7th Day Lecture AIMS 2012 4x Printing

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Introduction to Mathematical Finance:Part I: Discrete-Time Models

AIMS and Stellenbosch UniversityApril-May 2012

1 / 40 R. Ghomrasni Last updated: 9-5-2012

The Multi-Period Market Model: Dynamic Markets

One period: Staticcase, trades can be made only at current timet= 0. Revenues (i.e., profits or losses) can be realized only at thelater dateT(=1), while trades at intermediate times are not

allowed(i.e. you buy and hold), Multi-Period: Dynamic Markets. You may rebalance at the

beginning of each period


The Multi-Period Market Model

Let

t T ={0, 1, , T} representT+ 1 trading dates withT {2, 3, },

States belong to afinite sample space with K elementsK {2, 3, }:

= {1, 2, , K} F= P() =2

F= {Ft}{0,1, ,T} and F0 Ft FT Probability P with P()>0 for all

We have F0 = {,} and FT= F= P() =2

Filtered probability space(,F,F={Ft}{0,1, ,T},P)


Multi-Period Market Model

We consider a market model in which

(d+1) financial assets are priced at timest T ={0, 1, , T},withd {1, 2, }

The price of theith asset at timetis modelled as a non-negativerandom variable Sit on(,F,P)

The random vectorSt= (S0t,S

1t, , S

dt) is assumedmeasurable

w.r.t to thefiltrationFt, i.e. S0,S1, , Sd areadapted to thefiltrationF ={Ft}{0,1, ,T}.

The asset indexed by 0 is the riskless asset and we setS00 = 1 andS0t >0 for t T ={0, 1, , T}.

Ifthe return of the riskless asset over one period is constant andequal tor, we will obtain

S0t = (1+r)t



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Multi-Period Market Model

Often it is assumed that the 0th asset plays the role of alocallyriskless bond. In this case, one takesS00 1 and one letsS

0t evolve

according to aspot rate rt 0: At time t, an investmentx madeat time t 1 yields the payoffx(1+rt). Thus, a unit investment attime 0 produces the value

S0t =

tk=1

(1+rk)

at time t. An investment inS0 islocally riskless if the spot ratertis known before hand at timet 1. This idea can be madeprecise by assuming that the processr ispredictable:

rk Fk1, for k= 1, 2, , T


Strategies:

At time t= 0, we construct a portfolio0 1 with a costV0() = 0 S0 = 1 S0 =

di=0

i0S

i0 =d

i=0 i1S

i0, in other

words, investors initial wealth is equal to

V0() =d

i=0i1S

i0 .

(01, 11, ,

d1 ) F0 i.e. are constant.

When at time t= 1, the new stocks value are revealed, our portfoliovalue becomesV1() = 1 S1 =

di=0

i1S

i1.

With this wealth, we construct a new portfolio2 for the period(1, 2], i.e. V1+() =

di=0

i2S

i1, This portfolio is hence hold till

t= 2, when the stocks becomesS2.

at time t= 2 we have V2() =d

i=0 i2S

i2 and so on...


Definition (Portfolio Modelling: Strategies)A trading strategy or portfolio is defined as a stochastic process

= ((0t, 1t, ,

dt))0tT

in R d+1, where itdenotes the number of shares of assetiheld (in theportfolio) att-th trading period (t 1, t] (i.e. between t 1 and t).Exceptionally,0 = (

00,

10, ,

d0 ) is hold only at time t= 0.

The sequence is assumed to be predictable, i.e.

i {0, 1, , d}

i0 is F0-measurable,

and, for t 1, it isFt1-measurable.

RemarkIt is not always sensible to define the value that a predictableprocess has a time zero: Concretely, we let 0 1.


Strategies: Why Predictable?

The predictability of = (0t, 1t, ,

dt)0tTexpresses the fact that

investments must be allocated at the beginning of each trading period,without anticipating future price increments. In other words, the

predictable assumption means that the positions in the portfolio attime t, namely(0t,

1t, ,

dt) are decided with respect to the

information available at time t 1 and kept until time t, when newquotations are available.



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Strategies

Let us summarize,

Definition (Trading Strategy)A trading strategy is a predictable R d+1-valued process= ((0t,

1t, ,

dt))0tT with

0 1.

Remarkit represents the number of shares of the asset i that at held at time t.We do not place any restrictions on the signof it, which means thatshort sales and borrowing are allowed.

If0t 0, i= 1, nwhere ciis a constant (does not depend on time tand ). This strategyensures a gain when stock price(s) is increasing.


Definition (Wealth Process)The value of the portfolio or trading strategy at time tis the scalarproduct

Vt() := t St=d

i=0

itSit.

The value processV() = {Vt(), t= 0, 1, , T}is also called thewealth process.

RemarkThe r.v. Vt() isFt-measurable. (why?)

Definition (Gain Process)The gain process G() of a trading strategy is given by

Gt() =t

i=1

i (Si Si1) =t

i=1

i Si, t= 1, 2, , T


Self-Financing Strategy: Balancing condition

Definition (Self-Financing Strategy)A trading strategy is calledself-financing if the following equationknown as the Balancing condition is satisfied

t St= t+1 St for all t {0, 1, , T 1}

The interpretation is the following: at time t, once the new pricesS0t , , S

dt are quoted, the inve stor readjusts his/her positions fromtto

t+1 withoutbringing or consuming any wealth.



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Remark (Self-Financing Strategy)The equality

t St= t+1 St for t= 0, 1, , T 1

is equivalent to

t+1 (St+1 St) = t+1 St+1 t St,

or toVt+1() Vt() = t+1 (St+1 St)

At time t+1, the portfolio is worth t+1 St+1 andt+1 (St+1 St) = t+1 St+1 t+1 St is thenet gain caused bythe price changes between times t and t+1.


Discounted Value

Using the 0th asset as anumeraire, we form the discounted priceprocesses

Sit:= SitS0t =tSit, t= 0, , T, i= 0, , dwhere the coefficientt=

1S0t

is interpreted as the discount factor (from

timetto time 0): if an amount tis invested in the riskless asset at timet= 0, then one dollar will be available at timet.

RemarkTheeconomic reason for working with the discounted price processes isthat one should distinguish between one unit of a currency (e.g. $) attime t=0 and one unit at time t= 1, 2, , T. Usually people tend toprefer a certain amount today over the same amount which is promisedto be paid at a later time: this effect is referred to asthe time value ofmoney.


Definition (Discounted Wealth Process)The process S:= 1

S0S=

1,

S1

S0, ,

Sd

S0

is calleddiscounted price process. The discounted wealth process is

V() := SWe have

Vt() = Vt()S0t

=d

i=0

itSitS0t

=V0() +d

i=1

it Si


Definition (Discounted Gain)The discounted gain is defined by

Gt() := di=1

it Si NB in generalGt()= Gt()

S0t

RemarkIf two portfolios and differ only in their bank accounts, i.e. if theytake identical positions in the risky securities,= ((0t,

1t, ,

dt))0tT and = ((

0t,

1t, ,

dt))0tT. Then,

they have the same discounted gain

Gt() =Gt(), 0 t T.



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PropositionThe following areequivalent:

(i) The strategy isself-financing

(ii) For any t {1, , T},

Vt() = V0() +t

j=1

j Sj = V0() +t

j=1

di=0

ij Sij

= V0() +

tj=1

di=0

ij (S

ij S

ij1)

whereSjis the vector Sj Sj1.

(iii) For any t {1, , T},

Vt() = V0() + tj=1

j SjwhereSjis the vectorSj Sj1 = jSj j1Sj1.


Definition (Arbitrage Opportunity)An arbitrage opportunity is aself-financing strategy such that

1. V0() = 0;

2. VT() 0;

3. P(VT())>0)> 0.

The investor has no capital initially, and at timeThe/she ends up with

non-negative wealth, which is positive on a set of positive measure(and so E [VT()]>0).

Remark

V0() = 0

VT()() 0

: VT()()> 0. (or equivalently,E [VT()]> 0).


Remark (Arbitrage Opportunities)An arbitrage opportunity is astrategy such that

1. is self-financing (do not forget me!) i.e.t St= t+1 St for allt {0, 1, , T 1}

2. V0() = 0;

3. VT() 0;

4. P(VT())>0)> 0.

Definition (arbitrage-free or viable)We say that a security market isarbitrage-free or viable if there are noarbitrage opportunities in the class of self-financing strategies.

RemarkThis is a global No-Arbitrage Condition: only conditions on theinitial and final time are expressed.


Characterization of Self-Financing Strategies

PropositionFor anyR d-valued predictable process((1t, ,

dt))0tT and any

F0-measurable r.v. V0, there exists aunique predictable process(0t)0tTs.t. the trading strategy = (

0, 1, , d) isself-financingwith initial value of the corresponding portfolioV0() = V0.

Proof. If is self-financing, then

Vt() = V0+ tk=1

k Sk= V0+

tk=1

(1k S1k+ + dk Sdk)

On the other hand,

Vt() = t St= 0t+ 1tS1t + + dtSdt20 / 40 R. Ghomrasni Last updated: 9-5-2012


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Equate these:

0t =V0+t

k=1

(1k S1k+ + dk Sdk)(1t

S1t + + dtSdt)which defines0tuniquely. We just have to check that

0 is predictable,

but this is obvious if we consider the equation

0t =V0+t1k=1

(1k S1k+ + dk Sdk)(1tS

1t1+ +

dtS

dt1)


Characterization of Self-Financing Strategies

RemarkThus if an investor follows aself-financing strategy, the discountedvalue of his/her portfolio, and hence its value, are completely defined bythe initial wealth and the strategy((1t, ,

dt))0tT(this is only

justified becauseS0k =0).This also imply, thanks to the existence of a riskless asset, we can alwaystransform anyself-financing strategy/portfolio to auniqueself-financing strategy/portfolio which takes thesame positions inrisky securitiesas does, but which has the bank account readjusted sothat V0() = 0.


Characterization of Self-Financing Strategies: Case V0 = 0

Remark (Case V0 = 0)In particular, to any predictable process:= (1t, ,

dt)0tT, we have

a discounted gain given by

Gt() :=d

i=1 it Si

for which we can associate the discounted value of theuniqueself-financing strategy with zero initial valueby adjoining asuitableprocess(0t)0tTof holdings in the riskless asset S

0 such that

Gt() =Vt(), andV0() = V0() = 0where := (0t,

1t, ,

dt)0tT


Proposition (Numeraire Invariance Principle)is a self-financing for V() in S0,S1, , Sd iff is a self-financing

forV() inS0, S1, , SdProof. Remember a trading strategy is self-financing if

d

i=0 itS

it= t St= t+1 St=

d

i=0 it+1S

it

SinceS0t >0 for all t= 0, 1, , T, we have

di=0

itSitS0t

=tStS0t

=t St= t+1 StS0t

=t+1 St= di=0

it+1SitS0t



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Martingale

Let(,F,P, (Ft)t=0,1, ,T) a a filtered (finite) probability space

Definition (Martingale)A stochastic processX= (Xt)t=0,1, ,T is a P-martingalew.r.t to thefiltration F = {(Ft)t=0,1, ,T} if

E [Xt|F

s] = X

s for all 0 s t T.

Remark

Since for s= t, we haveE [Xt|Ft] = Xt, (Xt)t=0,1, ,T isadapted.

for0 s< t T , Xs is thebest predictionof Xt givenFs The sum of two martingales is a martingale (useful property).


Martingale

Definition (Martingale)Let(,F,P, (Ft)t=0,1, ,T) a a filtered (finite) probability space. Astochastic processX= (Xt)t=0,1, ,T is a P-martingale w.r.t to thefiltration F = {(Ft)t=0,1, ,T} if

1. X is F-adapted, i.e. Xt Ft

2. for all s, t= 0, 1, , Twith 0 s< t Twe have

E [Xt|Fs] = Xs


Martingale: Fair Game

Remark (Fair Game)The notion of a martingale is, intuitively, aprobabilistic model of a fairgame: the r.v. Xtof the process X can be interpreted as the fortune attime t of a gambler playing a game. In this context a martingale can be

thought of as a fair game (a game which is neither in your favour noryour opponents), in the sense that the gamblers future fortune isexpected to be the same as the gamblers current fortune, given all thepast and present information on the game.

E [Xt Xs|Fs] = 0 for all 0 s t T.


Martingale: Fair Game

RemarkThefundamental property of a martingale process is that its futurevariations are completely unpredictable given the current informationFt.In other words, the directions of the future movements inmartingales are impossible to forecast

u> 0, E [Xu+t Xt|Ft] = E [Xu+t|Ft] Xt= 0

In particular, for all0 s t T

tE [Xt] = E [Xs] = E [X0](=X0)

i.e. itsexpectation function is constant.



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In our finite market model ,={1, , K} and P({i})>0 for

i= 1, , K withK

i=1 P({i}) = 1.

Definition (Equivalent Probability)A probability measure P on (,F) is equivalent to P if and only if

P({})>0 for all


Definition ((Equivalent) Martingales Measures)A probability measure Q on(,FT) is called amartingale measureif

the discounted price processS= (S1, , Sd) = (S1S0, , S

d

S0) is a

d-dimensional Q -martingale, i.e.: fori= 1, 2 , dwe have

Sis= E

Q [

Sit|Fs] 0 s t T.

A martingale measureP is called anequivalent martingale measure(EMM)if it is equivalent to the original measure P on FT= P() =F.The set of all equivalent martingales measures is denoted byP

Remark (risk-neutral probability measure)In particular, we haveE Q [

Sit+1Sit

|Ft] = S0t+1S0t

. This states that the expected

return on each asset equals the risk-free return under an EMM. Thus theother name risk-neutral probability measure.


Remark (Convexity of (Equivalent) Martingales Measures Set)The set of all equivalent martingales measures denoted byPis aconvex set since ifQ i P,i= 1, 2 then

Q

=Q

1

+ (1 )Q2

P [0, 1] .

In particular, if you have2 different EMM, then you haveinfinite ofthem!


Proposition (EMM and Martingales)We suppose that thereexists a Equivalent Martingale Measure, inother words, a probability measureQ that is equivalent toP, and such

that the vectorS:= 1S0

S=

1, S1

S0, , S

d

S0

is a martingale underQ .

Then, thediscounted wealth process ( Vt())t=0, ,T is a martingaleunderQ (relative to the filtrationF = {F0, , FT})

Proof. For0 s t T,

EQ [Vt()|Ft1] = t EQ [St|Ft1]= t St1 =Vt1() by the self-financing prop

It follows,

EQ [Vt()|Fs] =Vs()



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Arbitrage and Martingales

Proposition (Arbitrage and Martingales)We suppose that thereexists a Equivalent Martingale Measure, inother words, a probability measureQ that is equivalent toP, and such

that the vectorS:= 1S0

S=

1, S1

S0, , S

d

S0

is a martingale underQ .

Then, there areno arbitrage opportunities

Proof. Since ( Vt())t=0, ,T is a martingale under Q for anyself-financing strategy , we have the properties of the conditionalexpectation, and the fact thatV0() = V0()

E Q [VT()] = E Q [V0()] = E Q [V0()]If the strategy has zero initial value, we have E Q [VT()] = 0, withVT() 0. HenceVT() = 0 and cannot be an arbitrageopportunity.


Harrison-Pliska Theorem: Finite

The 1st fundamental theorem of asset pricing, which relates theabsence of arbitrage opportunities to the existence of equivalentmartingale measures.

Theorem (Harrison-Pliska Theorem (1981))The market model is arbitrage-free if and only if the setPof allequivalent martingale measures is non-empty.

No Arbitrage Opportunities P =

Remark

Finite

The= is not easy to proof.....(constructive proof)


Important Consequence of the 1st Fundamental Theorem

Corollary (Law of One Price)In an arbitrage-free market, let1 an d2 be two self-financingstrategies such that they have the same terminal valuei.e.VT(1) = VT(2) a.s.. Then

Vt(1) = Vt(2) a.s. for t= 0, 1, , T

Proof. By assumption we have arbitrage-free market, therefore thereexists an EMM P. We conclude by using the MG property:

Vt(1) = E P[VT(1)|Ft] = E P [VT(2)|Ft] =Vt(2)


Definition (European contingent claim)A European contingent claim is represented by anFT-measurable r.v.X. A European contingent claimX is called aderivative of theunderlying assets S0, S1, , Sd ifX is measurable w.r.t filtrationgenerated by the price processS= (S0, S1, , Sd). This is interpretedto mean that the value (or payoff) of the contingent claim at theexpiration time Tis given by X. On the other hand,a look-back

option usually depends on the recent history of a risky asset.

Example (Asian option)Asian option:

X:= max{S10 + S

11 + +S

1T

1+T K, 0}



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American contingent claim: to be studied later on

Maturities prior to the final trading periodTof our model are alsopossible. We will meet later on another class of derivatives securities, theso-calledAmerican contingent claims or Bermudian option.

RemarkMore than 90% of options traded on exchanges are of American type.


We will frequently refer to the r.v. X, which represents a Europeancontingent claim, as the European contingent claim (or ECC for short).For a ECCX, we let

X= XS0T

, the discounted value of X

Definition (Replicating Strategy for a ECC)A replicating (or hedging) strategy for a European contingent claimX

is aself-financing trading strategy such that

VT() = X.

If there exists such a replicating strategy, the European contingent claimis said to be attainable

DefinitionThe finite multi-period market model is said to becomplete ifallEuropean contingent claims are attainable.


Arbitrage and Martingales

Theorem (Valuation via Arbitrage)Suppose that the finite multi-periodmarket model is viable and X is areplicableEuropean contingent claim. Then the value process{Vt(), t= 0, 1, , T} is the same for all replicating strategies for X.Indeed, forany replicating strategy andany equivalent martingalemeasure(EMM)P, we have

Vt() = E P [X|Ft] = E P [XS0T

|Ft] , t= 0, 1, , T ()

The right member of() defines the same stochastic process for all suchEMMP.

Proof. By the 1st fundamental theorem of asset pricing, there is atleast one EMM. Let P be such a measure. Let be a replicatingstrategy for X


Then, by the martingale property ofV() under P and sinceVT() =X, we have fort= 0, 1, , T,Vt() = E P[VT()|Ft]

= E P

[X|Ft]Since the last expression above does not depend upon, it follows that{

Vt(), t= 0, 1, , T} and hence{Vt(), t= 0, 1, , T} does not

vary with the particular choice of replicating strategy. Furthermore,

since the left member ofVt() = E P [VT()|Ft]does not depend upon the particular choice of an EMMP, it followsthat a stochastic process defined by

E P

[X|Ft] , t= 0, 1, , Tdoes not depend on the particular choice ofP.


7th Day Lecture AIMS 2012 4x Printing

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