Abstract
We investigate the valuation problem of a mortgage contract with full prepayment and default risks using the reduced-form model with regime switching. The hazard rates of full prepayment and default are specified as linear functions of the risk-free interest rate and house prices, respectively, which are characterized by Ornstein-Uhlenbeck processes with regime switching. To derive the explicit valuation formula, we derive the distribution of the number of transitions for two-state Markov processes in finite time and the conditional joint probability density function of transition times using the uniformization technique. Finally, we analyze the effect of parameters on the valuation of the mortgage.
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All datasets supporting the results of this article are retrieved from public data of Federal Reserve Bank of St. Louis and the Data Analytics Department of Black Knight. URLs: https://fred.stlouisfed.org/series/DGS10; https://fred.stlouisfed.org/series/ASPNHSUS; https://www.blackknightinc.com/data-reports/?report-type=mortgage-monitor&report-year=2022; https://fred.stlouisfed.org/series/MORTGAGE15US.
Notes
The dataset is daily data. To correspond with the housing price data, the first data of each month was selected as the observation value. Source: Board of Governors of the Federal Reserve System (US), Market Yield on U.S. Treasury Securities at 10-Year Constant Maturity [DGS10], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/DGS10.
The dataset is monthly data. Unit: million dollars. Source:U.S. Census Bureau and U.S. Department of Housing and Urban Development, Average Sales Price for New Houses Sold in the United States [ASPNHSUS], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/ASPNHSUS.
Data on both the delinquency rate and SMM were obtained from monthly monitoring reports published by the Data Analytics department of Black Knight, with a sample size of 57 for each. URL:https://www.blackknightinc.com/data-reports/?report-type=mortgage-monitor&report-year=2022
The initial risk-free rate r(0) is the average yield of U.S. 10-year fixed term Treasury bonds from 2006 to 2020; h(0) is the average sales price of new home in U.S. from 2006 to 2020 (unit: million dollars); m is the average value of the 15-year fixed-rate mortgage for the United States from 2006 to 2020, Source: https://fred.stlouisfed.org/series/MORTGAGE15US; \(\delta\) is the average recovery rate given in Tsai and Chiang (2012).
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Funding
The research of Guojing Wang, Yinghui Dong and Congjin Zhou were supported by National Natural Science Foundation of China (Grant No.11771320, No.12071335 and No.12301609).
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Congjin Zhou and Guojing Wang contributed to the conception of the study and wrote the main manuscript text; Yinghui Dong provided constructive suggestions in the derivation of propositions; Pin Wang contributed to the numerical illustration. All authors reviewed the manuscript.
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Zhou, C., Wang, G., Dong, Y. et al. The Valuation at Origination of Mortgages with Full Prepayment and Default Risks. Methodol Comput Appl Probab 26, 12 (2024). https://doi.org/10.1007/s11009-024-10081-2
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DOI: https://doi.org/10.1007/s11009-024-10081-2