1− exp(−αt)) + y0exp(−αt) + σ

 t 0

exp(−2α(t − s))ds

2

2α[1− exp (−2αt)]

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Stochastic Processes for Finance Bibliography

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Stochastic Processes for Finance Bibliography

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stationary distribution, 20 transient, 19

no memory process, 48 process, 15, 48

Brownian motion, 59 transition matrix, 15 Martingale, 21, 49

Brownian motion, 59 Doob, 24

submartingale, 21, 49 super-martingale, 21, 49 Modi cation

stochastic process, 43 Novikov

condition, 89 Path, 40

càdlàg, 41 càglàd, 43 continuous, 40 Brownian motion, 58 LCRL, 43

nowhere differentiable, 59 RCLL, 41

Poisson distribution, 47 process, 47 Probability transition, 15 Snell envelope, 36 Stochastic differential equation, 91 linear, 93

solution conditions, 92

de nition, 91 Markov, 92 Stochastic integral, 71 calculation rules, 79 definition, 76 properties, 78 stochastic differential, 78

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