80-646-08StochasticcalculusGeneviŁveGauthier Introductiontodiscrete-timemarketmodels:thebinomialmodel

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Single-period binomial model Two-period binomial model

Introduction to discrete-time market models:

the binomial model

80-646-08 Stochastic calculus

Geneviève Gauthier

HEC Montréal

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Single-period binomial model

Prices Measurable space Portfolio Arbitrage Contingent claim Pricing Risk-neutral measure

Martingale measure Discount factor Two-period binomial model

The riskless security I

The market model

The stochastic process n

S_t⁽¹⁾ :t =0,1o

represents the evolution of the share price of a riskless security (a bond, for example).

Let’s assume that time is measured in periods (a period may be one year, one month, one day, one hour, one second, etc.)

And let’s assume that theperiodic interest rate r is constant during the process observation,

then

8^ω2Ω, S₁⁽¹⁾(ω) =S₀⁽¹⁾(ω) (1+r).

We can assume that8^ω2Ω,S₀⁽¹⁾(ω) =1 which implies that

8^ω2Ω, S₁⁽¹⁾(ω) = (1+r).

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The risky security I

The market model

The second stochastic process n

S_t⁽²⁾ :t =0,1o , for its part, represents the evolution of the share price of a risky security, S_t⁽²⁾ being the price of a share at time t.

Since the current share price of that security is known with certainty,

8^ω2 ^Ω^, ^S0⁽²⁾(ω) =s0 2^R⁺= (0,∞).

We will assume that, in the next period, the share price of the risky security can only take one of two values, say s11

ands₁₂ (s₁₁,s₁₂ 2^R⁺^,^s¹¹ <s₁₂).

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Measurable space I

Question. What is the …ltered measurable space that allows us to model such a situation?

Answer. In this example, the random experiment is not clearly de…ned. However, if we consider the

two-dimensional stochastic process

!_S

t =ⁿ S_t⁽¹⁾,S_t⁽²⁾ :t =0,1o ,

we note that such a process has two di¤erent paths only, i.e.

S₀⁽¹⁾,S₀⁽²⁾ ^> S₁⁽¹⁾,S₁⁽²⁾ ^>

trajectoire #1 (1,s₀)^> (1+r,s₁₁)^>

trajectoire #2 (1,s0)^> (1+r,s12)^>

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Measurable space II

As a consequence, if Card(_Ω)>2, then there would be some identical paths, i.e.

9^ω^1,^ω² 2^Ω^{such that} 8^t 2 f^0,¹g^, !_S

t (ω1) =!_S

t(ω2) and, in such a case, even our largest source of information,

F¹ =σ S₀⁽¹⁾,S₀⁽²⁾,S₁⁽¹⁾,S₁⁽²⁾ ,

could not distinguish between ω1 andω2 being realized.

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Measurable space III

For that reason, we choose Card(Ω) =2 and

ω S₀⁽¹⁾(ω),S₀⁽²⁾(ω) ^> S₁⁽¹⁾(ω),S₁⁽²⁾(ω) ^>

ω1 (1,s₀)^> (1+r,s₁₁)^>

ω2 (1,s0)^> (1+r,s12)^>

This is not the only possible form for Ω, but this is a convenient one.

Grouping all undi¤erentiated ω under the same name is common practice. This is also one of the reasons why many people mix up the concepts of sample space and random variable.

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Measurable space IV

By choosing such an Ω= f^ω¹^,^ω²g, we will be able to distinguish which of its elements has occured after observing the process until maturity, i.e. untilt =1.

That’s why,

F¹ = F

= fall events inΩg

= f?^,f^ω¹g^,f^ω²g^,^Ωg^. Since the process S is constant at time t =0,F⁰ =f?^,^Ωg^.

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The portfolio I

Let

φ₁ = the number of shares of the riskless security being held φ₂ = the number of shares of the risky security being held.

De…nition The pair !

φ = (φ₁,φ₂)2R Ris called aportfolio.

Ifφ_i is negative, its means a short sale of φ_i shares of securityi has occurred.

The value of such a portfolio at timet is

V_φ(t,ω) =φ₁S_t⁽¹⁾(ω) +φ₂S_t⁽²⁾(ω).

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The portfolio II

Since V_φ(0,ω) =φ₁+φ₂s0, we will need to pay out an amount of φ₁+φ₂s0 in order to acquire the portfolio ! φ ifφ₁+φ₂s₀<0, we will receive an amount of

(φ₁+φ₂s₀)in acquiring the portfolio ! φ.

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Arbitrage opportunity

De…nition We say that !

φ is an arbitrage opportunity if (A1) 8^ω 2^Ω^, ^V^φ(0,ω) =0 (A2) 8^ω 2^Ω^, ^Vφ(1,ω) 0 (A3) 9^ω2 Ω, V_φ(1,ω)>0,

i.e., starting from a zero investment(A1), we are certain not to incur a loss(A2) and we have a positive probability to make a gain (A3).

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What are the arbitrage opportunities in the model I

Condition (A1)implies that, for !

φ = (φ₁,φ₂)to be an arbitrage opportunity, we must have

φ₁ = φ₂s₀ (1) since 0=V_φ(0,ω) =φ₁+φ₂s₀.

It is to be noted as of now that condition (A3)ensures that (0,0)is not an arbitrage opportunity. As a result, ( φ₂s₀,φ₂)cannot be an arbitrage opportunity ifφ₂ =0.

So, let’s assume φ₂ 6=0.

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What are the arbitrage opportunities in the model II

From condition (A2), it follows that( φ₂s₀,φ₂)is an arbitrage opportunity only if

V_φ(1,ω1) = φ₂s0S₁⁽¹⁾(ω1) +φ₂S₁⁽²⁾(ω1)

= φ₂s₀(1+r) +φ₂s₁₁

= φ₂(s₁₁ s₀(1+r)) 0 and

V_φ(1,ω2) = φ₂s₀S₁⁽¹⁾(ω2) +φ₂S₂(1,ω2)

= φ₂s₀(1+r) +φ₂s₁₂

= φ₂(s12 s0(1+r)) 0.

Therefore, arbitrage will be possible only if

s₁₂ >s₁₁ s₀(1+r)or ifs₁₁ <s₁₂ s₀(1+r).

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What are the arbitrage opportunities in the model III

Adding the third condition (A3) implies that, if

s₁₂ >s₁₁ s₀(1+r),then, for allφ₂>0, the portfolio ( φ₂s₀,φ₂)is an arbitrage opportunity since

V_φ(1,ω1) = φ₂(s11 s0(1+r)) 0;

V_φ(1,ω2) = φ₂(s12 s0(1+r))>0 and if s11 <s12 s0(1+r)then, for allφ₂ <0,the portfolio ( φ₂s₀,φ₂)is an arbitrage opportunity since

V_φ(1,ω1) = φ₂(s11 s0(1+r))>0;

V_φ(1,ω2) = φ₂(s₁₂ s₀(1+r)) 0.

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What are the arbitrage opportunities in the model IV

In other words, if s₁₂ >s₁₁ s₀(1+r) then one just has to

sell shortns₀ shares of the riskless security (borrow an amount ofns₀ dollars)

and buynshares of the risky security for the same amount.

The amount to be paid out isV_φ(0) = ns0 1+n s0 =0.

Att =1, we sell the shares of the risky security (we get nS₁⁽²⁾(ω)dollars)

and pay back our loan, with interest (which amounts to ns₀(1+r)dollars).

We therefore get a net amount of

nS₁⁽²⁾(ω) ns₀(1+r) =V₍ _ns₀_,n)(1,ω)

= ^ns¹¹ ^ns⁰(1+r) =n(s11 s0(1+r)) 0 if ω=ω1

ns12 ns0(1+r) =n(s12 s0(1+r))>0 if ω=ω2.

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What are the arbitrage opportunities in the model V

Ifs₁₁ <s₁₂ s₀(1+r)then we can sell shortnshares of the risky security and

and invest the amount ofns0 thus obtained in purchasing ns₀ shares of the riskless security:

V_φ(0) =ns₀ 1 n s₀ =0.

At timet =1,an amount ofns₀(1+r)dollars is obtained from selling the riskless security and we buy at a cost of nS₁⁽²⁾(ω)thenshares of the risky security which we had sold short.

The net proceeds from such a transaction are

ns₀(1+r) nS₁⁽²⁾(ω)

= V_(ns₀_, _n)(1,ω)

= ^ns⁰(1+r) ns11=n(s0(1+r) s11)>0 siω=ω1

ns₀(1+r) ns₁₂=n(s₀(1+r) s₁₂) 0 siω=ω2.

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What are the arbitrage opportunities in the model VI

By contrast, if s₁₁ <s₀(1+r)<s₁₂,there are no portfolios providing arbitrage opportunities since

V_φ(0) =0)^φ1 = φ₂s₀.

Ifφ₂=0,then V_φ(1,ω1) =V_φ(1,ω2) =0 and,

ifφ₂ 6=0,then V_φ(1,ω₁) andV_φ(1,ω₂)cannot be both non-negative since

V_φ(1,ω1) = φ₂(s₁₁ s₀(1+r))

| {z }

<0

andV_φ(1,ω₂) = φ₂(s₁₂ s₀(1+r))

| {z }

>0

.

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What are the arbitrage opportunities in the model VII

In what follows, we will assume that our model contains no arbitrage opportunities, i.e.

s₁₁ <s₀(1+r)<s₁₂. (2)

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Contingent claim I

De…nition

Acontingent claim is a contract between two parties, a seller and a buyer, the value of which will depend on the state of the market during the contract validity period. It is like an

insurance contract.

Mathematically speaking, a contingent claim C is any non-negative (Ω,F) random variable.

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Contingent claim II

For example, a put option is a contingent claim entered into by the two parties at time t =0, and allowing the buyer to sell, at time t=1,a share of the risky security at a price determined as of now, sayK,which is called the strike price.

Thus, ifS₁⁽²⁾<K then the buyer will exercise his option and sell his risky security share for an amountK greater than the price he would obtain on the market.

By contrast, ifS₁⁽²⁾ K, the buyer will not exercise his option, since he can obtain on the market a better price than the one set by the contract.

Hence, the value of the put option is max K S₁⁽²⁾,0 . Obviously, the contingent claim seller will not o¤er such a bene…t to the buyer without being compensated. The amount paid out at timet =0 by the buyer to the seller in order to acquire the option is the contingent claim price.

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Pricing

Contingent claim

Question. What is the contingent claim price?

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Pricing I

The seller

Can the portfolio be chosen in such a way that its value at t=1 matches the contingent claim value? In the case when there are as many scenarios as securities, it is possible to obtain a single solution under certain conditions.

Ifci =C (ωi)then

8^ω 2Ω,V_φ(1,ω) =C(ω)

, ^φ1(1+r) +φ₂s₁₁ =c₁ andφ₁(1+r) +φ₂s₁₂ =c₂ , ^φ¹

φ₂ =

s12c1 s11c2

(s12 s11)(1+r) c2 c1

s12 s11

!

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Pricing II

The seller

Thus, the value at timet =0 of such a portfolio is V_φ(0,ω)

= ^s¹²^c¹ ^s¹¹^c²

(s₁₂ s₁₁) (1+r)+ ^c² ^c¹ s₁₂ s₁₁s₀

= ^c¹ 1+r

s12 s0(1+r) s12 s11

+ ^c² 1+r

s0(1+r) s11

s12 s11

. (3)

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Risk-neutral measure I

So far, we have been working on a …ltered measurable space, and it has not been spoken of probability measure yet.

Let’s set

q = ^s¹² ^s⁰(1+r) s₁₂ s₁₁

and note that the no-arbitrage condition is equivalent to the fact thatq is comprised between 0 and 1:

s₁₁ <s₀(1+r)<s₁₂ ,⁰<q <1.

Thus, if Q(ω1) =q and Q(ω2) =1 q then Qis a probability measure built on our measurable space (Ω,F).

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Risk-neutral measure II

It is important to note that Q(ω1)likely does not correspond to the ”real” probability that the event ω1

occurs.

Generally, that probability is not known.

Let’s call P the probability measure that associates to each ω in the sample space the ”real” probability that such an event occurs (P(ω1) =p and P(ω2) =1 p).

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Risk-neutral measure III

As we have established in equation (3), the contingent claim price C is

c1

1+r

s12 s0(1+r) s₁₂ s₁₁ + ^c²

1+r

s0(1+r) s11

s₁₂ s₁₁

= ^c¹ 1+r

s₁₂ s₀(1+r) s₁₂ s₁₁ + ^c²

1+r 1 s₁₂ s₀(1+r) s₁₂ s₁₁

= ^c¹

1+rq+ ^c²

1+r (1 q)

= E^Q C

1+r = ¹

1+rE^Q[C].

So, the price of any contingent claim C is the expectation, under the measure Q, of that contingent claim discounted value ₁^C₊_r

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Risk-neutral measure IV

Wonderful! Rather than having to solve an optimization problem in order to determine a contingent claim price, we just have to calculate an expectation, provided the

appropriate probability measure to be placed on our

…ltered measurable space is known.

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Risk-neutral measure I

Why such a name?

The return on a security during the time period going fromt 1 tot is de…ned as

R_S(t,ω) = ^S^t(ω) St 1(ω) S_t ₁(ω)

whereSt(ω)denotes the share price of the security at time t.

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Risk-neutral measure II

Why such a name?

For any probability measure Pb, the expected return on the riskless security over the time period[0,1] is

E^P^b[R_S(1)(1)] =

∑

ω2Ω

S₁⁽¹⁾(ω) S₀⁽¹⁾(ω) S₀⁽¹⁾(ω)

Pb(ω)

=

∑

ω2Ω

(1+r) 1 1 Pb(ω)

=

∑

ω2Ω

rPb(ω)

= r.

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Risk-neutral measure III

Why such a name?

The expected return on the risky security, under the measurePb, is

E^P^b[R_S(2)(1)] =

∑

2 i=1

S₁⁽²⁾(ωi) S₀⁽²⁾(ωi) S₀⁽²⁾(ωi)

Pb(ωi)

= ^s¹¹ ^s⁰

s₀ Pb(ω1) +^s¹² ^s⁰

s₀ Pb(ω2).

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Risk-neutral measure IV

Why such a name?

So, the ”true” expected return on the risky security is the one calculated under the measure P and it is equal to

E^P[R_S(2)(1)] = ^s¹¹ ^s⁰

s₀ P(ω1) + ^s¹² ^s⁰

s₀ P(ω2)

= ^s¹¹ ^s⁰ s0

p+^s¹² ^s⁰ s0

(1 p)

= ^s¹¹ ^s⁰ s₀

s12 s0

s₀ p+ ^s¹² ^s⁰ s₀

= ^s¹² ^s⁰ s0

s₁₂ s₁₁ s0

p

and this latter quantity will be, in general, di¤erent fromr.

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Risk-neutral measure V

Why such a name?

By contrast, the expected return on the risky security, calculated under the measure Q, is

E^Q[R_S(2)(1)]

= ^s¹¹ ^s⁰

s0 Q(ω1) +^s¹² ^s⁰

s0 Q(ω2)

= ^s¹¹ ^s⁰ s₀

s₁₂ s₀(1+r) s₁₂ s₁₁ +^s¹² ^s⁰

s0

1 s₁₂ s₀(1+r) s12 s11

= r

which means that, on the space (Ω,F^,^Q), there is no bene…t associated with risk, the expected return on the risky security is the same as the one on the riskless security.

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Martingale measure I

Why such a name?

Because, on the …ltered probability space(Ω,F^,^F^,^Q), the discounted price processes of the securities ^S⁽

i) t

(1+r)^t :t=0,1 are martingales.

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Martingale measure II

Why such a name?

Proof. In the case of the riskless security,

E^Q

"

S₁⁽¹⁾ 1+r F⁰

#

= E^Q 1+r 1+r F⁰

= 1

= ^S

(1) 0

(1+r)⁰^.

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Martingale measure III

Why such a name?

With respect to the risky security,

E^Q

"

S₁⁽²⁾ 1+r F0

#

= E^Q

"

S₁⁽²⁾ 1+r

#

=

∑2 i=1

S₁⁽²⁾(ω_i) 1+r Q(ωi)

= ^s¹¹

1+rQ(ω1) + ^s¹² 1+rQ(ω2)

= ^s¹¹ 1+r

s₁₂ s₀(1+r) s12 s11

+ ^s¹² 1+r

s₀(1+r) s₁₁ s12 s11

= s0

= ^S

(2) 0

(1+r)⁰^.

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Discount factor

We have seen that, if the model contains no arbitrage opportunities, then there exists a probability measureQ such that the contingent claim price at time 0 is the expectation of the discounted value of its future cash ‡ows

Price=E^Q C 1+r .

But, under the « true» probability measure P,there is no reason to discount the future cash ‡ows with a discount factor based on the riskless interest rate since the contingent claim is a risky asset.

The contingent claim price could be obtained by taking the expectation, under the measure P, of the discounted value of its future cash ‡ow.

The problem in this case is to determine the discount factor.

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The model Measurable space Example Predictable process Trading strategy Example Arbitrage Risk-neutral measure Pricing

The riskless security

The model

The stochastic process n

S_t⁽¹⁾ :t =0,1,2o

represents the evolution of the share price of a riskless security.

Let’s assume that the periodic interest rater is constant during the process observation and that

8^ω 2^Ω^, ^S0⁽¹⁾(ω) =1, then

8^ω 2^Ω^, ^St⁽¹⁾(ω) = (1+r)^t.

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The risky security I

The model

The second stochastic process n

S_t⁽²⁾ :t =0,1,2o , for its part, represents the evolution of the share price of a risky security, S_t⁽²⁾ being the share price of the security at time t.

Since the current share price of such as security is know with certainty,

8^ω2 ^Ω^, ^S0⁽²⁾(ω) =s0 2^R⁺= (0,∞).

We assume that, in the following period, i.e. at t =1, the share price of the risky security can only take one of two values, says₁₁ and s₁₂ (s₁₁,s₁₂ 2^R⁺^,^s¹¹ <s₁₂).

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The risky security II

The model

We assume that, in the second period, i.e. att =2, there are only four possible values for the share price of the security, says₂₁,s₂₂ (if S₁⁽²⁾ =s₁₁), s₂₃ ands₂₄ (if S₁⁽²⁾ =s₁₂) subject to the constraintss₂₁,s₂₂,s₂₃, s24 2R+,s21 <s22,s23 <s24.

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The risky security III

The model

ω S₀⁽¹⁾(ω) S₀⁽²⁾(ω)

!

S₁⁽¹⁾(ω) S₁⁽²⁾(ω)

!

S₂⁽¹⁾(ω) S₂⁽²⁾(ω)

!

ω1 (1,s0)^> (1+r,s11)^> (1+r)²,s21

>

ω2 (1,s₀)^> (1+r,s₁₁)^> (1+r)²,s₂₂ ^>

ω3 (1,s0)^> (1+r,s12)^> (1+r)²,s23

>

ω4 (1,s₀)^> (1+r,s₁₂)^> (1+r)²,s₂₄ ^>

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Filtered measurable space

The model

As we have previously argued, we can justify choosing as sample space Ω=f^ω¹^,^ω²^,^ω³^,^ω⁴g^.

The sigma-algebra F is the set of all events in Ω and the …ltrationF =fF^t ^:^t =0,1,2gconsists of the following sigma-algebras of F^:

F0 =f?^,Ωg^,

F1 =σff^ω1,ω2g^,f^ω3,ω4gg^and F2 =F^.

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Example

The model

Ifr =10%then ω S₀⁽¹⁾(ω)

S₀⁽²⁾(ω)

!

S₁⁽¹⁾(ω) S₁⁽²⁾(ω)

!

S₂⁽¹⁾(ω) S₂⁽²⁾(ω)

!

ω1 (1;2)^> (1,1;2)^> (1,21;1)^>

ω2 (1;2)^> (1,1;2)^> (1,21;3)^>

ω3 (1;2)^> (1,1;4)^> (1,21;1)^>

ω4 (1;2)^> (1,1;4)^> (1,21;5)^>.

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Notation

The riskless security

De…nition

A stochastic processX =f^X^t ^:^t =0,1,2, ...g ^{built on a} measurable space(Ω,F) is said to bepredictable with respect to the …ltrationfF^t ^:^t =0,1,2, ...g ^if^X⁰ ^{is a} (Ω,F⁰) random variable and 8^t 2 f^1,^{2, ...}g^,^X^t ^{is a} (Ω,F^t ¹) random variable.

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Trading strategy I

The predictable stochastic processes n

φ⁽_t¹⁾ :t=0,1,2o and n

φ⁽_t²⁾:t =0,1,2o

represent the evolution of our portfolio over time:

φ⁽¹⁾_t = number of shares of the riskless security held during the time period(t 1,t],

φ⁽²⁾_t = number of shares of the risky security held during the time period(t 1,t].

Exceptionally, the portfolio !

φ0 = φ¹₀,φ²₀ is only held at time t =0. In practice, we will set !

φ0 = ! φ1.