Presale Contract and its Embedded Default and ...faculty.baruch.cuny.edu/kwang/BioVita/pdf/Presale...118 S.H. Chan et al. the spot price dives below the contract price.4 The basic

J Real Estate Finan Econ (2012) 44:116–152DOI 10.1007/s11146-010-9289-5

Presale Contract and its Embedded Defaultand Abandonment Options

Su Han Chan · Ko Wang · Jing Yang

Published online: 15 October 2010© Springer Science+Business Media, LLC 2010

Abstract The presale contract is a popular property selling method that allowsa buyer to default on the remaining payment and/or a developer to abandona project. Using a simple two-period game theoretical model, we derive aclosed-form pricing equation for a presale contract that explicitly accounts fora developer’s abandonment option and a buyer’s default option. Although adeveloper has an abandonment option under either a spot sale or a presalemethod, the option is more valuable under a presale contract because ofan additional cash inflow from the presale downpayment. A presale alsoprovides a buyer a default option, which is valuable in a real estate marketwith uncertain demand and price risk. We analyze the implications of theabandonment option on a developer’s construction decision and choice ofselling method, as well as the implications of the default option on a buyer’spurchase decision. Furthermore, our model framework has implications to thepricing of futures contracts that involve both stochastic revenues and costs.

Keywords Property presale · Real option · Development decision

S. H. Chan · K. WangDepartment of Real Estate, Zicklin School of Business,Baruch College/CUNY, New York, NY, USA

S. H. Chane-mail: [email protected]

K. Wange-mail: [email protected]

J. Yang (B)Department of Finance, Mihaylo College of Business and Economics,California State University at Fullerton, Fullerton, CA, USAe-mail: [email protected]

Presale Contract and its Embedded Default and Abandonment Options 117

Introduction

Under a presale contract, a developer can sell a property before its completion(or, in some countries, sell it before its construction). Under such a contract, abuyer makes a series of payments according to a construction schedule and/orthe progress of construction. The developer delivers the property when it iscompleted and upon receipt of the final payment from the buyer. The presalecontract is complicated and has two embedded options. The first is a buyer’sdefault option, whereby a buyer can stop payment if the value of the propertydrops below that of the remaining payments.1 The second is a developer’sabandonment option, whereby the developer can halt construction when theremaining construction cost is higher than either the remaining payments fromthe buyer or the sale price of the property in the open market.2

Despite its complexity, the presale method has been a popular tool forselling properties in many Asian countries for at least five decades.3 With thesurge in residential property prices during the 2002–2006 period, the presalemethod started to gain attention in the U.S. and Canada. However, littleresearch has been devoted to the study of the risk-and-return trade-off tothe parties involved. There are only a few studies that address this issue. Theearly group of research treats a presale as a forward or futures contract. Thosestudies, in general, do not explicitly address a developer’s nor a buyer’s optionto default. Their main objectives are to address issues such as the pricingfactors of a futures contract (Chang and Ward 1993), the signaling effectof presale activities (Wang et al. 2000), the impact of a presale market onthe adjustment of housing supply (Hua et al. 2001), the relationship betweenpresale price and future spot sale price (Wong et al. 2006), and moral hazardproblems that might result in defective properties being delivered (Edelsteinet al. 2008).

The second strand of the literature treats a presale as an option contract, andaddresses issues such as its impacts on a developer’s development strategiesand its pricing considerations (Lai et al. 2004). In this framework, a presalecontract can be viewed as a call option. That is, buyers have the right to buya property at a pre-determined exercise price and to default on the contract if

1In an article titled “Fla. highrise has 32 stories, but just one tenant”, Christine Armariodescribes a situation in which many buyers exercised the default options of their presale contracts(Associated Press Writer—Sat, 1 Aug 2009, 3:37 p.m. ET). This article can be downloaded fromhttp://news.yahoo.com/s/ap/20090801/ap_on_re_us/us_lonely_highrise.2For an interesting article describing a developer’s abandonment decision, see HubbieSmith, “Las Vegas Valley development is latest casualty of financial crisis”, Las VegasReview-Journal (Nevada), 10 March 2009. This article can be downloaded from http://www.allbusiness.com/legal/property-law-real-property-building-housing-codes/12223056-1.html.3The presale method has been used in many cities around the world, but is particularly popularin several Asian countries/regions such as Hong Kong, Taiwan, Korea, Singapore and MainlandChina.

http://news.yahoo.com/s/ap/20090801/ap_on_re_us/us_lonely_highrise

http://www.allbusiness.com/legal/property-law-real-property-building-housing-codes/12223056-1.html

http://www.allbusiness.com/legal/property-law-real-property-building-housing-codes/12223056-1.html

118 S.H. Chan et al.

the spot price dives below the contract price.4 The basic story of the paper isthat, by giving buyers an option to default on the contract, the presale systemoffers developers an opportunity to attract capital and to share developmentrisks with buyers. Chan et al. (2008) extend the literature by providing asimple equilibrium model in a game-theoretical framework to throw light ona developer’s pricing and production decisions.5 They find that the presalemethod per se does not affect a developer’s production decision in equilibrium.Instead, developers will increase the presale price to compensate for the optionvalue they give to buyers. In other words, a developer may price the defaultoption he grants to a buyer by increasing the presale price. Furthermore, in aworld with financing constraints, developers will tend to use the presale systemto reduce financing cost. The gain from the reduction in financing cost canthen be used to lower the presale price to attract buyers to the presale market.Buttimer et al. (2008) also argue that, since a presale can reduce market risk, itallows developers to carry out projects with a lower expected return on equity.Both papers confirm the arguments advanced by Lai et al. (2004) that thepresale method can reduce a developer’s risk (by sharing it with the buyer)and/or lower financing costs (of both debt and equity).

At this moment, the literature on the option value of a presale contractseems to be one sided and incomplete. Since the property is yet to becompleted when a presale contract is signed, the developer has an option tonot complete it. For example, the popular press reports that since 2008, risingconstruction and financing costs have forced many developers to abandon theirprojects (some with refunds to the buyer, some without).6 The popular pressalso reports that, with many condominium sales at an all-time low, a lot ofdevelopment projects have been cancelled in cities such as Vancouver, Miamiand Las Vegas. Clearly, the evidence we observe in the real world indicatesthat, in a presale contract, not only can a buyer default on the contract, but adeveloper also can abandon the project. So far, however, the presale literatureis still silent on this issue.

To extend the literature (that emphasizes only on a buyer’s default option),this paper explicitly examines a developer’s option to abandon a project andthe associated construction cost risks.7 We develop a simple two-period gametheoretical model in a perfectly competitive development market, where bothproperty price and construction cost are uncertain at the time a buyer and

4Many practices in real estate markets are associated with the use of options, see Grenadier (1995,1996), Wang and Zhou (2006), Lai et al. (2007), and Buttimer et al. (2008).5Wang and Zhou (2000) and Wang et al. (2002), among others, also used a game-theoreticalapproach to model the seemingly irrational phenomena observed in real estate markets.6See, for example, Peggy Sito, “Ban on Pre-sale of Flats Proposed”, South China Morning Post,26 July 2007:1.7Geltner and Ling (2007) argue that construction cost is a major variable in the developmentindustry information set and should be encompassed in an ideal real estate index.


a developer sign a presale contract. In other words, when a buyer and adeveloper are signing a presale contract, they both know that the realizedproperty value and construction cost in the future could differ from theircurrent expectations (although they both have the same expectations on theprice and cost). In our model setup, a buyer can default on the payment ifthe market value of the property drops to a point where the pay-off fromdefaulting is higher than the property value. A developer can abandon theproject if the construction cost increases to a level that exceeds the remainingpayments to be received from the buyer. It is interesting to note that, whilethere are many factors affecting the exercise decisions of both options, themost important variables seem to be the percentage of downpayment and thepercentage of the first-phrase construction cost. If a buyer pays a very high(low) downpayment in a presale contract, the chance for the buyer to exercisethe default option will be low (high). If a developer incurs a very high (low)first-phase construction cost, the developer is less (more) likely to exercise theabandonment option. We will present our model setup in the second sectionand derive the equilibrium presale contract in the third section. The last sectionconcludes.

Model

In our model, a developer identifies a positive NPV project, which is to builda one-unit of property to sell to a buyer. Since it is a positive NPV project, wedo not need to consider if the developer should make an investment decision.The only decision the developer needs to make is on the selling method.8

The property will be constructed in two phases. The first phase starts at t = 0,the second phase starts at t = 1, and the property is completed and deliveredat t = 2. However, at t = 0 and before the developer starts construction, thedeveloper will select a property selling method, which is either a presale or aspot sale. The developer can either sell the property now (t = 0) using a presalecontract or in the spot sale market when the construction of the property iscompleted (t = 2).

At the time the developer makes a decision (t = 0), there is informationuncertainty regarding the possible level of total construction cost c as well asthe spot price of the property p.9 The exact cost and the future spot price

8Since the choice of a selling method will not affect a developer’s investment decision (nor thesupply in the market), the use of a particular selling method will not affect the selling price of aproperty in the spot market. Given this, we do not need to explicitly model the determination ofthe spot price.9Note that c measures the construction costs according to the blueprint. While in practice adeveloper might be allowed to substitute materials or change floor plans in some presale contracts,our model does not allow for this possibility. However, as long as the substitutions must be donewith equal costs, allowing for this practice will not affect the conclusions of our paper.


will not be realized until t = 1. We assume there is no information asymmetrybetween the developer and the buyer. In other words, both the buyer and thedeveloper have identical information on all the parameter values. To greatlysimplify the model development, we also assume the interest rate is zero.10

The detailed procedure of the model and the payoffs to the two players aresummarized in Figs. 1 and 2. At t = 0, knowing the probability distribution ofthe total construction cost c ∼ � [c − θ, c + θ ] and that of the spot price p ∼ �

[p − δ, p + δ] to be uniform,11 the developer chooses a selling method (eithera spot sale or a presale) for the property she/he is going to construct. c andp are the expected construction cost and the future spot price of the property,respectively. θ and δ determine the upper and lower bounds of the constructioncost distribution and the future spot price distribution, respectively. A largerθ (δ) represents a larger construction cost risk (future spot price risk). p − δ

(or p + δ) represents the estimated lowest (or highest) possible future spotprice while c + θ (or c − θ) represents the estimated highest (or lowest)possible construction cost.12

Before a presale contract can be offered, the developer needs to first decideon the presale price pp of the contract. (It should be noted that the spot saleprice is exogenously determined by the market and cannot be affected bythe buyer nor the developer). If a developer launches a presale, she/he incursadditional transaction costs s regardless of whether the presale is successful ornot. Given this presale offer (at t = 0), the buyer decides either to accept theoffer and immediately make a downpayment dpp (where d is the percentage ofdownpayment for the presale contract and is also used as the percentage of thefirst-phase construction cost of the project), or to reject the offer and wait untilt = 2 to buy an identical property in the spot market (and pay the realized spotprice p). At t = 0, the developer also incurs the first-phase construction cost,which for simplicity, is assumed to be likewise a portion d (the same as thepercentage of the downpayment in the presale contract) of the constructioncost (or dc).13

10An inclusion of the interest rate into our model will not change our model implications but willmake the presentation of the result much more complicated.11This specification assumes that the future spot price and the construction cost follow uniformdistributions. We had tried to use the normal distribution assumption instead; however, we cannotderive a closed-form solution under this assumption.12Wang and Zhou (2006) report that, historically, the construction cost volatility could be higherthan the price volatility in certain periods.13It might not be realistic to assume that a developer can spend exactly the amount dc during thefirst construction phase as the value of c will not be realized until at the end of the first constructionphase (or t = 1). However, instead of using dc, we can assume that the developer spends a fixedconstruction cost during the first-phase period (that is, dc) and all our model implications will notchange. The assumption of dc simplifies the model presentation greatly.


Buyer

Accept presale Reject presale

Pay (1-d)pp

pp

Down payment d pp

Buyer

PRC

Presale

NC

Sell property at market

price p

Developer

Development is completed

PACC

p~ is realized at value p;

c~ is realized at value c

PRQ NQ

1st-phase development

No presale

Predict that buyerwill commit

Predict that buyer will default

Abandon project

Continue construction


Abandon project

Developmentis completed


Abandon project


Abandon project




price p


price p

PACQ PADC PADQ





Developer Developer

1st-phase development 1st-phase development

t=0

t=1

Spot price ],[~~ δδ +−Ω ppp ;

Total construction cost ],[~~ θθ +−Φ ccc

t=2

Fig. 1 Decision flows for developer and buyer

At t = 1, the actual construction cost and the spot price are realized andknown to both players. After observing the information, the developer decideson whether to continue the construction (based on the amount of remaining


1. Presale Case

t=0 t=1 t=2

Information status:],[~~],,[~~

θθδδ

+−Φ+−Ω

ccc

ppp pp~ and cc~

Developer's cash flows when:

No party defaults/abandons scddpp −− ~ cd )1( ppd )1(

Developer abandons scddpp −

− −

− −

−

− −

−

−

−

− ~

Buyer defaults scddpp −− ~ cd )1( p

Both parties default/abandon scddpp −−

−

−

−

−

−

−

−

−

− −

−

−

~

Buyer's cash flows when:

No party defaults/abandons pdp ppd)1(

Developer abandons pdp p

Buyer defaults pdp p

Both parties default/abandon pdp p

2. Spot Sale Case

t=0 t=1 t=2

Information status:],[~~],,[~~

θθδδ

+Φ+Ω

ccc

ppp pp~ and cc~

Developer's cash flows when:

Developer continues cd~ cd )1( p

Developer abandons cd~

Buyer's cash flows when:

Developer continues p

Developer abandons p

Fig. 2 Information content and cash flows at each decision point for both presale and spot salemethods

presale contract payment (1 − d)pp, the value of the realized spot price p, andthe amount of second-phase construction cost (1 − d)c with c being the realizedtotal construction cost) to fulfill the obligation of a presale contract or to sell


the property in the spot market at t = 2.14 If the developer chooses to continuethe construction, the whole project will be completed at t = 2. At t = 1,if the buyer has accepted the presale contract, she/he will choose to eitherpay the remaining presale payment (1 − d)pp and own the property, or defaulton the payment and buy an identical property from the market at the realizedspot price p. If the developer decides to abandon the project (with the buyerstill owning the presale contract), the buyer will lose the downpayment andhave to purchase a property from the market at the spot price p. It should benoted that the developer can abandon the project under either a presale or aplaned spot sale. However, the abandonment option is more valuable undera presale than under a spot sale because of the additional revenue from thepresale downpayment.

Since our model has two stochastic factors ( p and c) that can fall within dif-ferent ranges, we have to set some boundary conditions to make sure that theirparameter values are reasonably in line with each other. To do this, we assume

p > (1 − d)c. (1)This indicates that it is, on average, profitable for a developer to continueto the second-phase construction because the expected spot price p is higherthan the average second-phase construction cost (1 − d)c. It is unreasonable toassume that a buyer will enter into a presale contract if she/he expects that thedeveloper will abandon the project (which is true if p < (1 − d)c). It should benoted that Eq. 1 also implies

p > (1 − d)c > (1 − d)(c − θ). (2)Under this circumstance, the minimum second-phase construction cost (1 −d)(c − θ) will be lower than the expected spot price p. This indicates that thedeveloper will have a chance to continue the project even if the price volatilityδ is low or set to zero (note that p ∼ � [p − δ, p + δ]).

To summarize, there are four steps involved in our model framework. First,a developer makes a decision at t = 0 on the selling method to use and onthe presale price, pp, if a presale contract is used. Second, a buyer decides att = 0 on whether or not to buy the property at the presale stage or in the spotmarket. Third, the developer decides at t = 1 on whether or not to abandonthe project (i.e., by stopping construction). Fourth, the buyer decides at t = 1on whether or not to commit on the presale payment if she/he has accepted thepresale contract. Those decision flows are illustrated in Fig. 1. The figure endswith eight terminal nodes that correspond to the decision rules adopted by thebuyer and the developer. The decision rules and the payoffs to the buyer andthe developer at each terminal node are summarized in Tables 1 and 2,respectively.

14For the analyses presented in this paper, the downpayment ratio and the first-phase constructionratio are assumed to be the same. We also conducted analyses allowing these two ratios to differfrom each other. However, the mathematical presentation is much more complicated, while the im-plications of the model are exactly the same. Given this, we decided to report the result that usesonly one ratio. The results of using two different ratios are available from the authors upon request.


Tab

le1

Dec

isio

nru

les

Dev

elop

er’s

Selli

ngD

ecis

ion

Buy

er’s

Rea

lized

cost

Dev

elop

er’s

aban

donm

ent

Buy

er’s

defa

ult

Ter

min

alpa

yoff

sm

etho

dru

lede

cisi

on&

pric

ede

cisi

onde

cisi

onno

de

E(π

(p∗ p

))| pr

esal

e≥

Pre

sale

g(p

p)≤

0A

ccep

tc

≤c,

p≥

pC

onti

nue

Com

mit

PA

CC

E(π

(p)

)| spo

tc

>c,

p≥

pA

band

onC

omm

itP

AC

Qc

≤c′

,p

<p

Con

tinu

eD

efau

ltP

AD

Cc

>c′

,p

<p

Aba

ndon

Def

ault

PA

DQ

g(p

p)>

0D

eclin

ec

≤c′

Con

tinu

e–

PR

Cc

>c′

Aba

ndon

–P

RQ

E(π

(p∗ p

))| pr

esal

e<

Spot

–A

ccep

tc

≤c′

Con

tinu

e–

NC

E(π

(p)

)| spo

tSa

lec

>c′

Aba

ndon

–N

Q

Thi

sta

ble

sum

mar

izes

the

deci

sion

rule

sfo

rth

ede

velo

per

and

the

buye

r.P

AC

C,

PA

CQ

,P

AD

C,

PA

DQ

,P

RC

,P

RQ

,N

Can

dN

Qar

eth

ein

dexe

sfo

rou

tcom

es(s

eeF

ig.1

);E

(π)

isth

ede

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per’

sex

pect

edin

crem

enta

lpa

yoff

from

offe

ring

prop

erty

sale

s;p∗ p

isth

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uilib

rium

pric

eun

der

apr

esal

e;p

isth

ere

aliz

edsp

otpr

ice

unde

ra

spot

sale

;cis

the

real

ized

tota

lco

nstr

ucti

onco

st;g

(p

p)≤

0an

dg(

pp)>

0ar

eth

eco

ndit

ions

for

the

buye

rto

acce

ptor

reje

ctth

epr

esal

eof

fer;

and

c=

pp

isth

ece

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ofth

eco

nstr

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onco

st.A

deve

lope

rw

illab

ando

nth

epr

ojec

tif

the

cons

truc

tion

cost

ishi

gher

than

this

ceili

ngun

der

apr

esal

em

etho

d.c′

=p

1−d

isth

esa

me

ceili

ngun

der

asp

otsa

le.

p=

(1−

d)p

pis

the

ceili

ngsp

otpr

ice.

Abu

yer

will

not

cont

inue

tom

ake

paym

ent

ifth

epr

ice

isbe

low

this

ceili

ng.I

nter

estr

ate

isas

sum

edto

beze

ro


Tab

le2

Out

com

esan

dpa

yoff

s

Ter

min

alD

evel

oper

’sB

uyer

’sD

evel

oper

’sse

cond

-pha

seF

inal

selli

ngB

uyer

’sD

evel

oper

’sno

deof

fer

type

acti

onco

nstr

ucti

onm

etho

dco

stpa

yoff

PA

CC

Pre

sale

Acc

ept&

com

mit

Con

tinu

eP

resa

lep

pp

p−

c−

sP

AC

QP

resa

leA

ccep

t&co

mm

itA

band

onSp

otsa

led

pp

+p

dp

p−

dc−

sP

AD

CP

resa

leA

ccep

t&de

faul

tC

onti

nue

Spot

sale

dp

p+

pd

pp

+p

−c

−s

PA

DQ

Pre

sale

Acc

ept&

defa

ult

Aba

ndon

Spot

sale

dp

p+

pd

pp

−dc

−s

PR

CP

resa

leD

eclin

eC

onti

nue

Spot

sale

pp

−c

−s

PR

QP

resa

leD

eclin

eA

band

onSp

otsa

lep

−dc

−s

NC

Spot

sale

Acc

ept

Con

tinu

eSp

otsa

lep

p−

cN

QSp

otsa

leA

ccep

tA

band

onSp

otsa

lep

−dc

Thi

sta

ble

sum

mar

izes

allt

hepo

ssib

leou

tcom

esan

dth

eco

rres

pond

ing

payo

ffs

toth

ebu

yer

and

the

deve

lope

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PA

CQ

,P

AD

C,

PA

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RC

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chm

atch

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ith

ase

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s;p

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est

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pric

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der

asp

otsa

le;c

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est

ocha

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cons

truc

tion

cost

;sis

the

addi

tion

altr

ansa

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soci

ated

wit

ha

pres

ale;

and

dis

the

dow

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men

tra

tio

orth

efi

rst-

phas

eco

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stra

tio.

Inte

rest

rate

isas

sum

edto

beze

ro


Using a backward induction procedure, we solve the decision rules of thedeveloper and the buyer backwards. This can be done in four steps. First, wesolve for a buyer’s decision (at t = 1) on whether or not to exercise the defaultoption (or to pay the remaining presale payment). Second, we solve for adeveloper’s decision (at t = 1) on whether or not to exercise the abandonmentoption (or to continue the construction of the project). Third, we solve fora buyer’s decision (at t = 0) on whether or not to accept a presale contractoffered by a developer. Finally, we solve for the presale price that is acceptableto both the developer and the buyer given the decision rules discussed in thefirst three steps.

At t = 1: Buyer’s Default Decision

This decision is made at t = 1 and is contingent upon the fact that the buyer hasaccepted a presale offer and made the downpayment, and that the developerwill complete the construction. At this time, the values of the total constructioncost and the spot price are realized and known to both parties as c and p,respectively. At this moment, the buyer will decide if she/he wants to pay theremaining payment of the presale contract (1 − d)pp. If the buyer decides todefault on the presale contract, she/he will not pay (1 − d)pp to the developernor take title of the property. Instead, the buyer will pay the spot price p tobuy an identical property in the spot market.

Given the fact that a downpayment is a sunk cost, the buyer’s decision willbe based on a comparison of the incremental cost from committing, �C|commit,with that from defaulting, �π |default, where

�C|commit = (1 − d)pp, (3)

�C|default = p. (4)

The necessary and sufficient condition to commit the presale payment is�C|default ≥ �C|commit, or

p ≥ p = (1 − d)pp, (5)

where p is the lowest spot price that makes the buyer commit and pay theremaining presale payment. Note that

∂ p∂d

= −pp < 0, and (6)

∂ p∂pp

= 1 − d > 0. (7)

Holding the future spot price distribution constant, Eqs. 6 and 7 indicate thata higher downpayment ratio and/or a lower presale price will reduce a buyer’s


incentive to default on the remaining presale payment.15 The cashflows to boththe developer and the buyer under a presale contract are summarized in thefirst panel of Fig. 2.

At t = 1: Developer’s Abandonment Decision

A buyer’s decision to default on a presale payment is based on the threevariables, d, pp and p, which are known to both buyer and developer at t = 1.Therefore, at t = 1, a developer can clearly see if a buyer will default. Togetherwith the realized value of the construction cost, the developer can make adecision to either abandon the project or continue the construction. There area total of four situations that we need to consider here. First, the developermade a presale offer and the buyer accepted it at t = 0. The buyer alsodecides to pay the remaining presale payment at t = 1. Second, the developermade a presale offer which the buyer accepted, but the buyer will default onthe remaining presale payment and force the developer to sell the propertyin the spot market. Third, the developer made a presale offer but the offerwas rejected by the buyer at t = 0. As a result, the developer decided to sellthe property in the spot market when it is completed at t = 2. Fourth, thedeveloper decided to sell the property in the spot market at t = 0. It shouldbe noted that, given our model setup, the third and the fourth situations willlead to the same decisions and outcomes. Furthermore, since the downpay-ment and the first-phase construction cost are both sunk costs at t = 1, thedecisions and outcomes of the second situation should also be similar to thatin the third and fourth situations. To simplify our discussion, we combine thesecond, third, and fourth situations and term it as a “spot sale” situation.

We first discuss a developer’s construction decision in the first situationwhere a buyer accepted a presale and will continue the payment. If thedeveloper continues the construction, she/he will incur the second-phaseconstruction cost (1 − d)c at t = 1 but will receive the remaining presalepayment (1 − d)pp from the buyer. If the developer abandons the project(that is, stops the construction work), her/his payoff will be zero. Given thatthe downpayment income and the first-phase construction cost are sunk,the developer will compare her/his incremental payoff from continuing the

15It should be noted that we assume that the buyer will not receive a refund of the deposit whenthe developer abandons the project. In practice, depending on the developer’s strategies andcontract enforceability, it is sometimes possible for the buyer to receive a full or partial refund.The inclusion of this refund parameter will not affect our qualitative conclusions (it only changesthe magnitude of the cash flow under the abandonment scenario) but will significantly increase themathematical presentation in our model. This is true because the refund option only takes effectwhen the developer abandons and not when he continues the project. Under this circumstance, wehave to increase the number of equations to describe both scenarios. To simplify the presentation,we do not include the refund possibility in this paper. However, the results with a refund possibilityare available from the authors upon request. Clearly, a refund privilege will decrease a developer’sincentive to abandon a project and hence reduce the value of an abandonment option.


construction, �π |continue, with that from abandoning the project, �π |abandon,or will compare

�π |continue = −(1 − d)c + (1 − d)pp = (1 − d)(pp − c) with (8)

�π |abandon = 0. (9)

From Eqs. 8 and 9, we know that the necessary and sufficient condition for thedeveloper to continue the construction under a presale contract is �π |continue

≥ �π |abandon. This can be specified as

c ≤ c = pp, (10)

where c is the highest total construction cost under which a developer is willingto continue with the construction of the pre-sold property.16 It is interesting tonote that

∂ c∂pp

= 1 > 0. (11)

In other words, a lower presale price will reduce the developer’s incentive tocontinue the property construction. This is consistent with our observation thata consumer who signs a contract (for a product to be delivered at a future date)at a low price (relative to the cost) might face a problem at the delivery date.

In the spot sale situation (second, third, and fourth situations), the developeris willing to sell the property in the spot market at t = 2. At t = 1, after theamount of the total construction cost c is realized, the developer will comparethe incremental payoff from continuing the construction �π ′|continue with thatfrom stopping the construction �π ′|abandon in order to make a decision, or willcompare

�π ′|continue = −(1 − d)c + p, with (12)

�π ′|abandon = 0. (13)

16It should be noted that we assume that the developers will lose nothing if they decide toabandon the property. In practice, there is a reputation cost to the developer or buyers might beable to go after the developer’s other assets (if there are any). The inclusion of this reputationparameter will not affect our qualitative conclusions unless the reputation cost is larger thanthe abandonment benefits (this is a trivial implication). However, to include the reputation costinto the model will significantly lengthen the mathematical presentation in our paper. This istrue because the reputation cost only takes effect when the developer abandons the project andnot when the developer continues the project. Under this circumstance, we have to increase thenumber of equations to describe both scenarios. To simplify the presentation, we do not includethe reputation cost in this paper. However, the results with a reputation cost (which are the sameas in this paper) are available from the authors upon request. It is clear that, with a reputationcost, it is less likely for a developer to abandon a project and hence will reduce the value of anabandonment option. When the reputation cost is high enough, the value of the abandonmentoption approaches zero.


From Eqs. 12 and 13, we know that the necessary and sufficient condition for adeveloper to continue the construction when she/he plans to sell the propertyin the spot market is �π ′|continue ≥ �π ′|abandon. This can be specified as

c ≤ c′ = p1 − d

, (14)

where c′ is the highest total construction cost under which a developer iswilling to continue the construction so that she/he can sell the property in thespot market. The second panel of Fig. 2 summarizes the cash flows when thedeveloper decides not to offer a presale contract at t = 0.

At t = 0: Buyer’s Decision on Presale Contract

At t = 0, after a developer decides on a presale price and the use of the presalemethod, the buyer will make a decision on whether to sign the presale contractat t = 0 or to buy the property at the spot price at t = 2. At t = 0, the totalconstruction cost c and the future spot price p are not realized yet. The buyerknows that there is a possibility that the developer will abandon the project andthat she/he has an option to default on the contract. Given this, the expectedtotal cost for a buyer to accept the presale offer is

E(C)|accept =∫ p+δ

p

∫ c

c−θ

ppd�(c)d�( p) +∫ p+δ

p

∫ c+θ

c(dpp + p)d�(c)d�( p)

+∫ p

p−δ

(dpp + p)d�( p). (15)

The three terms in Eq. 15 represent the expected cost if no party defaults, theexpected cost if only the developer abandons the project, and the expectedcost if the buyer defaults on the remaining payment (regardless of whether thedeveloper abandons the project), respectively.

If the buyer decides not to take the presale offer at t = 0 but wait till t = 2to buy the property in the spot market, the buyer’s expected cost will be

E(C)|reject =∫ p+δ

p−δ

∫ c+θ

c−θ

pd�(c)d�( p). (16)

The necessary and sufficient condition for a buyer to accept the presale offerat t = 0 is

E(C)|accept ≤ E(C)|reject. (17)

Equation 17 can be written as a condition on a given price of a presale contractpp, or

g(pp) = E(C(pp)|accept − E(C)|reject ≤ 0. (18)


At t = 0: Developer’s Decision on Presale Price

With backward induction, which takes into account (1) a buyer’s decision onwhether to accept the presale contract, (2) a developer’s decision on whetherto abandon the project, and (3) a buyer’s decision on whether to pay theremaining presale payment, the developer chooses a selling method (a presaleor a spot sale) for the positive NPV development project and selects the presaleprice she/he is willing to offer for the presale contract. The developer knowsthat the buyer will accept the offer only if the offer satisfies g(pp) ≤ 0 (seeEq. 18). Given this, the developer will not make a presale offer that is deemedto be rejected (as there are associated costs for launching a presale). Therefore,a developer’s decision for offering an acceptable presale contract is

E(π)|presale = max{pp}

−s +∫ p+δ

p

∫ c

c−θ

(

pp − c)

d�(c)d�( p)

+∫ p

p−δ

∫ c′

c−θ

(dpp + p − c)d�(c)d�( p)

+∫ p+δ

p

∫ c+θ

c(dpp − dc)d�(c)d�( p)

+∫ p

p−δ

∫ c+θ

c′(dpp − dc)d�(c)d�( p),

s.t. g(pp) ≤ 0,

where c = pp, c′ = p1 − d

, p = (1 − d)pp and

g(pp) = E(C(pp))|accept − E(C)|reject. (19)

Equation 19 defines E(π)|presale as the developer’s expected payoff fromoffering a presale contract, where s is the cost associated with a presale activity.The first double-integral term of Eq. 19 is the expected payoff when bothbuyer and developer do not default on the contract. The second double-integral term is the expected payoff if the buyer is the only party who exercisesthe default option in a presale contract. The third double-integral term isthe expected payoff if the developer is the only party who exercises theabandonment option. The last double-integral term is the expected payoffif both the developer and the buyer exercise their options. Finally, Eq. 19indicates that an acceptable presale price is subject to the constraint g(pp) ≤ 0.

A developer also has an option to sell the project in the spot market whenthe property is completed at t = 2. Given any perceived spot price p, theexpected payoff to the developer for a spot sale is

E(π)|spot =∫ p+δ

p−δ

{

∫ c′

c−θ

( p − c) d�(c) +∫ c+θ

c′(−dc)d�(c)

}

d�( p), (20)

where c′ = p1 − d

.


Equation 20 defines E(π)|spot as the developer’s expected payoff if she/hedecided at t = 0 that she/he will offer a presale (which the buyer rejected) andwill sell the property in the spot market at t = 2. The first integral inside theouter integral is the expected payoff to the developer if the developer decidesto complete the project after the construction cost is realized at t = 1. Thesecond integral is the developer’s expected payoff if she/he decides to abandonthe project after the construction cost is realized at t = 1.

The developer will compare the expected payoffs of these two choices,E(π(p∗

p))|presale versus E(π)|spot, and chooses the presale method if andonly if

E(π(p∗p))|presale ≥ E(π)|spot. (21)

Table 1 summarizes the decision rules for the developer (based on Eqs. 10,14 and 21), and for the buyer (based on Eqs. 5 and 18), as well as thecorresponding terminal nodes (as summarized in Fig. 1). The detailed payoffsof the eight terminal nodes are summarized in Table 2.

The Equilibrium Contract

Since a buyer can purchase a property at either the presale market or thespot market, the presale price chosen by the developer must be the one thatmakes the buyer indifferent between buying the property at the presale stageor buying it in the spot market. This equilibrium price, by definition, must bethe one that maximizes the developer’s profit. This is true because, if a presaleprice is set higher than the price the buyer is willing to pay, the buyer willnot accept the presale contract and the developer incurs a presale cost. If thepresale price is set below the price the buyer is willing to accept, the developerleaves money on the table. Given the decision rules derived from Eqs. 10, 14,21, 5 and 18, Proposition 1 reports the conditions for the determination of theequilibrium price in a presale contract.

Proposition 1 In equilibrium, the price of a presale contract is

p∗p(c, θ, p, δ, d, s) = arg max

{pp}1

24θδ(1 − d)

(

L0 + L1 pp + L2 p2p − 2(1 − d)3 p3

p

)

s.t. g(pp) = 18θδ

(

(c − θ)(p + δ)2 + Y1 pp + Y2 p2p − (1 − d)2 p3

p

)

≤ 0,

where L0 = 6δ(1 − d)2 (

c2 + θ2) + 3(c − θ)(1 − d)(p − δ)2 − (p − δ)3

− 12θδ(1 − d)[

(1 + d)c + 2s]

,

L1 = 6(1 − d)[−c(p + δ)(1 − d) + θ (δ (1 + 3d) + p (1 − d))

]

,

L2 = 3(1 − d)2 [

(p + δ) + (1 − d)(c − θ)]

,

Y1 = −(p + δ)[

p + δ + 2(1 − d)c] + 2θ

[

(1 − d)p + (1 + 3d)δ]

,

Y2 = (1 − d)[

2(p + δ) + (1 − d)(c − θ)]

. (22)


Proof The results can be derived directly from Eq. 19.17 ��

It is challenging to derive a closed-form solution for the equilibrium pricep∗

p from Eq. 22 due to the complicated form. We will now simplify Eq. 22,by imposing boundary conditions, to analyze the default and abandonmentoptions associated with presales. We will analyze two situations explicitly. Thefirst situation is when the buyer’s default option value is zero, which allows usto focus on the developer’s abandonment option. The second situation is whenthe developer’s abandonment option value is zero, which allows us to focus onthe buyer’s default option.

Case 1: When the Buyer’s Default Option Value is Zero

We first focus on a developer’s abandonment option by analyzing the casewhen the buyer’s default option value is zero. It can be noted that as long as

(1 − d)p∗p ≤ p − δ, (23)

the buyer’s default option is out of the money and the buyer will not exercisethis option. This is true because the remaining presale payment is, withcertainty, always lower than the spot price. Given this, Eq. 23 indicates thatwhen the lower bound of the spot price distribution p − δ is very high and/orthe presale downpayment ratio d is very large, a buyer’s default probabilitycould be negligible.

Furthermore, to ensure that an abandonment option has value, we assume

(1 − d)(c + θ) > p − δ. (24)

Under this circumstance, the maximum remaining construction cost (1 −d)(c + θ) will be higher than the lower bound of the future spot price p − δ.

17We realize that the determination of the equilibrium presale price ignores two factors. First, wedid not explicitly analyze the impact of the presale cost, defined as s in our model, on the presaleprice. The presale cost must be included in the model because, without it, the developer can offera presale at any price without penalty. Second, the benefits of a presale contract are not modeledexplicitly. Lai et al. (2004) and Chan et al. (2008) speculate that a presale contract can reducethe development risk and borrowing cost of a development project. In reality, a developer shouldshare some of the benefits with a buyer (by lowering the presale price) to attract the buyer intothe presale market. However, to include these two factors into our model requires us to makemore behavioral related assumptions that will make the model more complicated. Given this, wesimply assume that the presale benefit equals the presale cost, so that s = 0. It should be notedthat, while this assumption (s = 0) simplifies our model development greatly, it also prohibits usfrom formally addressing interesting questions such as why a presale contract exists, what are thecosts and benefits of presale contracts, and how can the costs and benefits be allocated. Clearly,intuition tells us that we are more likely to see presale contracts when markets have tight credit (orhigh interest rate), when developers have low equity and face high bankruptcy cost, and/or whenbuyers have low risk tolerance. We plan to address these important issues in a future research witha formal behavioral model.


This means that the developer will have a chance to exercise the abandonmentoption since the remaining construction cost could be larger than the propertyvalue.

Combining Eqs. 23 and 24, we can write the required condition for this no-buyer-default scenario as

(1 − d)p∗p ≤ p − δ < (1 − d)(c + θ) . (25)

We can derive a closed-form solution for the equilibrium presale pricingdecision with the boundary conditions set by Eq. 25. Proposition 2 summarizesthe equilibrium outcomes.

Proposition 2 In equilibrium, the price of a presale contract under a no-buyer-default condition is

p∗p = A + √

A2 − 4(c − θ)(1 − d)p2(1 − d)

if (1 − d)p∗p ≤ p − δ < (1 − d)(c + θ),

A = p + (1 − d)c − (1 + d)θ . (26)

The presale price is lower than the expected spot price, or

p∗p < p. (27)

The buyer’s expected cost of purchasing the property is the same regardless ofthe selling method accepted, or

E(C(p∗p))|accept = E(C(p∗

p))|reject = E(C(p))|spot = p. (28)

Proof See Appendix A. ��

Equation 26 indicates that the presale pricing equation under the no-buyer-default condition still takes a complicated form and is basically determined bythe distribution of the construction cost c ∼ � [c − θ, c + θ ], the distributionof the spot price p ∼ � [p − δ, p + δ], and the downpayment ratio d. The mostinteresting finding reported in Proposition 2 is that the equilibrium presaleprice is always lower than the expected spot price, or p∗

p < p. Correspondingly,the price difference between the expected spot price and the presale price, or

�Price = p − p∗p, (29)

should reflect the value of the developer’s abandonment option and the valueof the buyer’s default option. However, under the condition (1 − d)p∗

p ≤ p − δ,the value of the buyer’s default option is zero. Consequently, �Price = p −p∗

p > 0 reflects only the value of the abandonment option. In other words, thedeveloper is willing to offer a presale price lower than the spot price becauseof the abandonment option the developer has.


The popular press frequently cites that one of the most important reasonsfor a buyer to rush into a presale market is “to catch the last bus”. Theargument is that, since the future spot price will increase at a fast rate, a buyerwill not be able to afford to buy a property if she/he does not act now in thepresale market. Casual observations sometimes also indicate that propertiespurchased from the presale market appreciated more than those bought from aspot market. Under this assertion, a buyer’s motivation for accepting a presalecould be driven by the belief that it is the only alternative to buy a property ina “hot” property market.

On the contrary, our model result indicates that the anecdotal evidencereported in some of the popular press could be an illusion. Since the presaleprice includes the value of a developer’s abandonment option, the presale priceshould be higher if there is no abandonment option. (The presale price shouldbe the expected future price if there are no default and abandonment optionsin a presale contract.) Given this, the property appreciation rate betweentwo periods (t = 0 and t = 2) should be higher if calculated using the presaleprice as the base price at t = 0 than using the spot price as the base price att = 0. Our result indicates that, while there are reasons for a buyer to buy aproperty in a presale market, the “catch the last bus” mentality might not bea valid one if buyers have perfect information about the market. If buyers donot have perfect information, then we might observe the “catch the last bus”phenomenon due to this market imperfection.18

It should be noted that this conclusion holds only when the no-buyer-defaultcondition (1 − d)p∗

p ≤ p − δ is satisfied. It is clear that the no-buyer-defaultcondition (1 − d)p∗

p ≤ p − δ is more likely to hold when the downpaymentratio d is large, the expected future price p is high, or the price dispersionδ is low. This means that our model result should hold in a “hot” market inwhich all players predict a growth in property price (a high value of p) with ahigh level of confidence (which is reflected by a lower value of δ). Intuitively,it is more likely for a buyer to accept a large downpayment ratio d when thebuyer believes, with a high level of confidence, that the future price is high (orthe market is “hot”).

Our result will change if the no-buyer-default condition (1 − d)p∗p ≤ p − δ

does not hold. In this case, the price difference, �Price = p − p∗p, reflects

both the value of a buyer’s default option and the value of a developer’sabandonment option. Under this circumstance, whether the price difference,�Price, is positive or negative depends on which option dominates.

We now analyze explicitly the functional forms of the equilibrium priceequation (Eq. 26) and the value of a developer’s abandonment option(�Price = p − p∗

p) under different parameter values. The analytical resultsare reported in Proposition 3.

18Also see Lai et al. (2004) and Chan et al. (2008) for some of the possible incentives for using thepresale method.


Proposition 3 Under a no-buyer-default condition, the equilibrium presaleprice p∗

p is decreasing in the expected construction cost c, the construction costrisk factor θ , and the downpayment ratio d. On the other hand, it is increasing inthe expected spot price p. The spread between spot sale and presale price (or thevalue of the abandonment option, �Price = p − p∗

p) is increasing in c, θ and d.In other words,

∂p∗p

∂c< 0,

∂p∗p

∂θ< 0,

∂p∗p

∂d< 0,

∂p∗p

∂ p> 0,

∂�Price∂c

> 0,∂�Price

∂θ> 0, and

∂�Price∂d

> 0. (30)

Proof See Appendix B. ��

Proposition 3 demonstrates that the equilibrium presale price p∗p is decreas-

ing in the expected construction cost c and the construction cost risk factorθ . On the opposite side, the value of a developer’s abandonment option(�Price = p − p∗

p) is increasing in the expected construction cost c and theconstruction cost risk factor θ .

Basic option theory posits that the higher the variance, the higher the valueof an option. This explains what happens here. When the construction costrisk factor θ is large, the distribution of the construction cost (c ∼ � [c − θ ,c + θ ] ) is wider. Consequently, the probability for a developer to exercise anabandonment option is higher. As such, the value of the abandonment optionincreases. The expected future construction cost c has a similar impact onthe value of a developer’s abandonment option because a higher value of theexpected future construction cost c moves the distribution of the constructioncost (c ∼ � [c − θ, c + θ ]) upward, which increases the chance for a developerto exercise the abandonment option. It should be noted that, in equilibrium,a developer will have to compensate a buyer for the abandonment optionthat the developer has embedded into a presale contract. Consequently, ahigher (lower) value of the abandonment option will decrease (or increase)the equilibrium presale price p∗

p of a presale contract.The downpayment ratio d has a significant impact on the value of a

buyer’s default option and a developer’s abandonment option. When thedownpayment ratio is lower, it is easier for the buyer to exercise the defaultoption. Given this, the value of a buyer’s default option is decreasing in thedownpayment ratio d. To attract a buyer, the developer should reduce thepresale price if the buyer has a lower value of the default option becauseof a high downpayment ratio d. On the other hand, when the downpaymentd is high, it is easier for a developer to exercise the abandonment option.Given this, the value of a developer’s abandonment option is increasing in thedownpayment ratio d. In other words, the equilibrium presale price shoulddecrease if a developer has a higher abandonment option value. The effects ofa large downpayment ratio d on the equilibrium presale price are consistentlynegative. In the case where (1 − d)p∗

p ≤ p − δ (the value of the buyer’s default


option is zero), we only need to consider the impact of the developer’sabandonment option on the equilibrium presale price. In this case, a largedownpayment ratio d reduces the equilibrium presale price p∗

p and increasesthe value of the developer’s abandonment option.

Our analysis indicates that the downpayment ratio d should be an important(if not the most important) factor in the design of a presale contract. It canshift the risk from the developer to the buyer, or vice versa. The player (eitherthe developer or the buyer) who needs to have the most flexibility in her/hisinvestment strategy can always pay an appropriate price to buy the option(default or abandonment) from the other party using a presale contract. Ourpaper provides a simple model (Eq. 26) for pricing the abandonment optionwhen a buyer and a developer are negotiating a presale contract.

Proposition 4 Under a no-buyer-default condition, a developer is more likely toabandon a project under a presale contract than under a spot sale method, or

Pr {abandon}|presale > Pr {abandon}|spot.

The probability to abandon a project under a presale method, Pr{abandon}|presale, is an increasing function of the expected development costc, the construction cost risk factor θ , and the downpayment ratio d. Theprobability to abandon a project is decreasing in the expected spot price p. Theresults (except that for the downpayment ratio) also apply to the probabilityto abandon a project under a spot sale method where the abandonment optionvalue is positive, or

∂ Pr {abandon}|presale

∂c> 0,


∂θ> 0,


∂d> 0,


∂ p< 0.

∂ Pr {abandon}|spot

∂c> 0,


∂θ> 0, and


∂ p< 0. (31)

Proof See Appendix C. ��

Proposition 4 demonstrates that the choice of a selling method will affectthe probability of a developer’s abandonment decision. With a spot sale, adeveloper will abandon a project only if the remaining construction cost of theproject is higher than the price at which the developer can sell it in the spotmarket. However, the chance for a developer to exercise the abandonmentoption under a spot sale is low. This is true because, at time t = 1, when thevalue of the spot price is known, the developer most likely would have alreadyincurred a significant amount of the cost for the first-phase construction. Given


this, the magnitude of the remaining construction cost is rarely higher than theproperty spot price to justify a developer’s abandonment decision.

On the other hand, a developer is more likely to exercise the abandonmentoption under a presale contract than under a spot sale method. This is truebecause the downpayment from a presale contract provides an additionalincentive for a developer to abandon a project. With a presale contract, adeveloper will abandon the project if the remaining construction cost is higherthan the remaining presale payment or the spot price of the property. It shouldbe noted that when the value of a buyer’s default option is zero (as is the caseunder consideration here), the remaining presale payment must be lower thanthe spot price of the property (otherwise the buyer will exercise the defaultoption). Given this, the developer’s abandonment option is more valuablebecause of a lower exercise price (remaining presale payment as opposed tothe spot price) embedded in a presale contract.

Proposition 4 also indicates that, regardless of the selling method used,a developer is more likely to abandon a project with a higher expectedconstruction cost c or a lower expected spot price p. This means that whenthe expected profit (p− c) is low, a developer is more likely to abandon aproject. It should be noted that this condition is more likely to hold when thereis competition among developers for positive NPV projects. This situationnormally happens at the end of a boom market. During that time, moredevelopers are attracted to the market because of the perceived high profits.Buyers of properties should be aware of this problem and get into the presalemarket with this risk in mind.

As we just discussed, a developer’s abandonment option is more valuablewith a presale contract than with a spot sale method because of the downpay-ment the developer will receive from a buyer in a presale. Given this, a higherdownpayment ratio d will increase the likelihood that a developer exercises theabandonment option (and, therefore, increases the value of the abandonmentoption and decreases the equilibrium presale price). The results suggest that,when purchasing properties from developers using a presale contract, buyersshould be cautious about the downpayment ratio. An offer of a low presaleprice coupled with a high downpayment might not be suitable for all buyers asthe developer may have less incentive to commit on the contract. Although ahigh downpayment ratio should be compensated with an attractive low presaleprice, buyers have to assess their risk tolerance levels to decide if it is the bestcontract for them. This is especially true when construction cost is expected tobe high or volatile.19

19This result conforms to a personal experience of a co-author of this paper. After signing acontract to remodel a house with a large downpayment, the contractor abandoned the projectonce he discovered that the realized cost would be on the high end of his estimate. Of course, thisdeveloper offered a very low price because of the high downpayment received.


Case 2: When the Developer’s Abandonment Option Value is Zero

We next focus on the value of a buyer’s default option. This can be done byanalyzing the case where the developer’s abandonment option value is zero,or when

(1 − d)(c + θ) ≤ min{(1 − d)p∗p, p − δ}. (32)

Intuitively, we know that under a presale, a developer will not default ifthe maximum expected remaining construction cost (1 − d)(c + θ) is lowerthan the remaining presale payment (1 − d)p∗

p. Similarly, under a spot sale,a developer will not default if the maximum remaining construction cost(1 − d)(c + θ) is lower than the lower bound of the spot price distribution(p − δ). Equation 32 summarizes these two intuitions to form the boundarycondition that the developer will not abandon the project.

From Eq. 5, we know that a buyer’s default option has value when thelower bound of the spot price distribution (min{ p} = p − δ) is lower than theremaining presale payment ( p = (1 − d)p∗

p). Given p = (1 − d)p∗p > min{ p} =

p − δ, we know

(1 − d)p∗p > p − δ. (33)

Applying this inequality to Eq. 32, the no-developer-abandonment condition(while the value of the buyer’s default option is larger than zero) is

(1 − d)(c + θ) ≤ p − δ < (1 − d)p∗p . (34)

Equation 33 reports that p − δ < (1 − d)p∗p is the condition that the value of

the buyer’s default option is larger than zero. It is easy to see that the condition(1 − d)(c + θ) ≤ p − δ holds when the maximum construction cost c + θ is low,the lower bound of the spot price p − δ is high, or the first-phase constructioncost ratio d is high.

We now can provide a closed-form solution for the equilibrium presalepricing decision under this no-developer-abandonment condition (as specifiedby Eq. 34). Proposition 5 summarizes the equilibrium outcomes.

Proposition 5 In equilibrium, the price of a presale contract under a no-developer-abandonment condition is

p∗p = (1−d)p+(1+d)δ−2

√dδ[(1−d)p+δ]

(1−d)2 , if

(1 − d)(c + θ) ≤ p − δ < (1 − d)p∗p. (35)

The presale price is higher than the expected spot price, or

p∗p > p, or |�Price| = p∗

p − p > 0. (36)

The buyer’s expected cost of purchasing the property is the same regardless ofthe selling method eventually used, or

E(C(p∗p))|accept = E(C(p∗

p))|reject = E(C(p))|spot = p. (37)


Proof See Appendix D. ��

It should be noted that Eq. 35 holds only when the value of the buyer’sdefault option is positive (or when p−δ

1−d < p∗p) and the condition (c + θ)(1 −

d) − (p − δ) ≤ 0 is satisfied. It is clear that, once the condition for p−δ

1−d < p∗p

is satisfied, the condition (c + θ)(1 − d) ≤ p − δ is more likely to hold whenthe first-phase construction cost ratio d is large, the expected spot price p ishigh, or the expected construction cost c is low. This means that our modelresults should hold in a market where all players predict a high spread betweenthe expected spot price (p) and the expected construction cost (c) with a highlevel of confidence (which is reflected by a lower value of δ and θ). Given this,it is clear that a developer would prefer to use the pricing strategy shown inEq. 35 when the real estate market is “stable” with low uncertainty on theconstruction cost.

In addition, when a developer adopts a high first-phase construction coststrategy (which means that the remaining construction cost (c + θ)(1 − d) islow), Eq. 36 indicates that the developer can set a presale price at a levelhigher than the expected spot price (to compensate herself/himself for thebuyer’s default option embedded in the contract). Consequently, we are morelikely to observe this type of pricing strategy (p∗

p > p) in a market whereconstruction cost is low and relatively stable, while there is still a reasonablelevel of uncertainty about the future spot price.20

Unlike the equilibrium presale price derived under the no-buyer-defaultcondition, the equilibrium presale price derived under the no-developer-abandonment condition is higher than the expected spot price. The differencebetween the expected spot price and the presale price is now negative, orp − p∗

p < 0. Given this, we now redefine the price spread as |�Price| = p∗p −

p > 0. (It should be noted that the price spread defined under the no-buyer-default condition is �Price = p − p∗

p.) This means that the developer nowcharges a premium (over the expected spot price) for the presale price. Asdiscussed before, in a presale contract, premiums are required to compensatefor the value of a buyer’s default option and the value of a developer’sabandonment option embedded in the presale contract. In this case, however,since the developer’s abandonment option value is zero, the increase in thepresale price must be due to the buyer’s default option. This finding is in linewith the finding of Chan et al. (2008). While their research emphasizes on adeveloper’s production decision, they also find that developers will increasethe presale price to compensate for the default option they give to the buyers.

20Lai et al. (2004) and Chan et al. (2008) argue that the presale method can reduce potentialbankruptcy cost and financing cost when compared to the spot sale method. The benefit of usingthe presale method, therefore, can be shared by the developer and the buyer. If we include thebenefits from a reduction in potential bankruptcy cost and financing cost into our model, it ispossible that the result p∗

p > p does not hold under this no-developer-abandonment condition.


We now analyze how the presale price and the value of a buyer’s defaultoption (measured as |�Price| = p∗

p − p > 0) are affected by other parametervalues. The results are reported in Proposition 6.

Proposition 6 Under a no-developer-abandonment condition, the equilibriumpresale price p∗

p is increasing in the expected spot price p and the dispersionof spot price δ, and decreasing in the downpayment ratio d. The size of thespread between presale price and expected spot price, |�Price| = p∗

p − p > 0,is decreasing in p and d, and increasing in δ. In other words,

∂p∗p

∂ p> 0,

∂p∗p

∂δ> 0,

∂p∗p

∂d< 0,

∂|�Price|∂ p

< 0,∂|�Price|

∂δ> 0, and

∂|�Price|∂d

< 0. (38)

Proof See Appendix E. ��

Proposition 6 reports that the equilibrium presale price p∗p is increasing

in the expected spot price p and the price dispersion factor δ, while thevalue of a buyer’s default option (|�Price| = p∗

p − p > 0) is decreasing in theexpected spot price p and increasing in the price dispersion factor δ. Whenthe price dispersion factor δ is large, the distribution of the spot price p ∼ �

[p − δ, p + δ] is wider. Consequently, the value of a buyer’s default option ishigher and hence, a higher price for the presale contract. A higher expectedspot price increases the presale price but decreases the spread between thepresale price and the expected spot price. This is true because, while thepresale price increases with an increase in the expected spot price, the speedof its increase is less than the speed of increase in the spot price. This is why∂|�Price|

∂ p < 0.When the first-phase construction cost ratio d is low, a developer has a

higher chance to exercise the abandonment option and, therefore, the valueof the abandonment option is higher. The downpayment ratio d also has asignificant impact on the value of a buyer’s default option. It should be notedthat, similar to a developer’s abandonment option (where the value of theoption decreases with an increase in the first-phase construction cost ratiod), the value of a buyer’s default option decreases with an increase in thedownpayment ratio d. To compensate the buyer for a lower default optionvalue, a developer will have to reduce the presale price. In other words, whilea developer can offer a high downpayment ratio as a condition for a low presaleprice to attract buyers in the presale market, the developer can also use a lowfirst-phase construction cost ratio together with a low presale price.

Numerical Examples

In this section, we use two numerical examples to illustrate our model impli-cations. According to the National Association of Realtors, the mean median


Table 3 Numerical results of two hypothetical cases

Case 1 Case 2c = 165 c = 120θ = 25 θ = 5

Panel A: presale price p∗p 176.84 180.21

Presale price p∗p versus expected spot price p p∗

p < p p∗p > p

Pr {abandon}|presale 0.26 –Pr {abandon}|spot 0.01 –

Panel B: presale price sensitivity· to construction cost mean c −0.47 –· to construction cost risk θ −0.22 –· to spot price mean p 1.44 0.99· to downpayment ratio d −91.25 −27.54

Panel C: presale abandonment probability sensitivity· to construction cost mean c 0.03 –· to construction cost risk θ 0.01 –· to spot price mean p −0.03 –· to downpayment ratio d 1.83 –

This table reports the numerical results of two hypothetical cases, which differ in the constructioncost mean c and risk θ . For both cases, the spot price has a mean p = 180 and risk δ = 12. Thedownpayment ratio (which is also the first-phase construction ratio) is d = 5%

sales price of condominiums and co-ops units in the US during the 2000–2008period is around $180,000, with an average two-year standard deviation of$12,000 (the average condominium construction period is around two years).Given this, for the spot price distribution p ∼ � [p − δ, p + δ], we set the meanas p = 180 and the boundary range parameter as δ = 12. According to Leunget al. (2007), one of the property presale payment schedules used in the US isa 5:95 system (that is, 5% deposit and 95% for later payments). We hence setthe presale downpayment ratio as d = 5%.

For the construction cost distribution c ∼ � [c − θ, c + θ ], we have toassume construction costs that range from low to high so that the developer’sabandonment option value can vary from “in the money” (Case 1) to “out ofthe money” (Case 2). We start with a scenario (Case 1) that assumes c = 165and θ = 25, which satisfies all the parameter constraints to ensure that thedeveloper’s presale abandonment option is in the money, the buyer’s defaultoption is out of the money (see Eq. 25), and the developer’s abandonmentoption is in the money under a spot sale. Note that we need to assign a largeconstruction cost mean c and a high construction risk factor θ to make theseconditions hold. We also study an alternative scenario (Case 2) that assumesc = 120 and θ = 5, satisfying all the parameter constraints to ensure that thedeveloper’s presale abandonment option is out of money, the buyer’s defaultoption is in the money (see Eq. 34), and the developer’s abandonment optionis out of the money under a spot sale. As expected, we need to assign a lowconstruction cost mean c and a small construction risk factor θ to make theseconditions hold.

With these parameter values, the numerical results of these two cases aresummarized in Table 3. Cases 1 and 2 of Panel A report the numerical results


predicted by our Propositions 2, 4 and 5. Cases 1 and 2 of Panel B reportthe numerical results predicted by our Propositions 3 and 6, respectively whileCase 1 of Panel C reports the numerical results predicted by our Proposition4. All the results are consistent with our model predictions. As predicted byPropositions 2 and 5, a presale contract under Case 1 of Panel A (the buyerwill not default while developer might abandon the project) is priced belowthe expected spot price, while the opposite holds true under Case 2 of Panel A(the developer will not abandon the project while the buyer might default). Aspredicted by Proposition 4, when a buyer will not default on a presale contract(Case 1), the developer has a higher probability to abandon the project undera presale contract than under a spot sale.21 The comparative static resultsof presale price and abandonment probability are also consistent with thepredictions of Propositions 3, 4, and 6.

Conclusions

This study is the first to examine in detail a developer’s abandonment optiontogether with a buyer’s default option. We show that a developer has an optionto abandon a project when the realized construction cost (compared to theexpected construction cost) is too high. Furthermore, while the developercan also abandon a project even if she/he plans to sell the property in thespot market when the project is completed (using the spot sale method), thedeveloper is more likely to abandon a project (or that the abandonment optionis more valuable) under a presale contract than under a spot sale method.This is the case because the downpayment from a presale contract providesan additional cash flow to the developer. Given this, in an environment withhighly volatile construction costs, a conservative buyer might want to lower thedownpayment ratio on a project to minimize the probability that a developerwill abandon the project. We also show that, when the value of the remainingpayment is higher than the spot price of the property, a buyer has an optionto default on the payment. Given this, in an environment with high pricevolatility, it might be wise for a conservative developer to minimize the first-phase construction cost so that she/he can also have an option to abandon theproject should a buyer default on the presale contract.

Our model shows that, when a presale contract uses a high downpaymentratio and/or when the estimated lowest future spot price is quite high, thebuyer’s estimated default probability could be very low and the value of thebuyer’s default option can approach to zero. Under this circumstance, thepresale price is always lower than the spot price because the developer needs tocompensate the buyer for the abandonment option the buyer grants her/him.Since the presale contract price is reduced from the expected spot price tocompensate for the value of the abandonment option, the nominal property

21See Eqs. 56 and 57 for the exact formulas on how to calculate these two probabilities.


appreciation rate (from a current presale price to an expected future spotprice) could be higher than its actual property appreciation rate (from a currentspot price to an expected future spot price). It is interesting to note that buyersare always optimistic about the distribution of future prices in a “hot” market,and it is exactly in such a period that presale contracts are most popular. Assuch, the widespread belief that buyers can do better in a presale market thanin a spot market could be misleading in a “hot” market. Of course, whenthe value of a buyer’s default option is also taken into consideration, thisconclusion might not hold. However, as we just discussed, the value of a buyer’sdefault option should be very low in a “hot” market.

However, it is well known that real estate assets are not liquid. Con-sequently, the lack of transactions makes it difficult to estimate the assetvalue with the same level of precision as financial assets. In other words,the observed values might contain noise. Under this condition, there mightbe heterogeneous beliefs in the marketplace (and our homogeneous beliefassumption might not hold). On a similar line of thought, Childs et al. (2002)argue that options are valued and exercised based on the best estimate of theunderlying asset value when that value can only be observed with noise. Thissuggests that our model results reported here may only reflect the estimatedvalues of these options.

Our model can be generalized in three important ways. First, our modelcurrently captures a simple equilibrium game between two players (one de-veloper and one buyer). It is not clear what the equilibrium presale contractsare like when there are multiple buyers and multiple developers in the market.Second, with a slight modification, our model can also be used for the pricingof futures contracts where the production cost and the price of the underlyingasset are both stochastic. (One good example would be the futures contractof timber.) Finally, our model framework can also be revised for studying theequilibrium contract of a multi-period project involving two players and withstochastic revenues and costs.

Acknowledgements The research on which this paper is based was partially funded by theGlobal Social Science Institute. We acknowledge helpful comments from Jared Delisle, DavidFrame, William Hardin, Austin Jaffe, Sheridan Titman, Gerd Welke, Eric Zhang, two anonymousreferees, the editor, and participants of the 2010 Florida State University Research Symposium on“The Economics of Information on Real Estate Markets” and the 2010 Asian Real Estate SocietyAnnual Conference.

Appendix A: Proof for Proposition 2

When the buyer’s default probability is negligible, the buyer’s expected totalcost function if she/he accepts the presale offer (as shown in Eq. 15) can bereduced to

E(C)|accept =∫ p+δ

p−δ

{∫ c

c−θ

ppd�(c) +∫ c+θ

c(dpp + p)d�(c)

}

d�( p), (39)


where the first term inside the outer integral represents the buyer’s expectedcost if the developer continues the construction and the second term of thesame integral represents the payoff if the developer abandons the project.Given the expected total cost if the buyer rejects the presale offer (as shownin Eq. 16), the pricing constraint to ensure that the buyer accepts the presalecontract (as defined in Eq. 18) is reduced to

ppl ≤ pp ≤ pp

u, (40)

where ppu = A + √

A2 − 4(c − θ)(1 − d)p2(1 − d)

, (41)

ppl = A − √

A2 − 4(c − θ)(1 − d)p2(1 − d)

, (42)

and A = p + (1 − d)c − (1 + d)θ . (43)

The developer’s decision problem associated with offering an acceptablepresale price (as shown in Eq. 19) is reduced to


−s +∫ p+δ

p−δ

{∫ c

c−θ

(

pp − c)

d�(c)

+∫ c+θ

c(dpp − dc)d�(c)

}

d�( p),

s.t. ppl ≤ pp ≤ pp

u,

where c = pp,

ppu = A + √

A2 − 4(c − θ)(1 − d)p2(1 − d)

,

ppl = A − √

A2 − 4(c − θ)(1 − d)p2(1 − d)

,

and A = p + (1 − d)c − (1 + d)θ . (44)

The first term inside the outer integral in Eq. 44 represents the developer’sexpected payoff if she/he continues the construction. The second term repre-sents the situation if the developer abandons the project. From the developer’sunconstrained objective function (see Eq. 44), the expected payoff from apresale can be derived as

E(π)|presale = 14θ

{

(1 − d)(pp − c)2 + θ2(1 − d)

+ 2θ[

(1 + d)pp − (1 + d)c − 2s]}

. (45)

It should be noted that Eq. 45 is a convex function of pp (since d2 E(π)|presale

dp2p

=1−d2θ

> 0). Using the first-order condition dE(π)|presale

dpp= 0, we can derive the


lowest point of the convex curve, which is the price that minimizes E(π)|presale,or

pbp = c(1 − d) − θ(1 + d)

(1 − d). (46)

Comparing pbp to the two boundaries of pp given by the constraint pp

l ≤ pp ≤pp

u, we find

ppu − pb

p = T1 + T2

2(1 − d)> 0, given

T1 = p − (1 − d)c + (1 + d)θ

= p − (1 − d)(c − θ) + 2dθ > 0,

T2 =√

A2 − 4(c − θ)(1 − d)p > 0,

where A = p + (1 − d)c − (1 + d)θ > 0, (47)

and

ppl − pb

p = T1 − T2

2(1 − d)> 0, given

T1 > 0, T2 > 0, and T21 − T2

2 = 8θdp > 0. (48)

Equations 47 and 48 indicate that an acceptable presale price must be locatedon the right half of the convex curve, where the developer’s expected payoff ismonotonically increasing in the presale price. Given this acceptable range, theequilibrium presale price that maximizes the developer’s expected incrementalpayoff, p∗

p, must be the upper boundary of the presale price ppu. From Eq. 25

we know that the boundary condition to ensure that a buyer will not defaultwhile a developer still has an option to abandon a project is (1 − d)p∗

p ≤ p −δ < (1 − d)(c + θ). Given this, a comparison between the presale price and theexpected spot price generates

p∗p − p = T2 − T3

2(1 − d)< 0, given

T3 = p(1 − 2d) − c(1 − d) + θ(1 + d) > 0, and

T23 − T2

2 = 4dp(1 − d)[(c + θ) − p] > 0. (49)

This is true because, from Eq. 25, we know that (1 − d)p∗p < (1 − d)(c + θ)

or c + θ − p∗p > 0. Since c + θ − p∗

p = T4−T22(1−d)

(with T4 = c(1 − d) + θ(3 − d) −p > 0 and T2 > 0) and T2

4 − T22 = 8θ(1 − d)[(c + θ) − p] > 0, we know that

(c + θ) − p > 0. Equation 49 demonstrates that the presale price is lower thanthe expected spot price.

Finally, substituting pp in the buyer’s expected presale cost function(Eq. 39) with p∗

p and comparing it with a buyer’s expected spot sale cost


function (see Eq. 16), which is also the buyer’s expected cost if the developeruses the spot sale method, we find

E(C(p∗p))|accept = E(C)|reject = E(C)|spot = p. (50)

Under this condition, a buyer should be indifferent between a presale anda spot sale offer because her/his expected cost is the same under these twocontracts. In other words, the equilibrium p∗

p is the presale price that makesthe buyer indifferent between accepting and rejecting the presale offer. ��

Appendix B: Proof for Proposition 3

Under a no-buyer-default condition, since the equilibrium presale price is theone that makes a buyer indifferent between accepting and rejecting the presaleoffer, we know that

E(C(p∗p))|accept − E(C)|reject = 0, (51)

hence we can write a function

f = −[

E(C(p∗p))|accept − E(C)|reject

]

=(p∗

p − c)[

p − (1 − d)p∗p

]

+ θ[

p − (1 + d)p∗p

]

2θ= 0. (52)

Taking partial derivatives of f (in Eq. 52), with respect to p∗p, c, θ , d and p, we

derive

∂ f∂p∗

p= p − 2p∗

p(1 − d) + c(1 − d) − θ(1 + d)

2θ= −T2

2θ< 0,

∂ f∂c

= − p − (1 − d)p∗p

2θ< − (p − p∗

p)

2θ< 0,

∂ f∂θ

= −(p∗

p − c)[

p − (1 − d)p∗p

]

2θ2 < −(p∗

p − c)[

p − p∗p

]

2θ2 < 0,

∂ f∂d

= −p∗

p

(

c + θ − p∗p

)

2θ< 0 (Eq. 25 indicates c + θ − p∗

p > 0),

and∂ f∂ p

= p∗p − (c − θ)

2θ> 0. (53)


Applying Roy’s identity, we have

∂p∗p

∂c= −

∂ f∂c∂ f∂p∗

p

< 0,

∂p∗p

∂θ= −

∂ f∂θ

∂ f∂p∗

p

< 0,

∂p∗p

∂d= −

∂ f∂d∂ f∂p∗

p

< 0, and

∂p∗p

∂ p= −

∂ f∂ p∂ f∂p∗

p

> 0. (54)

When the spot sale–presale price spread is defined as �Price = p − p∗p,

we have

∂�Price∂c

= −∂p∗p

∂c> 0,

∂�Price∂θ

= −∂p∗p

∂θ> 0, and

∂�Price∂d

= −∂p∗p

∂d> 0. (55)

This is true because p is unaffected by the parameters c, θ and d. ��

Appendix C: Proof of Proposition 4

Under a no-buyer-default condition, the probabilities for a developer toabandon a project under a presale contract and a spot sale method are

Pr {abandon}|presale =∫ p+δ

p−δ

∫ c+θ

cd�(c)d�( p) = 1

2θ

(

c + θ − p∗p

)

= G − T2

4θ(1 − d), where G = c(1 − d) + θ(3 − d) − p (56)

and

Pr {abandon}|spot =∫ p+δ

p−δ

∫ c+θ

c′d�(c)d�( p) = 1

2θ

[

c + θ − p(1 − d)

]

, (57)


respectively. From Eqs. 56 and 57, we know

Pr {abandon}|presale − Pr {abandon}| spot

=∫ p+δ

p−δ

(∫ c+θ

cd�(c) −

∫ c+θ

c′d�(c)

)

d�( p) = 12θ

[

−p∗p + p

(1 − d)

]

= 12θ(1 − d)

[

p − (1 − d)p∗p

]

>1

2θ(1 − d)

[

p − p∗p

]

> 0 (see Proposition 2). (58)

Furthermore,


∂c= 1

2θ

[

1 − ∂p∗p

∂c

]

> 0, given

∂p∗p

∂c< 0 (see Proposition 3). (59)


∂θ= HT2 + J

4c2(1 − d)T2> 0, given

T2 > 0,

H = p − (1 − d)c > 0,

and J = [

p−(1−d)c]2+θ

[

(1−3d)p−c(1−d2)]

>0. (60)


∂d= 1

2θ

(

−∂p∗p

∂d

)

> 0, given

∂p∗p

∂d< 0 (see Proposition 3). (61)


∂ p= 1

2θ

(

−∂p∗p

∂ p

)

< 0, given∂p∗

p

∂ p> 0 (see Proposition 3).


∂c= 1

2θ> 0. (62)


∂θ= p − (1 − d)c

2θ2(1 − d)> 0. (63)


∂ p= − 1

2θ(1 − d)< 0. (64)

��


Appendix D: Proof for Proposition 5

When the probability of a developer abandoning a project is negligible, thebuyer’s expected total cost function if she/he accepts a presale offer (Eq. 15)can be reduced to

E(C)|accept =∫ c+θ

c−θ

{∫ p+δ

pppd�( p) +

∫ p

p−δ

(dpp + p)d�( p)

}

d�(c), (65)

where the first term inside the outer integral represents the buyer’s expectedcost if she/he commits on the payment and the second term of the same integralrepresents the payoff if she/he defaults. Given the expected total cost if thebuyer rejects the presale offer (Eq. 16), the pricing constraint to ensure thatthe buyer accepts the presale contract (Eq. 18) can be reduced to

pp ≤l

︷︸︸︷

pp or pp ≥u

︷︸︸︷

pp ,

wherel

︷︸︸︷

pp = (1 − d)p + (1 + d)δ − 2√

dδ[δ + (1 − d)p](1 − d)2 ,

u︷︸︸︷

pp = (1 − d)p + (1 + d)δ + 2√

dδ[δ + (1 − d)p](1 − d)2 . (66)

Intuitively, we know that all buyers will accept a presale contract with a lowerpresale price. Between the two binding constraints reported in Eq. 66, themeaningful one should be

pp ≤l

︷︸︸︷

pp . (67)

Consequently, the developer’s decision problem (Eq. 19) is reduced to


−s +∫ c+θ

c−θ

{∫ p+δ

p

(

pp − c)

d�( p)

+∫ p

p−δ

(dpp + p − c)d�( p)

}

d�(c),

s.t. pp ≤l

︷︸︸︷

pp ,

where p = (1 − d)pp,

l︷︸︸︷

pp = (1 − d)p + (1 + d)δ − 2√

dδ[δ + (1 − d)p](1 − d)2 . (68)

The first term inside the outer integral is the objective function that representsthe developer’s expected payoff if the buyer commits on the payment. The


second term represents the situation if the buyer defaults. The developer’sunconstrained objective function can be derived as

E(π)|presale = 14δ

{−(1 − d)2 pp2 + 2pp[(1 − d)p + (1 + d)δ]

− (p − δ)2 − 4(c + s)δ}

. (69)

This is a concave function of pp (since d2 E(π)| presale

dp2p

= − (1−d)2

2δ< 0). Using the

first-order condition dE(π)|presale

dpp= 0, we can derive the price that maximizes

E(π)|presale as

pmp = (1 − d)p + (1 + d)δ

(1 − d)2 . (70)

Comparing pmp to the upper boundary of pp given by the constraint pp ≤

l︷︸︸︷

pp ,we find

l︷︸︸︷

pp −pmp = −

2√

dδ[

(1 − d)p + δ]

(1 − d)2 < 0. (71)

This indicates that an acceptable presale price must be located on the left halfof the concave curve (before it reaches the top point), where the developer’sexpected payoff is monotonically increasing in the presale price. Given thisacceptable range, the equilibrium presale price that maximizes the developer’sexpected incremental payoff p∗

p must be the upper boundary of the presale

pricel

︷︸︸︷

pp .On the other hand, the developer’s expected payoff under a spot sale

method (Eq. 20) is reduced to

E(π)|spot =∫ c+θ

c−θ

∫ p+δ

p−δ

( p − c) d�( p)d�(c). (72)

A comparison between the presale price and the expected spot price indicates

p∗p − p = G1 − 2

√

dδ[δ + (1 − d)p](1 − d)2 > 0, where

G1 = d(1 − d)p + (1 + d)δ > 0, given

G21 −

(

2√

dδ[δ + (1 − d)p])2 = (1 − d)2(δ − dp)2 > 0. (73)

Equation 73 demonstrates that the equilibrium presale price is higher thanthe expected spot price. Substituting pp in the buyer’s expected presale cost


function (Eq. 65) with p∗p and comparing it to the buyer’s expected spot sale

cost function (Eq. 16), which is also the buyer’s expected cost if the developeruses the spot sale method, we find

E(C(p∗p))|accept = E(C)|reject = E(C)|spot = p. (74)

��

Appendix E: Proof for Proposition 6

From Eq. 33, we derive (1 − d)p∗p − (p − δ) > 0. Since (1 − d)p∗

p − (p − δ) =2(δ−G2)

1−d , where G2 = √

dδ[(1 − d)p + δ] > 0, we know that δ2 − G22 = δ(1 −

d) (δ − dp) > 0 or δ − dp > 0. From Eq. 35, we derive

∂p∗p

∂ p= G2 − dδ

(1 − d)G2> 0, given

G22 − (dδ)

2 = dδ(1 − d) (p + δ) > 0,

∂p∗p

∂δ= (1 + d)G2 − d

[

2δ + (1 − d)p]

(1 − d)2G2> 0, given

(1 + d)2G22 − d2 [

2δ + (1 − d)p]2 = (1 − d)2d (p + δ) (δ − dp) > 0,

∂p∗p

∂d= G2G3 − G4

(1 − d)3G2< 0, given

G3 = (1 − d)p + (3 + d)δ > 0,

G4 = δ[

(1 + d − 2d2)p + (1 + 3d)δ]

, and

G22G2

3 − G24 = −(1 − d)3δ(δ − dp)(δ + p)2 < 0. (75)

Given that p∗p > p (see Proposition 5) and that the spot sale and presale price

spread is defined as |�Price| = p∗p − p, we derive

∂|�Price|∂ p

= ∂p∗p

∂ p− 1 = d (G2 − δ)

(1 − d)G2< 0, given

G22 − δ2 = −(1 − d)δ (δ − dp) < 0,

∂|�Price|∂δ

= ∂p∗p

∂δ> 0, and

∂|�Price|∂d

= ∂p∗p

∂d< 0. (76)

��


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Presale Contract and its Embedded Default and ...faculty.baruch.cuny.edu/kwang/BioVita/pdf/Presale...118 S.H. Chan et al. the spot price dives below the contract price.4 The basic