# American Institute of Mathematical Sciences

December 2018, 8(4): 481-492. doi: 10.3934/naco.2018030

## A stochastic approach to model housing markets: The US housing market case

 Institute of Applied Mathematics, Middle East Technical University, 6800 Ankara, Turkey

* Corresponding author: Bilgi Yilmaz

Received  September 2017 Revised  December 2017 Published  September 2018

This study aims to estimate the price changes in housing markets using a stochastic process, which is defined in the form of stochastic differential equations (SDEs). It proposes a general SDEs system on the price structure in terms of house price index and mortgage rate to establish an effective process. As an empirical analysis, it applies a calibration procedure to an SDE on monthly S&P/Case-Shiller US National Home Price Index (HPI) and 30-year fixed mortgage rate to estimate parameters of differentiable functions defined in SDEs. The prediction power of the proposed stochastic model is justified through a Monte Carlo algorithm for one-year ahead monthly forecasts of the HPI returns. The results of the study show that the stochastic processes are flexible in terms of the choice of structure, compact with respect to the number of exogenous variables involved, and it is a literal method. Furthermore, this approach has a relatively high estimation power in forecasting the national house prices.

Citation: Bilgi Yilmaz, A. Sevtap Selcuk-Kestel. A stochastic approach to model housing markets: The US housing market case. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 481-492. doi: 10.3934/naco.2018030
##### References:
 [1] N. Apergis and A. Rezitis, Housing prices and macroeconomic factors in Greece: prospects within the EMU, Applied Economics Letters, 10 (2003), 799-804. [2] M. Barari, N. Sarkar, S. Kundu and K. B. Chowdhury, Forecasting house prices in the United States with multiple structural breaks, International Econometric Review (IER), 6 (2014), 1-23. [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, (1973), 637-654. doi: 10.1086/260062. [4] M. J. Brennan and E. S. Schwartz, Evaluating natural resource investments, Journal of Business, (1973), 135-157. doi: 10.1086/296288. [5] J. P. Cohen, Y. M. Ioannides and W. W. Thanapisitikul, Spatial effects and house price dynamics in the USA, Journal of Housing Economics, 31 (2016), 1-13. doi: 10.1016/j.jhe.2015.10.006. [6] J. C. Cox, J. E. Ingersoll Jr and S. A. Ross, A theory of the term structure of interest rates, Econometrica: Journal of the Econometric Society, (1985), 385-407. doi: 10.2307/1911242. [7] J. C. Cox, J. E. Ingersoll Jr and S. A. Ross, The valuation of options for alternative stochastic processes, Journal of Financial Economics, 3 (1976), 145-166. doi: 10.1016/0304-405X(76)90023-4. [8] G. W. Crawford and M. C. Fratantoni, Assessing the forecasting performance of regime-switching, ARIMA and GARCH models of house prices, Real Estate Economics, 31 (2003), 223-243. doi: 10.1111/1540-6229.00064. [9] Y. Demyanyk and O. Van Hemert, Understanding the subprime mortgage crisis, Review of Financial Studies, 24 (2011), 1848-1880. [10] W. E. Diewert, A. O. Nakamura and L. I. Nakamura, The housing bubble and a new approach to accounting for housing in a CPI, Journal of Housing Economics, 18 (2009), 156-171. doi: 10.1016/j.jhe.2009.07.008. [11] M. I. Dröes and W. H. Hassink, House price risk and the hedging benefits of home ownership, Journal of Housing Economics, 22 (2013), 92-99. [12] E. Eerola and T. Lyytikäinen, On the role of public price information in housing markets, Regional Science and Urban Economics, 53 (2015), 74-84. doi: 10.1016/j.regsciurbeco.2015.05.006. [13] M. Fletcher, J. Mangan and E. Raeburn, Comparing hedonic models for estimating and forecasting house prices, Property Management, 22 (2004), 189-200. doi: 10.1108/02637470410544986. [14] J. Gallin, The long-run relationship between house prices and income: Evidence from local housing markets, Real Estate Economics, 34 (2006), 417-438. doi: 10.1111/j.1540-6229.2006.00172.x. [15] R. Gibson and E. S. Schwartz, Stochastic convenience yield and the pricing of oil contingent claims, Journal of Finance, 45 (1990), 959-976. doi: 10.1111/j.1540-6261.1990.tb05114.x. [16] C. Goodhart and B. Hofmann, House prices, money, credit, and the macroeconomy, Oxford Review of Economic Policy, 24 (2008), 180-205. doi: 10.1093/oxrep/grn009. [17] H. S. Guirguis, C. I. Giannikos and R. I. Anderson, The US housing market: asset pricing forecasts using time varying coefficients, The Journal of real estate finance and economics, 30 (2005), 33-53. doi: 10.1007/s11146-004-4830-z. [18] K. L. Guntermann and S. C. Norrbin, Empirical tests of real estate market efficiency, The Journal of Real Estate Finance and Economics, 4 (1991), 297-313. doi: 10.1007/BF00161931. [19] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327. [20] J. Hull and A. White, Pricing interest-rate-derivative securities, Review of Financial Studies, 3 (1990), 573-592. doi: 10.1093/rfs/3.4.573. [21] M. Iacoviello, House prices, borrowing constraints, and monetary policy in the business cycle, The American Economic Review, 95 (2005), 739-764. doi: 10.1257/0002828054201477. [22] D. Igan, A. Kabundi, F. N. De Simone, M. Pinheiro and N. Tamirisa, Housing, credit, and real activity cycles: Characteristics and comovement, Journal of Housing Economics, 20 (2011), 210-231. doi: 10.1016/j.jhe.2011.07.002. [23] J. B. Kau and D. C. Keenan, An overview of the option-theoretic pricing of mortgages, Journal of Housing Economics, 6 (1995), 217-244. [24] T. Kauko, On current neural network applications involving spatial modelling of property prices, Journal of Housing and the Built Environment, 18 (2003), 159-181. [25] K. Kim and J. Park, Segmentation of the housing market and its determinants: Seoul and its neighboring new towns in Korea, Australian Geographer, 36 (2005), 221-232. doi: 10.1080/00049180500150019. [26] R. Kouwenberg and R. Zwinkels, Forecasting the US housing market, International Journal of Forecasting, 30 (2014), 415-425. doi: 10.1016/j.ijforecast.2013.12.010. [27] P. Linneman, An empirical test of the efficiency of the housing market, Journal of Urban Economics, 20 (1986), 140-154. doi: 10.1016/0094-1190(86)90003-3. [28] S. Malpezzi, A simple error correction model of house prices, Journal of Housing Economics, 8 (1999), 27-62. doi: 10.1006/jhec.1999.0240. [29] S. Malpezzi and S. Wachter, The role of speculation in real estate cycles, Journal of Real Estate Literature, 13 (2005), 141-164. doi: 10.2139/ssrn.2585241. [30] S. Malpezzi and S. Wachter, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. [31] W. Miles, Boom-bust cycles and the forecasting performance of linear and non-linear models of house prices, The Journal of Real Estate Finance and Economics, 36 (2008), 249-264. doi: 10.1007/s11146-007-9067-1. [32] D. E. Rapach and J. K. Strauss, Differences in housing price forecastability across US states, International Journal of Forecasting, 25 (2009), 351-372. doi: 10.1016/j.ijforecast.2009.01.009. [33] E. S. Schwartz, The stochastic behavior of commodity prices: Implications for valuation and hedging, The Journal of Finance, 52 (1997), 923-973. doi: 10.1111/j.1540-6261.1997.tb02721.x. [34] D. E. Sommervoll, T.-A. Borgersen and T. Wennemo, The stochastic behavior of commodity prices: Implications for valuation and hedging, Journal of banking & finance, 34 (2010), 557-567. [35] S. Stevenson, New empirical evidence on heteroscedasticity in hedonic housing models, Journal of Housing Economics, 13 (2004), 136-153. doi: 10.1016/j.jhe.2004.04.004. [36] K. Tsatsaronis and H. Zhu, What drives housing price dynamics: cross-country evidence, BIS Quarterly Review, (2004). [37] O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5 (1977), 177-188. doi: 10.1002/9781119186229.ch6. [38] Z. Yang and S. Wang, Permanent and transitory shocks in owner-occupied housing: A common trend model of price dynamics, Journal of Housing Economics, 21 (2012), 336-346. doi: 10.1016/j.jhe.2012.08.001. [39] Z. G. Zhou, Forecasting sales and price for existing single-family homes: a VAR model with error correction, Journal of Real Estate Research, 14 (2009), 155-167. [40] Z. G. Zhou, Determinants of house prices: a quantile regression approach, The Journal of Real Estate Finance and Economics, 37 (2008), 317-333.

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##### References:
 [1] N. Apergis and A. Rezitis, Housing prices and macroeconomic factors in Greece: prospects within the EMU, Applied Economics Letters, 10 (2003), 799-804. [2] M. Barari, N. Sarkar, S. Kundu and K. B. Chowdhury, Forecasting house prices in the United States with multiple structural breaks, International Econometric Review (IER), 6 (2014), 1-23. [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, (1973), 637-654. doi: 10.1086/260062. [4] M. J. Brennan and E. S. Schwartz, Evaluating natural resource investments, Journal of Business, (1973), 135-157. doi: 10.1086/296288. [5] J. P. Cohen, Y. M. Ioannides and W. W. Thanapisitikul, Spatial effects and house price dynamics in the USA, Journal of Housing Economics, 31 (2016), 1-13. doi: 10.1016/j.jhe.2015.10.006. [6] J. C. Cox, J. E. Ingersoll Jr and S. A. Ross, A theory of the term structure of interest rates, Econometrica: Journal of the Econometric Society, (1985), 385-407. doi: 10.2307/1911242. [7] J. C. Cox, J. E. Ingersoll Jr and S. A. Ross, The valuation of options for alternative stochastic processes, Journal of Financial Economics, 3 (1976), 145-166. doi: 10.1016/0304-405X(76)90023-4. [8] G. W. Crawford and M. C. Fratantoni, Assessing the forecasting performance of regime-switching, ARIMA and GARCH models of house prices, Real Estate Economics, 31 (2003), 223-243. doi: 10.1111/1540-6229.00064. [9] Y. Demyanyk and O. Van Hemert, Understanding the subprime mortgage crisis, Review of Financial Studies, 24 (2011), 1848-1880. [10] W. E. Diewert, A. O. Nakamura and L. I. Nakamura, The housing bubble and a new approach to accounting for housing in a CPI, Journal of Housing Economics, 18 (2009), 156-171. doi: 10.1016/j.jhe.2009.07.008. [11] M. I. Dröes and W. H. Hassink, House price risk and the hedging benefits of home ownership, Journal of Housing Economics, 22 (2013), 92-99. [12] E. Eerola and T. Lyytikäinen, On the role of public price information in housing markets, Regional Science and Urban Economics, 53 (2015), 74-84. doi: 10.1016/j.regsciurbeco.2015.05.006. [13] M. Fletcher, J. Mangan and E. Raeburn, Comparing hedonic models for estimating and forecasting house prices, Property Management, 22 (2004), 189-200. doi: 10.1108/02637470410544986. [14] J. Gallin, The long-run relationship between house prices and income: Evidence from local housing markets, Real Estate Economics, 34 (2006), 417-438. doi: 10.1111/j.1540-6229.2006.00172.x. [15] R. Gibson and E. S. Schwartz, Stochastic convenience yield and the pricing of oil contingent claims, Journal of Finance, 45 (1990), 959-976. doi: 10.1111/j.1540-6261.1990.tb05114.x. [16] C. Goodhart and B. Hofmann, House prices, money, credit, and the macroeconomy, Oxford Review of Economic Policy, 24 (2008), 180-205. doi: 10.1093/oxrep/grn009. [17] H. S. Guirguis, C. I. Giannikos and R. I. Anderson, The US housing market: asset pricing forecasts using time varying coefficients, The Journal of real estate finance and economics, 30 (2005), 33-53. doi: 10.1007/s11146-004-4830-z. [18] K. L. Guntermann and S. C. Norrbin, Empirical tests of real estate market efficiency, The Journal of Real Estate Finance and Economics, 4 (1991), 297-313. doi: 10.1007/BF00161931. [19] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327. [20] J. Hull and A. White, Pricing interest-rate-derivative securities, Review of Financial Studies, 3 (1990), 573-592. doi: 10.1093/rfs/3.4.573. [21] M. Iacoviello, House prices, borrowing constraints, and monetary policy in the business cycle, The American Economic Review, 95 (2005), 739-764. doi: 10.1257/0002828054201477. [22] D. Igan, A. Kabundi, F. N. De Simone, M. Pinheiro and N. Tamirisa, Housing, credit, and real activity cycles: Characteristics and comovement, Journal of Housing Economics, 20 (2011), 210-231. doi: 10.1016/j.jhe.2011.07.002. [23] J. B. Kau and D. C. Keenan, An overview of the option-theoretic pricing of mortgages, Journal of Housing Economics, 6 (1995), 217-244. [24] T. Kauko, On current neural network applications involving spatial modelling of property prices, Journal of Housing and the Built Environment, 18 (2003), 159-181. [25] K. Kim and J. Park, Segmentation of the housing market and its determinants: Seoul and its neighboring new towns in Korea, Australian Geographer, 36 (2005), 221-232. doi: 10.1080/00049180500150019. [26] R. Kouwenberg and R. Zwinkels, Forecasting the US housing market, International Journal of Forecasting, 30 (2014), 415-425. doi: 10.1016/j.ijforecast.2013.12.010. [27] P. Linneman, An empirical test of the efficiency of the housing market, Journal of Urban Economics, 20 (1986), 140-154. doi: 10.1016/0094-1190(86)90003-3. [28] S. Malpezzi, A simple error correction model of house prices, Journal of Housing Economics, 8 (1999), 27-62. doi: 10.1006/jhec.1999.0240. [29] S. Malpezzi and S. Wachter, The role of speculation in real estate cycles, Journal of Real Estate Literature, 13 (2005), 141-164. doi: 10.2139/ssrn.2585241. [30] S. Malpezzi and S. Wachter, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. [31] W. Miles, Boom-bust cycles and the forecasting performance of linear and non-linear models of house prices, The Journal of Real Estate Finance and Economics, 36 (2008), 249-264. doi: 10.1007/s11146-007-9067-1. [32] D. E. Rapach and J. K. Strauss, Differences in housing price forecastability across US states, International Journal of Forecasting, 25 (2009), 351-372. doi: 10.1016/j.ijforecast.2009.01.009. [33] E. S. Schwartz, The stochastic behavior of commodity prices: Implications for valuation and hedging, The Journal of Finance, 52 (1997), 923-973. doi: 10.1111/j.1540-6261.1997.tb02721.x. [34] D. E. Sommervoll, T.-A. Borgersen and T. Wennemo, The stochastic behavior of commodity prices: Implications for valuation and hedging, Journal of banking & finance, 34 (2010), 557-567. [35] S. Stevenson, New empirical evidence on heteroscedasticity in hedonic housing models, Journal of Housing Economics, 13 (2004), 136-153. doi: 10.1016/j.jhe.2004.04.004. [36] K. Tsatsaronis and H. Zhu, What drives housing price dynamics: cross-country evidence, BIS Quarterly Review, (2004). [37] O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5 (1977), 177-188. doi: 10.1002/9781119186229.ch6. [38] Z. Yang and S. Wang, Permanent and transitory shocks in owner-occupied housing: A common trend model of price dynamics, Journal of Housing Economics, 21 (2012), 336-346. doi: 10.1016/j.jhe.2012.08.001. [39] Z. G. Zhou, Forecasting sales and price for existing single-family homes: a VAR model with error correction, Journal of Real Estate Research, 14 (2009), 155-167. [40] Z. G. Zhou, Determinants of house prices: a quantile regression approach, The Journal of Real Estate Finance and Economics, 37 (2008), 317-333.
Development of US National Home Price Index ($1975-2016$) with respect to the selected financial market indicators
Simulated SDEs compared with observed S&P Case-Shiller Home Price Indices (1975-2015)
Observed and predicted S&P Case-Shiller Home Price Indices (2015-2016)
Descriptives of house price index, $h$, and mortgage rate, $r$, (1975-2015).
 Min Max Mean Std Skewness Kurtosis $h$ 25.2 184.62 96.26 47.55 0.36 1.84 log-$h$ -0.023 0.02 0.004 0.006 -0.76 4.84 $r$(%) 3.32 18.44 8.38 3.24 0.79 3.33
 Min Max Mean Std Skewness Kurtosis $h$ 25.2 184.62 96.26 47.55 0.36 1.84 log-$h$ -0.023 0.02 0.004 0.006 -0.76 4.84 $r$(%) 3.32 18.44 8.38 3.24 0.79 3.33
Estimates of the parameters using calibration
 $\hat{\lambda}$ $\hat{\mu}_h (\%)$ $\hat{\sigma}_h (\%)$ $\hat{\kappa}$ $\hat{\mu}_r$ (%) $\hat{\sigma}_r$ (%) $\rho$ 16.30 5.23 6.23 7.74 -0.01 0.31 -0.77
 $\hat{\lambda}$ $\hat{\mu}_h (\%)$ $\hat{\sigma}_h (\%)$ $\hat{\kappa}$ $\hat{\mu}_r$ (%) $\hat{\sigma}_r$ (%) $\rho$ 16.30 5.23 6.23 7.74 -0.01 0.31 -0.77
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