Black-Scholes and Pricing Fundamentals€¦ · CHAPTER 1 Black-Scholes and
TheoryApplication Discrete Continuous - - Stochastic differential equations - - Ito’s formula - -...
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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