Semi-Continuous Models

Semi-Continuous Models12.1 Discrete-time Brownian Motion LetfYjgnj=1be a collection of independent, standard normal random variables defined on ;F;P, where IP is the market measure... Th

Semi-Continuous Models

12.1 Discrete-time Brownian Motion

LetfYjgnj=1be a collection of independent, standard normal random variables defined on ;F;P),

where IP is the market measure As before we denote the column vector( Y1;::: ;Yn ) T byY We therefore have for any real colum vectoru = ( u1;::: ;un ) T,

IEeuTY

= IE exp

8

<

:

n

X

j=1 ujYj

9

=

;= exp

8

<

:

n

X

j=1

1

2 u2j

9

=

;:

Define the discrete-time Brownian motion (See Fig 12.1):

B0 = 0 ;

Bk = Xk

j=1 Yj ; k = 1 ;::: ;n:

If we knowY1;Y2;::: ;Yk, then we knowB1 ;B2;::: ;Bk Conversely, if we knowB1;B2;::: ;Bk, then we knowY1 = B1;Y2 = B2,B1;::: ;Yk = Bk ,Bk,1 Define the filtration

F0 = f; g;

Fk =  ( Y1;Y2;::: ;Yk ) =  ( B1;B2;::: ;Bk ) ; k = 1 ;::: ;n:

Theorem 1.34 fBkgnk=0is a martingale (under IP).

Proof:

IE [ Bk+1jFk ] = IE [ Yk+1 + BkjFk ]

= IEYk+1 + Bk

= Bk :

131

Y

Y

Y 1

2

3

4

k

B k

Figure 12.1: Discrete-time Brownian motion.

Theorem 1.35 fBkgnk=0is a Markov process.

Proof: Note that

IE [ h ( Bk+1 )jFk ] = IE [ h ( Yk+1 + Bk )jFk ] :

Use the Independence Lemma Define

g ( b ) = IEh ( Yk+1 + b ) = 1p

2 

Z

1

,1

h ( y + b ) e,1 2y2

dy:

Then

IE [ h ( Yk+1 + Bk )jFk ] = g ( Bk ) ;

which is a function ofBk alone

12.2 The Stock Price Process

Given parameters:

 2IR, the mean rate of return.

  > 0, the volatility.

 S0 > 0, the initial stock price

The stock price process is then given by

Sk = S0 expn

Bk + ( ,1 2  2 ) ko

; k = 0 ;::: ;n:

Note that

Sk+1 = Sk expn

Yk+1 + ( ,1 2  2 )o

;

IE [ Sk+1jFk ] = SkIE [ eY k+1jFk ] :e ,

1 22

= Ske 1 2 2

e ,1 22

= e  Sk:

Thus

 = log IE [ Sk+1jFk ]

Sk = log IE



Sk+1 Sk

Fk



;

and

var

log Sk+1 Sk



= var

Yk+1 + ( ,1 2  2 )

=  2 :

12.3 Remainder of the Market

The other processes in the market are defined as follows

Money market process:

Mk = e rk ; k = 0 ; 1 ;::: ;n:

Portfolio process:


0;
1;::: ;
n,1;

 Each
k isFk-measurable

Wealth process:

 X0given, nonrandom



Xk+1 =
kSk+1 + e r ( Xk,
k Sk )

=
k ( Sk+1,e r Sk ) + e r Xk

 EachXk isFk-measurable

Discounted wealth process:

Xk+1 Mk+1 =
k



Sk+1 Mk+1 ,Mk Sk 

+ Mk Xk :

Definition 12.1 LetIPfbe a probability measure on ;F), equivalent to the market measure IP If

n

Sk

Mk

on

k=0is a martingale underfIP, we say thatfIP is a risk-neutral measure.

Theorem 4.36 IffIP is a risk-neutral measure, then every discounted wealth process

n

Xk Mk

on k=0is

a martingale underIPf, regardless of the portfolio process used to generate it.

Proof:

f

IE

Xk+1 Mk+1

Fk



= fIE

k



Sk+1 Mk+1 , Mk Sk 

+ Mk Xk

Fk



=
k



f

IE

Sk+1 Mk+1

Fk



,Mk Sk 

+ Mk Xk

= Mk Xk :

12.5 Risk-Neutral Pricing

LetVnbe the payoff at timen, and say it isFn-measurable Note thatVnmay be path-dependent Hedging a short position:

 Sell the simple European derivative securityVn

 ReceiveX0at time 0

 Construct a portfolio process
0;::: ;
n,1 which starts withX0and ends withXn = Vn

 If there is a risk-neutral measureIPf, then

X0 =fIE X Mn n = IE VfMn n :

Remark 12.1 Hedging in this “semi-continuous” model is usually not possible because there are

not enough trading dates This difficulty will disappear when we go to the fully continuous model

Definition 12.2 An arbitrage is a portfolio which starts withX0 = 0and ends withXnsatisfying

IP ( Xn0) = 1 ; IP ( Xn > 0) > 0 :

(IP here is the market measure)

Theorem 6.37 (Fundamental Theorem of Asset Pricing: Easy part) If there is a risk-neutral

mea-sure, then there is no arbitrage.

Proof: LetfIP be a risk-neutral measure, letX0 = 0, and letXnbe the final wealth corresponding

to any portfolio process Since

n

Xk Mk

on k=0is a martingale underIPf,

f

IE X Mn n =fIE X0

SupposeIP ( Xn0) = 1 We have

IP ( Xn0) = 1 =)IP ( Xn < 0) = 0 =)

f

IP ( Xn < 0) = 0 =)

f

IP ( Xn0) = 1 :

(6.2)

(6.1) and (6.2) implyfIP ( Xn = 0) = 1 We have

f

IP ( Xn = 0) = 1 =)

f

IP ( Xn > 0) = 0 =)IP ( Xn > 0) = 0 :

This is not an arbitrage

12.7 Stalking the Risk-Neutral Measure

Recall that

 Y1 ;Y2;::: ;Ynare independent, standard normal random variables on some probability space

;F;P)

 Sk = S0 expn

Bk + ( ,1 2  2 ) ko



Sk+1 = S0 expn

 ( Bk + Yk+1 ) + ( ,

1

2  2 )( k + 1)o

= Sk expn

Yk+1 + ( ,1 2  2 )o

:

Therefore,

Sk+1 Mk+1 = Mk Sk : exp

n

Yk+1 + ( ,r,1 2  2 )o

;

IE

Sk+1 Mk+1

Fk



= Mk Sk :IE [expfYk+1g jFk ] : expf,r,1 2  2

g

= Mk Sk : expf

1

2  2

g: expf,r,

1

2  2

g

= e ,r : S Mk k :

If = r, the market measure is risk neutral If = r, we must seek further

Sk+1 Mk+1 = Mk Sk : exp

n

Yk+1 + ( ,r, 1 2  2 )o

= Mk Sk : expn

 ( Yk+1 + ,r

 ),

1

2  2o

= Mk Sk : expn

 Yk+1 ~ , 1 2  2o

;

where

~

Yk+1 = Yk+1 + ,r

 :

The quantity,r

 is denotedand is called the market price of risk.

We want a probability measurefIP under whichY1 ~ ;::: ; Yn ~ are independent, standard normal ran-dom variables Then we would have

f

IE

Sk+1 Mk+1

Fk



= Mk Sk : IEf h

expf Yk+1 ~ gjFk

i

: expf,1 2  2

g

= Mk Sk : expf1 2  2g: expf,1 2  2g

= Mk Sk :

Cameron-Martin-Girsanov’s Idea: Define the random variable

Z = exp

2

4

n

X

j=1 (,Yj ,1 2  2 )

3

5:

Properties ofZ:

 Z 0



IEZ = IE exp

8

<

:

n

X

j=1 (,Yj )

9

=

;

: exp ,

n

2  2

= exp

n

2  2

: exp ,

n

2  2

= 1 :

Define

f

IP ( A ) =Z

A Z dIP 8A2 F:

ThenfIP ( A )0for allA2 Fand

f

IP IEZ = 1 :

In other words,IPfis a probability measure

We show thatfIP is a risk-neutral measure For this, it suffices to show that

~

Y 1 = Y1 + ; ::: ; Yn ~ = Yn + 

are independent, standard normal underfIP

Verification:

 Y1 ;Y2;::: ;Yn: Independent, standard normal under IP, and

IE exp

2

4

n

X

j=1 ujYj

3

5= exp

2

4

n

X

j=1 1 2 u 2j

3

5:

 Y ~ = Y1 + ; ::: ; Yn ~ = Yn + :

 Z > 0almost surely

 Z = exph

P

nj=1 (,Yj,1 2  2 )i

;

f

IP ( A ) =Z

A Z dIP 8A2 F;

f

IEX = IE ( XZ )for every random variableX

 Compute the moment generating function of(~ Y1;::: ; Yn ~ )underIPf:

f

IE exp

2

4

n

X

j=1 uj Yj ~

3

5 = IE exp

2

4

n

X

j=1 uj ( Yj +  ) +Xn

j=1 (,Yj ,1 2  2 )

3

5

= IE exp

2

4

n

X

j=1 ( uj, ) Yj

3

5: exp

2

4

n

X

j=1 ( uj,

1

2  2 )

3

5

= exp

2

4

n

X

j=1

1

2 ( uj, ) 2

3

5: exp

2

4

n

X

j=1 ( uj , 1 2  2 )

3

5

= exp

2

4

n

X

j=1



( 1 2 u 2j,uj  + 1 2  2 ) + ( uj ,1 2  2 )

3

5

= exp

2

4

n

X

j=1

1

2 u2j

3

5:

12.8 Pricing a European Call

Stock price at timenis

Sn = S0 expn

Bn + ( ,1 2  2 ) no

= S0 exp

8

<

:

Xn

j=1 Yj + ( ,1 2  2 ) n

9

=

;

= S0 exp

8

<

:

Xn

j=1 ( Yj + ,r

 ),( ,r ) n + ( ,1 2  2 ) n

9

=

;

= S0 exp

8

<

:

Xn

j=1 Yj ~ + ( r, 1 2  2 ) n

9

=

;

:

Payoff at timenis( Sn,K ) + Price at time zero is

f

IE ( Sn,K ) +

2

4e,rn

0

@S0 exp

8

<

:

Xn

j=1 Yj ~ + ( r,1 2  2 ) n

9

=

; ,K

1

A

+3 5

= Z 1

,1

e,rn

S0 expn

b + ( r,1 2  2 ) no

,K+ :p1

2 ne, b2

2n2 db

since

P

nj=1 Yj ~ is normal with mean 0, variancen, underfIP

This is the Black-Scholes price It does not depend on

Remark 12.1 Hedging in this ? ?semi-continuous? ?? model is usually not possible because there are

not enough trading dates