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Journal of Real Estate Finance and Economics, Vol. 16:3, 257±267 (1998)
# 1998 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
A Contingent Claims Analysis of Real EstateListing Agreements
RICHARD J. BUTTIMER, JR.
Department of Finance and Real Estate, UTA Box 19449, University of Texas at Arlington, Arlington, TX76013, e-mail: [email protected]
Abstract
This article examines real estate brokerage contracts as contingent claims. It derives terminal and boundary
conditions for this contract. A Nelson and Ramaswamy (1990) style lattice is then developed to solve the partial
differential equation for the value of the listing agreement. After identifying various parameters that determine
the contract's value, those parameters are varied to examine their relative impact on the contract.
Key Words:
Brokerage is perhaps the most studied area of real estate. Interest in brokerage research
originates with a variety of sources. Home sellers want to know whether brokers can raise
the sales prices of their houses. Governments and consumer groups want to know whether
brokers exercise monopoly power in the market. Brokers themselves wish to determine
their optimal pro®t maximizing behavior.
Academics tend to examine these issues in the context of market ef®ciency. The general
explanation for the existence of brokers is that they increase the ef®ciency of the market by
increasing the information available to market participants. An alternative way of saying
this is that they reduce the search costs faced by home buyers and sellers. Much of the
literature argues that there is, however, some collusion between brokers through the
Multiple Listing Services (MLS)1 and that this collusion may allow brokers to extract
some rents from the market. The primary evidence normally presented is the near-
uniformity of commission rates in a given market. A common argument is that the effort
required to sell a house is not a linear function of the sales price and that if there is no
collusion among brokers, there should be, at the very least, variation in commission rates
across house price ranges within a given market.
Focusing on commission rates ignores a key aspect of the agreement: listing contracts
are contingent, not certain, claims. The broker receives a payout only if the property sells.
At the origination of the listing agreement no money changes hands, so the broker's
compensation for marketing the property is the present value of the future expected cash
¯ows. As this article shows, a number of factors in¯uence that present value. Some factors,
such as the commission rate or term of the contract, the parties can specify contractually.
Other factors, such as the rate at which the broker shows the property or the seller's
reservation price, they cannot specify contractually. Given that the homeowner controls
some of these factors, they can, at origination, decide how much the broker's services are
worth and, even after contracting to a commission rate, limit the listing agreement's value.
Conversely, the broker can act to increase the present value of the contract by increasing
his or her selling effort. By increasing their selling efforts, of course, brokers behave in a
manner consistent with the homeowner's interests. The important point is that even if
brokers collude to ®x commission rates, home sellers can control the value of the listing
contract by altering other parameters, thus largely negating the effects of collusion.
This article develops a model for valuing real estate listing contracts and examines how
both the broker and seller can in¯uence this value. It is organized as follows. Section 1
describes the listing contract and the economic environment this model assumes. Section 2
examines the listing contract as a contingent claim and develops the appropriate boundary
conditions needed to value it. Section 3 presents the pricing algorithm. Numerical analysis
in section 4 shows the relative impact of various parameters on the value of a hypothetical
listing agreement. Section 5 summarizes the article.
1. The Contract
The listing contract between the broker and homeowner speci®es a number of parameters.2
The parties agree to a commission rate C, which will be paid if the home sells during the
term T of the contract. The homeowner states a listing price PL. The listing price obligates
the seller to pay a commission to the broker if a ready, willing, and able buyer is found who
will pay PL for the property, even if the homeowner chooses not to sell. Note, however,
that the listing price is neither a lower nor an upper boundary for the sales price of the
home. The homeowner still retains the right to sell the home at lower prices, and properties
occasionally sell for more than their list price. While there may not exist an upper bound
on the sales price, a true lower bound does existÐthe owner's reservation price PR (that is,
the lowest price at which the owner is willing to sell the property).3 The broker, of course,
has no way of knowing this reservation price. At best the broker might assume correlation
between PL and PR, but as Cauley (1994) and Yavas and Ying (1995) show, there is not a
simple relationship between a seller's reservation price and list price. In any event, it is the
reservation price, not the list price, that determines the lowest possible selling price for the
home, and this is what ultimately determines the boundary conditions for the list contract.
The price at which the homeowner can sell is not constant but evolves over time. De®ne
H0 as the price at which the homeowners could sell at the moment they enter into the
listing contract. This can be thought of as the price the property could bring with minimal
search efforts. This value of H0 is presumably less than PR, since if it were not, the owner
would sell immediately. Borrowing from the mortgage pricing literature (see Kau et al.,
1992), this article assumes that the house price evolves according to
dH
H� �aÿ s� dt� sHdzh; �1�
where a represents the total returns to the housing, s is the housing service ¯ow, sh is the
258 BUTTIMER
volatility of house prices, and dzh is a standard Wiener process. The service ¯ow
represents those bene®ts derived from possessing the house during the current periodÐ
that is, the implicit rent consumed by the homeowner. Since future owners cannot capture
these current bene®ts, they must be treated like a dividend in the current period.
Consistent with Merton (1973), the service ¯ow is simply subtracted from the total
returns.
Since the house price evolves over time and can rise or fall, the homeowner wishes to
poll this price as frequently as possible. The real service the broker provides is reducing
the cost to the owner of this polling, thus allowing more frequent polling for a given cost.4
That is, the broker can market the property more often than the homeowner could. Denote
the number of time steps between polling periods as G. One interpretation of this
parameter is that it is a measure of the search intensity by the seller or their broker, where
lower values of G represent higher levels of search effort. By hiring a broker, the
homeowner essentially reduces the value of GÐthat is, increases the frequency of polling,
holding all else constant.5
Given the above parameters, it is possible to determine the value of the listing contract
at its origination. As with most contingent claims, the way to value the listing contract is to
begin with the termination of the contract and work backward. The next section develops
such a method.
2. Listing Agreement Boundary Conditions
A listing contract is somewhat unusual in that the contract's payoffs are determined, at
least in part, by the writer of the contract (the owner) and not by the holder (the broker).
Because of this, any valuation method must take into account the homeowner's incentives.
The owners will always seek to maximize the value of their net position, and they
ultimately decide when to sell their house. Indeed, this is a critical point: by retaining at
least some control over the timing and size of the contract payouts, the homeowner has a
mechanism with which to control the listing agreement's value.
As shown by Cox, Ingersoll, and Ross (1985), the price f of any contingent contract that
is dependent on H must satisfy the following partial differential equation:
qf
qt� Hr
qf
qHÿ 1
2s2
HH2 q2f
qH2� rf ; �2�
where r is the risk-free rate, subject to the boundary conditions of the asset.6 Thus, the
®rst step in developing a valuation procedure is determining the appropriate terminal and
boundary conditions for the listing contract. These conditions directly follow from the
homeowner's incentives, and any model must explicitly incorporate these incentives.
Consider the situation facing the homeowner at time T, the termination of the listing
contract. There are two possible terminal states of the world. First, the market value of the
property HT, may be less than the seller's reservation price PR. In this case, the owner will
not sell, and the listing contract will expire without value.7 Second, the property value may
A CONTINGENT CLAIMS ANALYSIS OF LISTING AGREEMENTS 259
exceed PR, in which case the owner will sell it at price HT , and the broker will receive
HT � C. Denoting the payoff to the seller at termination of the listing agreement as ST , the
terminal boundary conditions for the owner's option to sell the property are
ST �0 if HT < PR:HT if PR � HT
��3�
These translate into the following boundary conditions for the listing contract:
LT �0 if HT < PR;
C*HT if PR � HT
(�4�
where LT is the value of the listing agreement to the broker at time T.
Moving backward through time, the owner must take into account not only the current
value of the house but also the expected future selling price of the house if it does not sell
in the current time period. Property owners face a compound option problem: if they sell
the property at time t, this precludes their selling at time t� 1. Thus, when selling the
property the owner gives up not only the property but also the option to sell the property in
the future. This option has value, and the owner will sell only when the present value of the
property exceeds both the seller's reservation price and the discounted expected value of
selling in the future.
To see this, consider the seller's position at time t � TÿG that is, at the last time they
can sell prior to the termination of the listing agreement. Let ST denote the discounted
expected payoff from selling the property at time T. The homeowner faces several possible
situations at this time. First, the property value H may be less than PR, in which case the
owner will not sell. Second, Ht might be greater than PR but less than ST (that is,
PR � Ht � ST). In this case, the homeowner would not sell the property since this would
require giving up the option to sell at time T, which is more valuable. Third, Ht might
exceed ST , and the property owner will sell at price Ht.
Once again, the boundary conditions for the homeowner's option to sell and the listing
contract interwine. A general way to state the payoff to the homeowner at time t < T is
St �0 if Ht < max�PR; St�G�;Ht if max�PR; St�G� � Ht
(�5�
where St�G is the discounted present value of the option to sell the property in the future.
The value of the contract to the broker at time t < T is
Lt �Lt�G if Ht < max�PR; St�G�;C*Ht if max�PR; St�G� � Ht
(�6�
where Lt�G is the discounted value of the contract at time t� G.
260 BUTTIMER
3. The Valuation Procedure
Since the listing contract is an American style claim, no closed-form solution to (2) exists,
and the price must be found via a numerical method. Nelson and Ramaswamy (1990)
present a general approach to modeling diffusion processes with simple binomial lattices.
This is the technique used here, and it applies directly.
The risk-neutral stochastic process for the house value is the same as that in equation
(1), except that the risk-free rate replaces the total returns to housing:
dH
H� �r ÿ s�dt� sHdzh: �7�
The variance of this process is nonconstant, which poses a problem for simple binomial
lattices. To alleviate this problem, transform this process into one with a constant
volatility by taking logs. That is, de®ne a new variable W, such that W� ln(H). Note that
from Ito's lemma W evolves according to
dW � r ÿ sÿ s2h
2
8>>: 9>>;dt� shdzh: �8�
Recovering the value of H at any node in the lattice is a simple matter of exponentiation.
With the above boundary conditions and the Nelson and Ramaswamy (1990) lattice,
valuing the listing contract is a straightforward exercise in backward pricing. The
algorithm starts at the terminal time step and determines the house value at each node of
that time step. It then applies the boundary conditions in equations (3) and (4) to determine
the value of ST and LT at each of the terminal nodes.
The algorithm then proceeds to time step T ÿ G. At every node on this time step, the
algorithm ®rst determines the homeowner's discounted expected value of selling the
property at time T, denoted as ST . This discounted expected value along with the current
house value allows the algorithm to apply the boundary conditions of equations (5) and (6)
to determine the payoffs, at each node of this time step, to the seller and broker,
respectively. If the broker receives a payout at a particular node, the value of the listing
contract at that node is the amount of the payout. If, however, the broker does not receive a
payout, the value of the listing contract is equal to the discounted expected value of the
listing agreement at the time T.
Working in this manner the algorithm proceeds backward through time to the
origination of the contractÐthat is, when t� 0. At this point it determines the value of L0,
which is the present value of the listing agreement at its origination.
4. Numerical Results
Given that section 3 shows how to solve for the value of a listing contract, this section
proceeds to price a hypothetical listing contract given a set of base parameters. These
A CONTINGENT CLAIMS ANALYSIS OF LISTING AGREEMENTS 261
parameters are then varied to determine their partial effects on the value of the listing
contract. These partial effects provide insights into the relationship between the broker and
the homeowner.
4.1. Listing Agreement Parameters
The pricing model presents several parameters that affect the value of the contract. For the
purpose of this analysis it is convenient to separate these parameters into two groupsÐ
those parameters that the parties can specify contractually and those that they cannot
specify contractually. The parameters that they can specify via contract include the
commission rate C and the term of the listing agreement T. The parameters that they cannot
specify via contract include the initial house value H0, the housing service ¯ow s, the risk-
free rate r, housing volatility sH, the polling frequency parameter G, and the seller
reservation price PR. Table 1 presents the base values for each of these parameters. Where
appropriate, the values for these parameters have been taken from the literature.
4.2. Contractually Speci®ed Parameters
The ®rst part of this analysis focuses on the impact that the contractually speci®ed
parameters have on the listing agreement value. In particular, table 2 examines the listing
agreement value for a variety of commission rates and contract terms. As this table shows,
increasing the commission rate while holding T constant results in a proportional increase
in the value of the listing contract. That is, if the commission rate doubles, so does the
value of the listing agreement, holding all else constant. Given the boundary conditions
(equations (4) and (6)) this is not surprising since the contract value is linearly
homogenous with respect to the commission rate.
Unlike the commission rate, changing the term of the contract does not produce a linear
change in the value of the listing agreement. Shorter-term contracts are more sensitive to
changes in T than are longer-term contracts. For example, doubling the term of a two-
month contract to four months results in a 12.6% increase in its value. Doubling the term
Table 1. Parameter values.
Parameter Notation Value
Initial house value H0 $100,000
Housing service ¯ow s 3%
Housing volatility sH 15%
Risk-free rate r 8%
Seller reservation price PR $100,500
Commission rate C 6%
Contract term T 3 months
Time between polling of house value G 1 week
Size of time binomial lattice dt 1 day
262 BUTTIMER
of a six-month contract, however, results in only a 6.5% increase in the value of the
contract. It is interesting to note that increasing the term of the contract beyond roughly
four months has little impact on the value of the listing agreement.
Table 2 shows that contractually speci®ed parameters do affect the value of the listing
agreement. It would be surprising if they did not. The more interesting question is whether
the noncontractually speci®ed parameters have as signi®cant an impact.
4.3. Noncontractually Speci®ed Parameters
The second part of this analysis examines the impact that the noncontractually speci®ed
parameters have on the listing agreement value. In particular this section is concerned with
the effects of housing volatility sH and seller reservation price PR on the listing agreement
value.
The analysis begins by examining the effect of the seller's reservation price PR. As table
3 shows, this parameter has an economically signi®cant in¯uence on the value of the
listing agreement. Changing the value of PR does not invoke a linear change in the listing
agreement value, nor does the scale of the change remain constant over volatility regimes.
This is due to the fact that the reservation price largely determines the probability of the
house selling. In this way the reservation price is analogous to the strike price of a call
option on a stock. Raising the reservation price decreases the probability of a sale and thus
lowers the broker's expected payoff, while decreasing the reservation price increases the
probability of a sale and increases the broker's expected payout. While raising the
reservation price will always reduce the probability of sale, the scale of that change
depends on the volatility of housing. The larger the volatility, the smaller the impact of a
given change in the reservation price. This is why in Table 3 increasing PR from $101,000
to $102,000 results in a 67% decrease in listing agreement value when housing volatility is
5% but only an 11% decrease when housing volatility is 30%.
Table 3 also examines the importance of housing volatility on the value of the listing
agreement. Note that there is not a monotonic change in the value of the contract for a
given change in housing volatility. Consider the ®rst row of table 3Ðthat is, where
PR� $101,000. As volatility increases from 5% through 20%, the value of the listing
contract increases. Above 20%, however, increasing the volatility actually lowers the
Table 2. Listing agreement values at origination for a variety of commission rates (C), contract terms (T).
T
C 2 Months 3 Months 4 Months 5 Months 6 Months 9 Months 1 Year
4% $2910 $3175 $3277 $3400 $3446 $3573 $3620
6% 4365 4763 4916 5099 5169 5360 5430
8% 5820 6351 6555 6799 6893 7146 7241
Parameters: H0� $100,000, sH � 15%, s� 3%, r� 8%, PR � $100,500, G�weekly.
A CONTINGENT CLAIMS ANALYSIS OF LISTING AGREEMENTS 263
value of the contract. At ®rst glance this may seem somewhat counterintuitive. A close
examination of the underlying housing process, however, reveals the source of this
behavior.
Recall equation (8), the transformed housing process. Notice that this process differs
from those usually used to represent stock or interest rates in that the volatility parameter
appears twiceÐonce in the stochastic portion of the process and once, as a variance, in the
drift term. Increasing sH, therefore, has two opposite effects on the value of the contract.
The ®rst effect is that it increases the house value dispersion, which, if the drift rate were
held constant, would increase the value of the contract. This is because although the
contract would ®nish in the money with the same frequency, when it did terminate in the
money, it would tend to do so at higher house values, resulting in higher payoffs.
Essentially the probability of a payoff remains constant, but the actual payoff amount
increases. This is the same reason increasing the volatility of a stock increases the value of
a call option.
The volatility also has a secondary effect. The drift rate of the transformed process is
given by
r ÿ sÿ s2H
2
8>>: 9>>;dt� �9�
Increasing the volatility, therefore, reduces the drift rate, which reduces the probability of
the contract terminating in the money, which reduces the value of the contract.
The results in table 3 show this mixed effect. Again consider the ®rst row, where
PR� $101,000. As long as the volatility remains below 25% the net effect on the value of
the contract is positiveÐthat is, increasing the volatility increases the contract value. At
volatility levels greater than or equal to 25%, however, the marginal decrease in the drift
rate outweighs the marginal increase in the house value dispersion, and the net value of the
contract falls.
The conclusion to draw from table 3 is that noncontractual parameters do, in fact, affect
the listing agreement value. Given that PR is one of these parameters and that the
homeowner controls that parameter, a logical question to ask is whether the broker has a
similar mechanism available. The next section examines this issue.
Table 3. Listing contract values at origination for various housing volatilities (sH) and reservation prices (PR).
sH
PR 0.05 0.10 0.15 0.20 0.25 0.30
$101,000 $3139 $4082 $4279 $4708 $4673 $4650
102,000 1026 2601 3132 3854 3807 4136
103,000 213 1455 2425 3044 3343 3771
104,000 32 815 1816 2334 2985 3301
105,000 1 342 1052 1722 2263 2517
Parameters: H0� $100,000, s� 3%, C� 6%, r� 8%, T� 3 months, G�weekly.
264 BUTTIMER
4.4. Polling and Contract Term
After signing the listing agreement, the only parameter over which brokers have any
control is the frequency with which they poll the house value G. Even here, the broker's
control over G is probably limited. While a broker may be able to control the marginal
level of G, other factors, such as overall market conditions, might also in¯uence
the general level of G. Nevertheless, this is the one parameter that brokers can in¯uence
after they enter into the listing agreement. Table 4 examines the effect this parameter
has on the value of the listing agreement. Given that frequency is intrinsically related
to time, the table also includes the contract term T, even though it also appears in
other tables.
Table 4 presents listing agreement values under a variety of T and G values. For any
given term, reducing the number of time steps between polling periods (that is, reducing
G) increases the value of the listing contract. Note that this polling frequency can be
thought of as the frequency with which the house is shown to quali®ed potential buyers.
The increase is not linear, however. To see this, consider the middle row of table 4 (that is,
where T� 5). Notice that with one month between pollings the value of the contract is
$4464, with two weeks between pollings it is $4818, while with one week between
pollings it is $5204. For illustrative purposes, table 4 provides a daily polling number,
although this would be dif®cult to achieve in practice.
It is interesting to note that the magnitude of G's effect decreases as the term of the
contract increases. Based on a three-month contract, increasing G from monthly to weekly
increases the value of the contract by over 15%. With a one-year contract, however, the
same increase in polling frequency results in only a 1.9% increase. This would tend to
indicate that shorter-term listing agreements to a better job of aligning the incentives of the
broker and seller than longer-term contracts. This result is consistent with Miceli's (1989,
1995) observation that shorter listing contracts tend to increase sellers' welfare relative to
longer listing contracts by increasing the cost of shirking to the broker.
From the homeowner's viewpoint, the relationship between the contract value and G is
welfare improving. If the broker sets G in such a way as to maximize the value of the
listing agreement, this will lead the broker to maximize the frequency with which he or she
shows the property. This, of course, is precisely what the homeowner wishes to happen.
Table 4. Listing agreement values at origination for various contract terms (T) and polling frequencies (G).
G
T Monthly Biweekly Weekly Daily
3 months $4223 $4802 $4869 $5237
4 months 4506 4972 5027 5367
5 months 4464 4818 5204 5456
6 months 4635 4939 5277 5520
1 year 5430 5490 5538 5714
Parameters: H0� $100,000, sH � 10%, s� 3%, PR � $102,500, C� 6%, r� 8%.
A CONTINGENT CLAIMS ANALYSIS OF LISTING AGREEMENTS 265
From this perspective, the listing contract appears to align the incentives of the broker and
the homeowner.
5. Summary
This article shows that it is possible to treat a listing agreement as a type of contingent
claim. When a broker and a homeowner enter into such a listing agreement, the payment
the broker receives for entering into this agreement is the discounted expected payoffs
from the contract. It is possible to express these payoffs in the form of terminal and
boundary conditions, which can then be applied to a pricing solution. This article develops
these boundary conditions and then uses them to price a hypothetical listing contract.
The goal of this study was to establish a model that can value a listing agreement. Since
the main focus of this article was pricing, it has necessarily left some topics unexplored.
For example, while this study looked at the most common fee structure, the percentage of
sales price commission, it is possible to modify the boundary conditions to study other free
arrangements such as either the net or the ¯at commission structures. The model also could
be used to examine public policy issues such as the ability of real estate brokers to extract
economic rents from the market via collusion. Finally, this article assumed the listing and
selling brokers were the same entity. It should also be possible to use this model to
evaluate the relative position of each broker within the contract and the relative incentives
presented to each broker.
Acknowledgments
I would like to thank Brent Ambrose, Ron Rutherford, Carlos Slawson, and an anonymous
referee for their comments and suggestions.
Notes
1. For example, see Crockett (1982) or Wachter (1987).
2. This article assumes that the listing broker and selling broker are the same entity. Provided that all of the
parameters are constant, this does not affect the value of the listing contract to the homeowner, although it
will affect the value of the contract to the broker. This article does not address the issue of which strategies
the listing and selling brokers should pursue to maximize their stake in the listing agreement.
3. This de®nition of reservation price is consistent with that of Miceli (1995).
4. Note that this cost of polling may well be in the form of an opportunity cost to the seller.
5. For simplicity and clarity, this article treats G as an exogenous parameter. In reality, however, brokers will set
G to maximize their pro®ts given their entire portfolio of listings. Assuming the broker faces a binding
resource constraint, increasing search efforts for one property (that is, decreasing G) reduces their search
efforts for other properties. Thus, for any particular contract, the value of G will be set by the opportunity
costs faced by the brokerÐthat is, it is essentially exogenous.
6. In the interest of simplicity this article uses a constant risk-free rate. Given that none of the boundary
conditions directly involve the risk-free rate and that listing contracts have a relatively short-term nature, any
error induced by a nonstochastic interest rate will not have an economic impact on the price.
266 BUTTIMER
7. This assumes PR is set by the homeowner such that it incorporates the value of the homeowner's option to sell
the property even after the expiration of the listing contract.
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A CONTINGENT CLAIMS ANALYSIS OF LISTING AGREEMENTS 267
