Valuation of Mortgages by Using Lévy Models

Abstract

In a mortgage valuation model, the early termination (i.e., prepayment and default) hazard rates and the recovery rate can be specified as multivariate affine functions that include the correlated stochastic state variables. For good capturing of the distributions for state variables, we specify that the state variables follow Lévy models. Accordingly, the early termination hazard rates and the recovery rate also follow Lévy models. Three popular Lévy models, the normal, Variance Gamma (VG), and Negative Inverse Normal (NIG), were used to obtain the closed-form pricing formula for a mortgage. We also conduct numerical applications. Our results reveal the following findings: first, VG model is better than the normal and NIG models in fitting the actual distributions of the interest rate and the housing return rate. Thus, mortgage valuation using a VG model should be better than that using the other two models. Second, the mortgage value estimated by the normal model is the lowest among the three Lévy models. Third, a prepayment hazard rate with a deterministic value (e.g., using the PSA prepayment model) could result in an unreasonable mortgage duration. Fourth, the mortgage convexity calculated by the normal model is higher than that in the other two Lévy models. Our general pricing formula for a mortgage as described in this study can help market participants accurately value mortgages and effectively manage their risks.

Keywords:

• Valuation
• Mortgage
• Prepayment Risk
• Default Risk
• Lévy Process

Acknowledgments

We thank the Ministry of Science and Technology for providing funding in support of this study (The project number: Most 106-2410-H-017-004-).

Availability of Data and Materials

The datasets of the housing prices and the short-term interest rates adopted during the current study are available at https://fred.stlouisfed.org/series/SPCS10RSA, and https://fred.stlouisfed.org/series/TB3MS.

Notes

1 Please refer to the SIFMA (Securities Industry and Financial Markets Association) Statistics website: http://www.sifma.org/research/statistics.aspx.

2 According to the definitions in Schoutens (2003), an adapted process $X(t)={(X(s))}_{t\le s<\infty }$ is a Lévy process if the following three conditions are satisfied: (1) $X(t)$ has increments independent of the past; (2) $X(t)$ has stationary increments; and (3) $X(t)$ is continuous in probability.

3 According to the Lévy-Khintchine formula, the characteristic function (denoted as ${\varphi }_X(b)$) for each Lévy process with triple $(\gamma ,{\zeta }^2,\nu (\mathit{\text{dx}})),$ where $\gamma \in R,$ $\zeta >0$ and $\nu (\cdot ),$ is a measure of $R\backslash \left\{0\right\},$ such that $\underset{R}{\int }(1\wedge x^2)\nu (\mathit{\text{dx}})<\infty$ can be expressed as follows: $${\varphi }_X(b)=\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}(i\gamma b-\frac{{\zeta }^2}{2}{b}^2+\underset{R}{\int }(1-e^{\mathit{\text{ibx}}}+\mathit{\text{ibx}}{I}_{\left\{\right|x|<1\}})\nu (\mathit{\text{dx}}),$$

where $i=\sqrt{-1},$ $b\in R$ and $I_{\left\{\right\}}$ is an indicated function, $I_{\left\{\omega \right\}}=1,$ if state $\omega$ occurs, otherwise $I_{\left\{\omega \right\}}=0.$ The subordinator is defined as $\gamma =0,$ ${\zeta }^2=0$ and $\nu (\mathit{\text{dx}})$ is defined on $(0,\infty )$ in the triple $(\gamma ,{\zeta }^2,\nu (\mathit{\text{dx}}))$ of the Lévy process.

4 The MGFs for the VG and NIG models are obtained from their characteristic functions.

5 We extend the model of OU process to consider the trend of time. If one considers that there is no trend in the OU processes of interest rate and housing return rate, ${\theta }_r^1$ and ${\theta }_H^1$ can be set to be zero values. That is the general specification in traditional model of OU process.

6 Please see the web site: https://fred.stlouisfed.org.

7 In this data, the monthly prepayment probability, default probability, and recovery rate were taken from the CoreLogic LoanPerformance Securities Database. The housing prices were the Standard and Poor’s Case-Shiller 10-City Home Price Index. For the short-term interest rates were for a 3-month U.S. treasury bill. The sample period is from September 2001 to October 2010.

8 The percentage of difference is calculated by the following formula: $\frac{V{(t)}^L-V^N(t)}{V^N(t)},$ where $V{(t)}^L$ is the theoretical mortgage value in VG model or in NIG model; and $V^N(t)$ the theoretical mortgage value in normal model.

9 The 100% PSA works as follows: the prepayment rate is 0.2% in the first month and increases 0.2% each month until it reaches 6% in the thirtieth month. From the thirtieth month on, the prepayment rate is assumed to be 6%.

10 HAMP can assist borrowers to avoid default. It can adjust the mortgage terms (such as: through extending the loan term, lowering the interest rate, or payment forbearance) until the monthly payment for borrower is no more than 31 percent of her/his monthly gross income. HARP can help homeowners to refinance their mortgage at a lower interest rate. For detailed description, see the web site:

https://www.hsh.com/finance/government/hamp-versus-harp-which-is-right-for-you.html

Additional information

Funding

Ministry of Science and Technology, Taiwan (Most 106-2410-H-017-004-) provided funding in support of this study.