Discrete Time Finance pot

Let hS1 be an option in our elementary single pe-riod market model, and let x, φ be a replicating strategy for hS1, then x is the only price for the option at time t = 0, which does not

Discrete Time Finance

Dr Christian-Oliver Ewald

School of Economics and Finance

University of St.Andrews

These are my Lecture Notes for a course in Discrete Time Financewhich I taught in the Winter term 2005 at the University of Leeds I amaware that the notes are not yet free of error and the manuscrip needsfurther improvement I am happy about any comment on the notes.Please send your comments via e-mail to ce16@st-andrews.ac.uk

1.1 The most elementary Market Model 3

1.2 A general single period market model 14

1.3 Single Period Consumption and Investment 37

1.4 Mean-Variance Analysis 52

1.5 Exercises 65

2 Multi period Market Models 67 2.1 General Model Specifications 67

2.2 Properties of the general multi period market model 77

2.3 The Binomial Asset Pricing Model 89

2.4 Optimal Portfolios in a Multi Period market Model 97

Chapter 1

Single Period Market Models

Single period market models are the most elementary market models.Only a single period is considered The beginning of the period is usu-ally denoted by the time t = 0 and the end of the period by time t = 1

At time t = 0 stock prices, bond prices,possibly prices of other financialassets or specific financial values are recorded and the financial agentcan choose his investment, often a portfolio of stocks and bond At time

t = 1 prices are recorded again and the financial agent obtains a payoffcorresponding to the value of his portfolio at time t = 1 Single periodmodels are unrealistic in a way, that in reality trading takes place overmany periods, but they allow us to illustrate and understand many

of the important economic and mathematical principles in FinancialMathematics without being mathematically to complex and challeng-ing We will later see, that more realistic multi period models can in-deed be obtained by the concatenation of many single period models.Single period models are therefore the building blocks of more compli-cated models In a way one can say :

Single period market models are the atoms of Financial

Mathematics.

Within this chapter, we assume that we have a finite sample space

Ω := {ω1, ω2, , ωk}

We think of the samples ωi as possible states of the world at time t = 1.The prices of the financial assets we are modeling in a single periodmodel, depend on the state of the world at time t = 1 and therefore

on the ωi’s The exact state of world at time t = 1 is unknown at time

t = 0 We can not foresee the future We assume however that we aregiven information about the probabilities of the various states Moreprecisely we assume that we have probability measure P on Ω with

P(ω) > 0 for all ω ∈ Ω This probability measure represents the beliefs

of the agent Different agents may have different beliefs and thereforedifferent P’s However in the following we choose one agent who is in away a representative agent

The most elementary but still interesting market model occurs when

we assume that Ω contains only two states We denote these two states

byω1 = H and ω2 = T We think of the state at time t = 1 as determined

by the toss of a coin, which can result in Head or Tail,

Ω = {H, T }

The result of the coin toss is not known at time t = 0 and is thereforeconsidered as random We do not assume that the coin is a fair coin,i.e that H and T have the same probability, but that there is a number

0 < p < 1 s.t

P(H) = p, P(T ) = 1− p

We consider a model, which consists of one stock and a money ket account If we speak of one stock, we actually mean one type ofstock, for example Coca Cola, and agents can buy or sell arbitrary many

mar-shares of this stock For the money market account we think of a ings account The money market account pays a deterministic ( nonrandom ) interest rate r > 0 This means that one pound invested intothe money market account at time t = 0 yields a return of 1 + r pounds

sav-at time t = 1 The price of the stock at time t = 0 is known and denoted

by S0 The price of the stock at time t = 1 depends on the state of theworld and can therefore take the two values S1(H) and S1(T ), depend-ing whether the coin toss results inH or T It is not known at time t = 0and therefore considered to be random S1 is a random variable, takingthe value S1(H) with probability p and the value S1(T ) with probability

We assume that 0 < d < 1 < u This means that the stock price caneither go up or down, but in any case remains positive The stock canthen be represented by the following diagram :

S0u

S0

p p88p p p p p p

1 −p

&&N N N N N N N

S0d

To complete our first market model we still need trading strategies.The agents in this model are allowed to invest in the money market ac-count and the stock We represent such an investment by a pair (x, φ)where x gives the total initial investment in pounds at time t = 0 and

φ denotes the numbers of shares bought at time t = 0 Given the vestment strategy (x, φ), the agent then invests the remaining money

in-x− φS0 in the money market account We assume that φ can take anypossible value, i.e φ ∈ R This allows for example short selling as well

as taking arbitrary high credits At the end of this section we will givesome remarks on the significance of these assumptions.

The value of the investment strategy (x, φ) at time t = 0 is clearly x,the initial investment The agent has to pay x pounds in order to buythe trading strategy (x, φ) Within the period, meaning between time

t = 0 and time t = 1 the agent does nothing but waiting until time t = 1.The value of the trading strategy at time t = 1 is given by its payoff.The payoff however depends on the value of the stock at time t = 1 and

is therefore random In fact it can take the two values :

Definition 1.1.1 The value process of the trading strategy (x, φ) in

our elementary market model is given by(V0(x, φ), V1(x, φ)) where V0(x, φ) =

x and V1 is the random variable

V1(x, φ) = (x− φS0)(1 + r) + φS1

An essential feature of an efficient market is that if a trading strategycan turn nothing into something, then it must also run the risk of loss

Definition 1.1.2 An arbitrage is a trading strategy that begins with

no money, has zero probability of losing money, and has a positive ability of making money.

prob-This definition is mathematically not precise It does not refer to thespecific model we are using, but it gives the basic idea of an arbitrage

in words A more mathematical definition is the following :

Definition 1.1.3 A trading strategy (x, φ) in our elementary market

model is called an arbitrage, if

1. x = V0(x, φ) = 0 (i.e the trading strategy needs no initial

invest-ment)

2. V1(x, φ) ≥ 0 (i.e there is no risk of losing money)

3 E(V1(x, φ)) = pV1(x, φ)(H) + (1 − p)V1(x, φ)(T ) > 0 (i.e a strictly

positive payoff is expected).

A mathematical model that admits arbitrage cannot be used for ysis Wealth can be generated from nothing in such a model Realmarkets sometimes exhibit arbitrage, but this is necessarily fleeting;

anal-as soon anal-as someone discovers it, trading takes actions that remove it

We say that a model is arbitrage free, if there is no arbitrage in the

model To rule out arbitrage in our elementary model we must assumethatd < 1 + r < u, otherwise we would have arbitrages in our model, aswee will see now :

If d ≥ (1 + r), then the following strategy would be an arbitrage :

• begin with zero wealth and at time zero borrow S0 from the moneymarket in order to buy one share of the stock

Even in the worst case of a tail on the coin toss, i.e S1 = S0d, the stock

at time one will be worth S0d ≥ S0(1 + r), enough to pay off the moneymarket debt and the stock has a positive probability of being worthstrictly more since u > d > 1 + r, i.e S0u > S0(1 + r)

If u ≤ 1 + r, then the following strategy is an arbitrage :

• sell one share of the stock short and invest the proceeds S0 in themoney market

Even in the best case for the stock, i.e S1 = S0u the cost S1 of replacing

it at time one will be less than or equal to the value S0(1 + r) of the

money market investment, and since d < u < 1 + r, there is a positiveprobability that the cost of replacing the stock will be strictly less thanthe value of the money market investment.

We have therefore shown :

Proposition 1.1.1 The elementary single period market model

dis-cussed above is arbitrage free, if and only if d < 1 + r < u.

Certainly, stock price movements are much more complicated than dicated by this elementary model We consider it for the following tworeasons:

in-1 Within this model, the concept of arbitrage pricing and its relation

to risk-neutral pricing can be clearly illuminated

2 A concatenation of many single period market models, gives a quiterealistic model, which is used in practice and provides a reasonablygood, computationally tractable approximation to continuous-timemodels

Let us now introduce another financial asset into our elementary ket model :

mar-Definition 1.1.4 A European call option is a contract which gives

its buyer the right ( but not the obligation ) to buy a good at a future time T for a price K The good, the maturity time T and the strike price

K are specified in the contract.

We will consider such European call options in all of our financial ket models, which we are going to discuss in this lecture European calloptions are frequently traded on financial markets A central questionwill always be:

mar-What price should such a European call option have ?

Within our elementary market model we do not have so many choices.First, we assume that the good is the stock, and second that the ma-turity time is T = 1, the end of the period This is the only nontrivialmaturity time The owner of a European call option can do the follow-ing:

• if the stock price S1 at time1 is higher than K, buy the stock at time

t = 1 for the price K from the seller of the option and immediatelysell it on the market for the market price S1, leading to a profit of

S1 − K

• if the stock price at time 1 is lower than K, then it doesn’t makesense to buy the stock for the price K from the seller, if the agentcan buy it for a cheaper price on the market In this case the agentcan also do simply nothing, leading to a payoff of 0

This argumentation shows, that a European call option is equivalent to

an asset which has a payoff at time T = 1 of

max(S1 − K, 0)

This payoff, is what the option is worth at time t = 1 Still the question

is, what is the option worth at time t = 0 ? We will answer this tion in the remaining part of this section, by applying the replication

ques-principle To do this, we consider more general an option, which is ofthe type h(S1) where h : R → R is a function Please note that since S1

is random, h(S1) is also random The European call option from above,

is then given by choosing the functionh as h(x) := max(x− K, 0) Thereare many other possible choices for h leading to different options Wewill discuss some of them in a later section The replication principlesays the following :

Replication principle : If it is possible to find a trading

strategy which perfectly replicates the option, meaning that the trading strategy guarantees exactly the same payoff as the option

at maturity time, then the price of this trading strategy must

coincide with the price of the option.

What would be, if the replication principle would not hold ? Assumethat the price of the option would be higher, than the price of a repli-cating strategy Then, with zero initial investment, one could sell theoption and buy the replicating strategy Since one earns more fromselling the option than paying for the replication strategy, one has apositive amount of money at hand at time t = 0 This money can then

be used to invest into the savings account ( or another riskless asset )

At maturity, one may have to pay for the obligation from the option, butthe replicating strategy which one owns will pay exactly for this obli-gation On the other hand, the money invested in the savings accounteven pays interest and one obtains a strictly positive payoff at matu-rity time of the option This is an arbitrage If the price of the optionwould instead be lower than the price of the replicating strategy, then asimilar strategy as above, where the option is bought and the replicat-ing strategy is sold short would lead to an arbitrage We therefore see,that under the assumption that there is no arbitrage in the market, theonly possible price for the option is the price of the replicating strategy.Let us formulate these ideas more mathematically in our elementarymodel

Definition 1.1.5 A replicating strategy or hedge for the optionh(S1)

in our elementary single period market model is a trading strategy(x, φ)

which satisfies V1(x, φ) = h(S1) which is equivalent to

(x− φS0)(1 + r) + φS1(T ) = h(S1(T )) (1.2)The following proposition follows from the argumentation above :

Proposition 1.1.2 Let h(S1) be an option in our elementary single

pe-riod market model, and let (x, φ) be a replicating strategy for h(S1), then

x is the only price for the option at time t = 0, which does not allow

arbitrage.

One way to find a price for an option is therefore to look for a replicatingstrategy and take the initial investment for this replicating strategy asthe price How to find the replicating strategy, and does a replicatingstrategy always exist ? The two equations (1.1) and (1.2) represent asystem of two linear equations, with two unknown variables x and φ

We simply have to solve it Solving for φ is easy We just subtract (1.2)from (1.1) and obtain

complete We will soon see, that there are also models which are not

complete and where the technique of pricing by the replication ciple does not work Formula (1.3) is often called the Delta hedging

prin-formula.

It is interesting to note, that the price x for the option computed above,does not depend on the probabilities p and 1− p at all In particular itdoes not coincide with the discounted expectation of the payoff of theoption using the probability measure P, i.e in general

x 6= EP

1

1 + rh(ST)



1 + r[ph(S1(H)) + (1 − p)h(S1(T ))]

The latter equation is only true ifp = ˜p and hence also 1−p = 1− ˜p, or

ifh(S1(H)) = h(S1(T )) in which the payoff of the option is deterministic,i.e non random This however is hardly the case in reality On theother side, if we define another measure ˜Pon our underlying probabilityspace Ω ={H, T }, by

The measure ˜P is often called a risk neutral measure, since under

this measure the option price only depends on the expectation of thepayoff, not on its riskiness Such measures will play a major role in thefollowing lecture As we will see risk neutral measures, or equivalentmartingale measures how they are also called, will enable us to com-pute prices for options, also in incomplete markets, where pricing byreplication as above is not applicable We spend the rest of this sectionwith some examples

Example 1.1.1 Assume the parameters in our elementary market model

are given by r = 13, S0 = 1, u = 2, d = 12 as well as p = 34 and we want

to compute the price of a European call option with strike price K = 1and maturity at time t = 1 In this case ˜p

Example 1.1.2 Using the same parameters as in the previous

exam-ple, we compute the price of a European put option, which has the lowing payoff

fol-h(S1) := max(K − S1, 0)The price of the European put is then given by

This relationship does not only hold for the special parameters chosen

in the Examples, but holds in general, whenever the underlying model

is arbitrage free The relationship is called the Put-Call parity

Exer-cise 2 will deal with this

We will now consider a general single period market model, in whichthe agent is allowed to invest in a money market account ( i.e savingsaccount ) and a finite number of stocks S1, , Sn The price of the i-thstock at time t = 0 resp t = 1 is denoted by Si

0 resp Si

1 The moneymarket account is modeled in exactly the same way as in section 1.1.The prices of the stocks at time t = 0 are known, but the prices thestocks will have at time t = 1 are not known at time t = 0 and areconsidered to be random We assume that the state of the world at time

t = 1 can be one of the k states ω1, , ωk which we all put together into

a set Ω, i.e

We assume that on Ω there is defined a probability measure P whichtells us about the likelihood P(ωi) of the world being in the the i-thstate at time t = 1 ( as seen from time t = 0 ) The stock prices Si

1 cantherefore be considered as random variables

Si

1 : Ω → R

Then S1i(ω) denotes the price of the i-th stock at time t = 1 if the world

is in state ω ∈ Ω at time t = 1 For technical reasons, we assume thateach state at time t = 1 is possible, i.e

P(ω) > 0 for all ω ∈ Ω

Let us now formally define the trading strategies which our agents aregoing to use.

Definition 1.2.1 A trading strategy for an agent in our general single

period market model is a pair (x, φ), where φ = (φ1, , φn) ∈ Rn is an

n-dimensional vector, specifying the initial total investment x at time t = 0

and the number of shares φi bought from the i-th stock.

Given a trading strategy (x, φ) as above, we always assume that therest money

x−

nX

i=1

φiS0i

is invested in the money market account As in section 1.1, we definethe corresponding value process to a trading strategy

Definition 1.2.2 The value process of the trading strategy (x, φ) in

our general single period market model is given by (V0(x, φ), V1(x, φ))

where V0(x, φ) = x and V1(x, φ) is the random variable

V1(x, φ) = x−Pni=1φiS0i
(1 + r) +Pni=1φiS1i (1.11)

It is often useful to consider an additional process, the so called gainsprocess G(x, φ), which in a single period market model consists only ofone random variable which is defined by

As the name indicates, G represents the gains ( or losses ) the agentobtains from his investment A simple calculation then gives

Often it is only necessary to study the prices of the stocks in relation

to the money market account For this reason we introduce the

dis-counted stock prices ˆSi

t defined as follows :

ˆ

S0i := S0iˆ

S1i := 1

1 + rS

i 1

for i = 1, , n We also define the discounted value process

corre-sponding to the trading strategy (x, φ) via

ˆ

V0(x, φ) := xˆ

V1(x, φ) := (x−

nX

i=1

φiS0i) +

nX

Example 1.2.1 We consider the following model featuring two stocks

S1 and S2 as well as states Ω = {ω1, ω2, ω3} The prices of the stocks attime t = 0 are given by S01 = 5 and S02 = 10 respectively At time t = 1the prices depend on the state ω and are given by the following table

The increments ∆Si are given by the following table

ω1 ω2 ω3

∆S11 53 53 −59

∆S12 103 −109 −109

and the gains process G by

Now consider the discounted prices of the stock at time t = 1:

ω1 ω2 ω3ˆ

ˆ

V1(x, φ)(ω1) = (x− 5φ1 − 10φ2) + 6φ1 + 4φ2ˆ

V1(x, φ)(ω2) = (x− 5φ1 − 10φ2) + 6φ1 + 8φ2ˆ

V1(x, φ)(ω3) = (x− 5φ1 − 10φ2) + 4φ1 + 8φ2Finally the increments of the discounted stock prices ∆ ˆSi are given by

ω1 ω2 ω3

∆ ˆS11 1 1 −1

∆ ˆS12 2 −2 −2and the gains process of the discounted prices ˆG

ˆG(x, φ)(ω1) = 1φ1 + 2φ2ˆ

G(x, φ)(ω2) = 1φ1 − 2φ2ˆ

G(x, φ)(ω3) = −1φ1 − 2φ2

Given the definition of the wealth process (1.11) in the general singleperiod market model, the definition of an arbitrage in this model looksalmost the same as in Definition 1.1.3 :

Definition 1.2.3 A trading strategy (x, φ) in our general single

pe-riod market model, where x denotes the total initial investment and

φ = (φ1, , φn), with φi denoting the number of shares from stock Si

obtained, is called an arbitrage, if

1. x = V0(x, φ) = 0

2. V1(x, φ) ≥ 0

3 E(V1(x, φ)) = Pki=1P(ωi)V1(x, φ)(ωi) > 0

Here V1(x, φ) is given by equation (1.11)

Often the following remark is very helpful :

Remark 1.2.1 Given that the trading strategy (x, φ) satisfies conditions

2 in Definition 1.2.3, condition 3 in Definition 1.2.3 is equivalent to condition

3.′ There exists ω ∈ Ω s.t V1(x, φ)(ω) > 0.

The definition of an arbitrage can also be formulated by using the counted value process or the discounted gains process This is some-times very useful, when one has to check whether a model has arbi-trage or not The following propositions gives us such a statement Theproof is left as an exercise

dis-Proposition 1.2.1 A trading strategy (x, φ) in the general single period

market model is an arbitrage if and only if one of the following lent conditions hold :

equiva-1 Conditions equiva-1.-3 in Definition equiva-1.2.3 are satisfied with ˆVt(x, φ)

in-stead of Vt(x, φ) for t ∈ {0, 1}.

2. x = ˆV0(x, φ) = 0 and conditions 2.-3 in Definition 1.2.3 are satisfied

with ˆ G(x, φ) instead of V1(x, φ).

Furthermore condition 3 can be replaced by condition 3.′.

We will now come back to the subject of risk neutral measures which

we shortly indicated in section 1.1.1 (see equation (1.8))

Definition 1.2.4 A measure ˜P on Ω is called a risk neutral ity measure for our general single period market model if

1 + rS

i 1



= S0i

We therefore see, that in the case that Ω only consists of two elementsand there is only one tradeable stock on the market, we have exactlywhat we have called a risk neutral measure in section 1.1.1 As alsoindicated in section 1.1.1, risk neutral measures are closely connected

to the question whether there is arbitrage in the model The followingTheorem is one of the cornerstones of Financial Mathematics

Theorem 1.2.1 Fundamental Theorem of Asset Pricing : In the

general single period market model, there are no arbitrages if and only

if there exist a risk neutral measure for the market model.

The proof of this proposition is essentially a geometric proof and needssome preparation First of all it is very useful to think of random vari-ables on Ω as vectors in the k dimensional euclidean space Rk This ispossible by the following identification :

X ⇔ (X(ω1), X(ω2), , X(ωk))⊤ ∈ Rk

The identification means that every random variable can be interpreted

as a vector in Rk and on the other side, every vector in Rk defines arandom variable on Ω We can therefore identify the set of randomvariables on Ω with the set Rk A probability measure Q on Ω can aswell be identified with a vector in Rk The identification is formallyidentical to the one above :

These two properties follow from the properties Q(ω) ≥ 0 for all ω ∈ Ω,

Q(Ω) = 1 and Q(A∪ B) = Q(A) + Q(B) for disjoint sets A and B, whichevery probability measure has to satisfy The subset of Rk consisting of

the vectors with properties 1 and 2 above is often called the standard

simplex in Rk We see, that not every vector in Rk belongs to a bility measure, however we can identify the set of probability measures

proba-on Ω with the standard simplex in Rk

In the following we will always use this identification, writing X forthe vector representing the random variable X and Q for the vectorrepresenting the probability measure Q Using this interpretation wecan for example write the expectation value of a random variable withrespect to a probability measure Q on Ω as a scalar product in the eu-clidean space Rk as follows :

EQ(X) = Pki=1X(ωi)Q(ωi) =< X, Q >

where < ·, · > denotes the standard scalar product in Rk Let us nowconsider the following set :

W = {X ∈ Rk|X = ˆG(x, φ) for some trading strategy (x, φ)} (1.17)

One should think of the elements of W as the possible discounted values

at time t = 1 of trading strategies starting with an initial investment

x = 0 Note that W is a sub vectorspace of Rk Next we consider the set

P+ = {X ∈ Rk|Pki=1Xi = 1, Xi > 0} (1.21)

This set can be identified with the set of probability measures on Ωwhich satisfy property 1 from Definition 1.2.4 We then have the fol-lowing lemma :

Lemma 1.2.1 A measure ˜P is a risk neutral probability measure on Ω

if and only if ˜P ∈ P+∩ W⊥.

Proof. Assume first that ˜P is a risk neutral probability measure on Ω.Then by property 1 in Definition 1.2.4 ˜P ∈ P+ On the other side usingproperty 2 in Definition 1.2.4 as well as the definition of the discountedgains process ˆG(x, φ) in (1.14) we have for X = ˆG(x, φ) ∈ W

< X, ˜P >= EP˜( ˆG(x, φ)) = EP˜

kX

and therefore ˜P ∈ W⊥ Together with the first part this gives ˜P ∈ P+ ∩

W⊥ On the other side if ˜P is an arbitrary vector in P+ ∩ W⊥ then

G(x, φ) = ∆ ˆSi On the other side ˆG(x, φ) ∈ W and since ˜P ∈ W⊥ we have

0 =< ˆG(x, φ), ˜P >= E˜P



∆ ˆSiTherefore ˜P also satisfies condition 2 in Definition 1.2.4 which com-pletes the proof of Lemma 1.2.1

Definition 1.2.5 We denote with M = W⊥ ∩ P+ the set of risk neutral measures.

We are now ready to prove Proposition 1.2.2

Proof. Assume first that the no arbitrage condition holds Let us definethe set

A+ = {X ∈ A| < X, P >= 1}

This is a closed, bounded and convex subset of RK Since A+ ⊂ A itfollows from (1.19) that

no arbitrage ⇒ W ∩ A+ = ∅.

By the separating hyperplane theorem, there exists a vector Y ∈ W⊥,s.t

< X, Y > > 0 for all X ∈ A+ (1.22)For each i = 1, , k define the vector Xi as the vector in Rk whose i-

th component is 1/P(ωi) and the other remaining components are zero.Then

< Xi, P >= 1

P(ωi)P(ωi) = 1and hence Xi ∈ A+ Denoting with Yi the i-th component of Y it thenfollows from (1.22), that

0 << Xi, Y >= 1

P(ωi)Yiand therefore Y (ωi) = Yi > 0 for all i = 1, , k Let us now define Q by

Q(ωi) = Y (ωi)

Y (ω1) + + Y (ωk).Then Q ∈ P+ On the other side, since Q is merely a scalar multiple of

Y , and W⊥ is a vectorspace, we follow that Q ∈ W⊥ Therefore

Q ∈ P+∩ W⊥and by Lemma 1.2.1 we have that Q is a risk neutral measure on Ω

We have therefore shown, that the condition no arbitrage implies theexistence of a risk neutral measure Let us now show the converse Weassume there exists a risk neutral measure Q Let(x, φ) be an arbitrarytrading strategy Then as in the proof of Lemma 1.2.1

EQ( ˆG(x, φ)) = 0

If we assume that ˆG(x, φ) ≥ 0, then the last equation clearly impliesthat ˆG(x, φ)(ω) = 0 for all ω ∈ Ω Hence by Proposition 1.2.1 there cannot be any trading strategy, which satisfies all conditions of an arbi-trage.

Example 1.2.2 We continue with example 1.2.1 Remind that the

in-crements of discounted prices in this example were given as displayed

in the following table :

φ1 = 12(X1 + X2) and φ2 = 14(X1 − X2) and obtain

The latter clearly implies that W⊥∩ P+ = ∅ and therefore that there is

no risk neutral measure for this model By Theorem 1.2.1 we already

know at this point that there must be arbitrage strategy in the model.

If we compare W and A we see that



we know that there must be a trading strategy (x, φ) s.t

ˆG(x, φ) =





0

X20

0− φ1S01 − φ2S02 = −X2

2 · 5 − (−X2

4 )· 10 = 0

in the money market The arbitrage we computed is therefore a egy which only invests in the risky assets, i.e the stocks.

strat-We now come back to the question ”What should the price of an option

in our model be ?” In section 1.1 we considered options of the typeh(S1)whereh is a payoff profile, a function of the single stock S1 at time t = 1

In our general model we now have more than one stock and the payoffprofiles may look more complicated For this reason we generalize ourdefinition of an option We call this more general product contingentclaim

Definition 1.2.6 A contingent claim in our elementary single period

market model is a random variable X on Ω representing a payoff at time

t = 1.

To price a contingent claim, we may follow the same approach as taken

in section 1.1 and apply the replication principle

Proposition 1.2.2 Let X be a contingent claim in our general single

period market model, and let (x, φ) be a hedging strategy for X, i.e a

trading strategy which satisfies V1(x, φ) = X, then the only price of X

which complies with the no arbitrage principle is x = V0(x, φ).

The proof of this proposition follows along the same argumentation as

in section 1.1 from the no arbitrage principle A crucial difference

to the elementary single period model as discussed in section 1.1 ishowever, that in the general single period market model, a replicatingstrategy might not exist This can happen, when there are more effec-tive sources of randomness, than there are stocks to invest in Let usconsider the following example, which represents an elementary ver-sion of a so called stochastic volatility model

Example 1.2.3 We consider the following market model It consists

of two tradeable assets, one money market account Bt and one stock

St ( t=0,1 ), as well as third object which we call the volatility v Thevolatility determines whether the stock price can make big jumps or

small jumps In this model the volatility is assumed to be random, or in

other words stochastic Such models are called stochastic volatility

models To be a bit more precise, we assume that our state space

S1(ω) :=

(1 + v(ω))· S0 if ω = ω1, ω3(1− v(ω)) · S0 if ω = ω2, ω4where S0 denotes the initial stock price The stock price can thereforejump up or jump down, as in the first elementary single period marketmodel from section 1.1 The difference to this model is that the amount

by which it jumps is itself random, determined by the volatility Finallythe money market account is modeled by

time t = 1 is given by ω = ω1 The contingent claim X can thereforealternatively be written as

X(ω) =



1 if ω = ω1

0 if ω = ω2, ω3, ω4Let us now see whether there exists a replicating strategy for this con-tingent claim, i.e a trading strategy (x, φ) satisfying

which cannot be hedged, since the volatility is not tradeable The ematical explanation is just, that the system of linear equations abovehas no solution.

math-To take account of this difficulty we introduce the following Definition

Definition 1.2.7 A contingent claim X is called attainable, if there

exists a trading strategy (x, φ) which replicates X, i.e satisfies V1(x, φ) =

X.

For attainable contingent claims the replication principle applies and

it is clear how to price them, namely by the total initial investmentneeded for a replicating strategy There might be more than one repli-cating strategy, but it follows again from the no arbitrage principle,that the total initial investment for replicating strategies is unique

In equation (1.8) we established a way, to use a risk neutral measure tocompute the price of an option in the elementary single period marketmodel This approach works fine in the general single period marketmodel as well, at least as attainable contingent claims are considered

Proposition 1.2.3 Let X be an attainable contingent claim and ˜P be

an arbitrary risk neutral measure Then the price x of X at time t = 0

defined via a replicating strategy can be computed by the formula

1

1 + rX



= EP˜

ˆ

Remark 1.2.2 Proposition 1.2.3 tells us in particular, that for all risk

neutral measures the model may contain, we get the same value when computing the expectation in equation (1.23).

The following example shows that the situation changes dramatically

if the contingent claim is not attainable

Example 1.2.4 Let us compute the set of risk neutral measures for

our stochastic volatility model from Example 1.2.3 For simplicity weassume r = 0 In this case the discounted processes and the originalprocesses coincide We have

∆ ˆS(ω) :=

v(ω)· S0 if ω = ω1, ω3

For a trading strategy (x, φ) the discounted gains process is given by

ˆG(x, φ) = φ· ∆ ˆSand we see that the vectorspace W is one dimensional, spanned by thevector ∆ ˆS, i.e

The orthogonal complement of W is then given by

On the other side(q1, q2, q3, q4)⊤ ∈ P+ if and only ifq1+q2+q3+q4 = 1 and

qi > 0 Since the set of risk neutral measures is given by M = W⊥ ∩ P+

qi > 0, q1 + q2 + q3 < 1h(q1 − q2) = l(1− (q1 + q2 + 2q3))

Clearly, this set is not empty, and we can conclude that our stochasticvolatility model is arbitrage free In particular, as is easy to see, forevery 0 < q1 < 1 there exists Q ∈ M s.t Q(ω1) = q1 Let us nowcompute the ( discounted ) expectation of a Digital call under a measure

Q = (q1, q2, q3, q4)⊤ ∈ M :

EQ(X) = q1 · 1 + q2 · 0 + q3 · 0 + q4 · 0 = q1.Since q1 is arbitrary, except that it has to lie between zero and one,

we see that we obtain many different values as discounted tion under a risk neutral measure, in fact every value x which satis-fies 0 < x < 1 The situation is therefore completely different than inProposition 1.2.3 The reason is, that as we showed in Example 1.2.3the contingent claim X is not attainable

expecta-Let us now consider a general contingent claim X, attainable or not

Definition 1.2.8 We say that a price x for the contingent claim X plies with the no arbitrage principle, if the extended model, which

com-consists of the original stocks S1, , Sn and an additional asset Sn+1

which satisfies S0n+1 = x and S1n+1 = X is arbitrage free.

The additional asset Sn+1 defined in the proposition above may not beinterpreted as a stock, since it can take negative values if the contin-gent claim takes negative values For the general arbitrage and pricingtheory developed so far, positiveness of asset prices was however notessential

In regard of the previous example the following proposition might besurprising It says that whenever one uses a risk neutral measure toprice a contingent claim by formula (1.23) one obtains a price whichcomplies with the no arbitrage principle Even, if prices differ, whenusing different risk neutral measures

Proposition 1.2.4 Let X be a possibly unattainable contingent claim

and ˜Pa risk neutral measure for our general single period market model Then

x = EP˜

1

1 + rX



(1.24)

defines a price for the contingent claim at time t = 0 which complies

with the arbitrage principle.

Proof. By the Fundamental Theorem of Asset Pricing ( Theorem 1.2.1) it is enough to show that there exists a risk neutral measure for thecorresponding model which is extended by Sn+1 as in Definition 1.2.8

By assumption ˜P is a risk neutral measure for the original model, sisting of the stocks S1, , Sn, i.e ˜P satisfies 1 and 2 in Definition1.2.4 for i = 1, , n For i = n + 1 the second condition translates into

con-EP˜(∆ ˆSn+1) = EP˜

1

1 + rX − x



= EP˜

1

1 + rX



− x

= x− x = 0

By Definition 1.2.4 ˜P is a risk neutral measure for the extended model

In the situation of Example 1.2.4, this proposition applied to a Digitalcall says, that any price between zero and one is a price, which doesnot allow arbitrage and can therefore be considered as fair This nonuniqueness of prices is a serious problem which until today is not com-pletely resolved We will later discuss some of the methods which aretrying to resolve this issue For now let us characterize the models, inwhich the problem of non uniqueness of prices does not occur :

Definition 1.2.9 A financial market model is called complete, if for

any contingent claim X there exists a replicating strategy (x, φ) A model

which is not complete is called incomplete.

By Proposition 1.2.3 the issue of computing prices in complete marketmodels is completely solved But how to recognize complete models ?The following proposition gives us a criterion for completeness.

Proposition 1.2.5 Assume a general single period market model

con-sisting of stocks S1, , Sn and a money market account modeled on the state spaceΩ = {ω1, , ωk} is arbitrage free Then this model is complete

if and only if the k × (n + 1) matrix A given by

has full rank, i.e rank(A) = k.

Proof. By standard linear algebra the matrix A has full rank, if andonly if for every X ∈ Rk the equation AZ = X has a solution Z ∈ Rn+1

On the other side we have

Example 1.2.5 We have seen already, that the stochastic volatility

model discussed in Examples 1.2.3 and 1.2.4 is not complete Anotherway to see this is by using Proposition 1.2.5 The matrix A in this casehas the form

The previous proposition presents a method how to determine whether

a model is complete, without computing replicating strategies Now, ifthe model is not complete, is there a method how to determine whether

a specific contingent claim is attainable, without trying to compute thereplicating strategy ? Yes there is The following proposition showshow

Proposition 1.2.6 The contingent claim X is attainable, if and only if

EQ 1+r1 X
takes the same value for all Q ∈ M.

The proof of this result can be found in Pliska ([1], page 23) We omitthe proof here An important consequence of this proposition is thefollowing theorem:

Theorem 1.2.2 Under the assumption that the model is arbitrage free,

it is complete, if and only if M consists of only one element, i.e there is only one risk neutral measure.

Proof. Since the model is arbitrage free, it follows from Theorem 1.2.1that there is at least one risk neutral measure, i.e M 6= ∅ Assumefirst that there is only one risk neutral measure Then the condition inProposition 1.2.6 is trivially satisfied for all contingent claims X and sothe market model is complete On the other side, assume the marketmodel is complete and consider two risk neutral measures Q1 and Q2 in

M For each i = 1, , k consider the contingent claim Xi given by

Since the model is complete, Xi is an attainable contingent claim Itfollows from Remark 1.2.2 that

Q1(ωi) = EQ1

1

In this section we continue our study of the general single period ket model which consists of one money market account and n stocks,based on the underlying state space Ω = {ω1, , ωk} and probabilitymeasure P In the last two sections we were mainly interested into set-ting up this model and the computation of prices for financial deriva-tives such as options and contingent claims We will come back to theseissues, but before we want to study a different question, which is by nomeans less important

mar-What is the optimal way to invest money into the market ?

The answer of this question naturally depends on the choice of themodel, and in this section we will stick to our general single periodmarket model from the last section We will later consider the samequestion in more advanced models

Before we are going to answer this question, we have to specify, whatexactly we mean by optimality, i.e what is our measure for the perfor-mance of a trading strategy In order to define such a performance mea-sure three fundamental characteristics of financial markets, or moreprecisely of the agents trading on the market, have to be included :

1 Agents prefer higher payoffs to less payoffs

This idea is so innate to financial markets, that it doesn’t need anydiscussion However we cannot judge trading strategies purely by thischaracteristic As we saw in the previous sections, the payoffs of finan-cial assets are generally modeled as random variables Assume now,

we have two trading strategies which performance we want to pare It can then well be, that in one state of the world the first tradingstrategy yields a higher payoff and in another state of the world thesecond trading strategy yields a higher payoff In order to compare theperformances of the two trading strategies, we must somehow take anaverage over the states of the world, but this is nothing else then takingexpectations

com-2 Agents assess trading strategies by average performance

The second characteristic leads directly to the use of expectationalvalues, which fits in very good into our probabilistic models There ishowever a third characteristic which has to be included To illustratethis, consider the following simple example : Assume an agent is of-fered to choose between two alternatives If he chooses the first one

he will be paid 10 million pound If he chooses the second one, a faircoin will be tossed If the coin shows head, he will be paid 20 millionpound instead of the 10 million, but if the coin shows tail, he will bepaid nothing What would the agent choose ? If the agent is not yet abillionaire, he would probably go for the first alternative, which giveshim 10 million for sure and financial safety for the rest of his life De-noting the payoff of the first alternative with X1 and the payoff of thesecond alternative with X2 we see that if we simply judge the two pay-offs by its expectation, the agent would be indifferent between the twoalternatives :