Self-…nancing trading strategies I

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Attainable contingent claims and replication strategies

80-646-08 Stochastic calculus I

Geneviève Gauthier

HEC Montréal

th.

Self-…nancing strategies

A riskless asset and a risky asset Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Self-…nancing trading strategies I

Assume that B = f ^B

^{: 0 t T} g ^and

S = f ^S

^{: 0 t T} g represent the evolutions of the riskless asset price and the risky asset price respectively.

Moreover, the riskless asset evolution is described by the di¤erential equation

dB

= rB

dt .

th.

Self-…nancing strategies

A riskless asset and a risky asset Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Self-…nancing trading strategies II

A trading strategy is a predictable portfolio process f ( φ

, ψ

) : 0 t T g

where

φ

= the number of riskless asset shares held at time t and ψ

= the number of risky asset shares held at time t . The value of such a trading strategy at time t is

V

= φ

B

+ ψ

S

.

th.

Self-…nancing strategies

A riskless asset and a risky asset Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Self-…nancing trading strategies III

By applying the multiplication rule, we have that dV

= φ

dB

+ B

d φ

+ d h ^{φ, B} i

+ ψ

dS

+ S

d ψ

+ d h ^{ψ, S} i

. De…nition

A trading strategy is said to be self-…nancing if dV

= φ

dB

+ ψ

dS

,

i.e. the variations in the strategy value are only caused by the asset price variations and do not depend on the portfolio weight ‡uctuations. A strategy is therefore self-…nancing, if

B

d φ

+ S

d ψ

+ d h ^{φ, B} i

+ d h ^{ψ, S} i

= 0.

th.

Self-…nancing strategies

A riskless asset and a risky asset Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Self-…nancing trading strategies IV

De…nition

Let X be contingent claim. X is therefore a

non-negative-valued, F

measurable, random variable. The contingent claim X is attainable if there exists a self-…nancing trading strategy f ( φ

, ψ

) : 0 t T g ^{such that}

V

= φ

B

+ ψ

S

= X .

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Martingale representation theorem I

Let W = f ^W

^{: 0 t T} g be a Brownian motion constructed on a …ltered probability space

( Ω , F ^, fF

g ^{, Q} ) such that the …ltration fF

g ^{is the one} generated by the Brownian motion, plus it includes all zero-probability events, i.e. for all t 0,

F

= σ ( N ^{and W}

^{: 0 s t} ) .

Let Z = f ^Z

^{: 0 t T} g be an Itô process such that Z

=

Z

+

Z

0

H

dW

where the fF

g adapted process H = f ^H

^{: 0 t T} g

satis…es the conditions

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Martingale representation theorem II

E

h R

0

H

ds i

< ∞ and

Q f ^ω 2 Ω : 9 ^t 2 [ 0, T ] such that H

( ω ) = 0 g = 0.

Theorem

Martingale representation theorem.

Z is a ( _Q , fF

g ) martingale (Z is sometimes called a Brownian martingale, since the …ltration is generated by the Brownian motion);

moreover, for any other ( _Q , fF

g ) martingale M, there exists a predictable process φ = f ^φ

: 0 t T g ^{such that} M

= M

+ R

0

φ

dZ

.

That theorem will be useful when we must prove that a contingent claim is attainable.

Lamberton and Lapeyre, page 74, or Baxter and Rennie, page 78.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication

Example Multidimensional References Appendix

Steps in a contingent claim replication I

Find a martingale measure Q under which the process Z = B

S for the present value of the underlying asset price is a martingale.

Construct the process M where

M

E

B

X jF

^{, 0 t T .}

Such a process is a martingale.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication

Example Multidimensional References Appendix

Steps in a contingent claim replication II

Using the martingale representation theorem, we know there exists a predictable process ψ such that

dM

= ψ

dZ

= ψ

S

dB

+ B

dS

= ψ

rB

S

dt + B

dS

.

Baxter and Rennie, pages 84-85.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication

Example Multidimensional References Appendix

Steps in a contingent claim replication III

So, let’s set

ψ

= the number of risky asset shares held at time t and φ

= the number of riskless asset shares held at time t where

φ

M

ψ

B

S

. The present value of such a strategy at time t is

B

V

= B

( φ

B

+ ψ

S

)

= φ

+ ψ

B

S

= M

ψ

B

S

+ ψ

B

S

= M

. Note that, at time T ,

V

= B

M

= B

E

B

X jF

= X .

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication

Example Multidimensional References Appendix

Steps in a contingent claim replication IV

Recall :

dM

= ψ

rB

S

dt + B

dS

φ

M

ψ

B

S

and V

= B

M

. Is such a strategy self-…nancing?

dV

= dB

M

= M

dB

+ B

dM

= φ

+ ψ

B

S

dB

+ B

ψ

rB

S

dt + B

dS

= φ

dB

+ ψ

B

S

dB

rψ

S

dt + ψ

dS

= φ

dB

+ rψ

B

S

B

dt r ψ

S

dt + ψ

dS

= φ

dB

+ ψ

dS

which implies that the strategy indeed is self-…nancing.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example I

Consider the classic case. An asset price is modeled using a geometric Brownian motion

dS

= µS

dt + σS

d W f

where W f is a ( _Ω , F ^, fF

^{: t 0} g ^{, P} ) Brownian motion.

The riskless interest rate r is set constant.

The riskless asset is therefore

B

= exp ( rt ) .

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example II

In a risk-neutral world, the asset price is dS

= rS

dt + σS

dW

where W is a ( _Ω , F ^, fF

^{: t 0} g ^, Q ) Brownian motion. Since the risk-neutral measure is unique, the market is complete and, as a consequence, all contingent claims C can be replicated.

Recall that a contingent claim is replicated if there exists a

self-…nancing trading strategy, the value of which at time

T is C .

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example III

The present value of a contingent claim is a Q martingale. Indeed, if

M

= E

[ β

C jF

] ,

then M

is the present value at time t of the contingent

claim C and f ^M

^{: t 0} g ^{is a Q} martingale.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example IV

Moreover, the present value of the risky asset is a Q martingale since

dB

S

= σB

S

dW

and

E

Z

0

σB

S

dt

=

Z

E

h

σB

S

i

dt (Fubini thm)

= σ

Z

0

E

S

exp σ

t + 2σW

dt

= S

σ

Z

0

exp σ

t dt = S

e

1 < ∞ .

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example V

Since B

S

: t 0 is a martingale, the di¤usion coe¢ cient of which, σB

S

, is always strictly positive, the representation theorem yields that there exists a predictable process ψ such that

dM

= ψ

dB

S

. Note that

dB

S

= σB

S

dW

implies that

dM

= ψ

σB

S

dW

. (1)

We now have the existence of a predictable process, but

we don’t know yet how to determine that process.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example VI

On the other hand, if we assume that the value of the contingent claim C can be expressed as a twice continuously di¤erentiable function of time and the risky asset price, i.e.

f ( t, S

) , then Itô’s lemma implies that df ( t, S

)

= ^∂f

∂t ( t, S

) dt + ^∂f

∂s ( t, S

) dS

+ ¹ 2

∂

f

∂s

( t, S

) d h ^S i

= ^∂f

∂t ( t, S

) dt + ^∂f

∂s ( t, S

) ( rS

dt + σS

dW

) + ¹

2

∂

f

∂s

( t , S

) σ

S

dt

= ^∂f

∂t ( t, S

) + rS

∂f

∂s ( t, S

) + ^σ

2

2 S

∂

f

∂s

( t, S

) dt + σS

∂f

∂s ( t, S

) dW

.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example VII

Using Itô’s lemma again (multiplication rule), we …nd the stochastic di¤erential equation for the present value of the contingent claim:

dM

= dB

f ( t, S

)

= f ( t, S

) dB

+ B

df ( t, S

)

= rf ( t, S

) B

dt + B

df ( t, S

)

=

∂t

( t, S

) + rS

∂f

∂s

( t, S

) +

2

2

S

f

∂s²

( t, S

) rf ( t, S

) B

dt +

S

∂s

( t, S

) dW

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example VIII

Recall that in equation (1), we had

dM

= ψ

σB

S

dW

. We now have

dM

=

∂t

( t, S

) + rS

∂f

∂s

( t, S

) +

2

2

S

f

∂s²

( t, S

) rf ( t, S

) B

dt +

S

∂s

( t, S

) dW

So the drift coe¢ cients and the di¤usion coe¢ cients in these two equations must be the same, i.e.

∂f

∂t ( t, S

) + rS

∂f

∂s ( t, S

) + ^σ

2

2 S

∂

f

∂s

( t, S

) rf ( t, S

) = 0 and

σB

S

∂f

∂s ( t, S

) = ψ

σB

S

.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example IX

By isolating ψ

in the equation σB

S

∂f

∂s ( t, S

) = ψ

σB

S

, we …nd

ψ

= ^∂f

∂s ( t, S

) .

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example X

We must therefore hold ψ

=

( t, S

) risky asset shares and

φ

= M

ψ

B

S

riskless asset shares.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example XI

Now assume that the contingent claim is a call option C = max ( S

K , 0 ) . We know that the value of such a contingent claim at time t is

f ( t, S

) = S

N ( d ( S

)) Ke

N d ( S

) σ p T t where N ( ) is the cumulative distribution function of the standard normal law and

d ( s ) =

ln s ln K + r +

( T t )

σ ( T t )

^.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example XII

Its present value is M

= B

f ( t, S

)

= e

S

N ( d ( S

)) Ke

N d ( S

) σ p

T t

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example XIII

But, the numbers of risky asset and riskless asset shares are respectively

ψ

= ^∂f

∂s ( t, S

)

= N ( d ( S

)) and φ

= M

ψ

B

S

= Ke

N d ( S

) σ p

T t

The derivatives are calculated in the Appendix 1.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example XIV

The value of the strategy at time t is V

= ψ

S

+ φ

B

= N ( d ( S

)) S

Ke

N d ( S

) σ p

T t .

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example

Multidimensional References Appendix

An example XV

It is possible to verify that the strategy does replicate the call option since, at time T ,

d ( S

) = lim

t!^T

ln S

ln K + r

( T t ) σ ( T t )

= 8 <

:

∞ if S

K

∞ if S

< K So, the value of our strategy is

V

=

t!T

N ( d ( S

)) S

Ke

N d ( S

) +

p T t

=

:

S

K

S

K

S

< K

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional

References Appendix

Martingale representation theorem I

Obviously, all of this exists in a multidimensional version!

Let W = W

, ..., W

be a Brownian motion in n dimensions constructed on a …ltered probability space ( Ω , F ^, fF

g ^{, Q} ) such that the …ltration fF

g ^{is the one} generated by the Brownian motion, plus it includes all zero-probability events, i.e. for all t 0,

F

= σ N ^{and W}

^{, ..., W}

^{: 0 s t .}

The components of such a Brownian motion are

independent from each other.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional

References Appendix

Martingale representation theorem II

Let Z

= ⁿ Z

: 0 t T o

be an Itô process such that

Z

=

Z

+

∑

Z

H

dW

where, for all i 2 f ^{1, ..., n} g ^{and j} 2 f ^{1, ..., n} g ^{, the} fF

g adapted process H

= f ^H

( t ) : 0 t T g satis…es the conditions

E

Z

0

H

ds < ∞ and

Q (

ω 2 Ω : 9 ^t 2 [ 0, T ] such that H

( ω ) H

( ω ) is singular

)

= 0.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional

References Appendix

Martingale representation theorem III

Theorem

The martingale representation theorem

For all i 2 f ^{1, ..., n} g ^{, Z}

^{is a} ( _Q , fF

g ) martingale.

Moreover, for any other ( Q , fF

g ) martingale M, there exist n predictable processes

φ

= ⁿ φ

: 0 t T o such that

M

= M

+

∑

Z

0

φ

dZ

.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional

References Appendix

Martingale representation theorem IV

Such a theorem will be useful when we must prove that a contingent claim is attainable in a market model

comprising multiple risky assets.

Baxter and Rennie, page 162

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional

References Appendix

In general I

X

, ...., X

represent the M risk factors in our model.

Some of these sources of risk may be tradable assets (stocks, bonds, etc.), some others are not (volatility, foreign exchange rate, interest rate, etc.). Assume their dynamics are described as follows:

dX

= µ

X

dt +

∑

σ

X

dW

where the K risk-neutral Brownian motions are independent,

µ

and σ

are predictable processes meeting certain

regularity conditions that ensure the existence of a solution.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional

References Appendix

In general II

There are N tradable assets, S

= f

( X

) , which are functions of risk factors. If the functions f

are su¢ ciently smooth, Itô’s lemma implies that

dS

=

∑

∂fn

∂x^(m)

(

) dX

+

∑

∑

∂²

f

∂x^(m)∂x^(j)

(

) d

X

X

t

=

∑

∂fn

∂x^(m)

(

)

X

dt +

∑

∂fn

∂x^(m)

(

)

∑

σ^(m,k)_t

X

dW

+

∑

∑

∂²

f

∂x^(m)∂x^(j)

(

)

∑

σ^(m,k_t ⁾σ^(j,k_t ⁾

X

X

dt.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional

References Appendix

In general III

Since S

is a tradable asset, its risk-neutral dynamics require the drift to satisfy

r

S

=

∑

∂fn

∂x^(m)

(

)

X

+

2

∑

∑

∂²

f

∂x^(m)∂x^(j)

(

)

∑

σ^(m,k_t ⁾σ^(j,k_t ⁾

X

X

As a consequence, dB

S

= B

∑

∑

∂f

∂x

( X

) σ

X

!

dW

. where B

= exp R

0

r

ds . If

E^Q 2 4^ZT

0

∑

∂fn

∂x^(m)

(

)

X

!2

ds

<

n

B

S

o

0 t T

is a Q martingale.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional

References Appendix

In general IV

The goal is to hedge a derivative product, the value of which at time t is g ( X

) . If g is su¢ ciently smooth, Itô’s lemma implies that

dB

g (

) = B

∑

∑

∂g

∂x^(m)

(

)

X

!

dW

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional

References Appendix

In general V

If the number of tradable assets is equal to the number of Brownian motions, N = K , and if

Q 8<

:9^t2[0,T] such that Ht

∑

∂fn

∂x^(m)(Xt)σ⁽t^m,k)X_t^(m)

!

n,k

is singular 9=

;=0,

then the martingale representation theorem implies the existence of predictable processes ϕ

such that

dB

g (

)

=

∑

ϕ⁽ⁿ_t ⁾

dB

S

= B

∑

ϕ⁽ⁿ⁾_t

∑

∑

∂fn

∂x^(m)

(

)

X

!

dW

= B

∑

∑

∑

ϕ⁽ⁿ⁾_t ∂fn

∂x^(m)

(

)

!

σ^(m,k_t ⁾

X

!

dW

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional

References Appendix

In general VI

We obtain two equations for dB

g ( X

) . By setting equal the processes in front of each Brownian, we …nd

∑

∂g

∂x^(m)

(

)

X

=

∑

ϕ⁽ⁿ⁾_t

∑

∂fn

∂x^(m)

(

)

X

! ,

which is equivalent to

∑

"

∂g

∂x

( X

)

∑

ϕ

∂f

∂x

( X

)

#

σ

X

= 0. (2) When x

= f

, the problem reduces to

∑

∂g

∂x

( X

) ϕ

( X

) σ

X

= 0 and we conclude that ϕ

=

n

( X

) .

More generally, the system of linear equations (2) needs to be

solved.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

References

Martin Baxter and Andrew Rennie (1996). Financial Calculus, an introduction to derivative pricing, Cambridge university press.

Damien Lamberton and Bernard Lapeyre (1991).

Introduction au calcul stochastique appliqué à la …nance,

Ellipses.

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Annexe I

In this appendix, we show the following:

ψ

= ^∂f

∂s ( t, S

)

= N 0

@ ln S

ln K + r +

( T t ) σ ( T t )

1 A

and

φ

= M

ψ

B

S

= Ke

N d ( S

) σ p

T t

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Annexe II

Recall that

f ( t, S

) = S

N ( d ( S

)) Ke

N d ( S

) σ p T t where

d ( s ) = ^ln ( s ) ln ( F

) + r +

σ

( T t ) σ p

T t .

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Annexe III

As a consequence

∂f

∂s

( t, s )

= N ( d ( s )) + s

∂s

N ( d ( s )) Ke

∂s

N d ( s )

p T t

= N ( d ( s )) + sn ( d ( s ))

∂s

d ( s ) Ke

n d ( s )

p

T t

∂s

d ( s )

= N ( d ( s )) +

sn ( d ( s )) Ke

n d ( s )

p

T t

∂s

d ( s )

where n ( ) is the probability density function of a standard

normal random variable. We will show that the term inside [ ]

is nil

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Annexe IV

Indeed,

n d ( s )

p T t

= p

2πexp 1

2

d ( s )

p T t

= p

2πexp 1

2

( d ( s ))

+

p

T td ( s )

2σ²

( T t )

= p

1

2

( d ( s ))

+

( s )

( K ) + r +

( T t )

1

2σ²

( T t )

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Annexe V

which implies that

Ke

n d ( s ) σ p T t

= p ¹

2π exp 1

2 ( d ( s ))

+ ln ( s )

= p ^s

2π exp 1

2 ( d ( s ))

= sn ( d ( s )) .

th.

Self-…nancing strategies Martingale representation th.

Steps in a contingent claim replication Example Multidimensional References Appendix

Annexe VI

Finally, φ

= M

ψ

B

S

= B

S

N ( d ( S

)) B

Ke

N d ( S

) σ p T t N ( d ( S

)) B

S

= Ke

N d ( S

) σ p

T t