Pricing Mortgage Insurance with Asymmetric Jump Risk and Default Risk: Evidence in the U.S. Housing Market
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DOI: 10.1007/s11146-011-9307-2
- Cite this article as:
- Chang, C., Huang, W. & Shyu, S. J Real Estate Finan Econ (2012) 45: 846. doi:10.1007/s11146-011-9307-2
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Abstract
This study provides the valuation of mortgage insurance (MI) considering upward and downward jumps in housing prices, which display separate distributions and probabilities of occurrence, and the mortgage insurer’s default risk. The empirical results indicate that the asymmetric double exponential jump diffusion performs better than the log-normally distributed jump diffusion and the Black-Scholes model, generally used in previous literature, to fit the single-family mortgage national average of all home prices in the US. Finally, the sensitivity analysis shows that the MI premium is an increasing function of the normal volatility, the mean down-jump magnitudes, the shock frequency of the abnormal bad events, and the asset-liability structure of the mortgage insurer. In particular, the shock frequency of the abnormal bad events has the largest effect of all parameters on the MI premium. The asset-liability structure of the mortgage insurer and shock frequency of the abnormal bad events have a larger effect of all parameters on the default risk premium.
Keywords
Mortgage insurance contractAsymmetric double exponential jump diffusion processDefault riskJEL Classification
G1G2Introduction
Private mortgage insurance (MI) guarantees that if a borrower defaults on a loan, a mortgage insurer will pay the mortgage lender for any loss resulting from a property foreclosure up to 20–30% of the claim amount. Many previous studies indicate that the housing price change is a crucial factor in determining MI premiums (e.g., Kau et al. 1992, 1993, 1995; Chen et al. 2010). Therefore, how to properly model the dynamic process of US housing price and price the MI contracts is an important issue. Chen et al. (2010) show empirical evidence of the jump magnitude phenomenon when using US monthly national average new home data. In contrast to the assumption of lognormality in the jump magnitude generally made in previous literature, we would like to investigate whether the jump risk is symmetric or asymmetric and how the asymmetric jump risk affects the value of MI premiums if the jump risk of US housing prices exists to a significant extent. Furthermore, in view of the rising foreclosure rates of the borrowers and the mortgage insurers’ huge losses, the default risk of the mortgage insurer has drawn more attention to the valuation of MI contracts, especially during the subprime mortgage crisis. Therefore, it is also vitally important to assess the impact of default risk of mortgage insurer on the MI premiums.
In addition to the interest rate change, the change in housing prices plays a crucial role in the pricing of MI contracts. In the previous literature, housing price change is assumed to follow a traditional Black-Scholes model (BSM), and this assumption is reasonable for relatively stable housing prices (e.g., Kau et al. 1992, 1993, 1995; Kau and Keenan 1995, 1999; Bardhan et al. 2006). Kau and Keenan (1996) use a compound Poisson process to only consider the down-jump component of housing prices in the case of catastrophic events. Chen et al. (2010) assume that the housing price process follows the log-normally distributed jump diffusion (LJD) process, capturing important characteristics of abnormal shock events. This assumption is consistent with the empirical observation of the US monthly national average of new home returns from January 1986 to June 2008. However, the traditional lognormality assumption involves a “generic jump” whose magnitude fluctuates between minus one and infinity, thus allowing the generation of both downward and upward jumps. Although the lognormal distribution has many useful properties, one drawback of this approach is the constraining of upward jumps and downward jumps to both come from the same distribution as well as the lack of precise differentiation between the probabilities of the occurrence of each type of jump.
In view of the subprime mortgage crisis in the US, Mortgage Insurance Companies of America (MICA) reports that large mortgage insurers reported $2.6 billion in losses in 2008, sparking concerns that rising foreclosure rates of the borrowers could compel the industry into a money crisis. For instance, shares of Radian Guaranty, Triad, and PMI Mortgage Insurance Group lost 90 percent of their value in 2007; Triad Guaranty Insurance Corporation failed to meet capital requirement on March 31, 2008 and is even going out of business. MICA reports that Triad’s risk-to-capital ratio, 27.7:1, exceeded the maximum (25:1) generally allowed by insurance regulations and Illinois insurance law. As we know, the default risk of the mortgage insurer is generally not considered by the previous pricing model of MI contracts.
There are three contributions to the pricing of MI contracts in this paper. First, we use US housing price data to find that the asymmetric DEJD process is the best fit by using the quasi-Newton algorithm, Bayesian information criterion (BIC) and likelihood-ratio test (LR test). Next, to be consistent with the asymmetric jump behavior of US housing prices, the relationship between the interest rate and housing prices and mortgage insurers’ default risk, this paper develops a contingent-claim framework for valuing an MI contract. We adopt a structural approach to model the default probability of the mortgage insurer. The mortgage insurer’s total asset and liability value consists of two risk components: risk in interest rate and housing price. Finally, the sensitivity analysis examines how the asymmetric jump risk of housing prices and the default risk of the mortgage insurer impact the valuation of MI contracts and the default risk premium. We find that the shock frequency of the abnormal bad events has the most significant effect on the MI premium, and the asset-liability structure of the mortgage insurer and shock frequency of the abnormal bad events show the greatest effect of all parameters on the default risk premium. This implies that the insurer must carefully consider the impact of the shock frequency of the abnormal bad events when pricing the MI contracts.
The remainder of this paper is organized as follows. Section “Model” illustrates the model. Section “Valuation of Mortgage Insurance Contract” derives the pricing formulae for MI contracts under asymmetric DEJD. Empirical and numerical analyses are presented in Section “Empirical and Sensitivity Analysis”. Section “Conclusions” summarizes the paper and gives conclusions.
Model
This study adopts a structural approach to model the default probability of the mortgage insurer. Because the interest rate, housing prices and the mortgage insurer’s asset–liability structure specifications are crucial factors in determining the value of MI contracts, we assume that the interest rate, housing prices and the mortgage insurer’s liability are related and that the interest rate and the mortgage insurer’s assets are related. This section outlines the dynamic processes of the interest rate process, the borrower’s housing price, the mortgage insurer’s assets, and the mortgage insurer’s liability under the risk-neutral measure Q.^{3}
The Instantaneous Interest Rate Process
The Housing Price Process
- Case (1)
Suppose that \( {\hat{\eta }_u} = {\hat{\eta }_d} \) and \( {\hat{\lambda }_u} = {\hat{\lambda }_d} \) (i.e.,\( \hat{p} = 0.5 \)); then the distribution of jumps will be symmetrical with a higher peak and a positive kurtosis relative to normal.
- Case (2)
Suppose that \( {\hat{\eta }_u} = {\hat{\eta }_d} \) and \( {\hat{\lambda }_u} \ne {\hat{\lambda }_d} \) (i.e.,\( \hat{p} \ne 0.5 \)); then, relative to the geometric Brownian motion, the distribution of the return of the housing price will be skewed and have excess kurtosis, and the relative size of \( {\hat{\lambda }_u} \) and \( {\hat{\lambda }_d} \) will lead to negative or positive skewness.
- Case (3)
Suppose that \( {\hat{\eta }_u} \ne {\hat{\eta }_d} \) and \( {\hat{\lambda }_u} = {\hat{\lambda }_d} \); again, the resulting return of the housing price will be skewed and show excess kurtosis. However, the relative size of \( {\hat{\eta }_u} \) and \( {\hat{\eta }_d} \) will determine whether the distribution is negatively or positively skewed.
- Case (4)
Suppose that \( {\hat{\lambda }_u} = {\hat{\lambda }_d} = 0 \); again, the resulting return of the housing price will be reduced to the geometric Brownian motion.
The Mortgage Insurer’s Liability Process
The Mortgage Insurer’s Asset Process
Valuation of Mortgage Insurance Contract
- (a)If A(t)/L(t) → ∞ (i.e., the mortgage insurer would not default) and there is a constant interest rate, the closed-form solution of Eq. 17 using put-call parity is given by the following expressions:^{5}where$$ \begin{array}{*{20}{c}} {DL(t) = P\left( {t,{K_1}} \right) - P\left( {t,{K_2}} \right) = \frac{{{e^{{ - \alpha {k_1}}}}}}{{2\pi }}\int_{{ - \infty }}^{\infty } {{e^{{ - i\omega {k_1}}}}\frac{{{e^{{ - rt}}}{\phi_{{\tilde{H}(t)}}}\left( {\omega - i\left( {\alpha + 1} \right)} \right)}}{{{\alpha^2} + \alpha - {\omega^2} + i\left( {2\alpha + 1} \right)\omega }}} \,d\omega } \hfill \\{\quad \quad \quad \quad \quad \quad - \frac{{{e^{{ - \alpha {k_2}}}}}}{{2\pi }}\int_{{ - \infty }}^{\infty } {{e^{{ - i\omega {k_2}}}}\frac{{{e^{{ - rt}}}{\phi_{{\tilde{H}(t)}}}\left( {\omega - i\left( {\alpha + 1} \right)} \right)}}{{{\alpha^2} + \alpha - {\omega^2} + i\left( {2\alpha + 1} \right)\omega }}} \,d\omega + \left( {{K_1} - {K_2}} \right){e^{{ - rt}}}.} \hfill \\\end{array} $$(18)$$ {\phi_{{\tilde{H}(t)}}}\left( {\omega - i\left( {\alpha + 1} \right)} \right) $$$$ \begin{array}{*{20}{c}} { = { \exp }\left\{ {i\left( {\omega - i\left( {\alpha + 1} \right)} \right)\left( {\ln H(0) + \left( {r - \frac{1}{2}\sigma_H^2 - {\lambda^Q}\kappa } \right)t} \right) - \frac{1}{2}{{\left( {\omega - i\left( {\alpha + 1} \right)} \right)}^2}\sigma_H^2t} \right\} \times } \hfill \\{\quad \exp \left\{ {{\lambda^Q}t\left( {\frac{{\hat{p}{{\hat{\eta }}_u}}}{{{{\hat{\eta }}_u} - i\left( {\omega - i\left( {\alpha + 1} \right)} \right)}} + \frac{{\left( {1 - \hat{p}} \right){{\hat{\eta }}_d}}}{{{{\hat{\eta }}_d} + i\left( {\omega - i\left( {\alpha + 1} \right)} \right)}}{ - 1}} \right)} \right\},} \hfill \\{\quad \quad \quad \quad \quad \quad {k_1} \equiv \ln {K_1} = \ln B(t),\;{k_2} \equiv \ln {K_2} = \ln \left( {1 - {L_C}} \right)B(t),\;\tilde{H}(t) \equiv \ln H(t).} \hfill \\\end{array} $$
- (b)
If A(t)/L(t) → ∞ and a constant interest rate and a lognormal jump component of the housing price exist, the closed-form solution of Eq. 17 reduces to the closed-form formula of Chen et al. (2010).
- (c)
If A(t)/L(t) → ∞ and a constant interest rate and no jump component of housing prices exist, the closed-form solution of Eq. 17 reduces to the closed-form formula of Bardhan et al. (2006).
Because the housing price is independent of the unconditional probability of the borrower’s defaulting, the MI premium (FPA) with an asymmetric jump risk is given by the following expression:where q represents the gross profit margin, and \( P(t) = 1 - {e^{{ - {\lambda_b}\,t}}} \), λ_{b} denotes the default frequency of the borrower. Equation 19 implies that FPA is calculated by 1 + q multiples of the fair price, i.e., \( \sum\limits_{{t = 1}}^T {P(t)DL(t)} \), which is the summation of a series of the loss amounts of the insurer if the borrower defaults in each year from the beginning to expiration. Therefore, the insurer can decide for each year the probability that the borrower will default rather than at only maturity.$$ FPA = \left( {1 + q} \right)\sum\limits_{{t = 1}}^T {P(t)DL(t)}, $$(19)
Empirical and Sensitivity Analysis
Data and Empirical Results
Our data come from the Federal Housing Finance Agency and contain the term on conventional single-family mortgages and the monthly national average of all home prices in the US. We investigate the monthly average of the prices of all homes with adjustable-rate mortgages.^{6} Our sample period is from January 1986 to October 2008, leading to 274 observations for each variable. We use the asymmetric DEJD process (see Eq. 3) to compare the model’s fitness for the national average of single-family mortgaged home prices in the US with results of the LJD and BSM generally used in previous literatures.^{7}
Descriptive statistics and unit root tests
Panel A: Descriptive Statistics | |||||||
Mean | Median | Max. | Min. | Std. Dev. | Skew | Kurt | Jarque-Bera |
0.00411 | 0.00420 | 0.21385 | −0.24246 | 0.05526 | −0.51617 | 5.90417 | 108.062* |
Panel B: Augmented Dickey-Fuller test | |||||||
Intercept | Trend & Intercept | None | |||||
−12.9175* | −12.9164* | −15.1943* |
Maximum likelihood estimates and testing for the asymmetric DEJD, LJD, and BSM
Model | μ_{H} | σ_{H} | ϕ_{rH} | α | β | λ | η_{u} | η_{d} | p | BIC | LR |
---|---|---|---|---|---|---|---|---|---|---|---|
DEJD | 1.416 × 10^{−1*} | 1.402 × 10^{−1*} | −4.948 × 10^{−2*} | – | – | 3.933* | 25.099* | 22.647* | 2.799 × 10^{−1*} | −798.699* | L_{LJD}-L_{DEJD} = −3.4402* |
(9.617 × 10^{−3}) | (3.006 × 10^{−3}) | (2.094 × 10^{−3})* | (4.815 × 10^{−1}) | (1.779) | (7.365 × 10^{−1}) | (4.898 × 10^{−2}) | |||||
LJD | 8.795 × 10^{−2*} | 3.920 × 10^{−2*} | −1.695 × 10^{−1*} | −6.506 × 10^{−2*} | 1.159 × 10^{−1*} | 3.606 × 10^{−1*} | – | – | – | −797.428 | L_{BSM}-L_{LJD} = −11.8878* |
(8.988 × 10^{−4}) | (5.915 × 10^{−3}) | (1.565 × 10^{−3}) | (2.407 × 10^{−3}) | (1.164 × 10^{−3}) | (2.206 × 10^{−2}) | ||||||
BSM | 6.768 × 10^{−2*} | 1.657 × 10^{−1*} | −9.579 × 10^{−2*} | – | – | – | – | – | – | −790.481 | – |
(1.463 × 10^{−3}) | (2.278 × 10^{−3}) | (3.956 × 10^{−3}) |
Sensitivity Analysis for Mortgage Insurance Premiums
Base Parameters and Value of Mortgage Insurance Premiums
Base parameter values of mortgage insurance premium
Interest rate parameters | Values | |
r | Initial instantaneous interest rate | 0.05 |
η_{r} | Magnitude of mean-reverting force | 0.2 |
λ_{r} | Market price of interest rate risk | 0.01 |
θ | Long-run mean of interest rate | 0.1 |
v | Volatility of interest rate | 0.05 |
Liability parameters | ||
L | Liability value of the mortgage insurer | 10000000 |
ϕ_{rL} | Interest rate sensitivity of change rate of liability value | −0.1 |
ϕ_{HL} | House price sensitivity of change rate of liability value | −0.1 |
σ_{L} | Volatility terms of change rate of liability value | 0.05 |
Asset parameters | ||
A | Asset value of the mortgage insurer | V/L = 1.5, 1.75 and 2 |
σ_{A} | Volatility terms of change rate of asset value | 0.05 |
ϕ_{rA} | Interest rate sensitivity of change rate of asset value | −0.1 |
Housing price parameters | ||
H(0) | Initial housing price | 242300 |
ϕ_{rH} | Interest rate sensitivity of change rate of house price | −0.04948 |
σ_{H} | Volatility terms of change rate of house price | 0.1402 |
η_{u} | The parameter of up-jump size | 25.099 |
η_{d} | The parameter of down-jump size | 22.647 |
λ | The parameter of jump intensity | 3.933 |
p | The up-jump size probability | 0.2799 |
Other parameters | ||
L_{R} | Initial loan-to-value ratio | 0.85 |
L_{C} | Coverage ratio | 0.3 |
c | Installments | 16597 |
T | Term to maturity of mortgage contract | 30 years |
y | Contract interest rate of mortgages | 0.07 |
q | Gross profit margin | 0.05 |
λ_{b} | Default frequency of the borrower | 0.05 |
Parameters Values Matter: Sensitivity Analysis
Sensitivity analysis of mortgage insurance premium
λ_{d} | η_{d} | σ_{H} | No default risk | Default risk | |||||
---|---|---|---|---|---|---|---|---|---|
Asset-liability structure | Default risk premiums | ||||||||
1.5 | 1.75 | 2 | 1.5 | 1.75 | 2 | ||||
2.139 | 21.174 | 0.134 | 1561.16 | 1516.61 | 1534.97 | 1540.72 | 44.55 | 26.19 | 20.44 |
(−17.70%) | (−20.05%) | (−19.08%) | (−18.78%) | (26.68%) | (−25.52%) | (−41.86%) | |||
0.140 | 1629.66 | 1579.30 | 1598.01 | 1605.64 | 50.36 | 31.65 | 24.02 | ||
(−14.09%) | (−16.75%) | (−15.76%) | (−15.36%) | (43.22%) | (−10.00%) | (−31.69%) | |||
0.146 | 1700.46 | 1643.39 | 1664.16 | 1671.64 | 57.07 | 36.31 | 28.83 | ||
(−10.36%) | (−13.37%) | (−12.27%) | (−11.88%) | (62.29%) | (3.25%) | (−18.02%) | |||
22.647 | 0.134 | 1454.53 | 1418.54 | 1436.67 | 1441.99 | 35.99 | 17.86 | 12.54 | |
(−23.32%) | (−25.22%) | (−24.27%) | (−23.98%) | (2.36%) | (−49.22%) | (−64.34%) | |||
0.140 | 1525.61 | 1479.71 | 1499.80 | 1506.25 | 45.90 | 25.81 | 19.36 | ||
(−19.58%) | (−22.00%) | (−20.94%) | (−20.60%) | (30.53%) | (−26.61%) | (−44.95%) | |||
0.146 | 1598.54 | 1545.19 | 1565.75 | 1573.83 | 53.35 | 32.79 | 24.72 | ||
(−15.73%) | (−18.54%) | (−17.46%) | (−17.03%) | (51.72%) | (−6.75%) | (−29.71%) | |||
24.120 | 0.134 | 1340.15 | 1309.25 | 1324.73 | 1330.46 | 30.90 | 15.42 | 9.68 | |
(−29.35%) | (−30.98%) | (−30.17%) | (−29.86%) | (−12.13%) | (−56.15%) | (−72.47%) | |||
0.140 | 1413.71 | 1376.73 | 1394.52 | 1401.40 | 36.98 | 19.19 | 12.31 | ||
(−25.48%) | (−27.43%) | (−26.49%) | (−26.12%) | (5.15%) | (−45.44%) | (−64.99%) | |||
0.146 | 1488.81 | 1444.23 | 1464.35 | 1470.55 | 44.58 | 24.46 | 18.26 | ||
(−21.52%) | (−23.87%) | (−22.81%) | (−22.48%) | (26.78%) | (−30.43%) | (−48.07%) | |||
2.832 | 21.174 | 0.134 | 2054.57 | 1995.48 | 2018.25 | 2025.68 | 59.09 | 36.32 | 28.88 |
(8.31%) | (5.19%) | (6.39%) | (6.78%) | (68.02%) | (3.29%) | (−17.86%) | |||
0.140 | 2117.63 | 2051.36 | 2073.05 | 2080.06 | 66.27 | 44.58 | 37.58 | ||
(11.63%) | (8.14%) | (9.28%) | (9.65%) | (88.46%) | (26.78%) | (6.86%) | |||
0.146 | 2180.26 | 2111.08 | 2133.82 | 2140.53 | 69.18 | 46.44 | 39.74 | ||
(14.93%) | (11.29%) | (12.49%) | (12.84%) | (96.74%) | (32.05%) | (13.00%) | |||
22.647 | 0.134 | 1870.06 | 1820.42 | 1837.94 | 1843.54 | 49.64 | 32.12 | 26.51 | |
(−1.42%) | (−4.04%) | (−3.11%) | (−2.82%) | (41.16%) | (−8.67%) | (−24.60%) | |||
0.140 | 1932.14 | 1876.19 | 1896.98 | 1902.34 | 55.95 | 35.17 | 29.81 | ||
(1.85%) | (−1.10%) | (BVP) | (0.28%) | (59.10%) | (BVD) | (−15.24%) | |||
0.146 | 1996.94 | 1938.48 | 1959.09 | 1966.47 | 58.46 | 37.85 | 30.47 | ||
(5.27%) | (2.19%) | (3.27%) | (3.66%) | (66.24%) | (7.63%) | (−13.35%) | |||
24.120 | 0.134 | 1753.90 | 1704.26 | 1723.73 | 1730.00 | 49.64 | 30.17 | 23.90 | |
(−7.54%) | (−10.16%) | (−9.13%) | (−8.80%) | (41.15%) | (−14.22%) | (−32.03%) | |||
0.140 | 1820.92 | 1765.13 | 1786.98 | 1794.45 | 55.79 | 33.94 | 26.48 | ||
(−4.01%) | (−6.95%) | (−5.80%) | (−5.40%) | (58.64%) | (−3.48%) | (−24.71%) | |||
0.146 | 1891.18 | 1827.75 | 1849.96 | 1858.38 | 63.43 | 41.22 | 32.80 | ||
(−0.31%) | (−3.65%) | (−2.48%) | (−2.03%) | (80.37%) | (17.21%) | (−6.72%) | |||
3.526 | 21.174 | 0.134 | 2480.07 | 2408.95 | 2431.88 | 2440.10 | 71.12 | 48.19 | 39.97 |
(30.74%) | (26.99%) | (28.20%) | (28.63%) | (102.24%) | (37.04%) | (13.67%) | |||
0.140 | 2535.25 | 2458.83 | 2481.24 | 2490.41 | 76.43 | 54.01 | 44.85 | ||
(33.65%) | (29.62%) | (30.80%) | (31.28%) | (117.33%) | (53.59%) | (27.53%) | |||
0.146 | 2591.34 | 2510.10 | 2534.85 | 2542.46 | 81.24 | 56.49 | 48.88 | ||
(36.60%) | (32.32%) | (33.63%) | (34.03%) | (131.01%) | (60.65%) | (38.99%) | |||
22.647 | 0.134 | 2297.06 | 2228.21 | 2249.24 | 2257.38 | 68.85 | 47.82 | 39.68 | |
(21.09%) | (17.46%) | (18.57%) | (19.00%) | (95.79%) | (35.99%) | (12.84%) | |||
0.140 | 2350.02 | 2280.84 | 2304.16 | 2312.38 | 69.18 | 45.86 | 37.64 | ||
(23.88%) | (20.24%) | (21.46%) | (21.90%) | (96.72%) | (30.41%) | (7.04%) | |||
0.146 | 2413.99 | 2336.88 | 2360.14 | 2368.40 | 77.11 | 53.85 | 45.59 | ||
(27.25%) | (23.19%) | (24.42%) | (24.85%) | (119.26%) | (53.12%) | (29.63%) | |||
24.120 | 0.134 | 2144.86 | 2078.04 | 2097.23 | 2104.84 | 66.82 | 47.63 | 40.02 | |
(13.07%) | (9.54%) | (10.56%) | (10.96%) | (90.01%) | (35.43%) | (13.81%) | |||
0.140 | 2202.46 | 2131.42 | 2152.52 | 2160.54 | 71.04 | 49.94 | 41.91 | ||
(16.10%) | (12.36%) | (13.47%) | (13.89%) | (102.01%) | (42.02%) | (19.19%) | |||
0.146 | 2266.22 | 2188.37 | 2210.46 | 2217.96 | 77.86 | 55.76 | 48.26 | ||
(19.46%) | (15.36%) | (16.53%) | (16.92%) | (121.39%) | (58.57%) | (37.23%) |
Conclusions
The collapse of the subprime mortgage market and the mortgage insurers’ massivelosses has drawn more attention to the precise valuation of MI contracts. This study derives the formula for MI contracts considering the asymmetric DEJD and mortgage insurer’s default risk.
Furthermore, we use the national average of all single-family mortgaged home prices from January 1986 to October 2008 to estimate and test the asymmetric DEJD, the LJD and the BSM. The empirical results indicate that the DEJD is the best model to fit the national average of all single-family mortgaged home prices in the US. Finally, the sensitivity results show that the MI premium is an increasing function of the normal volatility, the mean down-jump magnitude, the shock frequency of abnormal bad events and the asset-liability structure of the mortgage insurer. Compared with the base valuation, when a housing crash occurs in the future, and if this crash causes the normal volatility σ_{H} to increase by two standard errors while all other parameters are fixed, the MI premium increases by 3.27%. Conversely, as the normal volatility decreases by two standard errors, the MI premium decreases by 3.11%. Furthermore, when the parameter of down-jump magnitudes, η_{d}, increases by two standard errors, the MI premium decreases by 5.80%. Conversely, as the parameter of down-jump magnitudes decreases by two standard errors, the MI premium should increase by 9.28%. Furthermore, when the shock frequency of the abnormal bad events, λ_{d}, increases by two standard errors, the MI premium increases by 21.46%. Similarly, the MI premium decreases by 20.94% if the shock frequency of the abnormal bad events decreases by two standard errors. Furthermore, as the asset-liability structure of the mortgage insurer increases by 0.25 standard errors, the MI premium increases by 0.28%. Conversely, if the asset-liability structure of the mortgage insurer is reduced by 0.25 standard errors, the MI premium decreases by 1.10%. Therefore, the shock frequency of the abnormal bad events has the most significant effect on the MI premium. Furthermore, we find that the asset-liability structure of the mortgage insurer and the shock frequency of the abnormal bad events show the greatest effect of all the parameters on the default risk premium. This implies that the insurer must carefully consider the impact of the shock frequency of abnormal bad events when pricing the MI contracts. Compare to Bardhan et al. (2006) model and Chen et al. (2010) model, we conclude that it is necessary to consider the asymmetric jump risk and mortgage insurer’s default risk when pricing the MI contract, particularly for higher housing prices. Other potential improvements and possible extensions are given. First, in addition to the jump diffusion process, the positive serial correlation of US housing-price movements is a relevant and significant issue (see Case and Shiller 1989). Additionally, the structured change in US housing prices is also an important issue when pricing MI contracts. Many studies discussed the potential structural change in housing prices by employing the threshold regression, Markov-switching and smooth-threshold autoregressive (STAR) models, among others.
Stochastic volatility usually has a larger impact on long-term options, whereas the presence of jumps mostly benefits the pricing of short-term near-the-money options.
In general, there are two types of Lévy processes: the finite-activity Lévy process and the infinite-activity Lévy process. The finite-activity Lévy process generates only a finite number of jumps during any finite time interval. Examples of such models are the Merton jump-diffusion model with Gaussian jumps and the Kou model with asymmetric double exponential jumps. On the other hand, the infinite-activity Lévy model can generate an infinite number of small jumps at any finite time interval. Because the liquidity of the real estate market is lower than that of the financial market, the number of jumps should be finite. Furthermore, in finite-activity Lévy processes, the dynamic structure of the process is easy to understand and describe because the distribution of jump sizes is known. Such processes are also easy to simulate, and it is possible to use efficient Monte Carlo methods for pricing path-dependent options. Hence, this paper uses a finite-activity Lévy model to describe asymmetric jumps in the housing price process.
As argued by Bardhan et al. (2006), the valuation of MI can be also obtained if the assumption of the risk neutrality of agents is relaxed and insurance contracts are assumed to be traded. Therefore, we can price MI contracts without assuming the risk neutrality of the agents and instead assuming that MI contracts are traded.
The detailed description of the Esscher transform and similar detailed proof of the martingale condition can see Carr and Madan (1999).
In addition to ARM loans, FRM loans are also available for the FHFA. US national average all home price returns for single-family FRM loans also seem to feature the asymmetric jump phenomenon. It shows that there were four occasions when the monthly housing price changed by more than two standard errors per month. And then it can be seen that there were six occasions when the monthly housing price changed by less than two standard errors per month. Hence, the asymmetric jump phenomenon in ARM loans seems to be higher than one in FRM loans. Because that this paper focuses on the asymmetric jump phenomenon of US national average all home price returns, for simplification, in the empirical study, only US national average all home price returns for single-family ARM loans were used. In further research, it could compare the asymmetric jump phenomenon of ARM loans and FRM loans, and then investigate the impacts of their asymmetric jump behaviors on MI premiums.
The limitation of using national average house prices is that, compared with individual house prices, national average house prices may reduce the level of volatility of housing prices and understate the degree of risk for mortgage insurers who do not operate nationwide.
Acknowledgments
We thank the editor and anonymous referees for their helpful comments. We also acknowledge the financial support from the National Science Council of Taiwan (NSC 98-2410-H-151-001).