Securitization and Heterogeneous-Belief Bubbles

with Collateral Constraints

Jun Maekawa

November 11, 2013

Abstract

Miller(1977) or Harrison and Kreps(1978) show asset price is higher inheterogeneous model than common prior model. They assume no budgetconstraint or no limitation of financial market. Recent study explore therole of financial technology in heterogeneous belief model. In this paper,I show that some financial technology make the asset price as high asHarrison and Kreps. Key technology is securitization and especially loanbacked security.

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1 Introduction

There are many bubble theory in exonomics. For example, fiat money in over-lapping generation model.(Samuelson(1958))In rational bubble model, investors are willing to hold a bubble asset becausethe price of the asset is expected to rise in the future. A bubble can be sustainedtoday because the bubble is expected to grow in the future, at least as long asthe bubble does not burst.However, robust implication of rational bubble theory is that the price of theasset has to glow explosively. bubble component has to glow in expectation atrate of r(interest rate). Rational bubbles can also ruled out by using a generalequilibrium zero-sum argument.(Kreps(1977),Tirole(1982))

Another class of models relies on heterogeneous beliefs among investors togenerate bubbles. Combining heterogeneous beliefs with short-sale constraintscan result in overpricing. Optimist push up the asset prices, while pessimistscannot couterbalance it because of short-sell constraints. (Miller(1977)) Ofekand Richardson(2003) link this argument to the internet bubble of the late1990s.In dynamic model, the asset price can even exceed the valuation of the mostoptimistic investor in the economy. The currently optimistic investors have theoption to resell the asset in the future at a high price whenever they becomeless optimistic.(Harrison and Kreps(1978))The existence of optimistic agent are the source of the bubble. Asset price isaffected by optimistic beliefs. Harrison and Kreps(1978) consider a dynamicasset pricing model with the heterogeneous beliefs. Investor whose belief aremost optimistic in the period buy or hold the asset. Because they knows thatthe asset will be able to be sold to other agent in future, their estimation ofholding the asset today is higher than their own value. Then the price of theasset can be higher than any investor’s value.However, in Harrison and Kreps(1978), each trader have no budget constraint.All investor have enough cash or they can borrow cash from other traders. Ifsecurity market is not advanced, the investor who have little cash may not par-ticipate in the trade. In this situation, these traders can not affect asset pricing.Optimistic trader, who is considered to be a some kind of irational trader, oftenlose cash in long term trade. Suppose optimistic investor have less cash thanothers, they can not hold assets in equilibrium. As a result, asset prices getlower than Harrison and Kreps(1978).Geanakoplos analyse heterogeneous belief model with collateral constraint. Intheir model, traders need some asset as collateral when they make loan con-tract with other trader. Geanakoplos(1997) and Geanakoplos and Zame(1997)introduce endogenous collateral/margin constraints into a general equlibriumframework of Arrow-Debreu. No payments in future periods/states cannot bepromised without duable assets as collateral. The margin/haircuts of collater-alized borrowing are derived endogenously interaction with equilibrium prices.A key implication of collateral equlibrium is that the market will be endoge-nously incomplete. If collateral is scarece, only a small subset of contracts will

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be traded in equlibrium. There are financial confriction in the market.Simsek(2012) assume asset pricing in heterogeneous belief trader model withcollateral constraint by using Genakoplos model. Optimistic trader evaluate arisky asset very high. He want to buy the asset but he has little cash. Pessimistictrader dont have incentive to buy the asset and has enough cash. Optimist willmake loan contract with pessimist. In Simsek(2012), optimist need to hold someasset as collateral like Geanakoplos model. So, optimist make the asset itselfcollateral and he borrow cash from pessimist.If the asset return is high, the optimistic investor can return cash to pessimist.However, If asset return is low, the asset will be hold by pessimist because theasset is used as collateral. This collateral contract make the asset price lowerthan Harrison and Kreps(1978).In equilibrium, good state is evaluated by optimistic belief and bad state isevaluated by pessimistic belief. In heterogeneous belief model like Harrison andKreps, all area of asset return is evaluated by most optimistic trader. Assetreturn is influenced by optimistic belief and asset price get high. Then incom-pletness of incomplete market allow the asset price to be lower than Harrisonand Kreps.This paper also assume the incomplete financial market. There are heteroge-neous beliefs, optimist and pessimist. Optimists have a little cash, so they mustlend some cash from other investors. Investor must lend cash with collateralcontract. But I asssume some securitization technology to the loan contract.Investor who hold the loan contract can make new security, that is, loan backedsecurity. Investor can receive loan payment at future date, so they can sell theright to receive the payment to others.In my paper, by introducing new financial technology, securitization, and simpledynamic structure, asset prices in Simsek(2012) economy is as high as HK(1978).Simsek model is static, one generation model. If there are many generation,many optimist and pessimist, traders may have some speculative incentive tohold asset. In HK(1978), traders hold assets by strong speculative incentiveand asset prices get higher than any trader’s expected return of assets. Assumemulti generation model, there must be some financial technology for participat-ing in trading. Both in Simsek(2012) and my paper, risky asset is only one typeand asset supply is one. Optimist hold asset and doesnot have incentive to sellit. Then, other optimist have nothing to do for their profit. There is essentiallyinter-temporal friction. Securitization is a financial technology that allow manytraders to participate in the market. In my paper, traders can sell the right toreceive a loan payment to other traders by securitization. Pessimist who makeloan contract with optimist can sell the security which have the return equalto loan payment to other optimist. New optimist also make loan contract withpessimist for buying this security. Pessimist can securitize this new loan con-tract and sell it to other new optimist and so on.This security make the loan contract itself riskless. Because the lender of theloan contract can sell that security to some more optimistic one, the lender neednot to hesitate to lend cash to optimistic trader. For lender of the loan contract,pessimist, this security is also raise their utility. They find new investment by

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securitization.However, this security introduce to the asset price, bubble. This system allowthe optimistic belief to evaluate the asset. I will insist that the asset price maybe as high as Harrison and Kreps(1978). Pessimist have speculative incentiveto lend cash to optimist and optimist buy the asset. These situation is repeatedin each generation. Finally, optimistic trader can buy the asset with very highprice.At first, I will explain one generation model like Simsek(2012) at section 2.In equilibrium, the price formula is caluculated. Asset price lies between pes-simistic expected return and optimistic expected return. This price is importantfor understanding multi generation.At section 3, multi generation model (with single oprimist type) will be ex-plained. In multi generation model, asset price gets higher than Simsek(2012).At subsection 3-1, two generation model is analyzed. Two generation model ishelpful to understand multi generation model. Securitization is introduced toeconomy and new optimist and new pessimist come to market. Asset price ishigher than one generation case. Multi generation model is dicussed at subsec-tion 3-2. Two generation case is easily repeated and developed to general multigeneration case. In this time, asset price is equal to optimistic expected return.At section 4, I will introduce various type optimists to multi generation model.In this model, there are many type of optimist who have different frequencyabout asset return state. At each generation, security is sold to optimist whohave a highest value about it. Because this scheme allow speculative action, as-set price gets high. Surprisingly, the asset price exceeds the optimistic expectedreturn like Harrison and Kreps(1978).

In my paper, optimistic trader behave like ”noise trader”(DeLong, Shleifer,Summers and Waldmann(1990)). Although there are many pessimistic traders,optimistic trader’s influence is very large. Pessimist can get high return by ex-ploiting optimistit by the financiual technology.My reserch is a part of theory that concerns the borrowing constraints on assetprices like Shleifer and Vishny(1992, 1997), Kiyotaki and Moore(1997),Gromband Vayanos (2002), Brunnermeier and Pedersen (2009). Financial technologycan eliminate the borrowing constraint by speculative incentive of traders. Inheterogeneous belief model, borrowing constraint is a important role for pre-venting bubble economy.The optimist pushes up the asset price in heterogeneous belief model(Miller(1977),Harrisonand Kreps(1978),Scheinkman and Xiong(2003)). Many paper by Genakop-los(1997,2003,2010), or Fostel and Genakoplos(2012) explore the role of ccreditin heterogeneous belief bubble. Simsek(2012) and this paper is also one of thestudy of the role of financial inovation.

4

lend cash

sell contract

pessimist 0optimist 0

buy asset

asset market

2 One Generation Model:Simsek(2013)

Assume there are only one period, no trader comes at date 1 like Simsek(2012).There is a continuum of states at date 1, denoted by s ∈ S = [0, smax]. Theasset pays s dollars at state s. There are two type of traders, optimist andpessimist. Both type agents have risk neutral utility function. But, they havedifferent belief about asset return. Trader type j have piror belief about states, Fj(s).(j = o, p) Note their expectation Eo[.], Ep[.]. Traders know each other’spriors, that is, optimists and pessimists agree to disagree.Optimists are optimistic about the asset return. Assume first order stochasticdominance. Optimist’s distribution function is lower than pessimistic one forall state s.

Assumption 1 Fp[s] ≥ Fo[s] for all s

Assumption imply optimists are natural buyers of the asset.Optimists are initially endowed with n > 0 dollars and zero unit of the asset.

Because n is not larege, optimists might want to borrow cash from pessimists.

Assumption 2 n < Eo[s]

Pessimists have a plenty of cash. They are natural lenders of cash. As thereare only one asset in this market, Lending contract is only way for pessimiststo earn cash in the economy.All borrowing contract in this economy is subject to a collateral constraint. Thatis, promise made by borrowers must be collateralized by the asset or the cashthat they own. Traders choose their positions in all contracts taking the pricesas given. Optimists promise the payment ϕ dollars at next period. Pessimistslend (φ) units of cash to optimists. In addition to the promise payment varphi,optimists must collateralized the asset itself. If low return state realize (s < ϕ),optimists give the asset to pessimists. Then, for pessimists, this loan contracthave payoff Ep[min(s, ϕ)].

Let a be the optimist’s demand of the asset and p the price of the asset. Atdate 0, the optimist buy the asset in the market and the pessimist lends cashto him. The optimist make the loan contract with collateral with the pessimist.So, optimist problem is:

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maxa,ϕ

a(Eo[s]− p− Eo[min(s, ϕ)]− φ)

s.t.ap = n+ aφ

Because there are many pessimists in the market and they have enoughmoney, pessimists competition imply that the quantity of cash lending φ isequal to the contract’s payoff.

φ = Ep[min(s, ϕ)]

So, Optimist solve their problem:

maxa,ϕ

aEo[s]− aEo[min(s, ϕ)]

s.t.ap = n+ aEp[min(s, ϕ)]

By calculating, asset price p is caluculated.

p =

∫ ϕ

0

sdFp + (1− Fp(ϕ))

∫ smax

ϕ

sdFo

1− Fo(ϕ)

Or equivalentry,

p = Fp(ϕ)Ep[s|s < ϕ] + (1− Fp(ϕ))Eo[min(s, ϕ)|s = ϕ]

Asset supplyn is one, so the budget constraint of the optimist characterizeshis asset demand a.

p = n+ Ep[min(s, ϕ)]

By these two equations, the equilibrium ((p, ϕ) are determinded.the first equation has important implication about asset price determinance.

If high return state is realized at period 1, the optimist hold the asset. If lowreturn state is realized and the optimist’s default occurs, the pessimist hold theasset. The asset return in low states is evaluated by pessimistic belief Fp andthe return in high states is evaluated by optimistic belief Fo.

In Miller(1977) and Harrison and Kreps(1978), the asset prices are deter-minded by the optimistic beliefs. In this case, Eo[s]. Because the optimist facesthe budget constraint, the asset price is influenced by his belief and by pes-simistic belief. If the optimist has enough cash or he can borrow cash withoutlimit, the price equal to Eo[s].

For the pessimist, their payoff from the loan contract is min(s, ϕ) at period1. For assumption, first order stochastic dominance, this payoff is more valuablefor optimists:

Eo[min(s, ϕ)] > Ep[min(s, ϕ)]

6

q∗

ϕ∗

q

ϕ

p =∫ ϕ

0sdFp + (1− Fp(ϕ))

∫ smax

ϕs dFo

1−Fo(ϕ)

p = n+ Ep[min(s, ϕ)]

n

γ

optimist

pessimist

state

return

0 s

γ

7

date 0

date 1

date 2

optimist and pessimist come

asset market open

new optimist and pessimst come

state s realize

Then, pessimists have incentive to sell the right to receive the payoff atperiod 1 to some optimists if possible. At next section, multi generation model,i will analyze the case.

3 Multi Generation Model

In this section, i will introduce the new financial technology, ”securitization”to Simsek(2013). Pessimists can sell securities, which payoff is backed by loancontracts. At first, i will explain the two generation model. In addition toSimsek’s model, there is one more generation. The pessimist who lent cashto the optimist at date 0 can sell a security whose payoff is min(s, ϕ) at thenext date. In the two generation model, i show the asset price is raised by thesecuritization technology.

3.1 Two Generation Model

The asset pays s dollars at state s. s ∈ S = [0, smax]. There are two type oftraders, optimists and pessimists. Trader type j have piror belief about states, Fj(s).(j = o, p) Both traders know each other’s priors, that is, optimists andpessimists agree to disagree. Assume first order stochastic dominance as onegeneration model.Time line is different from Simsek(2013). The asset return is realized at date 2.At date 0, there exists one optimist and chosen pessimist. The asset is suppliedto the economy.At date 1, a new optimist comes and he choose a new pessimist.At date 2, state s is revealed, the asset payoff is distributed a to traders ascontracted.

Optimists are initially endowed with n > 0 dollars and zero unit of the asset.At date 0, the date 0 optimist make a loan contract with the pessimist. The

loan contract is defined (ϕ0, φ0). ϕ0 is the promise payment by the optimistand the pessimist lends cash φ0 to the optimist.

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lend cash

sell contract

sell security

lend cash

sell contract

pessimist 1

pessimist 0optimist 0

optimist 1

buy asset

asset market

At date 1, there is no asset supply. But One unit of a security is sold in themarket.The date 0 pessimist can securitize the right to receive payoff min(s, ϕ) at date1 with price q1. Because Eo[min(s, ϕ)] > Ep[min(s, ϕ)], the date 1 optimist hasan incentive to buy this security from the date 0 pessimist. The optimist alsohas n units of cash.

Assumption 3 n < Eo[min(s, ϕ0)]

Because n < Eo[min(s, ϕ0)], the optimist cannot buy the security by hisown cash. The date 1 optimist can also borrow cash from the pessimist atdate 1 with collateral constraint. Let ϕ1 and φ1 be the promise payment andborrowing cash in the loan contract at date 1.At date 2, s is realized, each optimist receives the asset return and he pays thepromised payment to each pessimist as contracted before.

It is natural that ϕ0 > ϕ1 in equilibrium.The date 1 optimist borrow cash for buying the security whose payoffmin(s, ϕ0).This relation imply the asset return distribution in equilibrium.If a high return state is realized, the date 0 optimist and the date 1 optimistcan pay back the cash to pessimists.If a medium return state is realized, the date 1 optimist can return cash but thedate 0 optimist cannot.If a low return state realize, both optimists cannot.The security price q1 is determinded by the security payment min(s, ϕ0) and thecontract min(s, ϕ1) at date 1. The date 0 optimist chooses their asset positionand outstanding debt to solve:

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ϕ

state

return

0 s

ϕ

maxa0,ϕ0

a0(Eo[s]− p− Ep[min(s, ϕ0)] + φ0)

s.t.a0p = n+ a0φ0

The date 0 pessimist lend cash to the date 0 optimist. The optimist paymin(s, ϕ0) to the pessimist if state s is realized.

The date 1 optimist’s problem can be denoted like the date 0 optimist.The date 1 optimist’s problem is:

maxa1,ϕ1

a1(Eo[min(s, ϕ0)]− q1 − Eo[min(s, ϕ1)] + φ1)

s.t.a1q1 = n+ a1φ1

The problem is similar to Simsek’s model.In one generation model, the asset return is s. In this time, security payoffis min(s, ϕ0). By substituting min(s, ϕ0) to s and q1 to p, this is just theoptimist’s problem in Simsek(2013).

Because there are many pessimists in the market and they have enoughmoney at date 1, pessimists competition imply that the quantity of cash lendingφ1 is equal to the contract’s payoff.

φ1 = Ep[min(s, ϕ1)]

At date 0, there are many pessimists. The pessimist who lend cash to thedate 0 optimist can sell the security at date 1 with price q1. Pessimists’ compe-tition imply no arbitrage condition.

φ0 = q1

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Then, the date 0 optimist problem is rewritten.

maxa0,ϕ0

a(Eo[s]− p− Eo[min(s, ϕ0)] + q1)

s.t.a0p = n+ a0q1

The date 1 optimist problem is rewritten:

maxa1,ϕ1

a1(Eo[min(s, ϕ0)]− q1 − Eo[min(s, ϕ1)] + Ep[min(s, ϕ1)])

s.t.a1q1 = n+ a1Ep[min(s, ϕ1)]

By solving the each date optimist problem, equilibrium (ϕ0, ϕ1, q1, p) arecaluculated. From the date 1 optimist problem, the security price q1 is:

q1(ϕ0, ϕ1) =

∫ ϕ1

0

sdFp + (1− Fp(ϕ1))

∫ smax

ϕ1

min(s, ϕ)dFo

1− Fo(ϕ1)

Or equivalentry,

q1(ϕ0, ϕ1) = Fp(ϕ1)Ep[s|s < ϕ1] + (1− Fp(ϕ1))Eo[min(s, ϕ0)|s = ϕ1]

This formula shows that optimism is asymmetrically disciplined in equilib-rium like Simsek(2013). Pessimistic beliefs assess the value of the asset con-ditional on default, while optimistic beliefs are used to assess the value of theasset conditonal on no default. By considering asset payoff min(s, ϕ1) insteadof s, this is the same expression in Simsek(2013).If asset return exceed ϕ1, the date 1 optimist can repay the loan payment tothe date 1 pessimsit. But if not, the date 1 optimist must give the security tothe date 1 pessimist. So, this formula imply that low return state is evaluatedby pessimist and high return state is evaluated by optimist just like asset priceof one generation model.

The security supply is one, so the budget constraint of problem characterizesoptimists’ asset demand:

q1(ϕ0, ϕ1) = n+ Ep[min(s, ϕ1)]

These two equations determine the equilibrium q1 and ϕ1.By backward induction, the date 0 optimist problem can be solved.

p = q1 +1− Fp(ϕ0)

1− Fo(ϕ0)

∫ smax

ϕ0

sdFo

Because asset supply is one, supply function is written:

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ϕ0

state

return

0 s

ϕ0

ϕ1

ϕ1

optimist

pessimist

p = n+ q1

These two equation determine the equilibrium p and ϕ0.Let pS be the asset price of one generation model. Next proposition show

the asset price at two generation model gets higher than Simsek’s price.

proposition 1 In two generations case, the asset price exceeds the asset priceof one generation case: p > pS

Proof 3.1 From the date 0 optimist problem, the asset price at two generationmodel:

p = q1 +1− Fp(ϕ0)

1− Fo(ϕ0)

∫ smax

ϕ0

sdFo

From the date 1 optimist problem, the security price q1:

q1(ϕ0, ϕ1) =

∫ ϕ1

0

sdFp + (1− Fp(ϕ1))

∫ smax

ϕ1

min(s, ϕ)dFo

1− Fo(ϕ1)

These two equations imply that the asset price p is rewritten:

p =

∫ ϕ1

0

sdFp +1− Fp(ϕ1)

1− Fo(ϕ1)

∫ smax

ϕ1

sdFo

If ϕ1 < ϕ (ϕ is the loan promise from Simsek(2013)), the asset price ishigher than Simsek(2013).

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γ1

optimist0

pessimist

optimist1

γ0

γ0

γ1

From the date 1 optimist’s budget constraint:

q1(ϕ0, ϕ1) = n+ Ep[min(s, ϕ1)]

From the date 0 optimist’s budget constraint:

p = n+ q1

These two equations imply the asset price p:

p = 2n+ Ep[min(s, ϕ1)]

Assume ϕ1 ≥ ϕ. Two budget constraints imply:

p = 2n+ Ep[min(s, ϕ1)] ≥ 2n+ Ep[min(s, ϕ)] ≥ n+ Ep[min(s, ϕ)] = pS

This is contradicted by the price formulation. Then, ϕ1 is smaller than ϕand p > pS.

The proposition show the asset price at date 0 is larger than Simsek(2013).For the date 0 pessimist, lending cash to optimist is very profitable investment.By making loan contract, the pessimist can sell security at date 1. That is,pessimist has speculative incentive for loan contract.In one generation model, high asset return states(s > ϕ) are evaluated byoptimistic belief and low return states (s < ϕ)are evaluated by pessimisticbelief.In two generation model, ϕ1 < ϕ. High states upper than ϕ1 are evaluated byoptimists.Optimistic belief dominates wider area of asset return than Simsek(2013).

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ϕt−1

state

return

0 s

ϕt−1

3.2 Multi Generation Model

If security market is large and the security technology evolve, more investorparticipate this scheme. The scheme can be extended by a simple way. In thissection, i will introduce multi generation model.This is multi date or multi generation model. The asset state is realized at dateT (T is some large natural number). At date 0, the date 0 optimist buys theasset and he borrows cash from the pessimist. The date 0 pessimist securitizethe loan and make a new security that pay min(s, ϕ0). the date 1 optimst buythe security and he borrow cash from a new pessimist at period 1. The date 1pessimist also securitizes the loan contract and he sells the new security to newoptimist at period 2.From date (t=1,2,3,..), the date t optimist borrows cash from the date t pes-simist. He buys the security that is securitized by the date t-1 pessimist withprice qt. The date t pessimist also securitizes the loan contract that is madewith the date t optimist and he sells it to the date t+1 optimist with price qt+1.The date t loan contract is defined by (ϕt, φt). The date t optimist promisesthe payment ϕt at date T and the date t pessimist lends cash φt to him.Because the optimist collateralizes the security whose payoff is min(s, ϕt−1),the contract’s payment is min(s, ϕt).

At each date T − 1 > t ≥ 0, there are many pessimists and they competewith each other. The date t pessimist can sell the loan contract to the date t+1optimist with price qt+1. Then, no arbitrage condition determine the level oflending φt.

φt = qt+1

At date T − 1, the pessimist cannot sell the loan contract at next date T .So, his no arbitrage condition:

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φT−1 = Ep[min(s, ϕT−1)]

The each date optimist problem is written like two generation model.The date t optimist problem is written(t ≥ 1) :

maxat,ϕt

at[Eo[min(s, ϕt−1)]− qt − atEo[min(s, ϕt)] + φ]

s.t.atqt(ϕt−1, ϕt) = n+ atφt

The date 0 optimist problem:

maxa0,ϕ0

a0[Eo[s]− p− Eo[min(s, ϕ0)] + φ0]

s.t.a0p(ϕ0) = n+ a0φ0

Proposition 3.2 In equlibrium, the security price at date t:

qt = qt+1 +1− Fp(ϕT−1)

1− Fo(ϕT−1)

∫ smax

ϕt

min(s, ϕt)dFo

The security price at date T − 1:

qT−1 =

∫ ϕT−1

0

sdFp +1− Fp(ϕT−1)

1− Fo(ϕT−1)

∫ smax

ϕT−1

min(s, ϕT−2)dFo (1)

The equlibrium date t promise payment ϕt, the asset price p and the securityprice qt are caluculated by the date t optimist problem, the date t pessimistcompetition and the market clearing condition (at = 1).

Proof 3.3 From the date T-1 pessimist competition:

φT−1 = Ep[min(s, ϕT−1)]

The date T-1 optimist problem is rewritten:

maxaT−1,ϕT−1

aT−1[Eo[min(s, ϕT−2)]− qT−1 − Eo[min(s, ϕT−1)] + Ep[min(s, ϕT−1)]]

s.t.aT−1qT−1(ϕT−2, ϕT−1) = n+ aT−1Ep[min(s, ϕT−1)]

By solving the problem, the date T-1 security price is caluculated:

qT−1 =

∫ ϕT−1

0

sdFp +1− Fp(ϕT−1)

1− Fo(ϕT−1)

∫ smax

ϕT−1

min(s, ϕT−2)dFo

From the date t pessimist competition:

φt = qt+1

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ϕt−1

state

return

0 s

ϕt−1

ϕt

ϕt

optimist at date t

qt+1(ϕt)

The date t optimist problem is rewritten:

maxat,ϕt

at[Eo[min(s, ϕt−1)]− qt − Eo[min(s, ϕt)] + qt+1]

s.t.atqt(ϕt−1, ϕt) = n+ atqt+1

By solving the problem, date t price is calculated:

qt = qt+1 +1− Fp(ϕT−1)

1− Fo(ϕT−1)

∫ smax

ϕt

min(s, ϕt)dFo

The security price equation imply that the security return area is dividedinto two partitions. The return area above ϕt is evaluated by the date t optimistand the area under ϕt is evaluated by the next date security price qt+1.

The asset price is caluculated by solving the date 0 optimist problem. Fromthe date 0 pessimists competition:

quation φ0 = q1quationThe date 0 optimist problem is rewritten:

maxa0,ϕ0

a0[Eo[min(s, ϕ0)]− p− Eo[min(s, ϕ0)]− q1]

s.t.a0p(ϕ0) = n+ a0q1

This problem and market clearing condition at date 0 (a0 = 1) determinethe equlibrium asset price p.

In equlibrium, the asset price p:

p =

∫ ϕT−1

0

sdFp +1− Fp(ϕT−1)

1− Fo(ϕT−1)

∫ smax

ϕT−1

sdFo

16

Proof 3.4 From the date 0 optimist problem:

p = q1 +1− Fp(ϕ0)

1− Fo(ϕ0)

∫ smax

ϕ0

sdFo

From the security price at date 1:

q1 = q2 +1− Fp(ϕ1)

1− Fo(ϕ1)

∫ smax

ϕ1

min(s, ϕ0)dFo

By repeating the recursive calculation:

q1 =

∫ ϕT−1

0

sdFp +1− Fp(ϕT−1)

1− Fo(ϕT−1)

∫ smax

ϕT−1

min(s, ϕ0)dFo

By substituting the equation to q1, the asset price p:

p =

∫ ϕT−1

0

sdFp +1− Fp(ϕT−1)

1− Fo(ϕT−1)

∫ smax

ϕT−1

sdFo

The asset price p depends on the promise payment at date T −1, ϕT−1. Theasset return areas upper than ϕT−1 are evaluated by optimistic belief. If ϕT−1

is low, the asset price is high.The security payoff at date t is min(s, ϕt−1). The date t optimist has cash n.If Eo[min(s, ϕt−1)] ≤ n, the optimist can buy the security without the loancontract. Assume there is the date t when Eo[min(s, ϕt−1)] ≤ n in equilibrium.Let t′ be the first date when n exceeds Eo[min(s, ϕt′−1)]. After t′ + 1, thereis no security supply. The optimists need not to make loan contract with thepessimists. At date t′. the sequlity price is simply equal to the optimisticexpected return:

qt′ = Eo[min(s, ϕt′−1)]

Because the optimist does not need to make the loan contract, the promisepayment ϕt′ = 0. After date t′, there is no security supply in the market. Then,the security price t > t′ is equal to zero. The date t (t > t′) optimist does notmake the loan contract. The promise payment at date t ≥ t′:

ϕt = 0

If the date t′ exists, the maximum asset price is determinded. If T is large,the assset price is as high as the asset price of Harrison and Kreps(1978).

Proposition 3.5 If T is large enough(T −1 ≥ t′), the equlibrium asset price isequal to Harrison and Kreps(1978):

p = Eo[s]

17

Proof 3.6 At date t′, the optimist buy the security by his cash. The securityprice is equal to the optimistic expectation.

qt′ = Eo[min(s, ϕt′−1)]

The date t′ − 1 optimist problem:

maxat′−1,ϕt′−1

at′−1[Eo[min(s, ϕt′−2)]− qt′−1 − Eo[min(s, ϕt′−1)] + Eo[min(s, ϕt′−1)]

s.t.at′−1qt′−1(ϕt′−2) = n+ atEo[min(s, ϕt′−1)]

By solving the problem:

qt′−1 =

∫ ϕt′−1

0

sdFo + (1− Fo(ϕt′−1))

∫ smax

ϕt′−1

min(s, ϕt′−2)dFo

1− Fo(ϕt′−1)

= Eo[min(s, ϕt′−1)]

By repeating backward induction, the security price at date t′ > t > 0:

qt = Eo[min(s, ϕt−1)]

The date 0 optimist problem:

maxa0,ϕ0

a0[Eo[s]− p− Eo[min(s, ϕ0)] + Eo[min(s, ϕ0)]]

s.t.a0p(ϕ0) = n+ atEo[min(s, ϕ0)]

By solving the problem, the asset price p:

p = Eo[min(s, ϕ0)] +1− Fo(ϕ0)

1− Fo(ϕ0)

∫ smax

ϕ0

sdFo

= Eo[s]

From theorem 1, the asset price p:

p =

∫ ϕT−1

0

sdFp +1− Fp(ϕT−1)

1− Fo(ϕT−1)

∫ smax

ϕT−1

sdFo

By substituting ϕT−1 = 0 to this equation, the asset price is equal to Eo[s].The asset price is valued by the optimistic belief.

At time T , the state s is realized. smax > ϕ1 > ϕ2 > .... > ϕt′−1 > smin.Let t∗ satisfying ϕt∗−1 > s > ϕt∗ . The realized payoff for the date t optimist is:

1. t < t∗ : 0

2. t = t∗ : s− ϕt∗

18

lend cash

sell contract

sell security

lend cash

sell contract

pessimist 1

pessimist 0optimist 0

optimist 1

lend cash

sell contract

pessimist 2optimist 2

buy asset

γ1

optimist0

optimist1

optimist2other optimists

γ0 state

return

19

s

return

s

ϕ∗t

3. t′ ≥ t > t∗: ϕ∗t − ϕt∗+1

4. t > t′ : 0

The asset is evaluated by the each date optimist. As noted in two generationcase, the area which are evaluated by optimist gets wider in multi generationmodel.The asset price is the same as Harrison and Kreps(1978). In Harrison andKreps(1978), the asset price may be higher than any trader’s expectation. Be-cause the asset is hold by the most optimistic investor in each period, the assetprice is higher than anyone’s expectation. In this paper, the asset price is sharedamong optimistic traders. Harrison and Kreps assume complete market. So, anytrader have enough cash or rent cash in each period.

They can hold asset when their expectaion is highest in the market.In Simsek(2013), the optimist does not have enough cash, he must borrow cashfrom pessimist. Because loan contract is limited, he must collateralize the assetitself. So the asset price must be influenced by pessimistic belief.In my paper, the optimists are distributed among generations. They have in-centive to buy the asset, but they does not have enough cash. Without thesecurity market, they have no way to cooperate with each other. Security andloan contract play role for helping their cooperation.The loan contract causes the security and it causes the loan contract. Thescheme allows the each optimist to participate in the market.

Pessimistic belief vanishes in the asset price equation. However, if there isno pessimist, the date 0 optimist have no way to bring enough cash to buy theasset. There are many pessimists in the security market.The pessimists know that the new optimists will come to the market at the

20

next date. So, they have strong incentive to lend their cash to the optimists.Lending cash is speculative action for the pessimist. The pessimist can sell therisk of the optimist’s default to the other optimist by securitization. It is oneof risk-shifting problem.(Shleifer and Vishny(1992))As a result, the asset return is shared by the optimists and the asset price rises.This security raise the utility for both optimists and pessimists. The scuritytechnology compensate for incomplete financial market.The asset price is influenced by optimistic belief. High asset price imply muchoptimist participate in the markets. From budget constraint of the date 0 opti-mist:

ap = n+ aq1

Asset supply is one a = 1:

p = n+ aq1

From the budget constraint of optimist at date 1 and security supply is one:

q1 = n+ q2

q2 is security price at date 2. Then, budget constraint at date 0 imply:

p = n+ n+ q2

By repeating caluculation, budget constraint imply asset price is sum ofoptimist cash n.

p = n + n + n + ... + n + Eo[min(s, ϕt′)] = t′n+ Eo[min(s, ϕt′)]In equilibrium, asset price is Eo[s]. Then, the number of optimists who buy

asset or securities (that is t′) is:

t′ =Eo[s]− Eo[min(s, ϕt′)]

n

t′ optimists buy each area of asset return in equilibrium. If there is nosecurity market, only one optimist and one pessimist participate in the market.Many optimists can participate in economy by these securitization market.

In general, if there are heterogeneous beliefs exists, completeness of securitymarket make economy riskier. The advanced security market allow heteroge-neous investor to act freely, so their action cause variety effect. Optimist canact more optimistically, and pessimist can also act more pessimistically.In the next section, I will show that asset price exceeds any traders’ expectationin various optimist setting. The intution is very simple. In multi generationcase, the each date optimist buys the security and he evaluates each area of theasset return. If various optimists buy the each security, asset return is evaluatedby many optimistic belief. Because the each date security is bought by a traderwho has the most optimistic belief about the security return, the security priceis higher than single type optimist case.In this section, asset price is Eo[s]. This is one case of Harrison and Kreps(1978),

21

smax

1

0

F (s)

s

FpFou

Fod

one type optimist case. In the next section, asset price exceed optimist’s expec-tation:

p > Eo[s]

Like general case of Harrion and Kreps(1978), this is the bubble caused byheterogeneous belief.

4 Multi Generation with Various Optimists

In this section, I will show that the asset prices exceed any traders’ expectstionsin various optimistitic types case. Model settings are almost same as multigenerations case.For simplicity, I will give two type optimists case. It is helpful to understandmulti generation with various optimists model.

4.1 Example:Two Type Optimist Case

In addition to multi generation case, there are two type optimists in this model,upside optimist and downside optimist. Optimists type j have optimistic beliefFj(s).(j = Ou, Od)(Note expectation EOu ,EOd

) Optimists have same expecta-tion about the return of the asset (EOu [s] = EOd

[s] = Eo[s]). Upside optimiststhinks bad event is unlikely and downside optimists think good event is unlikely.(see figure.)

Both optimistic beliefs first order stochastic dominates pessimistic belief.

Assumption 4 Fp[s] ≥ Fj [s] for all s and j = Ou, Od

The asset state is realized at date T (T is some large natural number). Atdate 0, there exists two optimists and one pesimist. The pessimist is chosen byoptimist from many pessimists. There are three type of traders at each date..

22

The upside optimist, the downside optimist and the pessimist. Both optimistsare initially endowed with n > 0 dollars and zero unit of the asset.

All borrowing in this economy is subject to a collateral constraint. The loancontract made by optimists must be collateralized by the asset or the security.Because the asset or the security supply is one, only one optimist contracts withthe pessimist and he buys the asset.This scheme ends at the date when some optimist buys the security by his owncash. At period t, an optimist type j borrow cash and he buys the security. Thepessimist who lend cash to optimist at period t can securitize the loan contractand sell the security to optimist at date t+1 with price qt+1(ϕt). The pessimistscompetition imply:

φt = qt+1φT−1 = Ep[min(s, ϕT−1)]

Then, the optimist problem who buys the security at date t:

maxat,ϕt

at[EOj [min(ϕt−1, ϕt)]− qt − EOj [min(s, ϕt)] + qt+1(ϕt)

s.t.atqt(ϕt−1, ϕt) = n+ atqt+1(ϕt)

The security supply is one. By substituting at = 1 to budget constraint:

qt(ϕt−1, ϕt) = n+ qt+1(ϕt)

The optimist problem and this equation solve the sequirty price.Because asset prices are very heavily influenced by the belief of the asset

holder, who buys the asset or the securities at each date is very importantproblem. As seen the multi generation case, at some period t′, the optimistbuys the security by his cash. This security’s payoff is min(s, ϕt′).From the optimist type definition, EOd

[min(s, ϕt′)] > EOu [min(s, ϕt′)].The downside optimist have higher evaluation about min(s, ϕt′). Then, thedate t′ security must be bought by the downside optimist. For the same reason,we can guess that the asset is bought by the upside optimist at date 0.The upside optimist thinks upper return state is likely and downside optimistthinks opposite.Then some period t̄(0 < t̄ < t′) exists. For t < t̄, the asset and securities arebought by upside optimists. For t ≥ t̄, the securities are bought by downsideoptimists.

By backward induction, the original asset price is calculated. In this time,the asset price is higher than the multi-generation case with single optimisttype.

proposition 2 In multi-generation with two type optimist model, the asset priceexceeds any optimistic trader’s expecatation of asset return:

p > Eo[s]

23

optimisitOu

optimistod

optimistu or d

lend cash

sell contract

pessimistoptimist 0u

optimist Od

buy asset

buy security

at period 0

at period t̄

24

Proof 4.1 As noted above, the security price at date t′ is Eod [min(s, ϕt′−1)].Assume optimist type j buy the security at date t′ − 1. Optimist problem at

date t′ − 1 is:

maxat′−1,ϕt′−1

at′−1[EOj [min(s, ϕt′−2)]− qt′−1 − EOj [min(s, ϕt′−1)] + EOd[min(s, ϕt′−1)]]

s.t.at′−1qt′−1 = n+ at′−1EOd[min(s, ϕt′−1)]

Like muti-generation case with single optimist type, if downside optimistbuy the security at date t′ − 1, the security price qt′−1(ϕt′−2, ϕt′−1) is simplyEOd

[min(s, ϕt′−2)]. By simple backward induction, the security price at date t̄is downside optimist’s expected return.

qt̄(ϕt̄−1, ϕt̄) = EOd[min(s, ϕt̄−1)]

The date t̄ optimist problem:

maxat̄−1,ϕt̄−1

at̄−1[EOu [min(s, ϕt̄−2)]− qt̄−1 − EOu [min(s, ϕt̄−1)] + EOd[min(s, ϕt̄−1)]]

s.t.at̄−1qt̄−1 = n+ at̄−1EOd[min(s, ϕt̄−1)]

This equation is very similar to the problem of the single optimist type caseat period t̄−1. If both types’ expectations about min(s, ϕt̄−1) are the same value,this problem has very simple solution:qt̄−1(ϕt̄−2, ϕt − 1) = EOu

[min(s, ϕt̄−2)]By solving the problem, the security price at date t̄− 1 is calculated.

qt̄−1 =

∫ ϕt̄−1

0

sdFOd+ (1− FOd

(ϕt̄−1))

∫ smax

ϕt̄−1

min(s, ϕt̄−2)dFOu

1− FOu(ϕt̄−1)

The security supply is one. Substitute a = 1 to budget constraint, the supplyfunction is calculated.

qt̄−1(ϕt̄−2) = n+ EOd[min(s, ϕt̄−1)]

These two equations determine the security price at date t̄−1. From assump-tion, downside optimists evaluate min(s, ϕt̄−2) higher than upside optimists.That is:

EOd[min(s, ϕt̄−1)] > EOu [min(s, ϕt̄−1)]

By comaring to the single optimist type case, upside optimist at date t̄ − 1have bigger budget. Then, the security price qt̄−1(ϕt̄−2) is higher than singleoptimist type case:

qt̄−1(ϕt̄−2, ϕt̄−1) > Eou [min(s, ϕt̄−2)]

At date t̄− 2, the upside optimist buys the security. The date t̄− 2 optimistproblem:

25

maxat̄−2,ϕt̄−2

at̄−2[EOu [min(s, ϕt̄−3)]− qt̄−2(ϕt̄−3, ϕt̄−2)− Eou [min(s, ϕt̄−2)] + qt̄−1(ϕt̄−2, ϕt̄−1)]

s.t.at̄−2qt̄−2 = n+ at̄−2qt̄−1

By solving this problem, the security price at period t̄− 2 is calculated. Be-cause qt̄−1(ϕt̄−2) > Eou [min(s, ϕt̄−2)] , the security price at period t̄−2 is higherthan single optimist type case:

qt̄−2(ϕt̄−3) > Eou [min(s, ϕt̄−3)]

By repeating caluculation, the date 1 sequrity price:

q1(ϕ0) > Eou [min(s, ϕ0)]

Upside optimist problem at period 0 is:

maxa0,ϕ0

a0[Eou [s]− p− Eou [min(s, ϕ0)] + q1(ϕ0)]

s.t.a0p = n+ a0q1

Because q1(ϕ0) > EOu [min(s, ϕ0)], The upside optimist have bigger budgetthan single type optimist case. In single type optimist case, the asset price isEOu [s] = Eo[s]. So, in multi generation with two type optimist case, price exceedthe optimistic expectation:

p > Eo[s]

In multi-generation with single optimist type case, asset price is equal tothe optimistic expectation. The optimists holds the asset and the securities.Because there are two type optimists, upside and downside, high return statesare evaluated by upside optimist and low rturn states are evaluated by downsideoptimist.

4.2 Various Optimist Types Case:Setting

Asset price exceeds both optimists’ expectation in two type optimists case. Ianalyse the general case, multi generation with various optimists model.

There are J + 1 type of traders in this economy. J types of optimists andone type of pessimist. The optimist’s type j have optimistic belief FOj (s).All types of optimists have same expectation about the return of the asset(EOj [s] = Eo[s]∀j ∈ J).

Each optimistic belief Fj first order stochastically dominates pessimistic be-lief Fp.

Assumption 5 FOj [s] ≥ Fp[s] ∀s and ∀j ∈ J

26

upside optimist

downside optimist

s

return

0

smax

1

0

F (s)

s

Fp

various optimists

27

The asset, the securities and the loan contracts is the same as multi genera-tion model. At date 0, there exists J optimists and one pessimist. The pessimistis chosen by the optimist from many pessimists. Each optimist is initially en-dowed with n > 0 dollars and zero unit of the asset. Assume the pessimists haveenough cash to lend the optimists. All borrowing in this economy is subject toa collateral constraint. Because asset supply is one, only one optimist contractswith the pessimist and buys the asset.So, optimist type j problem at date 0 is:

maxa0,ϕ0

a0[EOj [s]− p− EOj [min(s, ϕ0)]− φ0]

s.t.a0p = n+ a0φ0

The optimist who bids the highest price to the asset buys it. Let pj be thebidding price of type j optimist. Note j∗0 be the optimist type who have thehighest pj . That is:

j∗0 = argmaxj pj

At date t(t = 1, 2...), the optimist buys the security with the loan contract.The type j optimist problem:

maxat,ϕt

at[EOj [min(s, ϕt−1)]− qt − EOj [min(s, ϕt)]− φt]

s.t.atqt(ϕt−1, ϕt) = n+ atφt

From each date t and T − 1 pessimists’ competition:

φt = qt+1φT−1 = Ep[min(s, ϕT−1)]

By solving the each date problem, p, qt and ϕt for all t are calculated. Likesingle type optimist model, the scheme ends at the time some optimist buy thesecurity by his own cash. Let this date be t′. At date t′, the security price issimply q(ϕt′−1) = EOj∗

t′[min(s, ϕt′−1)]. j

∗t′ is the type who has highest security

price in equilibrium.The optimist who bid highest price to the security buys it. Let the highest

type optimist be j∗t .

j∗t = argmaxj qt(ϕt−1, ϕt)

By solving the problem, the asset price p, the each security prices qt and theeach promise payment ϕt are calculated.

In this model, asset price is solvede by backward induction like multi gen-eration case. But in this case, we need to know what type of optimist get eachsecurity or the asset.From budget constraint of optimists’ problem at date t:

28

lend cash

sell contract

sell security

lend cash

sell contract

pessimist 1

pessimist 0optimist j∗0

lend cash

sell contract

pessimist 2

buy asset

optimist j∗1

optimist j∗2

29

ϕt−1

state

return

0 s

ϕt−1

ϕt

ϕt

optimist j∗t−1

qt+1(ϕt)

aqt(ϕt−1, ϕt) = n+ aqt+1(ϕt)

In equlibrium, asset supply is one:

qt(ϕt−1, ϕt) = n+ qt+1(ϕt)

At date t′, security price is expected return of j∗t′ .

q(ϕt′−1) = EOj∗t′[min(s, ϕt′−1)]

Then, security price at date t′ − 1 is:

qt′−1(ϕt′−1, ϕt′) = n+ qt′(ϕt′−1) = n+ EOj∗t′[min(s, ϕt′−1)]

Then, qt′−1(ϕt′−1, ϕt′) is increasing at ϕt′ . By repeating this calcultation,qt is increasing at ϕt.Then, type j∗t , which has higest qt in equilibrium, has the highest ϕt′ .From type j optimist’s problem at date t,

qt(ϕt−1, ϕt) = q′t+1(ϕt) + (1− Fp(ϕt))

∫ smax

ϕt

min(s, ϕt−1)dFoj

1− Foj (ϕt)

Asset supply is one:

qt(ϕt−1, ϕt) = n+ qt+1(ϕt)

This two equations determine equilibrium promise payment ϕt and securityprice qt. First equation depends on optimist type j.(see figure.) Pessimist sellthe security who has highest ϕt in equilibrium.

30

q∗t

ϕ∗t

qt

ϕt

qt = q′t+1(ϕ) + (1− Fp(ϕt))∫ smax

ϕtmin(s, ϕt−1)

dFoj

1−Foj(ϕt)

qt = n+ qt+1

Type j∗ has a belief that the state around ϕt−1 occur with high probability.Then he has high valuation about security at date t.By backward induction, asset price is caluculated. In this case, asset priceexceeds any trader’s expectation about asset return.

proposition 3 In multi-generation with various type optimist model, the assetprice exceeds optimistic trader’s expecatation of asset return:

p > Eo[s]

Proof 4.2 As noted above, the security price at date t′ is Eo∗j t [min(s, ¯ϕt′−1)].

The optimist type j∗t′−1 buys the date t′ − 1 security. The optimist problemat date t′ − 1 is:

maxat′−1,ϕt′−1

at′−1[EOj∗t′−1

[min(s, ϕt′−2)]− qt′−1 − at′−1EOj∗t′−1

[min(s, ϕt′−1)] + EOj∗t′[min(s, ϕt′−1)]

s.t.at′−1qt′−1(ϕt′−2, ϕt′−1) = n+ at′−1EOj∗t′[min(s, ϕt′−1)]

By solving this problem,

qt′−1(ϕt′−2, ϕt′−1) =

∫ ϕt′−1

0

sdFO∗j t′+(1−FO∗

j t′ (ϕt′−1))

∫ smax

ϕt′−1

min(s, ϕt′−2)dFOj∗

t′−1

1− FOj∗t′−1

(ϕt′−1)

The security supply is one:

qt−1(ϕt′−2, ϕt′−1) = n+ qt(ϕt′−1)

If j∗t′−1 = j∗t′ , that is, the same type optimists buy the securities at bothdates, the date t′ − 1 security price is simply EO∗

j t′−1[min(s, ϕt′−1)]. In the

31

various optimists model, the optimist type who buys the security can be differentat each date. As noted above, the security is bought by optimist who have thehighest value. If the different type optimist buys the security at date t′ − 1, thedate t′ − 1 optimist can borrow EOj∗ t′ [min(s, ϕt′−1)] units of cash. BecauseEOj∗ t′ [min(s, ϕt′−1)] ≥ EO∗

j t′−1[min(s, ϕt′−1], the date t′ − 1 optimist increase

his security demand by comparing with the sigle type optimist model. Then, thesecurity price is higher than the single optimist model. Let the security price ofsingle optimist type model be qsint

qt′−1(ϕt′−2, ϕt′−1) ≥ EOj∗ t′−1[min(s, ϕt′−2)] = qsint′−1(ϕt′−2, ϕt′−1)

The security price can be exceed the optimistic expected return. The samecaluculation imply the each date security price exceeds the single optimist typemodel.

qt(ϕt−1, ϕt) ≥ qsint = Eo[min(s, ϕt)]∀t, ϕt−1, ϕt

The type j∗0 optimist problem at date 0 is:

maxa0,ϕ0

a0[Eo[s]− p− EOj∗0[min(s, ϕ0)]− q1(ϕ0)]

s.t.a0p = n+ a0q1(ϕ0)

Because q1(ϕ0) ≥ Eo[min(s, ϕ0)], the type j∗0 optimist have bigger budgetthan single optimist type model. So, asset demand a0 increase and asset supplyis one, the asset price p exceeds the single type price Eo[s].

p ≥ Eo[s]

The existance of various optimist leads to the high asset price. This priceis similar to Harriso and Kreps(1978). In HK, there are various type traders.They have heterogenous beliefs about asset. At some point, trader x is optimistand trader y is pessimist. But at different point, x is pessimist and y is optimist.The holder of asset changes at each date and the most optimistic trader buysit. The asset holder knows that he can sell it to some other optimist at somefuture date. Then, asset price is higher than the asset holders’ expectations.

In my paper, the optimistic belief about asset return is same:Eo[s]. Thesecuritization technology divides the asset return and the each optimist evaluatethe each partition.

The each area are evaluated by the most optimistic trader. In Simsek(2013),asset return is evaluated by the optimist and the pessimist. In equilibrium, asnoted section 2, asset price:

p =

∫ ϕ

0

sdFp + (1− Fp(ϕ))

∫ smax

ϕ

sdFo

1− Fo(ϕ)

32

γ1

optimisit type j∗1

γ0 state

return

optimisit type j∗0

optimisit type j∗2

This price equation imply that optimist evaluate upper return area and pes-simist evaluate lower area. Simsek insists that only high state part of optimisticbelief is used for determining the asset price. This is the reason why asset priceis lower than Harrison and Kreps(1978). In my paper, the trader’s belief isused for the area which he evaluate higher than any other traders. Only hisoptimistic part of belief is used for evaluation of asset return. Because eachoptimist have his optimistic part of asset return, the asset price is much moreoptimitic than anyone’s expectation.Tranching and asset pricing are analized by Fostel and Geanakoplos(2012) andthey insist that tranching technology raise asset price in recent housing bubble.In my model, each state of asset are distributed to most optimistic trader.Then, equlibrium distribution is similar to some equlibrium with tranching inFG(2012).From budget constraint of optimist problem:

p = n+ n+ ...n+ EOj∗t

′ [min(s, ϕt′−1)]

Asset price is higher than expected return Eo[s] and n > EOj∗

t′[min(s, ϕt′−1)].

The number of optimists is:

t′ =p− E

Oj∗t′ [min(s, ϕt′−1)]

n

So, the number of optimist is larger than multi generation with single typeoptimist. Many optimists participate in the market and they buy the securitiesat each date.

33

5 conclusions

In my paper, by introducing securitization and simple dynamic setting to Sim-sek(2013), asset price will be as high as Harrison and Kreps.In homogeneous belief model, securitization improve market efficiency. But inheterogeneous belief model, securitization allow many optimists to participatein markets and asset prices will be raised.In heterogeneous belief model, financial friction can make asset price lower level.Like Harrison and Kreps(1978) or Miller(1977), optimist belief have heavy influ-ence on pricing. Financial friction like borrowing constraint can prevent thesetrader to participate in market. In these situations, financial technology, likesecuritization, loose this friction and asset price raise by noisy trader who havelittele cash.In my model, securitization is very important role. By securitization, loan con-tract is appealing for cash lender. Because of speculative reason, pessimist lendcash to optimist very easily.By dynamic securitization scheme, asset return is distributed to many parti-tions. Each partition is very small, but this samllness allow various optimist toget payoff of each partitions. Highest optimist evaluate each partition and totalasset evaluation is higher than any trader’s expected return of asset. Heteroge-neous Belief Bubble occurs in this model.This is very similar to Bubble in HK(1978). In their model, traders’ beliefschange in each date and most optimistic trader get the asset at each date.Then, asset price raise. In my paper, traders’ belief is different about the fre-quency of state and expected return is same among optimists. But only by theheterogeneity about frequency, asset price exceeds any traders’ expectation inmy model.Heterogeneous Belief are said to be eliminated in markets in economics. Wrongbelief traders may lose cash and they may stay out from markets in future. Iguess that it depends on financial technology. Financial technology progressallow many trader to participate in market before asset return realize. Beforeasset return realize, there are littele incentive to collect their belief for them. Ifthere are heterogeneity in markets, financial technology can amplify it.

34

6 Reference

Brunnermeier, M. K. and L. H. Pedersen (2009), ”Market Liquidity and FundingLiq- uidity,”Review of Financial Studies, Vol. 22(6), p.2201-2238.

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