Theory Application

Discrete

Continuous

- Stochastic differential equations

- Ito’s formula

- Derivation of the Black-Scholes equation

- Markov processes and the Kolmogorov equations

Why Ito’s formula?

• Model stock dynamics using stochastic differential equationsModel stock dynamics using stochastic differential equations

• Derive an option pricing formula in continuous timeDerive an option pricing formula in continuous time

• Compute the price of an optionCompute the price of an option

What is Ito’s formula?

• Differential representationDifferential representation

• Different from ordinal chain ruleDifferent from ordinal chain rule

• Additional term from quadratic variationAdditional term from quadratic variation

Ito’s formula:

- Basic idea:- Basic idea:

;)()()(0t

udButI

Taylor’s formula using “Ito’s rule”Taylor’s formula using “Ito’s rule”

““informally”informally”dttdBtdB )()(WriteWrite

dtttdItdI )()()( 2

- Provides a “shortcut”- Provides a “shortcut”

T

xx

T

x dttBftdBtBfBfTBf00

)(2

1)()()0()(

• Differential form for Ito’s formula:Differential form for Ito’s formula:

T

xx

T

x dttBftdBtBfBfTBf00

)(2

1)()()0()(

2)()(2

1)()()()( tBtBftBtBftBfttBf xxx

- Step 1: Use Taylor’s formula- Step 1: Use Taylor’s formula

)()()(2

1)()()( tdBtdBtBftdBtBftBdf xxx

- Step 2: Take - Step 2: Take tt sufficiently small, and write sufficiently small, and write

- Step 3: Apply Ito’s rule:- Step 3: Apply Ito’s rule: dttdBtdB )()(

dttBftdBtBftBdf xxx )(2

1)()()(

- Step 4: Integrate this from 0 to - Step 4: Integrate this from 0 to T…T…

Differential vs. Integral forms:

dttBftdBtBftBdf xxx )(2

1)()()(

T

xx

T

x dttBftdBtBfBfTBf00

)(2

1)()()0()(

• Ito’s formula in differential form:Ito’s formula in differential form:

- More convenient,- More convenient, - Easier to compute- Easier to compute

• Ito’s formula in integral formIto’s formula in integral form

- Mathematically well-defined- Mathematically well-defined

- Solid definitions for both the integrals- Solid definitions for both the integrals

Geometric Brownian motion:

dBdBtBtfdBtBtfdttBtftBtdf xxxt )(,2

1)(,)(,)(,

ttBStS 2

2

1)(exp)0()(

• Apply Ito’s formula to get differential formApply Ito’s formula to get differential form

txSxtf 2

2

1exp)0(),(

2),(2

1),(),( xxtfxxtftxtf xxxt),(),( xtfxxttf

- Ito’s formula:- Ito’s formula:

dBtBtfdttBtftBtf xxxt )(,)(,2

1)(,

dt

,2

1exp)0(),( 2

txSxtf

t

fft

),,(

2

1 2 xtf

x

ff x

),,( xtf 2

2

x

ff xx

),(2 xtf

))(,()( tBtftS

dBtSdttStS )()(2

1)(

2

1 22

)(tdS dBtSdttS )()( (G.B. motion in differential form)(G.B. motion in differential form)

TT

tdBtStdtSSTS00

)()()()0()(

dBtBtfdttBtftBtf xxxt )(,)(,2

1)(,

)(, tBtdf

Ito’s lemma:

• Differential form:Differential form:

ttBStS 2

2

1)(exp)0()(

)()()()( tdBtSdttStdS

)()()()()()()()( tdBtSdttStdBtSdttStdStdS

)()()()( tdBtStdBtS

dttS )(22

),0[on abledifferentily continuous twice:),( xtf

dBStSfdtStfSStftSStf xxxxt ,,2

1,)(, 22

Stdf ,

dttS )(2

dSdStStStfdStStfdttStfttSdf xxxt )()(,2

1)(,)(,),( 22

)()()( tdBtSdttS

dtSStfSdBSdtStfdtStf xxxt22,

2

1,,

dBStSfdtStfSStftSStf xxxxt ,,2

1,)(, 22

Stdf ,

(Ito’s formula of (Ito’s formula of ff((tt, , SS) in differential form)) in differential form)

dBSfdtfSSffSfTSTf x

T

xxxt

0

22

2

10,0,

Black-Scholes equation:

)()()()( tdBtSdttStdS • Stock:Stock:

dttrMtdM )()( • Money market:Money market:

• Self-financing portfolio:Self-financing portfolio:

)()()()1()()()( tSttXrtSttX ttt

ttt tSttXrtStSttXtX )()()()()()()()(

dttSttXrtdSttdX )()()()()()(

dttSttXrtdSttdX )()()()()()(

tt trMtMtM )()()( )(1)( tMrtM tt

ttSv ),(• Value of an option:Value of an option:

put)(European )(

call)(European )()(),(

TSK

KTSTSgTTSv

- Apply Ito’s lemma for - Apply Ito’s lemma for vv((xx, , tt):):

dBSvdtvSSvvdv xxxxt

22

2

1

dttSttXrtdSttdX )()()()()()(

dttSttXrdBtSdttSt )()()()()()(

dBtStdttSrttrX )()()())(()(

dBSvdtvSSvvdv xxxxt

22

2

1

SdBdtSrrXdX )(

• Values of option vs. portfolio: ???),()( ttSvtX

)(,)( 1. tStvt x

xxxt vSSvvSrrX 22

2

1)( 2.

xxxtx vSSvvSrvrv 22

2

1)(

),,(2

1),(),(),( 22 xtvSxtrSvxtvxtrv xxxt

)(),( xgxTv

(Black-Scholes partial differential equation)

,0 Tt 0x

- It can be solved with various boundary conditions- It can be solved with various boundary conditions

)(),( xgxtv

- For American derivative securities,- For American derivative securities,

- Black-Scholes PDE does not depend on - Black-Scholes PDE does not depend on

Theory Application

Discrete

Continuous

- Stochastic differential equations

- Markov processes and Feynman-Kac formula

Stochastic differential equations:

tdBtXtdttXttdX )(,)(,)(

tt

udBuXuudXuXuXtX00

)(,)()(,)0()(

• What are the properties of solutions?What are the properties of solutions?

• How can we solve a given such equation?How can we solve a given such equation?

• What are solutions?What are solutions?

Solution to SDE:

- A function of the underlying Brownian sample path - A function of the underlying Brownian sample path BB((tt))

- Adapted to the filtration generated by- Adapted to the filtration generated by

and of the coefficient functions and of the coefficient functions ((tt, , xx) and ) and ((tt, , xx))

tdBtXtdttXttdX )(,)(,)(

• A strong solution A strong solution

(SDE)(SDE)

],0[)( Ttt

Brownian motion Brownian motion BB((tt), ),

:)( ],0[ TttX

• Is there a strong solution?Is there a strong solution? Is it unique?Is it unique?

Tt ,0

Uniqueness of strong solutions:

(SDE)(SDE)

(SDE) has a unique strong solution (SDE) has a unique strong solution XX((tt) if the coefficient) if the coefficient

functions functions ((tt, , xx) and ) and ((tt, , xx) are Lipschitz continuous:) are Lipschitz continuous:

yxLytxtytxt ,,,,

:, ],,0[ yxTt

- There exists a constant - There exists a constant LL s.t. s.t.

tdBtXtdttXttdX )(,)(,)(

Linear Stochastic differential equation:

tdBtXdttXtdX 2121 )()()( (L-SDE)(L-SDE)

ytxtytxt ,,,,

,),( 21 xxt ,),( 21 xxt

yxL

yxyx 11

11 L

(Lipschitz condition)(Lipschitz condition)

• Solve (L-SDE) using Ito’s formula!Solve (L-SDE) using Ito’s formula!

tdBtXdttXtdX 2121 )()()(

tdBdtdBtXdttXtdX 2211 )()()(

==

?)()( tXtYd

tdBdttYdBtXdttXtdXtY 2211 )()()()()(

)0()()( XtXtY tt

udBuYduuY0 20 2 )()(

tt

udBuYduuYXtYtX0 20 2

1 )()()0()()(

• Multiply a geometric Brownian motionMultiply a geometric Brownian motion ::)( 21 tBMtMetY

s.t. and , Find 21 MM•

dBtXdttXtdXtYtXtYd )()()()()()( 11

t uBMuMt uBMuMtBMtM udBedueXetX

0 2)(

0 2)()( 212121 )0()(

2112

21

11 , ,2

MM

:)( 21 tBMtMetY

tdBtXdttXtdX 2121 )()()(

022 )(2 12

1)0()( tBteXtX

(Geometric Brownian motion)(Geometric Brownian motion)

:),( xyyxf

dXdYYXfdYYXfdXYXfYXdf xyyx ),(),(),(),(

,yf x ,xf y ,0xxf ,0yyf 1xyf

dXdYXdYYdX (Integration by parts)(Integration by parts)

• Ito formula for a functionIto formula for a function

)(XYd

ProofProof::

s.t. and , Find 21 MM•

dBtXdttXtdXtYtXtYd )()()()()()( 11

(Homework!)(Homework!)

Markov property:

• Brownian motion starting at Brownian motion starting at xx:: )(tBhx ;1),0( xBx

)()( stsBh )()( sBtsBh )()( tBhsB

s+t

BB((ss))

ss

t

BB((ss))

00

00

)0( with process Markov :)( - 0 StS t

property Markov - )()( 1,

01,0 00 tShttSh tSt

000t 1t

0tS

• Geometric Brownian motion: an example of Markov processGeometric Brownian motion: an example of Markov process

tdBtSdttrStdS )()()(

,2

1)()(exp)( 01

2011

ttrtBtBxtS xtS )( 0

tTSh )(,0

Martingale property:

,)(, , TShxtu xt

tdBtSdttrStdS )()()(

)()(, , TShtStu tSt

tTSh )(:

Tt 0

• Markov property:Markov property:

• Martingale property:Martingale property:

vtStu )(, vtTh )( vTh )(

Ttv 0

)(, TShvSv )(, vSvu