Stabilising House Prices: the Role of Housing Futures Trading

Abstract

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.

Appendix

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:

$$ h^{c}_{i, t}={1}-\frac{\mu (\delta_{R}-\delta_{O})}{b}; \quad h^{c}_{n, t}={1}+\frac{(1-\mu)(\delta_{R}-\delta_{O})}{b}. $$

(55)

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

(56)

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

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

(57)

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

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

(58)

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:

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

(59)

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

(60)

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

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

(61)

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:

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

(62)

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.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

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

(63)

where

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

(64)

House price variance is given as

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

(65)

Moments of the variables are calculated as follows:

$$ E(\theta)= {\int}^{2 \gamma {\sigma^{2}_{P}}\frac{(\delta_{R}-\delta_{O})}{b}}_{0} 1 f(\rho) d\rho+ {\int}^{\rho^{u}}_{2 \gamma {\sigma^{2}_{P}}\frac{(\delta_{R}-\delta_{O})}{b}} \mu f(\rho) d\rho, $$

(66)

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

(67)

$$ E(\eta)={\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{0} 1 f(\rho) d\rho+ {\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} \frac{1}{\mu} f(\rho) d\rho, $$

(68)

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

(69)

$$ E(\kappa \rho)= {\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{0} (\mu \rho) f(\rho) d\rho+ {\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} (1 \rho) f(\rho) d\rho, $$

(70)

$$ Var(\kappa \rho)={\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{0}[(\mu \rho)- E(\kappa \rho)]^{2} f(\rho) d\rho+ {\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} [(1 \rho)-E(\kappa \rho)]^{2} f(\rho) d\rho, $$

(71)

$$ E(\eta \kappa \rho)= {\int}^{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}}_{0} (\mu \rho) f(\rho) d\rho+ {\int}^{\rho^{u}}_{\frac{2 \gamma {\sigma^{2}_{P}}}{\mu}} \left( \frac{\rho}{\mu}\right) f(\rho) d\rho, $$

(72)

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

(73)

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

(74)

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.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

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

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Noise Trader Households

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

(76)

Sophisticated Investors

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

(77)

Noise Trader Investors

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

(78)

The introduction of the futures market impacts the welfare of households by causing changes in housing consumption,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],

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

• 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.