80-646-08StochasticCalculusIGeneviŁveGauthier Discrete-TimeMarketModels

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Introduction Notation Prop. 2.6 Lemma, p.228 Theorem 2.7 Cor. p. 228 Prop. 2.8 Prop. 2.9 Pricing QMeasures Complete market References Appendix

Discrete-Time Market Models

80-646-08 Stochastic Calculus I

Geneviève Gauthier

HEC Montréal

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Introduction

Notation Prop. 2.6 Lemma, p.228 Theorem 2.7 Cor. p. 228 Prop. 2.8 Prop. 2.9 Pricing QMeasures Complete market References Appendix

Introduction I

We will study in depth Section 2: ”The Finite Theory” in the article Martingales, Stochastic Integrals and

Continuous Trading by J. M. Harrison and S. R. Pliska.

The market model here is rather general. Indeed, a …nite number (which may however be enormous) of securities are modelled during a …nite number (which, again, may however be very large) of time periods.

The only restriction on the distributions of the unit prices

of the securities at each time is that such distributions

must be discrete and positive, i.e. for each security and at

each time, the unit price of the security at that time can

only take a …nite number of strictly positive values.

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Introduction

Introduction II

The …rst important result in that Section is Proposal 2.6 on page 227.

It is proven that, if there exists a probability measure under which all discounted price processes are martingales, then we can build, based on such a measure, a price system for attainable contingent claims that is consistent with the market model.

On the other hand, if there exists such a price system, then we can build, based on the latter, a probability measure that transforms our discounted price processes into martingales.

Such a proposal thus establishes a bijective correspondence

between the set of martingale measures and the consistent

price systems.

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Introduction

Introduction III

It should be noted that such a proposal is silent on

whether at least one martingale measure exists, or whether

at least one price system exists. It only states that, if

either one exists, then both exist, and it makes the link

between them explicit.

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Introduction

Introduction IV

Theorem 2.7 on page 228 sets the necessary and su¢ cient condition for at least one martingale measure, and as a consequence, at least one consistent price system, to exist.

That condition is that the market model contains no arbitrage opportunities.

Since there may be several price systems that are consistent with the market, we need to ensure that, whatever the price system that is used, the price

associated with any given attainable conditional claim X is

always the same, i.e. if π

1

and π

2

are two consistent price

systems, then for any attainable contingent claim X ,

π

1

( X ) = π

2

( X ) . Such a result is provided by a corollary

on page 228.

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Introduction

Introduction V

How can it be veri…ed that our market model contains no arbitrage opportunities? A necessary and su¢ cient condition for no arbitrage is established by a lemma on page 228.

Proposal 2.8, on page 230, proves that the discounted market value of an admissible strategy is a martingale under all martingale measures in a market model. Such a result is used in proving Proposal 2.9 (page 230), which shows how to determine the market value of an attainable contingent claim at any time.

The last part addresses the notion of market completeness.

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Introduction Notation

Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function

Prop. 2.6 Lemma, p.228 Theorem 2.7 Cor. p. 228 Prop. 2.8 Prop. 2.9 Pricing QMeasures Complete market References Appendix

The probability space

We are working with a …ltered probability space ( Ω , F ^, ^F ^, ^P ) .

The sample space Ω has a …nite number of elements, each of which is interpreted as a possible state of the world. We assume that the market participants agree on the fact that Ω represents all the possible states of the world.

Thus, since every state of the world ω 2 Ω is possible, the probability measure P applying on the measurable space ( Ω , F ) must be such that 8 ^ω 2 ^Ω ^, ^P ( ω ) > 0.

The measure P , that associates a probability to each state of the world, represents a particular investor’s vision, i.e.

two investors may associate di¤erent probabilities to the

same state of the world ω.

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The …ltration

Information structure

We choose to observe the system until a time horizon T , which is a terminal date for all economic activity under consideration.

Since time is considered discretely, the …ltration F = fF

^t

^: ^t 2 f ^0, ^{1, ...,} ^T gg represents information available at each time.

We set F

⁰

= f ^∅ ^, ^Ω g ^, the sigma-algebra that contains no information, and F

^T

= the set of all events in Ω , i.e. F

^T

allows us to distinguish each possible state of the world.

Since Card ( Ω ) < ∞ , for all t 2 f ^0, ^{1, ...,} ^T g , there exists a …nite partition

P

^t

= A

^t₁

, ..., A

^t_n_t

that generates the sub-sigma-algebra F

^t

^.

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The securities I

We model K + 1 securities.

De…nition

The multidimensional stochastic process

! _S

= ⁿ ! _S

t

: t 2 f ^0, ^{1, ...,} ^T g ^o

representing the evolution of the individual unit prices of all securities is made up of the random column vectors

! _S

t

= S

_t⁰

, S

_t¹

, ...S

_t^K ⁰

where

S

_t^k

= the unit price of the security k at time t.

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The securities II

Each of the components S

^k

= S

_t^k

: t 2 f ^0, ^{1, ...,} ^T g ^of the price process ! _S is a stochastic process adapted to the

…ltration F and it takes strictly positive values.

Since Card ( Ω ) < ∞ , we have that 8 ^t 2 f ^0, ^{1, ...,} ^T g ^and

8 ^k 2 f ^0, ^{1, ...,} ^K g ^, ^S

t^k

discretely distributed random

variable, i.e. it can only take a …nite number of values.

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The securities III

De…nition

The 0 th security plays a part a bit peculiar, in that it is a riskless security (we can think, for example, of a bank account or a bond). It implies that

8 ^ω 2 ^Ω ^and 8 ^t 2 f ^0, ^{1, ...,} ^T ¹ g ^, ^S

t⁰

( ω ) S

_t⁰₊₁

( ω ) . It can also be assumed, without loss of generality, that

8 ^ω 2 ^Ω ^, ^S

0⁰

( ω ) = 1.

The unidimensional stochastic process β

_t

= ¹

S

_t⁰

will be used as discount factor.

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The securities IV

The …ltration F describes how information is revealed to the investors at each time.

The fact that F

0

= f

^∅,^Ω

g implies that the components of !

_S

0

are constants, i.e. at time

t

= 0, all security prices are known with certainty.

Since F

t

is generated by the …nite partition

P

t

=

A^t₁, ...,A^t_n_t ,

then, at time

t

, every investor knows with certainty which of the cells of P

t

has occurred, but the investor is not able to distinguish between the elements in that cell.

Since F

T

is the sigma-algebra made up of all possible events of

Ω

, investors can, at time

T

, determine with certainty which state of the world

ω

has occurred.

This means that we group under a single name, say

ωi,

all

the states of the world that have the same e¤ect on the

market. This is one the reasons why it is warranted to

restrict the sample space to a …nite set (

Card

(

_Ω

) <

_∞

) .

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Trading strategy I

De…nition

A trading strategy is a predictable vector stochastic process

! φ = ⁿ !

φ

t

: t 2 f ^{1, ...,} ^T g ^o where the random row vector !

φ

t

= φ

⁰_t

, φ

¹_t

, ...φ

^K_t

represents the investor’s portfolio at time t :

φ

^k_t

= the number of shares of security k held at time t.

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Trading strategy II

By requiring that the strategy be a predictable process, we are allowing the investor to select his time t portfolio right after the share prices at time t 1 are announced. As a result, the random variable φ

^k_t

represents the number of shares of security k held during the time period ( t 1, t ] (except for φ

^k₁

that represents the number of shares of security k held during the time period [ 0, 1 ] ).

The market value of portfolio !

φ

t

, right after the security prices at time t 1 are available, is !

φ

t

! _S

t 1

, whereas the market value of the same portfolio, but at time t, is

! φ

t

! _S

t

.

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Trading strategy III

De…nition

We say a strategy is self-…nancing if no funds are added to or withdrawn from the value of the portfolio after time t = 0, i.e.

8 ^t 2 f ^{1, ...,} ^T ¹ g ^, ! φ

t

! _S

t

= ! φ

t+1

! _S

t

. So, !

φ

t

! _S

t

represents the amount we receive at time t + ε (ε represents a very small positive quantity) when the portfolio is liquidated !

φ

t

and ! φ

t+1

! _S

t

is the amount we must pay, again at time t + ε, to purchase portfolio !

φ

t+1

.

Note that there are no transaction costs.

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Trading strategy IV

The process V !

φ = ⁿ V

_t

!

φ : t 2 f ^0, ^{1, ...,} ^T g ^o represents the market value of the strategy at any time.

V

_t

! φ =

( !

φ

1

! _S

0

if t = 0

! φ

t

! _S

t

if t 2 f ^{1, ...,} ^T g ^.

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Trading strategy V

De…nition

A trading strategy is called admissible if it is self-…nancing and its market value is never negative.

So, an admissible strategy is such that an investor is never put into a situation of debt. This doesn’t mean however that short sales are prohibited.

Φ = the set of admissible strategies

= 8 >

> >

<

> >

> :

! φ

each of the components φ

^k

of !

φ is predictable,

! φ is self-…nancing and 8 ^t 2 f ^{0, ...,} ^T g ^, ^V

^t

!

φ 0

9 >

> >

=

> >

> ;

.

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Arbitrage opportunity I

De…nition

An admissible strategy !

φ is an arbitrage opportunity if V

0

!

φ = 0 and if E

^P

h

V

_T

! φ

i > 0.

Note that the latter condition implies that there exists ω 2 Ω for which V

T

!

φ , ω > 0.

Indeed,

E^Ph V_T

!

φ

i

= ∑

ω2Ω

V_T

!

φ,ω

| {z }

0

P

(

ω

)

| {z }

>0

So, in order to have

E^Ph V_T

!

φ

i

> 0, there must be at least one

ω

for which

V_T

!

φ,ω

> 0.

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Arbitrage opportunity II

So, the strategy !

φ is an arbitrage opportunity when, while investing nothing V

₀

!

φ = 0 , we are assured not to lose any money (V

_T

!

φ , ω 0 since ! φ is admissible) and we have a positive probability to make a gain ( 9 ^ω 2 Ω for which V

T

!

φ , ω > 0).

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Contingent claims I

De…nition

A contingent claim X is a ( Ω , F

^T

) non-negative random variable. It can be thought of as a contract between two parties, such that X ( ω ) will be paid by one party to the other if state ω pertains.

De…nition

X = the set of contingent claims

= X X is a ( Ω , F

T

) random variable

such that 8 ^ω 2 ^Ω ^, ^X ( ω ) 0 .

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Contingent claims II

De…nition

A contingent claim X is said to be attainable if there exists an admissible trading strategy !

φ that can replicate the cash ‡ow generated by X , i.e. V

_T

!

φ = X .

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Price systems I

De…nition

A price system π is a linear operator on X that returns non-negative values π : X ! [ 0, ∞ ) and that satis…es

π ( X ) = 0 , ^X = 0 and

8 ^a, ^b ^{0 and} 8 ^X

¹

^X

²

2 ^X, ^π ( aX

₁

+ bX

₂

) = aπ ( X

₁

) + bπ ( X

₂

) . As its name indicates, a price system is intended to

associate with each of the contingent claims, a price.

Since, for any admissible strategy !

φ , V

_T

! φ is a ( _Ω , F

^T

) random variable taking non-negative values, V

_T

!

φ 2 ^X.

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Price systems II

De…nition

A price system is said to be consistent with the market model if the price associated with the contingent claim V

_T

!

φ is its market value at time t = 0, V

₀

!

φ , i.e.

π V

_T

!

φ = V

₀

!

φ .

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Price systems III

De…nition

Π

= the set of price systems consistent with the market model

= 8>

>>

<

>>

>:

π:X![0,∞)

π(X) =0,^X =0 8^a,^b ^{0 and}8^X¹^,^X²2^X, π(aX₁+bX₂) =aπ(X₁) +bπ(X₂)

8!

φ 2^Φ,^π ^VT !

φ =V₀ ! φ

9>

>>

=

>>

>; .

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Risk-neutral measures I

We will build, if possible, a set of probability measures on ( Ω , F ) that we call equivalent martingale measures (also know as risk-neutral measures).

Such measures bear, a priori, no relationship to the actual probability that an event occurs, no more than they re‡ect the market knowledge of an investor.

They are nothing more than a very convenient calculation tool in pricing contingent claims.

De…nition

P = the set of martingale measures equivalent to P

= 8 <

: Q

Q is a probability measure on ( Ω , F ) , 8 ^ω 2 ^Ω ^, ^Q ( ω ) > 0 and

8 ^k 2 f ^0, ^{1, ...,} ^K g ^, ^βS

^k

^{is a} ^Q martingale.

9 =

;

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Indicator functions I

De…nition

Throughout this text, indicator functions will be denoted by I , i.e. 8 ^A 2 F ^, ^I

^A

^: ^Ω ! f ^0, ¹ g is de…ned by

I

A

( ω ) = ^{1 if} ^ω 2 ^A 0 if ω 2 / ^A ^.

To become familiar with such functions, the reader can verify that:

( i ) If A 2 F

^t

^then ^I

^A

^is F

^t

measurable.

( ii ) If A, B 2 F are disjoint, then I

A[^B

= I

A

+ I

B

.

( iii ) If Y is a random variable, then Y = _∑

_ω₂_Ω

Y ( ω ) _I

_ω

.

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Introduction Notation Prop. 2.6

Statement Interpretation Proof

Part I Part II

Lemma, p.228 Theorem 2.7 Cor. p. 228 Prop. 2.8 Prop. 2.9 Pricing QMeasures Complete market References Appendix

Statement

Proposal 2.6

Theorem

Proposal. There is bijective correspondence between the set Π of price systems that are consistent with the market model and the set P of martingale measures equivalent to P. Such as correspondence is

( i ) π ( X ) = E

^Q

[ β

_T

X ] , X 2 ^X

( ii ) Q ( A ) = π S

_T⁰

I

A

, A 2 F ^.

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Part I Part II

Interpretation I

Proposal 2.6

If we know a martingale measure Q equivalent to P, then we can build a consistent price system π by setting 8 ^X 2 ^X, ^π ( X ) = E

^Q

[ β

_T

X ] .

On the other hand, if a price system π consistent with the market model is available, then we can build a martingale measure Q equivalent to P by de…ning 8 ^A 2 F ^,

Q ( A ) = π S

_T⁰

I

A

.

Such a proposal tells us that, if there exists a martingale measure equivalent to P or if there exists a price system consistent with the market model, then both exist, and the proposal establishes the link between them.

However, nothing allows us to show that either of such

two objects exist.

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Part I Part II

Interpretation II

Proposal 2.6

The necessary and su¢ cient condition for a martingale

measure, and, as a consequence, for the price system to

exist, is addressed by Theorem 2.7.

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Part I Part II

De…nition

Proof

De…nition

Let P and Q be two probability measures that exist on the measurable space ( Ω , F ) . The measures P and Q are said to be equivalent if and only if the impossible events are the same under both measures, i.e.

8 ^A 2 F ^, ^P ( A ) = 0 , ^Q ( A ) = 0.

In our case, since 8 ^ω 2 Ω , P ( ω ) > 0, all measures Q equivalent to P shall satisfy the condition

8 ^ω 2 Ω , Q ( ω ) > 0.

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Part I Part II

First part I

Proof

To be shown. Let Q be a martingale measure equivalent to P. Then, the function π de…ned on the set of contingent claims X by π ( X ) = E

^Q

[ β

_T

X ] is a price system consistent with the market model.

Since Q 2 ^P ^then

1

Q is equivalent to P, i.e. 8 ^ω 2 Ω , Q ( ω ) > 0,

2

and the components of the discounted security prices,

β ! _S , are martingale under the measure Q.

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Part I Part II

First part II

Proof

We want to show that π is a consistent price system. We therefore need to verify that

(i) π ( X ) = 0 , ^X = 0, (ii) 8 ^a, ^b ^{0 and} 8 ^X

¹

^, ^X

²

2 ^X,

π ( aX

₁

+ bX

₂

) = aπ ( X

₁

) + bπ ( X

₂

) , (iii) 8 !

φ 2 Φ , π V

_T

!

φ = V

0

!

φ .

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Part I Part II

First part - Point (i)

Proof

Starting from the de…nition of π, we get π ( X ) = E

^Q

[ β

_T

X ] = ∑

ω2Ω

β

_T

( ω )

| {z }

>0

X ( ω )

| {z }

0

Q ( ω )

| {z }

>0

.

Note that β

_T

( ω ) > 0 since β

_T

=

_S¹0 T

and the market model assumes that security prices can only be strictly positive.

Therefore,

π ( X ) = 0 , ∑

ω2Ω

β

_T

( ω )

| {z }

>0

X ( ω )

| {z }

0

Q ( ω )

| {z }

>0

= 0

, 8 ^ω 2 Ω , X ( ω ) = 0.

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Part I Part II

First part - Point (ii)

Proof

8 ^a, ^b ^{0 and} 8 ^X

¹

^, ^X

²

2 ^X,

π ( aX

₁

+ bX

₂

) = E

^Q

[ β

_T

( aX

₁

+ bX

₂

)]

= aE

^Q

[ β

_T

X

₁

] + bE

^Q

[ β

_T

X

₂

]

= aπ ( X

₁

) + bπ ( X

₂

) .

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Part I Part II

First part - Point (iii) I

Proof

We want to show that 8 ! φ 2 ^Φ ^, π V

_T

!

φ = V

0

! φ .

But, from the de…nition of π, π V

_T

!

φ = E

^Q

h

β

_T

V

_T

! φ

i

, (1)

so we determine, in a …rst step, β

_T

V

_T

!

φ :

(36)

Part I Part II

First part - Point (iii) II

Proof

β_TV_T ! φ

= β_T! φT!_S

T from the de…nition ofVT, (ref: eq (2.4), p. 226).

= β_T! φT!_S

T+

T 1 i

∑

=1

β_i ! φi!_S

i !

φi+1!_S

i

| {z }

=0

since !

φ is self-…nancing.

= β_T!_φ

T!_S

T+

T 1 i

∑

=1

β_i!_φ

i!_S

i T 1

i

∑

=1

β_i!_φ

i+1!_S

i

= β_T!_φ

T!_S

T+

T 1 i

∑

=1

β_i!_φ

i!_S

i

∑

T i=2

β_i ₁!_φ

i!_S

i 1

= β_T! φT!_S

T+β₁! φ1!_S

1+

T 1 i

∑

=2

!φi β_i!_S

i β_i ₁!_S

i 1 β_T ₁!

φT!_S

T 1

= β₁! φ1!_S

1+

∑

T i=2

!φi β_i!_S

i β_i ₁!_S

i 1

(37)

Part I Part II

First part - Point (iii) III

Proof

Therefore, substituting the expression above into the equation (1),

π VT ! φ

= _E^Q^hβ_TVT !_φ ⁱ

= E^Q

"

β₁! φ1!_S

1+

∑

T i=2

!φi β_i!_S

i β_i ₁!_S

i 1

#

= E^Qh β₁!_φ

1!_S

1

i+

∑

T i=2

E^Qh!_φ

i β_i!_S

i β_i ₁!_S

i 1

i

= _E^Q^h_E^Q^hβ₁!_φ

1!_S

1jF⁰ⁱⁱ+

∑

T i=2

E^Qh E^Qh!_φ

i β_i!_S

i β_i ₁!_S

i 1 jFⁱ ¹ⁱⁱ

= E^Qh! φ1E^Qh

β₁!_S

1jF⁰ⁱⁱ+

∑

T i=2

E^Qh! φiE^Qh

β_i!_S

i β_i ₁!_S

i 1 jFⁱ ¹ⁱⁱ since!

φ is predictable,

(38)

Part I Part II

First part - Point (iii) IV

Proof

= _E^Q 2 66 66 4

!φ1E^Qh β₁!_S

1jF⁰ⁱ

| {z }

β₀! S₀

3 77 77 5

+

∑

T i=2

E^Q 2 66 66 64

!φiE^Qh β_i!_S

i β_i ₁!_S

i 1 jFⁱ ¹ⁱ

| {z }

=E^Qh β_i!_S

ijFi 1 i

β_i ₁!_S i 1=!₀

3 77 77 75

since the components ofβ!_Sare martingales under the measureQ,

= E^Qh! φ1β₀!_S

0

i

=β₀! φ1!_S

0

since β₀,! φ1 and!_S

0areF⁰ measurable, therefore constant,

= !_φ

1!_S

0 sinceβ₀= ¹ S₀⁰ =1

= V0 !

φ from the de…nition ofV0 (ref: eq (2.4), p. 226).

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Part I Part II

Proof I

Second part

To be shown. Let π be a price system consistent with the market model. Then, the function Q, de…ned on the set of events F ^{by Q} ( A ) = π S

_T⁰

I

A

martingale measure equivalent to P.

Since π 2 ∏ then

1

π ( X ) = 0 , ^X = 0;

2

8 ^a, ^b ^{0 and} 8 ^X

¹

^, ^X

²

2 ^X,

π ( aX

₁

+ bX

₂

) = aπ ( X

₁

) + bπ ( X

₂

) ;

3

8 !

φ 2 Φ , π V

_T

!

φ = V

₀

!

φ .

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Part I Part II

Proof II

Second part

We want to show that Q 2 ^P ^{, i.e.}

(i) Q is a probability measure;

(ii) Q is equivalent to P, i.e. 8 ^ω 2 ^Ω ^, ^Q ( ω ) > 0;

(iii) The components of the discount price process for the

securities β ! _S , are martingales under the measure Q.

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Part I Part II

Second part - Point (i) I

Proof

Let’s start by building a strategy that will be subsequently of use :

8 ^t 2 f ^{1, ...,} ^T g ^, ^let !

φ

t

= ( 1, 0, ..., 0 ) . This trading strategy is such that, at any time, we hold nothing more than a single share of the riskless security.

Note that at any time !

φt

is constant, i.e.

8

^ω

2

Ω,

!

φt

(

ω

) = ( 1, 0, ..., 0 )

.

A a consequence, !

φt

is F

0

measurable, therefore F

t 1

measurable, which implies that !

φ

is predictable.

It is also easy to show that !

φ

is self-…nancing since we

keep the same portfolio.

(42)

Part I Part II

Second part - Point (i) II

Proof

Now, V

_T

!

φ =

∑

K k=0

φ

^k_T

S

_T^k

from the de…nition of V ! φ

= S

_T⁰

from the de…nition of !

φ . (2) and

V

0

! φ =

∑

K k=0

φ

^k₁

S

₀^k

from the de…nition of V ! φ

= S

₀⁰

from the de…nition of ! φ .

= 1 since, by hypothesis, S

₀⁰

= 1. (3)

(43)

Part I Part II

Second part - Point (i) III

Proof

Let’s now show that Q ( Ω ) = 1.

Starting from the de…nition

Q,

Q ( Ω )

= π S

_T⁰

I

_Ω

= π S

_T⁰

= π V

_T

!

φ from Equality (2).

= V

₀

!

φ since π is consistent with the market model.

= 1 from Equality (3).

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Part I Part II

Second part - Point (i) IV

Proof

We must also ensure that for all disjoint events A

₁

, ..., A

n

2 F ^,

Q

[n

i=1

A

i

!

=

∑

n i=1

Q ( A

i

) .

(45)

Part I Part II

Second part - Point (i) V

Proof

But Q

[n

i=1

A

_i

!

= π S

_T⁰

I

^Sⁿ_i₌₁Ai

= π S

_T⁰

∑

n i=1

I

Ai

!

=

∑

n i=1

π S

_T⁰

I

Ai

from Property (2.5b), p. 226.

=

∑

n i=1

Q ( A

i

) from the very de…nition of Q.

(46)

Part I Part II

Second part - Point (ii)

Proof

We want to prove that 8 ^ω 2 Ω , Q ( ω ) > 0.

Let’s assume there exists at least one ω 2 Ω for which Q ( ω ) = 0 and let’s show this leads to a contradiction.

From the equivalence relation π ( X ) = 0 , ^X = 0, we have 0 = Q ( ω ) = π S

_T⁰

I

ω

, 8 ^ω 2 Ω , S

_T⁰

( ω ) I

ω

( ω ) = 0 ) ^S

T⁰

( ω ) = S

_T⁰

( ω ) _I

_ω

( ω ) = 0.

Since the prices are strictly positive, we have that 8 ^ω 2 Ω , S

_T⁰

( ω ) > 0. As a consequence

S

_T⁰

( ω ) I

ω

( ω ) = S

_T⁰

( ω ) > 0.

Contradiction! So, there cannot exist ω 2 Ω for which

Q ( ω ) = 0.

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Part I Part II

Second part - Point (iii) I

Proof

The objective is to show that each of the components in the vector of discounted security prices, β ! _S _{, is a}

Q martingale.

Let’s choose one of the components arbitrarily: let k 2 f ^0, ^{1, ...,} ^K g ^.

By the Optional Stopping Theorem, it will be su¢ cient to show that

E

^Q

h β

_τ

S

_τ^k

i

= E

^Q

h β

₀

S

₀^k

i for all ( Ω , F ^, ^F ) bounded stopping time τ ( 8 ^ω 2 ^Ω ^, ^τ ( ω ) T ).

So let’s choose arbitrarily a bounded stopping time τ.

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Part I Part II

Second part - Point (iii) II

Proof

Let’s consider the trading strategy de…ned as follows:

φ

^j_t

= 8 >

> <

> >

:

I

_ft τg

if j = k

S_τ^k

S_τ⁰

I

_ft>τg

= S

_τ^k

β

_τ

I

_ft>τg

if j = 0

0 if j 2 f ^{1, ...,} ^K g ^, ^j 6 = k .

Such a strategy consists in holding a share of security k

until the random time τ and then in exchanging it for a

random quantity S

_τ^k

β

_τ

of bonds (riskless security) until

maturity ( t = T ) .

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Part I Part II

Second part - Point (iii) III

Proof

Let’s verify that such as strategy is admissible, i.e. ! φ is a predictable process, that strategy is self-…nancing and its market value, V

_t

!

φ is non-negative at all times.

(50)

Part I Part II

Second part - Point (iii) IV

Proof

Let’s begin by verifying that !

φ is a predictable process : 8 ^t 2 f ^{1, ...,} ^T g ^,

1

φ

⁰_t

= S

_τ^k

β

_τ

I

_fτ<tg

= _∑

^t_i₌₀¹

S

_τ^k

β

_τ

I

_fτ=ig

=

∑

^ti=0¹

S |

_i^k

β

_i

{z I

_f_τ=ig

}

Fⁱ ^measurable

is F

^t ¹

measurable,

2

φ

^k_t

= _I

_f_τ _t_g

is F

^t ¹

measurable since f ^τ ^t g = f ^τ < t g

^c

=

0 B @

^S^ti=0¹

f _{| {z }} ^τ = i g

2Fi Ft 1

1 C A

c

2 F

^t ¹

^and

3

φ

^j_t

= 0 is F

⁰

measurable therefore F

^t ¹

measurable.

80-646-08StochasticCalculusIGeneviŁveGauthier Discrete-TimeMarketModels

Discrete-Time Market Models