Stabilising House Prices: the Role of Housing Futures Trading


This study investigates the effects of housing futures trading on housing demand, house price volatility and housing bubbles in a theoretical framework. The baseline model is an application of the De long, Shleifer, Summers and Waldman (1990) model of noise traders to the housing market, when the risky asset is housing. This adds new features to the model as households receive utility from housing services and cannot short-sell houses. The existence of noise traders in the housing market creates uncertainty about house prices, causes prices to deviate away from their fundamental values, and leads to a distortion in housing consumption. To investigate the impact of housing derivatives trading on the housing market, a new financial instrument, housing futures, is introduced into the baseline model. Housing futures trading affects house price stability through three channels: by (i) enabling households to disentangle their housing consumption decisions from investment decisions; (ii) allowing short-selling; and (iii) attracting an additional set of traders (pure speculators) looking for portfolio diversification opportunities. The results show that, for a large set of admissible parameter values, housing futures trading decreases the volatility of house prices and increases the welfare of households and investors.

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Fig. 1


  1. 1.

    In financial markets, short-selling is defined as the sale of a security or financial instrument which is not currently owned. However, this practice is not possible in the physical housing market.

  2. 2.

    Although the rental market enables households to separate their housing investment decisions from their housing consumption decisions, services from owner-occupied housing and rental housing are imperfect substitutes. Most households express a strong preference for owning rather than renting, which makes it difficult to disentangle investment and consumption decisions.

  3. 3.

    To my knowledge, there are only two theoretical papers (Voicu and Seiler 2013; De Jong et al. 2008) that study the role of housing futures in hedging house price risk. However, these two papers do not discuss the effects of the introduction of the housing derivatives market on house prices and its role in solving the imperfections and distortions in the housing market. In addition, Fan et al. (2012) develop a utility indifference model to study the pricing of forward house transactions. They discuss the role of forward house sales in hedging against house price risk in the spot market from the perspective of optimal investment strategy among the house, risk-free bond and traded stock. Although forward house sales and housing derivatives trading have some similarities, they have clear differences which give rise to discrepancies between authors’ study and this analysis. They don’t study the impact of forward house sales on the housing market, and the role of these transactions in solving the imperfections such as short-selling and distortions in the housing market.

  4. 4.

    Derivatives markets may reduce spot volatility by supporting price discovery and transferring risk (Kawai 1983; Turnovsky 1983; Sarris 1984; Demers and Demers 1989). On the other hand, it has been argued that derivatives markets may destabilise spot markets by attracting uninformed speculative investors through the higher degree of leverage, low transaction costs and low margins (Danthine 1978; Stein 1987; Newbery 1987; Chari et al. 1990).

  5. 5.

    Australia, Canada, China, France, Ireland, Italy, New Zealand, Norway, Russia, South Africa, Spain, the United Kingdom and the United States, among other countries.

  6. 6.

    The first class of models assumes that all investors have rational expectations: they can either have identical information (Samuelson 1958; Blanchard and Watson 1982; Tirole 1985; Santos and Woodford 1997; Martin and Ventura 2012) or be asymmetrically informed (Allen and Gorton 1993; Allen et al. 1993). In the second class of models, bubbles can occur due to heterogeneous beliefs among investors (Miller 1977; Harrison and Kreps 1978; Scheinkman and Xiong 2003) or due to the interaction between rational and behavioral traders (De Long et al. 1990; Shleifer and Vishny 1997; Abreu and Brunnermeier 2003).

  7. 7.

    In the model there is no fundamental uncertainty. House prices vary as a result of stochastic changes in noise traders’ opinions between generations.

  8. 8.

    Moral hazard problem occurs in the rental markets as it is hard to ensure a high standard of maintenance by tenants. Since the maintenance efforts of tenants cannot be observed by landlords, they assume tenants will choose low maintenance efforts. Hence, tenants pay a premium reflecting the additional maintenance cost.

  9. 9.

    With normally distributed returns, maximizing the expected value of the CARA utility function is equivalent to maximizing the mean-variance utility function. However, for tractability, this analysis assumes a uniform distribution, and uses explicitly a mean-variance preference as it gives closed-form solutions.

  10. 10.

    Henderson and Ioannides (1983) introduce an investment constraint, h lh c, which requires owner-occupiers’ housing investment to be at least as large as their housing consumption. Therefore, as consumption tenure can not be split, when h l < h c, households rent for their consumption and rent out their housing investment.

  11. 11.

    Heterogeneity in noise trader and sophisticated households’ house price expectations can give rise to an active rental market (without requiring additional heterogeneity in tastes or incomes). Relatively optimistic households invest more in housing for speculative purposes. When their housing investment demand is higher than their housing consumption demand, to avoid the higher maintenance cost, they can owner-occupy their housing investment up to their housing consumption and rent out the rest. Relatively pessimistic households, on the other hand, invest less in housing. When their housing investment demand is less than their housing consumption demand, they can rent their consumed housing since they cannot own only part of their consumption. Therefore, while relatively optimistic households can choose to be landlords, relatively pessimistic households can choose to be tenants depending on the dispersion in their beliefs.

  12. 12.

    To simplify the analysis noise traders are assumed to be optimistic. However, when this assumption is relaxed to allow noise traders to be either optimistic or pessimistic, the main results still hold. See Appendix for the extended analysis.

  13. 13.

    13 See Appendix for the housing investment demand of noise traders and sophisticated households (Eq. 56).

  14. 14.

    The assumption of positive housing consumption implies \(\frac {(\delta _{R}-\delta _{O})}{b}<\frac {1}{\mu }\). See Eq. 55 in Appendix.

  15. 15.

    House price variance is given as \({\sigma ^{2}_{P}}=\frac {\mu ^{2} \sigma ^{2}_{\rho }}{(1+r)^{2}}\), where \(\sigma ^{2}_{\rho }=\frac {(\rho ^{U})^{2}}{12}\) for a uniformly distributed ρ. For Case 1 to be valid, the following condition must be satisfied: \( \rho ^{U} \geq \frac {6}{\gamma }\frac {b}{(\delta _{R}-\delta _{O})}\left (\frac {1+r}{\mu }\right )^{2}. \)

  16. 16.

    Housing futures contracts are based on a house price index and settled in cash. Therefore, the buyer and the seller exchange the difference between the realised index on maturity and the contract price agreed upon.

  17. 17.

    In the first period, households trade housing futures by writing a contract without making any financial transaction. For simplicity the margin account requirement for futures trading is not taken into consideration. In the second period, households settle by paying (receiving) the loss (gain) related to the contract in cash.

  18. 18.

    Danthine (1978), Holthausen (1979) and Feder et al. (1980) show that separation result can also be derived in a general expected utility maximization framework.

  19. 19.

    Define the critical value as \(c=\frac {\Psi (\delta _{R}-\delta _{O})}{b}\)to simplify the notation for the following covariance expression:\(\sigma _{\theta , \rho }=E(\theta , \rho )-E(\theta ) E(\rho )=(1-\mu )\frac {c}{2} \left (\frac {c}{\rho ^{u}}-1\right )<0.\)

  20. 20.

    This case can be considered as analysing the effect of housing futures trading when shared equity schemes are available in the economy. Shared equity schemes allow households to receive utility from the full range of housing services in a property while only owning a fraction of it. They also give the resident household all the management controls and right to decide when to sell. Therefore, shared equity schemes help to eliminate the differences in services received from renting and owner-occupying (Caplin et al. 1997). In practice, shared ownership/equity schemes are not common. Either they are not available in many countries or only available for first time buyers and people with limited funds.

  21. 21.

    It is also shown in Appendix that if \(\rho ^{u}=\frac {\Psi }{\mu }\), house price volatility does not change with housing derivatives trading.

  22. 22.

    See Appendix for details.

  23. 23.

    Halket and Vasudev (2012) use the Current-cost Net Stock of Residential Fixed Assets and Current-cost Depreciation of Residential Fixed Assets tables in the National Income and Product Accounts to compute the rate of depreciation of non-farm owner-occupied housing and tenant-occupied housing.

  24. 24.

    The fundamental value of house prices, \(\frac {(a-b)-\delta _{O}}{r}\), is used to calculate the annual maintenance cost. Otherwise, fluctuations in house prices as a result of noise traders’ misperceptions create variations in maintenance costs as well.

  25. 25.

    The yield rate of 20-Year US Treasury Bond is taken as 2.7% in the calculations.

  26. 26.

    The volatility estimate is computed as the standard deviation of the annualised percentage change in a house price index over 20 years. In the analysis, the range of the noise traders’ misperception is chosen so that the baseline model (without the futures market) matches the house price volatility with the estimated volatility of Federal Housing Finance Agency’s House Price Index (4.9%) between 1994:Q1 and 2013:Q4. In fact, volatility measures may differ significantly with different house price indices. For example, the volatility of S&P/Case-Shiller House Price Index is 8% over the same period. However, the results are robust to a wide range of noise trader misperception between [0, 0.1] and [0,13] with respective volatility measures 0.2% and 25%.

  27. 27.

    A change in welfare is calculated as the difference between the expected utility received from housing consumption and terminal wealth with and without housing futures trading. The introduction of the derivatives market affects the welfare of households by causing changes in housing consumption, speculative investment demand, and the return on housing investment through variations in house price volatility and risk premium.

  28. 28.

    Checking whether \(\mathcal {M}=\frac {\mu ^{2} \chi ^{2} }{12 \left [-(1-\mu )^{2}\frac {4}{\chi ^{2}}+(2-\mu )(1-\mu )\frac {8}{3\chi }-2 (1-\mu )+\frac {\chi ^{2}}{12}\right ]}\gtreqless 1\), is equivalent to checking if (1 − μ 2)χ 4 − 24(1 − μ)χ 2 + (2 − μ)(1 − μ)32χ − 48(1 − μ)2 ⪌ 0. When μ → 1, the expression approaches zero, and when μ → 0, as χ > 2 by assumption, the expression is positive, indicating that \(\mathcal {M}<1\). However, if χ = 2 housing futures trading does not change the volatility of house prices, as \(\mathcal {M}=1\) in that case.

  29. 29.

    Additionally, whether marginal utility of consumption and housing consumption are positive, and whether the condition for the active rental market is satisfied are checked for the defined parameter values.

  30. 30.

    Housing futures trading leads a change in homeownership structure. It has a positive effect on welfare for sophisticated households, who are renters without the futures market, as they become homeowners and consume more when they are able to trade housing futures. On the other hand, it has a negative effect for noise trader households, who are owner-occupiers without the futures markets. Although, with the introduction of the futures markets they still owner-occupy housing, their housing consumption decreases as the reduction in the implicit cost of owner-occupied housing (due to the spread in maintenance costs) is eliminated.


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This paper is a chapter of my PhD dissertation at European University Institute. I am indebted to Russell Cooper, Nicola Gennaioli, Ramon Marimon and Jaume Ventura for their continuous advise and comments.

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Correspondence to Arzu Uluc.

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The author declares that she has no conflict of interest.



Required Conditions for the Rental Market

The rental market becomes active when optimistic households (noise traders) prefer to owner-occupy their housing consumption and rent out the rest of their housing investment, and relatively pessimistic households (sophisticated households) prefer to rent. The optimization problems of the noise trader landlords and sophisticated household tenants yield the following housing investment and consumption demands:

$$ h^{c}_{i, t}={1}-\frac{\mu (\delta_{R}-\delta_{O})}{b}; \quad h^{c}_{n, t}={1}+\frac{(1-\mu)(\delta_{R}-\delta_{O})}{b}. $$
$$ h^{l}_{i, t}=max\left\{{1}-\frac{\mu \rho_{t}}{\Psi}, 0\right\}; \quad h^{l}_{n, t}=max\left\{{1}+\frac{(1-\mu) \rho_{t}}{\Psi}, \frac{1}{\mu}\right\}. $$

where \({\Psi }=2 \gamma \sigma ^{2}_{P}\). For an active rental market, two inequalities must be satisfied: \(h^{l}_{n, t}>h^{c}_{n, t}\) and \(h^{l}_{i, t}<h^{c}_{i, t}\), which yields the following condition: \(\frac {(\delta _{R}-\delta _{O})}{b}< \left \{\begin {array}{rcl} \frac {\rho _{t}}{\Psi }& \text { if } &\rho _{t} < \frac {\Psi }{\mu } \\ \frac {1}{\mu } & \text { if } &\rho _{t}\geq \frac {\Psi }{\mu }. \\ \end {array}\right .\)

Additionally, the assumption of positive housing consumption requires that \(\frac {(\delta _{R}-\delta _{O})}{b}<\frac {1}{\mu }\). These necessary conditions indicate the rental market becomes active if \(\rho _{t}>\frac {\Psi (\delta _{R}-\delta _{O})}{b}\).

Proposition 4

If the noise traders’ misperception is uniformly distributed over [0, ρ u], where \(\rho ^{u}>\frac {\Psi }{\mu }\) and δ R = δ O , the equilibrium house price function without derivatives market is

$$ p_{t}=\frac{(a-b)-\delta_{O}}{r}+\frac{\kappa_{t} \rho_{t}}{(1+r)}+\frac{\overline{\kappa \rho}}{(1+r)r}-\frac{\eta_{t} {\Psi}}{(1+r)}-\frac{\overline{\eta} {\Psi}}{(1+r)r}, $$

where \({\Psi }=2 \gamma {\sigma ^{2}_{P}}\) and \( (\kappa _{t}, \eta _{t})= \left \{\begin {array}{rcl} (1, \frac {1}{\mu }) & \text {if} & \rho _{t} \geq \frac {\Psi }{\mu }\\ (\mu , 1) & \text {if} & 0 \leq \rho _{t} < \frac {\Psi }{\mu }. \end {array}\right .\) House price variance is determined by

$$ {\sigma^{2}_{P}}=\frac{1}{(1+r)^{2}}\left[Var(\kappa \rho)- 2 Cov(\kappa \rho, \eta) \left( 2 \gamma {\sigma^{2}_{P}}\right)+ Var(\eta) \left( 2 \gamma {\sigma^{2}_{P}}\right)^{2}\right]. $$

The aim of this analysis is not to solve for the house price variance but to compare house price variance with and without the futures market. Therefore, I try to simplify the analysis as much as possible in order to have an expression which allows this comparison. Once the house price variance is known, it is possible to denote the upper bound value as \(\rho ^{u}=\frac {\chi \gamma {\sigma ^{2}_{P}}}{\mu }\), where χ > 2 to guarantee that the short-selling constraint is binding for noise traders. After substituting in the respective expressions of the moments of variables, Eq. 58 can be expressed as follows:

$$ (1+r)^{2} {\sigma^{2}_{P}}=\left[-(1-\mu)^{2}\frac{4}{\chi^{2}}+(2-\mu)(1-\mu)\frac{8}{3\chi}-2 (1-\mu)\!+\frac{\chi^{2}}{12}\right] \frac{\gamma^{2}}{\mu^{2}} \left( {\sigma^{2}_{P}}\right)^{2}, $$
$$ {\sigma^{2}_{P}}=\frac{(1+r)^{2} }{\left[-(1-\mu)^{2}\frac{4}{\chi^{2}}+(2-\mu)(1-\mu)\frac{8}{3\chi}-2 (1-\mu)+\frac{\chi^{2}}{12}\right] \frac{\gamma^{2}}{\mu^{2}}}. $$

With the introduction of the futures market both type of households buy houses. Hence, the futures market increases participation in the housing market. The house price function is given by

$$ {p^{D}_{t}}=\frac{(a-b)-\delta_{O}}{r}+\frac{\mu (\rho_{t}-\overline{\rho})}{(1+r)}+\frac{\mu \overline{\rho}}{r}-\frac{2 \gamma \sigma^{2}_{P^{D}}}{r}, $$

where \(\sigma ^{2}_{p^{D}}=\frac {\mu ^{2} \sigma ^{2}_{\rho }}{(1+r)^{2}}\). The house price variance expression, when futures trading is available, can be rewritten by substituting in \(\sigma ^{2}_{\rho }=\frac {U^{2}}{12}=\frac {\left [\frac {\chi \gamma {\sigma ^{2}_{P}}}{\mu }\right ]^{2}}{12}\) and \({\sigma ^{2}_{P}}\) from Eq. 60 as follows:

$$ \sigma^{2}_{P^{D}}=\sigma^{2}_{P} \frac{\mu^{2} \chi^{2} }{12 \left[-(1-\mu)^{2}\frac{4}{\chi^{2}}+(2-\mu)(1-\mu)\frac{8}{3\chi}-2 (1-\mu)+\frac{\chi^{2}}{12}\right]}. $$

Since \(\mathcal {M}=\frac {\mu ^{2} \chi ^{2} }{12 \left [-(1-\mu )^{2}\frac {4}{\chi ^{2}}+(2-\mu )(1-\mu )\frac {8}{3\chi }-2 (1-\mu )+\frac {\chi ^{2}}{12}\right ]}<1\), for χ > 2 and ∀μ, the introduction of the futures market decreases the volatility of house prices.Footnote 28

Numerical Exercise

Calculating the variance of the house price function analytically would be complicated as both prices and participation in the housing market are determined in equilibrium, and moreover, participation depends on a critical value which is a function of the house price volatility. For this reason, a numerical exercise is conducted.

The price function in Theorem 1 is

$$\begin{array}{@{}rcl@{}} p_{t}=&&\!\!\frac{(a-b)-\delta_{R}}{r}+\frac{\theta_{t} (\delta_{R}-\delta_{O})}{(1+r)}+\frac{E(\theta) (\delta_{R}-\delta_{O})}{(1+r)r}+\frac{\kappa_{t} \rho_{t}}{(1+r)}\\ &&+\frac{E(\kappa \rho)}{(1+r)r}-\frac{\eta_{t} 2 \gamma \sigma^{2}_{P}}{(1+r)}-\frac{E(\eta) 2 \gamma {\sigma^{2}_{P}}}{(1+r)r}, \end{array} $$


$$\begin{array}{@{}rcl@{}} (\kappa_{t}, \eta_{t}, \theta_{t})= \left\{\begin{array}{rcl} (1, \frac{1}{\mu}, \mu) & \text{ if } & \rho_{t} \geq \frac{2 \gamma {\sigma^{2}_{P}}}{\mu}\\ (\mu, 1, \mu) & \text{ if } & 2 \gamma {\sigma^{2}_{P}}\frac{(\delta_{R}-\delta_{O})}{b}<\rho_{t} < \frac{2 \gamma {\sigma^{2}_{P}}}{\mu}\\ (\mu, 1, 1) & \text{ if } & 0 \leq \rho_{t} \leq 2 \gamma {\sigma^{2}_{P}}\frac{(\delta_{R}-\delta_{O})}{b}.\\ \end{array}\right. \end{array} $$

House price variance is given as

$$\begin{array}{@{}rcl@{}} {\sigma^{2}_{P}}&=&\frac{1}{(1+r)^{2}}\left[(\delta_{R}-\delta_{O})^{2} Var(\theta)+Var(\kappa \rho)+Var(\eta)\left( 2 \gamma {\sigma^{2}_{P}}\right)^{2}\right.\\ &&\qquad\qquad+2 (\delta_{R}-\delta_{O}) Cov(\theta, \kappa \rho)-2 (\delta_{R}-\delta_{O}) (2 \gamma {\sigma^{2}_{P}}) Cov(\theta, \eta)\\ &&\qquad\qquad\left.-2 \left( 2 \gamma {\sigma^{2}_{P}}\right) Cov(\eta, \kappa \rho)\right]. \end{array} $$

Moments of the variables are calculated as follows:

$$ E(\theta)= {\int}^{2 \gamma {\sigma^{2}_{P}}\frac{(\delta_{R}-\delta_{O})}{b}}_{0} 1 f(\rho) d\rho+ {\int}^{\rho^{u}}_{2 \gamma {\sigma^{2}_{P}}\frac{(\delta_{R}-\delta_{O})}{b}} \mu f(\rho) d\rho, $$
$$ Var(\theta)={\int}^{2 \gamma {\sigma^{2}_{P}}\frac{(\delta_{R}-\delta_{O})}{b}}_{0} [1-E(\theta)]^{2} f(\rho) d\rho+ {\int}^{\rho^{u}}_{2 \gamma {\sigma^{2}_{P}}\frac{(\delta_{R}-\delta_{O})}{b}} [\mu-E(\theta)]^{2} f(\rho) d\rho, $$
$$ E(\eta)={\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{0} 1 f(\rho) d\rho+ {\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} \frac{1}{\mu} f(\rho) d\rho, $$
$$ Var(\eta)={\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{0} [1- E(\eta)]^{2} f(\rho) d\rho+ {\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} \left[\frac{1}{\mu}- E(\eta)\right]^{2} f(\rho) d\rho, $$
$$ E(\kappa \rho)= {\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{0} (\mu \rho) f(\rho) d\rho+ {\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} (1 \rho) f(\rho) d\rho, $$
$$ Var(\kappa \rho)={\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{0}[(\mu \rho)- E(\kappa \rho)]^{2} f(\rho) d\rho+ {\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} [(1 \rho)-E(\kappa \rho)]^{2} f(\rho) d\rho, $$
$$ E(\eta \kappa \rho)= {\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{0} (\mu \rho) f(\rho) d\rho+ {\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} \left( \frac{\rho}{\mu}\right) f(\rho) d\rho, $$
$$ E(\theta \eta)={\int}^{2 \gamma {\sigma^{2}_{P}}\frac{(\delta_{R}-\delta_{O})}{b}}_{0} 1 f(\rho) d\rho + {\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{2 \gamma {\sigma^{2}_{P}}\frac{(\delta_{R}-\delta_{O})}{b}} \mu f(\rho) d\rho+ {\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} 1 f(\rho) d\rho, $$
$$ E(\theta \kappa \rho)\,=\,{\int}^{2 \gamma {\sigma^{2}_{P}}\frac{(\delta_{R}-\delta_{O})}{b}}_{0} (\mu \rho) f(\rho) d\rho +\! {\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{2 \gamma {\sigma^{2}_{P}}\frac{(\delta_{R}-\delta_{O})}{b}} (\mu^{2} \rho) f(\rho) d\rho+\! {\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} (\mu \rho) f(\rho) d\rho. $$

Since all of the moments can be written as a function of the house price variance, Eq. 65, a fourth-order polynomial with one unknown, \({\sigma ^{2}_{P}}\), is solved numerically in Matlab. After solving for \({\sigma ^{2}_{P}}\), whether the thresholds, \(2 \gamma {\sigma ^{2}_{P}}\frac {(\delta _{R}-\delta _{O})}{b}\) and \(\frac {2 \gamma {\sigma ^{2}_{P}}}{\mu }\), are within the range of noise traders’ misperceptions is checked.Footnote 29 Then, the effects of housing futures trading on the housing market via three channels are analysed.

Finally, a welfare analysis is conducted. A change in welfare is calculated as the difference between the expected utility received from housing consumption and terminal wealth with and without housing futures trading. The threshold for active rental market is defined as \(\zeta =2 \gamma {\sigma ^{2}_{P}}\frac {(\delta _{R}-\delta _{O})}{b}\). The changes in expected utility of households and investors with (EV D) / without (EV) derivatives trading are expressed as follows:

Sophisticated Households

$$\begin{array}{@{}rcl@{}} && E{V^{D}_{i}}-EV_{i}={\int}^{\zeta}_{0} \left( \gamma \left[\sigma^{2}_{P^{D}}\theta^{2}-{\sigma^{2}_{P}}\right]-\rho_{t}(\psi\theta-\mu)\right.\\ &&\,\,\,\,\left.+\frac{1}{2}\left[\frac{(\psi\rho_{t})^{2}}{2 \gamma \sigma^{2}_{P^{D}}}-\frac{(\mu\rho_{t})^{2}}{2 \gamma {\sigma^{2}_{P}}+b}\right]\right) f(\rho) d\rho \\ \,\, && +{\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{\zeta} \left( \mu (\delta_{R}-\delta_{O})\left[1-\frac{\mu (\delta_{R}-\delta_{O})}{2b}\right]+\gamma \left[\sigma^{2}_{P^{D}}\theta^{2}-{\sigma^{2}_{P}}\right]-\rho_{t}(\psi\theta-\mu)\right.\\ &&\,\,\,\,\left.+\frac{1}{2}\left[\frac{(\psi\rho_{t})^{2}}{2 \gamma \sigma^{2}_{P^{D}}}-\frac{(\mu\rho_{t})^{2}}{2 \gamma {\sigma^{2}_{P}}+b}\right]\right) f(\rho) d\rho \\ \,\, && +{\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} \left( \mu (\delta_{R}-\delta_{O})\left[1-\frac{\mu (\delta_{R}-\delta_{O})}{2b}\right]+\frac{\left[2\gamma \sigma^{2}_{P^{D}}\theta-\psi\rho_{t}\right]^{2}}{4 \gamma \sigma^{2}_{P^{D}}}\right) f(\rho) d\rho \end{array} $$

Noise Trader Households

$$\begin{array}{@{}rcl@{}} && E{V^{D}_{n}}-EV_{n}={\int}^{\zeta}_{0} \left( \gamma \left[\sigma^{2}_{P^{D}}\theta^{2}-{\sigma^{2}_{P}}\right]+\rho_{t}[(1-\psi)\theta-(1-\mu)]\right.\\ &&\,\,\,\,\left.+\frac{1}{2}\left[\frac{[(1-\psi)\rho_{t}]^{2}}{2 \gamma \sigma^{2}_{P^{D}}}-\frac{[(1-\mu)\rho_{t}]^{2}}{2 \gamma {\sigma^{2}_{P}}+b}\right]\right) f(\rho) d\rho \\ && +{\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{\zeta} \left( -(1-\mu) (\delta_{R}-\delta_{O})\left[1+\frac{(1-\mu) (\delta_{R}-\delta_{O})}{2b}\right]\right) f(\rho) d\rho \\ && +{\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{\zeta} \left( \gamma \left[\sigma^{2}_{P^{D}}\theta^{2}-{\sigma^{2}_{P}}\right]+\rho_{t}[(1-\psi)\theta-(1-\mu)]\right.\\ &&\,\,\,\,\left.+\frac{1}{2}\left[\frac{[(1-\psi)\rho_{t}]^{2}}{2 \gamma \sigma^{2}_{P^{D}}}-\frac{[(1-\mu)\rho_{t}]^{2}}{2 \gamma {\sigma^{2}_{P}}+b}\right]\right) f(\rho) d\rho \\ && +{\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} \left( -(1-\mu) (\delta_{R}-\delta_{O})\left[1+\frac{(1-\mu) (\delta_{R}-\delta_{O})}{2b}\right]\right.\\ &&\,\,\,\,\left.+\frac{[2\gamma \sigma^{2}_{P^{D}}\theta+(1-\psi)\rho_{t}]^{2}}{4 \gamma \sigma^{2}_{P^{D}}}-\frac{ \gamma {\sigma^{2}_{P}}}{\mu^{2}}\right) f(\rho) d\rho \end{array} $$

Sophisticated Investors

$$ EV^{D}_{is}-EV_{is}={\int}^{\rho^{u}}_{0} \frac{\left[2\gamma \sigma^{2}_{P^{D}}\theta-\mu\rho_{t}\right]^{2}}{4 \gamma \sigma^{2}_{P^{D}}} f(\rho) d\rho $$

Noise Trader Investors

$$ EV^{D}_{ns}-EV_{ns}={\int}^{\rho^{u}}_{0}\frac{\left[2\gamma \sigma^{2}_{P^{D}}\theta+(1-\mu)\rho_{t}\right]^{2}}{4 \gamma \sigma^{2}_{P^{D}}} f(\rho) d\rho $$

The introduction of the futures market impacts the welfare of households by causing changes in housing consumption,Footnote 30 speculative investment demand, and return on housing investment through variations in house price volatility and risk premium.

Noise Traders’ Misperceptions: Optimism & Pessimism

The baseline model is extended to allow noise traders be either optimistic or pessimistic in their house price expectation. The noise traders misperception is assumed to be uniformly distributed over [ρ L, ρ U], where ρ L < 0 and ρ U > 0. As shown in Fig. 2, the solutions to the optimisation problems yield an equilibrium consisting of five regions:

Propositions 2 and 3 are still valid after allowing pessimistic misperception of noise traders, while Proposition 4 has to be revised as follows:

Proposition 7

If the noise traders’ misperception is uniformly distributed over [ρ L, ρ U],

  1. i.

    where \(\rho ^{L} > -\frac {\Psi }{1-\mu }\) and \(\rho ^{u}>\frac {\Psi }{\mu }\), trading housing futures decreases the house price volatility;

  2. ii.

    where \(\rho ^{L} < -\frac {\Psi }{1-\mu }\) and \(\rho ^{u}>\frac {\Psi }{\mu }\), the volatility of house prices can increase or decrease with housing futures trading.

Fig. 2

Noise traders’ perception and equilibrium regions

For the defined interval of the misperception in (i), the short-selling constraint is binding for sophisticated households when \(\rho _{t} > \frac {\Psi }{\mu }\). Trading housing futures enables sophisticated households to participate to the housing market, and hence decreases the effect of the noise traders’ misperception on house prices and volatility. On the other hand, for the interval defined in (ii), the short-selling constraint can be binding also for noise traders \(\left (\text {when } \rho _{t} < -\frac {\Psi }{1-\mu }\right )\), and hence volatility might increase for some parameter values by allowing them to short housing futures and invest in housing. Indeed, the introduction of futures market can increase the volatility if a majority of the households are noise traders.

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Uluc, A. Stabilising House Prices: the Role of Housing Futures Trading. J Real Estate Finan Econ 56, 587–621 (2018).

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  • Housing derivatives market
  • Speculation
  • House price volatility
  • Short-selling
  • Noise traders