Pricing Mortgage Insurance with Asymmetric Jump Risk and Default Risk: Evidence in the U.S. Housing Market

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.

Appendix A

Maximum Likelihood Function of Asymmetric DEJD, LJD and BSM

Let H = {H(0),H(1),H(2),…,H(N)} denote the realizations of housing price at equally-spaced times t = 0,1,2,…,N. The one period rate of return R _H (t) = 1n H(t) - 1n H(t-1) is IID. ${\hbox{f}}{{\hbox{r}}_k}{ = }\frac{1}{{2\pi }}\int_{{ - \infty }}^{\infty } {{{\hbox{e}}^{{ - i\omega \left( {{R_{{H,k}}}} \right)}}}{\phi_{{{R_{{H,k}}}}}}\left( \omega \right)} \,d\omega,$ where k represents the asymmetric DEJD, LJD, or BSM, represents the density of return of housing price based on different model. And the characteristic function of return of housing price, ${\phi_{{{R_{{H,DEJD}}}}}}\left( \omega \right)$, under the asymmetric DEJD is given by:

$$ {\phi_{{{R_{{H,DEJD}}}}}}\left( \omega \right) = { \exp }\left\{ {t\left( {i\omega \,{{\tilde{\mu }}_H} - \frac{1}{2}{\omega^2}\tilde{\sigma }_H^2 + {\lambda^P}\left( {\frac{{p{\eta_u}}}{{{\eta_u} - i\omega }} + \frac{{\left( {1 - p} \right){\eta_d}}}{{{\eta_d} + i\omega }}{ - }1} \right)} \right)} \right\} $$

(A.1)

where ${\tilde{\mu }_H} = {\mu_H} - \frac{1}{2}\sigma_H^2 - \frac{1}{2}\phi_{{rH}}^2$, $\tilde{\sigma }_H^2 = \sigma_H^2 + \phi_{{rH}}^2$, and the characteristic function of return of housing price,${\phi_{{{R_{{H,LJD}}}}}}\left( \omega \right)$, under the LJD is given by:

$$ {\phi_{{{R_{{H,LJD}}}}}}\left( \omega \right) = { \exp }\left\{ {t\left( {i\omega \,{{\tilde{\mu }}_H} - \frac{1}{2}{\omega^2}\tilde{\sigma }_H^2 + {\lambda^P}\left( {{ \exp }\left( {i\,\omega \,\alpha { - }\frac{{1}}{{2}}{\omega^2}{\beta^2}} \right) - 1} \right)} \right)} \right\} $$

(A.2)

where α is the mean of jump size and β is the standard deviation of jump magnitudes. And the characteristic function of return of housing price, ${\phi_{{{R_{{H,BSM}}}}}}\left( \omega \right)$, under the BSM is given by:

$$ {\phi_{{{R_{{H,BSM}}}}}}\left( \omega \right) = { \exp }\left\{ {t\left( {i\omega \,{{\tilde{\mu }}_H} - \frac{1}{2}{\omega^2}\tilde{\sigma }_H^2} \right)} \right\}. $$

(A.3)

Denote k as the asymmetric DEJD, LJD, or BSM and suppose the k model, M _k , has parameter vector θ _k , where θ _k consists of j _k independent parameters to be estimated. Denote ${\hat{\theta }_k}$ as the MLE of θ _k . The parameter space of the asymmetric DEJD, LJD, and BSM is denoted as θ _DEJD = (μ _H , σ _H , ϕ _rH , λ, η _u , η _d , p), θ _LJD = (μ _H , σ _H , ϕ _rH , λ,, α, β), and θ _BSM = (μ _H , σ _H , ϕ _rH ), respectively. Hence, log-likelihood function based on different model is given by the following expression:

$$ \log f\left( {\left. H \right|{\theta_k},{M_k}} \right) = \sum\limits_{{t = 1}}^N {\ln \left( {{\hbox{f}}{{\hbox{r}}_k}\left( {\hbox{t}} \right)} \right)} . $$

(A.4)