Estimation of Housing Price Jump Risks and Their Impact on the Valuation of Mortgage Insurance Contracts

E

STIMATION OF

H

OUSING

P

RICE

J

UMP

R

ISKS AND

T

HEIR

I

MPACT ON THE

V

ALUATION OF

M

ORTGAGE

I

NSURANCE

C

ONTRACTS

Housing price jump risk and the subprime crisis have drawn more atten-tion to the precise estimaatten-tion of mortgage insurance premiums. This study derives the pricing formula for mortgage insurance premiums by assuming that the housing price process follows the jump diffusion process, capturing important characteristics of abnormal shock events. This assumption is con-sistent with the empirical observation of the U.S. monthly national average new home returns from 1986 to 2008. Furthermore, we investigate the impact of price jump risk on mortgage insurance premiums from shock frequency of the abnormal events, abnormal mean and volatility of jump size, and normal volatility. Empirical results indicate that the abnormal volatility of jump size has the most significant impact on mortgage insurance premiums.

INTRODUCTION

Mortgage insurance has an important role in the housing finance market since it transfers the borrower’s default risk exposure from the lenders to insurers and facil-itates the creation of secondary mortgage markets (see Canner and Passmore, 1994). When determining mortgage termination, the present value of amortizing mortgage payments and the ability of the borrower to release from the payments through either prepayment or default must be considered. Although the prepayment decision is significantly affected by the interest rate, the house price influences significantly the

Ming-Chi Chen is from National Sun Yat-sen University, Kaohsiung, Taiwan. Chia-Chien Chang is from National Kaohsiung University of Applied Science, Kaohsiung, Taiwan. Shih-Kuei Lin is from National University of Kaohsiung, Kaohsiung, Taiwan. So-De Shyu is from Na-tional Sun Yat-sen University, Kaohsiung, Taiwan. Ming-Chi Chen can be contacted via e-mail: mcchen@finance.nsysu.edu.tw. The authors are grateful for the valuable comments of the anonymous referees.

FIGURE1

U.S. National Average New Home Price Returns for Single-Family Mortgage

Note: The dashed (solid) line represents the mean of the housing price return plus (minus) three

standard deviations.

decision to default.1Some studies show that changes in the loan-to-value ratio and housing price produce a wider range of mortgage default (see Kau et al., 1992; Kau, Keenan, and Muller, 1993; Kau and Keenan, 1995, 1996, 1999). When the loan-to-value ratio is higher, the price of mortgage insurance is higher. Furthermore, the house price volatility parameter is important for mortgage insurance and the impact of increasing the house price volatility is significant. Hence, the housing price change is a crucial factor in determining mortgage insurance premiums.2

In the previous literature, housing price change is assumed to follow a traditional ge-ometric Brownian motion, and this assumption is reasonable under relatively stable housing prices. For example, all related studies on mortgage insurance pricing (see, e.g., Kau et al., 1992, 1995; Kau, Keenan, and Muller, 1993; Kau and Keenan, 1995, 1999; Bardhan et al., 2006) assume that the housing price process follows a geometric Brownian motion. However, Figure 1 shows the U.S. national average new home price returns for single-family mortgage from January 1986 to June 2008. It can be seen that there were 14 times when the monthly housing price changed more than 10 percent per month. In particular, the highest monthly housing price returns was 20.85 percent, in June 1992, while the lowest monthly housing price returns was−22.76 per-cent, in November 2007. Since 2007, with the higher interest rates and higher mortgage payments, subprime crisis occurred, which caused significantly downward jumps of 1_{Some empirical studies also indicate that the patterns of default and prepayment are}

signif-icantly explained by the economic risk factors, such as interest rate, housing price return, loan-to-value ratio, and unemployment rate (Campbell and Dietrich, 1983; Schwartz and Torous, 1989, 1993; Quigley and Van Order, 1990, 1995; Deng, Quigley, and Van Order, 2000; Lambrecht, Perraudin, and Satchell, 2003; Caselli, Gatti, and Querci, 2008).

2_{Some empirical studies also examine the effects of catastrophic events on property values}

housing price. On the other hand, other abnormal shocks, such as “Black Wednes-day” in September 1992 or the “Iraq disarmament crisis” in July 1993, caused the U.S. Federal Reserve to adapt an expansionary monetary policy. Previous studies have suggested a strong connection between real interest rates and housing prices. Har-ris (1989), Abraham and Hendershott (1996), Englund and Ioannides (1997), Sutton (2002), Borio and Mcguire (2004), and Kostas and Zhu (2004), among others, all report the significantly negative relationship between the real interest rates and housing prices. Since the U.S. Federal Reserve lowered the interest rate 23 times from 1990 to 1992, we can understand that the announcement of lower interest rates could cause the U.S. housing price to make greater upward jumps. Overall, the housing price seems to have made higher jumps and volatility spikes during these years.

In order to properly model the housing price process, most of the discrete time models have been of the generalized autoregressive conditional heteroskedastic (GARCH) type, while the continuous time models were based on diffusion models. Mizrach (2008) and Schloemer et al. (2006) address the existence of jumps in housing markets. Mizrach demonstrates the jump risk component from the returns on the Chicago Mercantile Exchange (CME) futures. The empirical result indicates that, on average, it requires about 69 jump risks to be significant in the 315-day sample. Although GARCH models are capable of capturing smooth persistent changes in volatility, GARCH models are not suited to explaining the large discrete changes due to the abnormal events found in housing price returns. Hence, rather than studying volatil-ity spikes, this article investigates the jump parameters of the housing price and their impacts on the mortgage insurance premium, when abnormal event informa-tion important to the housing market arrives, especially in the subprime mortgage crisis. Corresponding to the abnormal event of subprime mortgage crisis, the jump component of housing price represents systematic and nondiversifiable risk, which is, therefore, correlated with the market. To value the mortgage insurance contract, we use the Esscher transform technique developed by Gerber and Shiu (1994), which is a well-established technique in actuarial science and is suitable in cases where the log-returns of the underlying asset are governed by a process with independent and stationary increments. The behavior of the change of housing price can be divided into two parts: (1) continuous diffusion, which is responsible for the usual housing price movement and is described by a traditional Brownian motion, and (2) discontinuous jumps, which correspond to the arrival of new information important to the housing market.

This article contributes to the literature on mortgage insurance contract pricing in the following ways. First, this article estimates parameters of the jump diffusion model (JDM) using expectation-maximum (EM) gradient algorithms (based on the U.S. housing price data). EM gradient algorithms can be appropriate for latent-data problems because the total number of jumps for housing price is unobserved. The empirical results show that the likelihood ratio test (LRT) rejects the model without jumps at the significance level of 99 percent when using the national average for new home prices, but it does not reject the model without jumps when using the national average for previously occupied home prices. Second, to be consistent with the jump behavior of U.S. housing prices, this article applies a jump diffusion framework to derive the closed-form formula of mortgage insurance contracts by using Esscher transform technique. Our pricing formula for mortgage insurance contracts can also

be reduced to the closed-form formula of Bardhan et al. (2006). Finally, to investigate how the jump risk of housing price impacts the valuation of mortgage insurance premiums, numerical analysis shows the relationships among mortgage insurance premium, the shock frequency of the abnormal event, the abnormal volatility of jump size, and the abnormal mean of jump size. If the shock frequency of the abnormal event (the abnormal volatility of jump size) increases two standard deviations, while all other parameters are fixed, the mortgage insurance premium should increase 0.27 percent (11.36 percent), respectively. Meanwhile, as the abnormal mean of jump size decreases two standard deviations, the mortgage insurance premium increases 6.57 percent. Therefore, the abnormal variance of jump size has the most significant effect on the mortgage insurance premium, and this implies the necessity of considering the jump parameters when pricing mortgage insurance contracts whose collateral asset is new homes.

The remainder of this article is organized as follows. The second section illustrates the mortgage insurance contract and the model. The third section derives the pricing formulas for mortgage insurance contracts under JDMs. Estimation via EM algorithms is shown in the fourth section. Empirical and numerical analyses are presented in the fifth section. The sixth section summarizes the article and gives conclusions. For simplicity, most proofs are in the Appendix.

THECONTRACT ANDMODEL

For the related models used in previous work, Kau et al. (1992, 1995), Kau, Keenan, and Muller (1993), Kau and Keenan (1995, 1999), and others consider two state variables: the interest rate and the housing price process. Prepayments and defaults are also typically determined endogenously within the model. However, implementation of these models requires complex numerical procedures since no closed-form formulas exist. Furthermore, some articles, such as Hendershott and Van Order (1987), also find evidence that mortgage insurance premiums are not very sensitive to interest rate volatility. Hence, Schwartz and Torous (1993), Dennis, Kuo, and Yang (1997), and Bardhan et al. (2006), among others, model the unconditional probability of default exogenously. Closest to our model is the option pricing method proposed by Bardhan et al. (2006). We develop a closed-form option-pricing framework for pricing mortgage insurance premiums and model the unconditional probability of default exogenously. Bardhan et al. assume that the housing price follows a geometric Brownian motion process. In contrast, to capture the jump phenomenon of housing price as shown in the Figure 1, the dynamic process of housing price proposed in this article combines a Brownian motion and a compound Poisson process. Therefore, in this section, the structure of mortgage insurance contract is illustrated and then we use the JDM to allow for abnormal shock events in housing prices.

Mortgage Insurance Contract

At time t= 0, i.e., at origination, the lender issues a T-month mortgage, secured by the housing, for the amount of L(0)= L_VH(0). Let L_Vbe the initial loan-to-value ratio and H(0) be the initial housing price. We assume that the mortgage loan has a fixed

interest rate c and that installments x are paid monthly. Hence, with no prepayment or default prior to time t, the unpaid loan balance L(t) at time 0≤ t ≤ T is given by the following expression:

L(t)= x c  1− 1 (1+ c)T−t  . (1)

This equation shows that the unpaid loan balance is equal to the value of an ordinary annuity with a monthly payment equal to x and the discount rate equal to the contract rate c. In addition, at time t= 0, the insurer writes a mortgage insurance contract that promises to compensate the lender only when the borrower defaults. We follow the model of Bardhan et al. (2006) to consider that the realized loss for the insurer in case of the borrower’s default can be represented as a portfolio of put options on the borrower’s collateral. Thus, if a default occurs at time t, the insurer has to pay the lender the following amount:

L OSS(t)= max [0, min(L(t−) − H(t), LRL(t−))] , (2)

where LRdenotes the loss ratio. Equation (2) implies that if the housing price exceeds

the remaining loan balance, after the house is sold and the lender is compensated from the proceeds, the lender bears no loss, and hence, the loss for the insurer is also zero. On the other hand, if the housing price is not sufficient for a full repayment of the loan balance, the maximum loss to the insurer is equal to LRL(t−).

The previous studies on mortgage insurance pricing all assume that the housing price process follows a geometric Brownian motion. However, if the housing price process has an explicit jump risk that corresponds to the arrival of new important information to the housing market, the geometric Brownian motion will fail to capture important characteristics. Hence, it is necessary to use an appropriate model that considers jump risks to price the mortgage insurance contracts. In the following, we use jump diffusion framework to describe the housing price process with jump risks.

Model Structure of Housing Price With Jump Risks

We construct our model on a filtered probability space (, F, P) generated by these two processes; i.e., the process of housing price H(t), and the process of jump size in housing price Y(t). There exists a unique physical probability P such that capital markets are complete. The filtration F= (Ft)t≥0satisfies Ft = FtW∨ FtYfor any time t,

where F_tW= σ(W(u), 0 ≤ u ≤ t), and F_tY= σ (Y(u), 0 ≤ u ≤ t). Hence, F_tW∨ F_tY con-tains complete information on Brownian motions and jump sizes of the housing price returns. Further, there are three types of agents in the economy: the lender, the borrower, and the insurer.

Assume that the risk-free security is the money market account B(t)= er t, where r is constant continuously compounded return, r ∈ R++. Furthermore, let the process of

the housing price be the combination of a Brownian motion and a compound Poisson process as follows:3 H(t)= H(0) exp ⎛ ⎝μt + σHdWP(t)+ N(t)  n=0 Yn− λE(Y)t ⎞ ⎠ , (3)

where H(0) is the initial housing price,μ is the expected growth rate of housing price, and σH is the constant volatility of the Brownian component of the housing price

process. In addition, W(t) is a standard Brownian motion. The role of WP(t) with drift can be used to capture the unanticipated instantaneous change of housing price, which is the reflection of normal events, but it may not work so well for abnormal shocks. For example, the government may suddenly increase interest rates, or a mortgage crisis may arise. Thus, a compound Poisson process is constructed here to address the total number of jumps and jump size corresponding to the arrival of abnormal information. N(t) represents the total number of jumps (including the house price rise and drop event) during a time interval of (0, t], and it is based on a Poisson process with a intensity parameter λ . Notation Yn, n= 1, 2, . . . are i.i.d. random variables

representing the size of the nth size of the jumps with the density function f (dy) and the expectation E(Y)< ∞. In particular, we assume that the jump size is normal distributed with meanϕ and variance δ2, that is, Y∼ N(ϕ, δ2).ϕ > 0(ϕ < 0) represents the upward average jump size (downward average jump size) in housing price in case of abnormal events during a time interval of (0, t]. In addition, all three sources of randomness, WP(t) standard Brownian motion, N(t) Poisson process, and Y the jump size, are assumed to be independent.

Changes in the housing price have three components: (1) the expected instantaneous housing price change conditional on no abnormal events; (2) the unanticipated instan-taneous housing price change, which is the reflection of causes that have a marginal impact on the gauge; and (3) the instantaneous change due to an abnormal shock event. The housing price H(t) follows a geometric Brownian motion during time period (0, t] given that no information of abnormal event arrivals during the time period. When information on an abnormal event arrives at time t, the housing price changes instantaneously from H(t−) to exp(Yn) H(t−).

Note that accurately predicting the unconditional probability of borrower default is not our purpose, and so we assume that the unconditional probability of borrower default at time t∈ T is exogenously determined and is set equal to P(t).

MORTGAGEINSURANCE VALUATION

For the pricing methods for mortgage insurance contracts in the context of the liter-ature, Kau et al. (1992, 1995), Kau, Keenan, and Muller (1993), and Kau and Keenan (1995, 1999) use arbitrage principles of option pricing theory to rationally price a mortgage. The value of the mortgage can be described as the solution to a partial differential equation in backward time, whose terminal and boundary conditions embody the terms of the contract. Dennis, Kuo, and Yang (1997) propose the actuarial 3_{This process is a special case of the Levy process (see Ballotta, 2005).}

pricing method, in which a feasible premium structure is defined as one such that the present value of the expected loss (plus a gross margin) for the insurer is equal to that of the expected premium revenues. Bardhan et al. (2006) assume that the agents in the economy are risk neutral. In this case, the present value of the severity of loss would involve the expectation with respect to the risk neutral probability measure. Hence, they develop an option-pricing framework to price mortgage insurance contracts under the risk-neutral probability measure.

To compute a mortgage insurance contract, we also assume that financial markets are frictionless. There are no transaction costs or differential taxes, trading takes place continuously in time, borrowing and short selling are allowed without restriction and with full proceeds available, and borrowing and lending rates are equal. Furthermore, when the housing price process has jumps, the market becomes incomplete, and then there is no unique pricing measure. Because most shock events causing the housing price to jump, such as the subprime mortgage crisis, are systematic risks, the jump component of housing price represents both systematic and nondiversifiable risk. We make use of the Esscher transform measure developed by Gerber and Shiu (1994) to define the Radon-Nikodym processη(t) as follows:

η(t) = exp ⎛ ⎝−σH2 2 h 2_{t+ hσ} HWP(t)+ h N(t)  n=0 Yn− t R(e hy_{− 1) f (dy)} ⎞ ⎠ , (4)

whereη(t) is called the Esscher transform of parameter h. Hence, it is possible to select the risk-neutral Esscher measure as the measure Phso that housing prices discounted at the risk-free rate are Ph-martingales. Under the risk neutral probability measure Ph, the dynamic process of housing price becomes:

H(t)= H(0) exp ⎧ ⎨ ⎩ r−σ 2 H 2 − R ehy(ey− 1) f (dy)  t+ σHWh(t)+ N(t)  n=0 Yn ⎫ ⎬ ⎭. (5) Under Equations (2) and (5), the present value of the severity of loss, DL(t), can be given by the following expression:

DL(t)= e−rtEQ[L OSS(t)]

= e−rtEQ[max(K1− H(t), 0)] − e−rtEQ[max(K2− H(t), 0)] ,

(6)

where K1= L(t−), K2= (1 − LR)L(t−).

When a borrower’s default occurs at time t∈ T, Equation (6) implies that the present value of the severity of loss, DL(t), can be duplicated by a long position in a European put option with a strike price K1and a short position in a European put option with

Based on the Esscher measure, the expectation in (6) can be represented, for all t∈ T, as follows:4 DL(t)= ∞  m=0 exp(−λμh+1t) (λμh+1t)m m! [P(K1)− P(K2)] , (7) where P(Ki)= Kiexp(−rm;ht)(−d2m(Ki))− H(0)(−d1m(Ki)), i = 1, 2, d_1m,2m(Ki)= ln(H(0)/Ki)+ rm;h±σ 2 m 2  t  σ2 mt , rm;h=r −λ(μh+1− μh)+m t ln  μh+1 μh  , σ2 m= σH2 +mδ 2 t μh= exp  hϕ +1 2h 2_δ2  , μh+1= exp  (h+ 1)ϕ +1 2(1+ h) 2_δ2  .

Hence, volatility of the housing price processσ_m2t includes two components: normal

volatility of the Brownian componentσ_H2 t during time t and abnormal volatility of

jump size mδ2when an abnormal event occurs m times. Ifλ = 0, this means that no abnormal shock event occurs, and so the volatility of housing price process captures only the normal volatility, and so Equation (7) can be reduced to the standard closed-form closed-formula of Bardhan et al. (2006).

Since the housing price is independent of the unconditional probability of borrower default, the fair price (FP) of a mortgage insurance contract with jump risk is given as follows: FP= T  t=1 P(t)DL(t). (8)

Finally, across the mortgage life, the insurers are expected to earn a profit even though they may lose money in some periods. Thus we assume that the gross profit margin that the insurer requires is equal to q , and the mortgage insurance premium (FPA) with jump risk is given by the expression:

FPA= (1 + q)FP. (9)

Equation (8) implies that the fair price (FP) is calculated by the summation of a series of the loss amount of the insurer if the borrower defaults in each month from inception to expiration. Hence, the insurer can judge in each month the probability that the borrower will default rather than at only maturity.

4_{The detailed proof of a European put option using the Esscher transform technique can be}

∂2_{Q(θ | θ} n) ∂ϕ∂δ2 = ∞  mt=0 T  t=1  −(λ + mt) mt(Rt− μ − λ ϕ − mtϕ)  σ2 H+ mtδ2 2  P(Nt= mt| ˜R, θn), ∂2_{Q(θ | θ} n) ∂δ4 = ∞  mt=0 T  t=1 ⎡ ⎢ ⎢ ⎣ m2_t  1 2  σ2 H+ mtδ2  − (Rt− μ − mtϕ)2   σ2 H+ mtδ2 3 ⎤ ⎥ ⎥ ⎦P(Nt = mt| ˜R, θn), ∂2_{Q(θ | θ} n) ∂ϕ∂λ = 0, ∂2_{Q(θ | θ} n) ∂δ2_∂λ = 0, ∂2_{Q(θ | θ} n) ∂λ2 = ∞  mt=0 T  t=1 " −mt λ2 # P(Nt = mt| ˜R, θn). REFERENCES

Abraham, J. M., and P. H. Hendershott, 1996, Bubbles in Metropolitan Housing Mar-kets, Journal of Housing Research, 7(2): 191-207.

Andersen, T., F. X. Diebold, and T. Bollerslev, 2007, Roughing It Up: Including Jump Components in the Measurement, Modeling, and Forecasting of Return Volatility,

Review of Economics and Statistics, 89: 701-720.

Ballotta, L., 2005, A L’evy Process-Based Framework for the Fair Valuation of Partici-pating Life Insurance Contracts, Insurance: Mathematics and Economics, 37(2): 173-196. Bardhan, A., R. Karapandˇza, and B. Uroˇsevi´c, 2006, Valuing Mortgage Insurance Con-tracts in Emerging Market Economies, Journal of Real Estate Finance and Economics, 32: 9-20.

Barndor-Nielsen, O., and N. Shephard, 2006, Econometrics of Testing for Jumps in Financial Economics Using Bipower Variation, Journal of Financial Econometrics, 4: 1-30.

Bin, O., J. B. Kruse, and C. E. Landry, 2008, Flood Hazards, Insurance Rates, and Amenities: Evidence From the Coastal Housing Market, Journal of Risk and Insurance, 75: 63-82.

Borio, C., and P. Mcguire, 2004, Twin Peaks in Equity and Housing Prices? BIS

Quar-terly Review, (March): 79-93.

Canner, G. B., and W. Passmore, 1994, Private Mortgage Insurance, Federal Reserve

Bulletin, 9: 883-899.

Campbell, T. S., and K. Dietrich, 1983, The Determinants of Default on Insured Con-ventional Residential Mortgage Loans, Journal of Finance, 38: 1569-1581.

Case, K. E., and R. J. Shiller, 1989, The Efficiency of the Market for Single-Family Homes, American Economic Review, (March): 125-137.

Caselli, S., S. Gatti, and F. Querci, 2008, The Sensitivity of the Loss Given Default Rate to Systematic Risk: New Empirical Evidence on Bank Loans, Journal of Financial

Chernov, M., A. R. Gallant, E. Ghysels, and G. Tauchen, 2003, Alternative Models for Stock Price Dynamics, Journal of Econometrics, 116: 225-257.

Dempster, A. P., N. M. Laird, and D. B. Rubin, 1977, Maximum Likelihood From Incomplete Data via the EM Algorithm, Journal of the Royal Statistical Society: Series

B, 39: 1-38.

Deng, Y., J. M. Quigley, and R. Van Order, 2000, Mortgage Terminations, Heterogene-ity and the Exercise of Mortgage Options, Econometrica, 68: 275-307.

Dennis, B., C. Kuo, and T. Yang, 1997, Rationales of Mortgage Insurance Premium Structures, Journal of Real Estate Research, 14(3): 359-378.

Durham, G. B., and A. R. Gallant, 2002, Numerical Techniques for Maximum Like-lihood Estimation of Continuous-Time Diffusion Processes, Journal of Business and

Economic Statistics, 20: 297-316.

Englund, P., and Y. M. Ioannides, 1997, House Price Dynamics: An International Empirical Perspective, Journal of Housing Economics, 6(2): 119-136.

Eraker, B., M. Johannes, and N. G. Polson, 2003, The Impact of Jumps in Returns and Volatility, Journal of Finance, 53: 1269-1300.

Gerber, H. U., and E. S. W. Shiu, 1994, Option Pricing by Esscher Transforms (With Discussion), Transactions of the Society of Actuaries, 46: 99-140 (Discussion, 141-191).

Harris, J. C., 1989, The Effect of Real Rates of Interest on Housing Prices, Journal of

Real Estate Finance and Economics, 2(1): 47-60.

Hendershott, P., and R. Van Order, 1987, Pricing Mortgages: Interpretation of the Models and Results, Journal of Financial Services Research, 1(1): 19-55.

Huang, X., and G. Tauchen, 2005, The Relative Contribution of Jumps to Total Price Variance, Journal of Financial Econometrics, 3(4): 456-499.

Kau, J., and D. Keenan, 1995, An Overview of the Option-Theoretic Pricing of Mort-gages, Journal of Housing Research, 6(2): 217-244.

Kau, J., and D. Keenan, 1996, An Option-Theortic Model of Catastrophes Applied to Mortgage Insurance, Journal of Risk and Insurance, 63(4): 639-656.

Kau, J., and D. Keenan, 1999, Catastrophic Default and Credit Risk for Lending Institutions, Journal of Financial Services Research, 15(2): 87-102.

Kau, J., D. Keenan, and W. Muller, 1993, An Option-Based Pricing Model of Private Mortgage Insurance, Journal of Risk and Insurance, 60(2): 288-299.

Kau, J., D. Keenan, W. Muller, and J. Epperson, 1992, A Generalized Valuation Model for Fixed-Rate Residential Mortgages, Journal of Money, Credit and Banking, 24: 280-299.

Kau, J. B., D. C. Keenan, W. J. Muller, and J. E. Epperson, 1995, The Valuation at Origination of Fixed-Rate Mortgages With Default and Prepayment, Journal of Real

Estate Finance and Economics, 11: 3-36.

Kostas, T., and H. Zhu, 2004, What Drives Housing Price Dynamics: Cross-Country Evidence, BIS Quarterly Review, (March): 65-78.

Kuo, C. L., 1996, Housing Price Dynamics and the Valuation of Mortgage Default Options, Journal of Housing Economics, 5: 18-40.

Kvarnstrom, K. M., 2005, Estimation of the Diffusion Coefficient in a Mixture Model,

Kwantitatieve Methoden, 72: 1-23.

Lambrecht, B., W. Perraudin, and S. Satchell, 2003, Mortgage Default and Possession Under Recourse: A Competing Hazards Approach, Journal of Money, Credit, and

Banking, 35(3): 425-442.

Lange, K., 1995, A Gradient Algorithm Locally Equivalent to the EM Algorithm,

Journal of the Royal Statistical Society: Series B. Methodological, 57(2): 425-437.

Mizrach, B., 2008, Jump and Co-Jump Risk in Subprime Home Equity Derivatives, Working Paper, Department of Economics, Rutgers University.

Nityasuddhi, D., and D. Bohning, 2003, Asymptotic Properties of the EM Algorithm Estimate for Normal Mixture Models With Component Specific Variances,

Compu-tational Statistics and Data Analysis, 41: 591-601.

Pan, J., 2002, The Jump-Risk Premia Implicit in Options: Evidence From an Integrated Time-Series Study, Journal of Financial Economics, 63: 3-50.

Quigley, J. M., and R. Van Order, 1990, Efficiency in the Mortgage Market: The Bor-rower’s Perspective, Journal of the AREUEA, 18: 237-252.

Quigley, J. M., and R. Van Order, 1995, Explicit Tests of Contingent Claims Models of Mortgage Default, The Journal of Real Estate Finance and Economics, 11: 99-117. Schloemer, E., W. Li, K. Ernst, and K. Keist, 2006, Losing Ground: Foreclosures in

the Subprime Markets and Their Cost to Homeowners, Working Paper, Center for Responsible Lending.

Schwartz, E. S., and W. N. Torous, 1989, Prepayment and the Valuation of Mortgage-Backed Securities, Journal of Finance, 44: 375-392.

Schwartz, E. S., and W. N. Torous, 1993, Mortgage Prepayment and Default Decisions: A Poisson Regression Approach, Journal of the AREUEA, 21: 431-449.

Sutton, G. D., 2002, Explaining Changes in House Prices, BIS Quarterly Review, (September): 46-55.

Sutton, G. D., 1995, Price Inertia and the Valuation of Real Estate Options, Unpub-lished PhD Dissertation, Yale University.

Tanner, M. A., 1996, Tools for Statistical Inference (Heidelberg: Springer-Verlag). Zeng, D., and D. Cai, 2005, Asymptotic Results for Maximum Likelihood Estimators

in Joint Analysis of Repeated Measurements and Survival Time, Annals of Statistics, 33: 2132-2163.