Lecture 2 - Discrete Models 1

8/3/2019 Lecture 2 - Discrete Models 1

1/191

Single-Period Model

We consider N securities at times 0 and 1:

S(t)=[S1(t),S2(t),..,SN(t)], t=0,1, where

Si(1,) - r.vs defined on a sample space .

The time-1 prices can be written as

IfS1

is the bank account then

where i is the one-period interest rate (nonnegative).

At time t=0 an investor selects a portfolio.

j = the number of units of assetj held from 0 to 1.

The column vector = [1,..,N]T - a trading strategy.The value of the portfolio at time t:

=

),1(),1(),1(

),1(),1(),1(

),1(),1(),1(

),1(

21

22221

11211

MNMM

N

N

SSS

SSS

SSS

S

L

MLMM

L

L

).(...)()( 11 tStStS NN ++=

,,1),1(Sand1)0( 11 +== iS


2/192

Arbitrage. State Price Vector.

Def (5.2.1) An arbitrage opportunity is a

trading strategy such that

A security market model is arbitrage free if

there are no arbitrage opportunities.

Def (5.2.2) A state price vector is a strictlypositive row vector [(1),.., ()] for which

S(0) = S(1, ).Equivalently,

Def. The Arrow-Debreu security for outcome

m, m=1,..,M, is a security that at time 1 pays 1if outcome m occurs, and 0 otherwise.

It has the payoff vector

em=[0,..,0,1,0,..,0]T, with 1 in the m-th position.

.0)S(1,and0)0( > S

.,..,1),,1()()0( NjSS jj ==


3/193

Fundamental Theorem of Asset

Pricing. Risk-Neutral Measures.

Theorem (5.2.3) The single-period securitiesmarket model is arbitrage free if and only if

there exists a state price vector.

Suppose that S1 is a bank account. Define

From the definition of a state price vector :

In particular,

Hence, the price at t=0 can be interpreted as the

expected discounted price of the security with

respect to the probability measure Q, not P.

Thus, staring with a state price vector, we have

constructed what is called a risk-neutral

probability measure.

.),()1()( += iQ

N.1,..,jfor,1

),1()()0()1( =+=

iSQS jj

==

).(1)0(1 QS


4/194

Risk-Neutral Probability Measures.

Def. (5.2.4) A risk-neutral probability measureis a probability measure Q on such that

(i) Q() > 0, for all from .

(ii) Equation (1) holds for j=2,..,N.

Under this measure the expected discounted

price on any security equals the initial price of

the same security.

Theorem (5.2.5) Suppose security 1 is a bank

account. Then the following are equivalent:

(i) The single-period model is arbitrage free.

(ii) There exists a state price vector.

(iii) There exists a risk-neutral probability

measure.


5/195

Valuation Of Cash Flows

In the case of single-period models, any cash

flow at time 1 can be represented by a randomvariable, say X.

What is the time-0 value of X?

Definition We say that the trading strategy replicates the cash-flow X if

Definition We say that the cash flow X is

attainable if it can be replicated by some trading

strategy.

Theorem (5.2.6) In an arbitrage-free, single

period model, the time-0 value of an attainable

cash flow X is equal to the time-0 value of the

portfolio that replicates X.

.all),(),1( = XS jj

j


6/196

Valuation Of Cash Flows

Observe that if

S(1,)=X S(1,) =X.Since, S(0) = S(1,) , we have

S(0) = XTherefore, the time-0 price of the replicating

portfolio is equal to X, which can becomputed by knowing the state price vector.

Theorem (Risk Neutral Valuation) (5.2.7)

In an arbitrage-free, single period model, the

time-0 value of an attainable cash flow X is

equal to X, where is the state price vector.If, in addition, security 1 is a bank account with

short interest rate i, then

where Q is the risk-neutral probability measure.

,1

)()(

+=

i

XQX


7/197

Completeness in the Single-Period

Model

Definition (5.2.8) An arbitrage-free marketmodel is said to be complete if for every cash-

flow vector X there exists some trading strategy such thatS(1,) =X.

Theorem (5.2.9) Suppose a single-period

model is arbitrage free. Then the model iscomplete if and only if there is unique state

price vector.

Corollary (5.2.10) Suppose security 1 in asingle-period model is a bank account. Then

this model is arbitrage free and complete if and

only if the risk-neutral probability measure is

unique.


8/198

One-period model: two assets

Two assets S1, S2:

Assumptions: all the prices are strictly positive

the market is complete: returns on the

securities are linearly independent

We also have a derivative security with payoffat time 1:

2,1)),,1(),,1((),1( 21 == jSSgV jjj

),1( 11 S

),1( 21 S

)0(1

S

),1( 12 S

),1( 22 S

)0(2

S


9/199

One-period model (continued)

We can construct a portfolio of the two

primitive securities that has the same value as

the derivative security in each state at time 1.

To ensure the same payoff we must have:

A unique solution must exist because of the

complete market assumption. Important step: if there is no arbitrage, then

the time-0 value of the portfolio must be equal

to the time-0 price of the derivative. Hence,

Otherwise we would make arbitrage profits

with a zero initial investment.

).,1(),1(),1(

),,1(),1(),1(

2222112

1221111

SSV

SSV

+=+=

).0()0()0( 2211 SSV +=


10/1910


The key assumptions for the valuation

formula and their implications:

Market is

complete

Replication of the derivative

payoff is feasible

Current price of the

derivative = current value ofthe replicating portfolio

No

arbitrage


11/1911


Equivalent expression for the price of the

derivative security:

if

we have

where

In terms of expectations:

where Q1 - the probability measure {q1,1-q1}.

,)0(

),1(

1

111

S

Su

= ,

)0(

),1(

2

122

S

Su

=

,)0(

),1(1

211

SSd = ,)0( ),1(2

222

SSd =

,),1(

)1(),1(

)0(2

21

1

11

u

Vqu

VqV

+=

).1,0()(

2112

2111

=

dudu

dduq

],)1(

)1([

)0(

)0(

11

1

S

VE

S

V Q=


12/1912

The Multiperiod Model

We considerNsecurities at times k=0,1,..,T:

S(k)=[S1(k),S2(k),..,SN(k)],

where Si(k,) denotes time k price of security i if the

underlying state of the world is .

For each from we call the function

k -> Sj(k, )

a sample path of the stochastic process {Sj}.

We assume that S1 is the bank account:

S1(0, )=1 and k -> S1(k, ) is nondecreasing The r.vs

are called the one-period interest rate.In the special case where ik=i:

Example: Binomial Stock Price Model.

,1,..,1,0,01)(

)1(:

1

1=

+= Tk

kS

kSik

.,..,1,0,)1()(1 TkikS

k=+=


13/1913

The Multiperiod Model

By Pkwe denote the set of all events that are

known by the investors at time kby observingprices S(0),S(1),..,S(k). We can view Pkas a

partition of.

Definition (5.3.2.) A stochastic process

X={X(k):k=0,1,..,T} is said to be adapted to the

information submodel {Pk;k=0,1,..,T} if, for

each k, there is a function such thatX(k,) can

be expresses as this function of the time-k

history, for all from .

Definition (5.3.3.) A stochastic process

X={X(k):k=0,1,..,T} is said to be a martingale

if it is adapted and

E[X(k+1)| Pk]=X(k), k=0,1,..,T-1.

Remark: The conditional expectationE[X(k+1)| Pk] is

the same as the conditional expectation

E[X(k+1)| S(0),S(1),..,S(k)] .


14/1914

Self-Financing Trading Strategies

Let j(k,) denote the number of shares of

securityj held from time kto time k+1. Thenthe stochastic process

: = {(k); k=0,1,..,T-1}is referred to as a trading strategy.

Observe that {(k)} is adapted to Pk. The time-k value of the portfolio is given by

V(k, )=S(k,)(k, ), k=0,1,..,T-1. We should also consider the value of the

portfolio just before time-t transactions:V(k-, )=S(k,)(k-1, ), k=1,..,T.

The cash flow out of the portfolio at time k:

Definition A trading strategy is said to be

self-financing if c

(1)= c

(2)=..= c

(T-1)=0.

.

),,(

1,..,1),,(),(

0),,(

),(

=

=

=

=

TkkV

TkkVkV

kkV

kc


15/1915

The Fundamental Theorem of Asset

Pricing

Def (5.3.1) A multiperiod market model admits

arbitrage if there is a self-financing trading

strategy , called an arbitrage opportunity, s.t.:

Def (5.3.4.) A risk-neutral probability measure

is a probability measure Q on satisfying1. Q() > 0, for all from .2. S

j

/S1

is a martingale under Q for j=2,3,..,N.

Def (5.3.5.) A stochastic process

={();=0,1,..} is said to be a state priceprocess if the following conditions are true:

for all time-khistoriesH, all k=0,1,..,T-1, and allj=1,..,N.

++=H

j

H

j kSkkSkd

),,1(),1(),(),(.

0)0()0()0(.1 = SV

0)1()()(.2 >= TTSTV

= 1),0(.a

positivestrictlyisIt.c

adaptedisIt.b


16/1916

The Fundamental Theorem of Asset

Pricing.

Theorem (5.3.6.) For a multiperiod model of asecurities market the following are equivalent:

a. The model is arbitrage free

b. There exists a state price process.

c. There exists a risk-neutral probability measure.

Proposition (5.3.8.) If Q is a risk-neutral

probability measure and if is a self-financingtrading strategy, then the discounted value of

the corresponding portfolio is a martingale

under Q:

for all .0 Ttk

= k

QP

tStVE

kSkV |

)()(

)()(

11


17/1917

Valuation Of European Options

Suppose that X is a European option:

X=g(S1(T),,SN(T)).

We say thatXis attainable if there exists a self-

financing trading strategy such that

V(T)=X.

Theorem (5.3.9.) In an arbitrage free

multiperiod model, the time-0 value of an

attainable European option X is equal to thetime-0 value of the portfolio that replicates X.

Theorem (5.3.10.) In an arbitrage-free

multiperiod model having-risk neutral

probability measure Q, the time-0 value of an

attainable European option X is equal to

==

).(),(

),(

)()(

)( 11XT

TS

XQ

TS

XEQ


18/1918

Valuation Of Cash-flow Streams

Suppose that c is a cash-flow stream described by an

adapted process:

c={c(k,); k=1,2,,T}.

A trading strategy (not necessarily self-financing) andthe corresponding portfolio V are said to replicate c andto finance c if

c(k)=c(k), k=1,2,,T.

The cash-flow stream c is attainable if it can be replicated

by some trading strategy . Theorem (5.3.11.) In an arbitrage free

multiperiod model, the time-0 value of an

attainable cash-flow stream c is equal to the

time-0 value of the portfolio that finances c.

Theorem (5.3.12.) In an arbitrage-free

multiperiod model having-risk neutral

probability measure Q, the time-0 value of anattainable attainable cash-flow stream c is

==

=

.),(),()(

)(

11 1

T

k

T

k

Q kckkS

kcE


19/19

Completeness in the Multiperiod

Model

Definition (5.3.13.) An arbitrage-free market

model is said to be complete if every adapted

cash-flow stream c is financed by some trading

strategy .

Theorem (5.3.14.) For an arbitrage-free

multiperiod model, the following areequivalent:

a. The model is complete.

b. The state price is unique.

c. The risk-neutral probability measure Qis unique.

Lecture 2 - Discrete Models 1

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