A new proposition on the martingale representation theorem and on the approximate hedging of contingent claim in mean variance criterion

A new proposition on the martingale representation theorem and on the approximate hedging of contingent claim in mean-variance criterion April, 2006 Abstract: In this work we revisit th

A n e w pr oposit ion on t h e m a r t in ga le

r e pr e se n t a t ion t h e or e m a n d on t h e

a ppr ox im a t e h e dgin g of con t in ge n t cla im in

m e a n - v a r ia n ce cr it e r ion

A Fa r be r , N gu y e n V H a n d V u on g Q.H

I n t his w or k w e r evisit t he pr oblem of t he hedging of cont ingent claim using

m ean- squar e cr it er ion We pr ov e t hat in incom plet e m ar k et , som e pr obabilit y

m easur e can be Q ~ P ident ified so t hat { Sn} becom es { Fn} - m ar t ingale under Q

This is in fact a new pr oposit ion on t he m ar t ingale r epr esent at ion t heor em The new r esult s also ident ify a w eight funct ion t hat ser ves t o be an appr oxim at ion t o

t he Radon- Nikodým der ivat ive of t he unique neut r al m ar t ingale m easur e Q

JEL Classificat ions: G12; G13

Keyw or ds: Mar t ingale r epr esent at ion t heor em ; Hedging; Cont ingent claim ;

Mean- var iance

CEB Wor king Paper N° 06/ 004

Apr il 2006

Université Libre de Bruxelles – Solvay Business School – Centre Emile Bernheim

ULB CP 145/01 50, avenue F.D Roosevelt 1050 Brussels – BELGIUM

e-mail: ceb@admin.ulb.ac.be Tel : +32 (0)2/650.48.64 Fax : +32 (0)2/650.41.88

A new proposition on the martingale representation theorem and on the approximate hedging of contingent claim in mean-variance criterion

April, 2006

Abstract:

In this work we revisit the problem of the hedging of contingent claim using mean-square criterion We prove that in incomplete market, some probability measure can be identified so that becomes -martingale under This is in fact a new

proposition on the martingale representation theorem The new results also identify a weight function that serves to be an approximation to the Radon-Nikodým derivative of the

unique neutral martingale measure Q

P

Q ~

}

JEL Classification:

G12; G13

Keywords:

Martingale representation theorem; Hedging; Contingent claim; Mean-variance

1 Introduction

The activity of a stock market takes place usually in discrete time Unfortunately such markets with discrete time are incomplete, so the traditional pricing and hedging of

contingent claim are usually not applicable

The purpose of this work is to propose a simple method for hedging a contingent claim or

an option in mean-variance criterion

probability space ^ ` `, and ^

d n

S , 0,1,, , R

P

,

, F

information available up to the time n

(b) An ^ `F n -measurable random variable H is called a contingent claim that in the case

of a standard call option H max(S n K,0)

of securities of type j kept by the investor in the time interval

) , , ( n1 n2 n j

n

J

) , 1

-measurable (based on the information available up to the time

) 1 (n

F

1



n ) Thus, ^ ` Jn is said to

be predictable

(d) Suppose that 'S n S n S n1,HL2(P),

k

k k

G

1

)

j

j k j k k

1 J J

The traditional problem is to find constant cand J {Jn,n 1,2,.N}, such that

*

Centre Emile Bernheim, Université Libre de Bruxelles, 21 Ave F.D Roosevelt, 1050-Bruxelles, Belgium

†

Vietnam National University, Hanoi, 334 Nguyen Trai, Hanoi, Vietnam

^HcG N(J)`2 omin

Definition 1 minimizes the expectation in (1.1) is called an optimal

strategy in the mean square criterion corresponding to initial capital

)) ( ( ) (Jn* Jn* c

c

Problem (1.1) has been investigated in a number of works such as Föllmer and Schweiser (1991), Schweiser (1995, 1996), Schäl (1994), and Nechaev (1998) However, the solution for (1.1) has been very complicated as {S n} is not a {F n}-martingale under P.

When{S n} is {F n}-martingale under some measure Q ~ P, we can find c,J such that:

^H cG N(J)`2 omin

The solution of this problem may be simple enough, and the construction of an optimal strategy is much easier in practice

We notice that if L N d Q d/ P then

^H c G N ` E ^ ^H c G N ` L N`

N

G c

This is similar to the pricing of asset based on a neutral martingale measure

N L

In this work we give a solution of the problem (1.3) and a martingale representation theorem in the case of discrete time

2 Defining the optimal portfolio

Let Q be a probability measure such that Q is equivalent to P, and under Q,

is a martingale, then }

, ,

2

,

1

,

) ( ,

), (

)

Theorem 1 If ^ ` d is a

n

S , 0,1,, , R ^ `F n Q then

2 2

*

Where

(2.2)

}

)]

)[var(

{(

} )]

)[var(

{(

1 1

1 1

*

s a S

S H E

S S

H E

n n

n

n n

n n

n

 '

'

' '

'









P

J

with the convention that 0/0 = 0

Proof We shall prove the theorem only for the case d 1 We note that:

N

2 1 2 1

2 1

2

E  ' J '  '  J  ' ' J  '

n

J

Furthermore, we have:

¾

½

¯

®



 1 0

2

) (

)) ( (

N

n

n n N N

H

2

* 0

1

2

* 1

2 0

1

2 1

2 0 1

2 2

0

)}

( {

} {

) (

} {

) (

} {

)

(

J J

J J

N N

Q

N

n

n n n n

N

n

n n n n

N

n

n n n

G H H E

S H

E E c H

S H

E E c H S

H E c

H





'

 '





'

 '



 '

 '





¦

¦

¦



Q

Q Q

3 The martingale representation theorem

Theorem 2 Let {H n,n 0,1,}, {S n,n 0,1,} be random variables defined on the same probability space ^:, F,P`, F n S V(S0,,S n) We denote 3(S,P) a set of the probability measure Q such that Q ~ P, and that ^ `S n is ^ `S

n

F n S n n

k

n k k

H

1

where ^C n` is ^ `S

n

F -Q-martingale orthogonal to the martingale ^ `S n , that is

, whereas

^ ` 0, 0,1,2,

E n n n ^ ` Jn is ^ `S

n

k

N k

k

H

1

0

for all finite iff the set n 3(S,P) consists of only one element

Remark 1 By the fundamental theorem of mathematical finance, a stock market has no

arbitrage opportunity and is complete iff 3(S,P) consists of only one element and in this case we have (3.2) with J being defined by (2.2) Furthermore, in this case the conditional

probability distribution of ^S n` given ^ `S

n

F1 concentrates at d1 points of R (see [2]) d

4 Examples

Example 1 Let us consider a stock with the discounted price S0 at t 0, at , where:

1

S t 1

°

¯

°

®

­





! 3

0 2 1

3 2 1 3

2 1 2 0

1 0

2 3

1

prob

with

1 ,

0 , , prob

with

prob

with

p S

p p p p

p p p S

p S

S

Suppose that there is an option on the above stock with the maturity at and with strike price We shall show that there are several probability measures

1

t

0

S

Q S0, S1 EQ('S1) 0

In fact, suppose that Q is a probability measure such that under Q , S1 takes the values of

3S , S , 1S , with the positive probabilities q ,q ,q , respectively, then:

3 1 3

1 0

Therefore, Q is defined by q1,12q1,q1, 21

1

In the above market, the payoff of the option is:

) 0 , max(

) ( )

Apparently, it is feasible to construct an optimal portfolio with:

2 )

(

2

1 ) (

) (

0 1

2 1

1

*

S q H

E

S E

S H E

' '

Q

Q

Q

J

Example 2 A semi-continuous market model, which is discrete in time, but continuous in

state Now, let us consider a financial market with two assets:

(a) A risk-less asset {B n,n 0,1,,N} which exhibits a dynamics given by (4.1):

(4.1) 1

0 , exp

1

¿

¾

½

¯

®

n

k k

B

(b) A risky asset {S n,n 0,1,,N} given by the following dynamics

(4.2)

¿

¾

½

¯

®

­



n

k

k k k

S

1

1 1

directly from (4.2) that

} , , 1 , 0 ,

, 2 / ) ( ) ( ) (

, ) ( ) ( exp

1 2 1 1

1 1

1

















n n

n

n n n

n n

S S

a S

g S S

S S

V P

V

P

(4.3) with S0 given, and a(x),V(x) being some functions defined on [0,f )

The discounted price of risky asset S n S n/B n is:

~

(4.4)

¿

¾

½

¯

®

n

k

k k k

k

S

1

1 1

0

~

] ) ( )

( [

We now find a martingale measure Q for this model.

It is easy to see that EP{exp(Og k)} exp(O2/2), for g k ~ N(0,1), hence

(4.5)

exp

1

1 2 1

¿

¾

½

¯

®

­

¸

¹

·

¨

©

n

k

k k k k

Thus, putting

L

n

k

k k k k k

1

1 2

¹

·

¨

©

~

~

/ n

n S

addition, we see that

^ n n n n`

n

S / 1 exp ( 1) ( 1)

~

~





Denoting by E ,$ E expectations corresponding to P,Q, and ^ `S

n

choosing

) (

) ( 1

1



n

n n n

S

r S a

V

~

~ 1

~

~

1

¿

¾

½

¯

®

­

¿

¾

½

¯

®

­







S n n n n n

n

under Q Also, under Q , can be represented in the form

¿

¾

½

¯

®

­

n S

~

n S

1 1

*

n

or the weight function to find the optimal portfolio

n n n n

n

S  V  E 

1 2 1

*

, 2 / ) ( )

) , (S P

N L

Remark 2 The models (4.1), (4.2) are that of discretization of the following diffusion

model Let us consider a financial market with continuous time of two assets :

¿

¾

½

¯

®

­

³t

o

B exp ( )

(b) A risky asset: dS t S ta(S t)dtV(S t)dW t, S0 is given, or

S

t

u u t

o

u u

¿

¾

½

¯

®

­



exp

0

V

Putting

(4.11) 2

/ ) ( ) ( )

and dividing [ T0, ] into N equal intervals 0,',2',,N'}, where sufficiently large, it follows from (4.10),(4.11) that

' /T N

n

n n

n n

n

n

u u n

n u n

n

g S

S S

W W S

S S

dW S du

S S

S

2 / 1 ) 1 ( )

1 ( )

1 (

) 1 ( )

1 ( )

1 ( )

1 (

) 1 ( )

1 ( )

1 (

) (

) (

exp

] )[

( ) (

exp

) ( )

( exp

'

 '

#



 '

#

°¿

°

¾

½

°¯

°

®

­



'

 '

 '



'

 ' '

 '

 '



' '



' '

 '



V P

V P

V P

obtain the model:

`

N n

g n, 1,, NIID ~ N(0,1)

n

S ' ( 1)'exp P( ( 1)')'V( ( 1)')'1/2 (4.12) Similarly we have

^ n n

B ( 1)exp ' `

According to (4.10) ,the discounted price of the stock S t is

> @

¿

¾

½

¯

®

­



t

u u t

t

S

0 0

0

~

) ( )

( exp

The unique probability measure Q under which is a martingale is defined by

¿

¾

½

¯

®

­

Q

, ,

~

S t

t F S

 /  exp : ( )

0

2 2 1 0

Z E

T

u u S

F d d

¿

¾

½

¯

®

­

³

P

where Es (a(S s)r s)/V(S s), and under Q, then: is a Wiener

process It is obvious that can be approximated by:

³

 t u t

W

0

T

L

(4.15)

>

¿

¾

½

¯

®

¦N

k

k k k

L

1

2 2

/ 1

2 /

where

 ( (n1)') n'/ ( (n1)')

Therefore, the weight function (4.14) is an approximation to a Radon-Nikodým derivative

of the unique neutral martingale measure Q to , where can be used to price such derivatives

4 Further problems to be investigated

We realize that further problems that could arise in these models are the following:

1) We have to show that for the weight function (4.15)

0 )) ( (H H0 G N J* 2 o

EQ as N of or 'o0 2) Which neutral martingale measure is the nearest one with the subjective measure

in the semi-continuous model?

REFERENCES

[1] Föllmer H., and Schweiser M “Hedging of contingent claim under incomplete

information,” Applied Stochastic Analysis, Ed by M Davis, and R Elliot London:

Gordan & Breach, 1991, pp.389-414

[2] Jacod J., Shiryaev A.N “Local martingales and the fundamental asset pricing theorem

in the discrete case,” Finance Stochastics 2, pp.259-272

[3] Nechaev M.L “On mean-variance hedging,” Proceeding of Workshop on

Mathematical Finance, May 18-19, 1998 Institut Franco-Russe Liapunov, Ed by A

Shiryaev and A Sulem

[4] Nguyen V.H., and Tran T.N “On a generalized Cox-Ross-Rubinstein option market

model,” Acta Mathematica Vietnamica, 26(2), 2001, pp 187-204

[5] Schweiser M “Variance-optimal hedging in discrete time,” Mathematics of Operations

Research, 20(1), 1995, pp 1-32

[6] Schweiser M “Approximation pricing and the variance-optimal martingale measure,”

Annals of Probability, 24(1), 1996, pp 206-236

[7] Schäl M “On quadratic cost criteria for option hedging,” Mathematics of Operations

Research, 19(1), 1994, pp 131-141

in the discrete case,” Finance Stochastics 2, pp.259-272

[3] Nechaev M.L ? ?On mean- variance. .. Föllmer H., and Schweiser M ? ?Hedging of contingent claim under incomplete

information,” Applied Stochastic Analysis, Ed by M Davis, and R Elliot London:

Gordan & Breach, 1991,... V.H., and Tran T.N ? ?On a generalized Cox-Ross-Rubinstein option market

model,” Acta Mathematica Vietnamica, 26(2), 2001, pp 187-204

[5] Schweiser M ? ?Variance- optimal hedging