Computing the endogenous mortgage rate without iterations

A number of mortgage prepayment models require a specification of the mortgage rate process. Usually, ad-hoc models are used (e.g., a Treasury yield plus some constant). Recently, a number of papers have appeared where the authors have utilized a mortgage rate implied by the current yield curve (the so-called endogenous mortgage rate). However, the existing computational algorithms suffer from the curse of dimensionality and, consequently, are problematic to use for full-scale problems. A computational algorithm, proposed in this paper, is tractable in the sense that its complexity is equivalent to the problem of mortgage valuation. Moreover, the algorithm does not require iterations. The numerical example is based on a PDE computation. An implementation of a Monte Carlo method is also discussed.     

Evaluating Real Estate Valuation Systems

A framework for comparing real estate valuation systems (including automated valuation models (AVMs) and current appraisal methods) is proposed. The density estimation and profit simulation (DEPS) method measures quality of a valuation system by simulating benefits to the mortgage lender who uses this method in mortgage underwriting to limit mortgage portfolio losses due to default. Related simple measures relevant to the selection of a valuation system are also discussed: skewness of the distribution of errors, correlation of valuation errors with current selling price errors, correlation of errors of the valuation system with errors of valuation systems used by competing mortgage lenders, and other measures.     

Mortgage Contracts,Strategic Options and Stochastic Collateral

This paper offers a game-theoretic model for both the analysis and valuation of mortgage contracts in the context of an economy with complete information and complete contingent claims markets. We analyze the equilibrium strategy of the lender, who holds an option over the magnitude of mortgage credit extended per dollar of collateral offered, and the mortgagor, who holds options to default or prepay, in a class of intertemporal mortgage contracts collateralized by property evolving according to a random process which is common knowledge to both parties to the mortgage contract. Using continuous–time arbitrage valuation principles, we derive the value of the mortgage contract to both parties and show, through both analytical solutions and numerical simulations, that Markov perfect equilibria exist in which, among other properties, a lower flow of housing services accruing to the borrower, per dollar of initial house value, and a correspondingly lower rate of effective depreciation, will elicit a larger volume of funds offered by a lender; the amount of credit offered, the values of the contract to both lender and mortgagor, and the expected losses to both parties from costly bankruptcy are highly sensitive to the perceived volatility of the value of the property collateralizing the mortgage, even in an economy with complete markets or risk neutrality on the parts of lender and borrower; the upper limit on mortgage credit offered by a rational lender may be a small fraction of the current fair market value of the property, regardless of the contractual yield offered by the borrower, and will decrease, at each such yield, as bankruptcy costs or housing service flows increase; and under significant but plausible values for bankruptcy and costs of liquidating property under foreclosure, the flow of mortgage credit can become negatively related to the spread of the mortgage yield over the riskless rate, with the lender preferring a lower contractual yield to a higher one.     

A New Prepayment Model (with Default): An Occupation-time Derivative Approach

A new prepayment model is developed, which improves the modeling of the borrowers decision process by incorporating an occupation-time derivative in the valuation framework of a fixed-rate mortgage. This option-theoretic mortgage valuation model is based on stochastic house-price and interest-rate models, and requires a particularly subtle technique to incorporate a new type of occupation-time derivative, where the barrier (which activates the derivative) is in the value process and not the underlying process (as it is in standard occupation-time derivatives). This new model simulates a delay in prepayment by the borrower (beyond the time simple ruthless prepayment dictates), thus increasing the value of the mortgage to the lender, compared to the value gained using more basic models. This allows for a more advanced borrower decision process, where a rational exercise structure is retained in a modified form. Empirical evidence supports this theory, which should be beneficial for accurate mortgage-backed security pricing. The results in this paper explore thoroughly the effect on the mortgage value of a delay in prepayment by the borrower on the embedded options held and on the insurance component.

Valuation of Residential Mortgage-Backed Securities with Default Risk Using an Intensity-Based Approach

This paper develops a valuation model for fixed-rate mortgages, mortgage pools, and residential mortgage-backed securities (RMBS's) using an intensity-based approach. This model incorporates full prepayment, partial prepayment, and default in valuing a mortgage. Full prepayment is further classified into “refinancing” and “sale of a house” depending on the reason. The time of occurrence of each of these three types of prepayment and default is modeled as the first jump time of a Cox process. Under these conditions, the valuation formula for a mortgage as well as a partial differential equation (PDE) that the mortgage value satisfies is provided. As for implementation of the model, the short-term riskless interest rate and the house price are adopted as state variables. Each intensity process is specified in a manner that allows a jump in intensity depending on the state variables and the borrower's incentive for prepayment or default. Through such specifications, it is shown that our model has characteristics similar to some structural models in previous literature. As for the numerical method for valuation, we propose a simple backward induction technique on a tree instead of the commonly used Monte Carlo method. Additionally, the method for estimating the model is discussed, and the results of numerical simulations are reported.This paper represents the view of the author and does note necessarily the views of the Mitsubishi UFJ Securities Co., Ltd. or members of its staff.     

Strategic default,workout, and commercial mortgage valuation

This paper extends existing equilibrium commercial mortgage pricing models by endogenizing negotiated workout into the usual noncooperative lending game. Workout is a feasible subgame strategy for the lender to play whenever foreclosure transaction costs exist for either party to a loan transaction. In particular, negotiated workout solutions Pareto dominate the foreclosure alternative when default occurs. To obtain our results, we embed a cooperative bargaining game within a noncooperative mortgage loan/default game. We also address the valuation wedge problem that occurs when foreclosure transaction costs are introduced. Through the notion of replacement game equilibrium, we find symmetric mortgage pricing solutions that eliminate the valuation wedge and thus suggest that lending will occur in commercial real estate mortgage markets even when foreclosure transaction costs exist.     

Pricing commercial mortgages and their mortgage-backed securities

This article has taken considerable effort to accurately model the complexity of a commercial mortgage and its mortgage-backed security. In fact, it is the first example in the general literature on mortgage pricing to present a comprehensive set of numerical results in which the valuation of a mortgage-backed security is explicitly tied to that of the underlying mortgage. The conclusion we reach is that option pricing provides an accurate and flexible approach to valuing the complex mortgage instruments now being developed in the financial community.     

The financial reporting of fair value based on managerial inputs versus market inputs: evidence from mortgage servicing rights

This research examines whether the fair value of mortgage servicing rights (MSRs) based on managerial inputs (Level 3) better reflects the cash flow and risk characteristics of the underlying assets than the fair value of MSRs based on market inputs (Level 2). Using mortgage servicing fees as a proxy for the underlying cash flows, we find that the valuation multiples for MSRs based on Level 3 inputs are more positively associated with the persistence of future servicing fees compared with the fair value of MSRs based on Level 2 inputs. We also document that only the valuation multiples based on Level 3 fair values are negatively associated with proxies for risk factors. Our results suggest that, although unobservable inputs are subject to managerial discretions, managers can generate higher quality fair value estimates than market inputs due to their information advantage, especially when the market for the underlying asset is inactive.     

The valuation at origination of fixed-rate mortgages with default and prepayment

This paper develops a model to rationally price fixed-rate mortgages, using the arbitrage principles of option pricing theory. The paper incorporates amortization, prepayment and default in valuing the mortgage. Having completely specified the model, numerical procedures value the different features of the mortgage contract under a variety of economic conditions. The necessity of having both the interest rate and the house price as explanatory variables, due to the interaction of default and prepayment, is demonstrated. The numerical solutions presented center around mortgage pricing at origination. Thus, variations in the equilibrium contract rate are examined for differing economic conditions and changes in the contract. Finally, by presenting a complete model, the paper yields insights for the existence of common institutional practices.     

An Empirical Investigation of Alternative Contingent Claims Models for Pricing Residential Mortgages

Researchers have employed option pricing techniques to analyze mortgage financing and valuation. Alternative models (one-, two-, and three-variable models) employing different variables (short- and long-term interest rates and building value) have been designed to price mortgage securities. No prior research has addressed the question of whether the pricing accuracy of these contingent claims models improves as states increase or whether contingent claims models' valuation abilities generate reasonable estimates of primary mortgage market prices. The articles investigates the relative efficiency of each of these alternative mortgage valuation models in predicting primary market mortgage values. Our results show that a two-variable model (short rate and building value) is the most efficient. Valuation results indicate a positive pricing spread between the primary market and the theoretically estimated value.     

Unobservable heterogeneity and rational learning: Pool-specific versus generic mortgage-backed security prices

Previous mortgage prepayment and valuation models assume that two mortgage pools with the same observable characteristics should behave indistinguishably. However, even pools with apparently identical characteristics often exhibit markedly different prepayment behavior. The sources of this heterogeneity may be unobservable, but we can infer information about a pool from its prepayment behavior over time. This article develops a methodology for using this information to calculate pool-specific mortgage-backed security prices. Knowledge of these prices is important both for portfolio valuation and for determining the cheapest pool to deliver when selling mortgagebacked securities. We find that unobservable heterogeneity between mortgage pools is statistically significant, and that pool-specific prices, calculated for a sample of outwardly identical mortgage pools between 1983 and 1989, may differ greatly from any single representative price.     

Optimal Mortgage Contract Choice Decision in the Presence of Pay Option Adjustable Rate Mortgage and the Balloon Mortgage

The unprecedented run-up in global house prices of the 2000s was preceded by a revolution in U.S. mortgage markets in which borrowers faced a plethora of mortgages to choose from collectively known as nontraditional mortgages (NTMs), whose poor performance helped ignite the global financial crisis in 2007. This paper studies the choice of mortgage contracts in an expanded framework where the menu of contracts includes the pay option adjustable rate mortgage (PO-ARM), and the balloon mortgage (BM), alongside the traditional long horizon fixed rate mortgage (FRM) and the short horizon regular ARM. The inclusion of the PO-ARM is based on the fact it is the most controversial and perhaps the riskiest of the NTMs, whereas the BM has not been analyzed in the literature despite its different risk-sharing arrangement and long vintage. Our inclusive model relates the structural differences of these contracts to the horizon risk management problems and affordability constraints faced by the households that differ in terms of expected mobility. The numerical solutions of the model generates a number of interesting results suggesting that households select mortgage contracts to match their horizon, manage horizon risk and mitigate liquidity or affordability constraints they face. From a risk management and welfare perspectives, we find that the optimal contract for households with shorter horizons, specifically households who expect to move house once every 1 to 2 years, is the PO-ARM. Beyond 2 years the welfare advantage of the PO-ARM diminishes and BM becomes the more optimal contract up to 5-year horizon. Overall, the results suggest that households are neither as risk averse as the selection of the FRM would suggest, nor are they as risk-seeking as the selection of PO-ARM or regular ARM would suggest. The results also suggest that the exuberance demonstrated for NTMs by borrowers, especially PO-ARMs, may be both rational and irrational.     

Valuing Mortgage Insurance Contracts in Emerging Market Economies

We develop a new option-based method for the valuation of mortgage insurance contracts in closed form in an economy where agents are risk neutral. While the proposed valuation method is general and can be used in any market, it may be particularly useful in emerging market economies where other existing methods may be either inappropriate or are too difficult to implement because of the lack of relevant data. As an application, we price a typical Serbian government-backed mortgage insurance contract.     

Geometric Asian option pricing in general affine stochastic volatility models with jumps

In this paper, we present some results on Geometric Asian option valuation for affine stochastic volatility models with jumps. We shall provide a general framework into which several different valuation problems based on some average process can be cast, and we shall obtain closed form solutions for some relevant affine model classes.     

An exact and explicit solution for the valuation of American put options

In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, is the optimal exercise boundary being elegantly and temporarily removed in the solution process of each order, and, consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical and exact series-expansion solution for the optimal exercise boundary and the option price of American put options.     

Interest Rate Derivatives in a Duffie and Kan Model with Stochastic Volatility: An Arrow-Debreu Pricing Approach

Simple analytical pricing formulae have been derived, by different authors and for several derivatives, under the Gaussian Langetieg (1980) model. The purpose of this paper is to use such exact Gaussian solutions in order to obtain approximate analytical pricing formulas under the most general stochastic volatility specification of the Duffie and Kan (1996) model. Using Gaussian Arrow-Debreu state prices, first order stochastic volatility approximate pricing solutions will be derived only involving one integral with respect to the time-to-maturity of the contingent claim under valuation. Such approximations will be shown to be much faster than the existing exact numerical solutions, as well as accurate.     

Anatomy of an ARM: The Interest-Rate Risk of Adjustable-Rate Mortgages

This article analyzes the dynamics of the commonly used indices for adjustable rate mortgages and systematically compares the effects of their time-series properties on the interest-rate sensitivity of adjustable-rate mortgages. Our ARM valuation methodology allows us simultaneously to capture the effects of index dynamics, discrete coupon adjustment, mortgage prepayment, and both lifetime and periodic caps and floors. We can, moreover, either calculate an optimal prepayment strategy for mortgage holders or use an empirical prepayment function. We find that the different dynamics of the major ARM indices lead to significant variation in the interest-rate sensitivities of loans based on different indices. We also find that changing assumptions about contract features, such as loan caps and coupon reset frequency, has a significant, and in some cases unexpected, impact on our results.     

Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results

Stochastic volatility (SV) and local stochastic volatility (LSV) processes can be used to model the evolution of various financial variables such as FX rates, stock prices and so on. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes. Many issues remain, though, including the efficacy of the standard alternating direction implicit (ADI) numerical methods for solving SV and LSV pricing problems. In general, the amount of required computations for these methods is very substantial. In this paper, we address some of these issues and propose a viable alternative to the standard ADI methods based on Galerkin-Ritz ideas. We also discuss various approaches to solving the corresponding pricing problems in a semi-analytical fashion. We use the fact that in the zero correlation case some of the pricing problems can be solved analytically, and develop a closed-form series expansion in powers of correlation. We perform a thorough benchmarking of various numerical solutions by using analytical and semi-analytical solutions derived in the paper.     

Macroeconomic News and Mortgage Rates

The study analyzes the influence of macroeconomic news announcements on (a) interest rates for commercial mortgages, residential mortgages, 10-year Treasury notes, and Baa-rated corporate bonds; and (b) corresponding mortgage spreads. It is both interesting and highly relevant from a policy and portfolio management standpoint to examine the implications of the influence of macroeconomic news announcements on mortgage markets. Some important results are reported. First, consistent with the notion of market integration, mortgage rates are found to be co-integrated with other capital market instruments. Second, of the 22 types of periodic macroeconomic news releases considered, 13 of them have a significant influence on at least one of the interest rates, and notably changes in hourly earnings and housing starts significantly influence all debt-security yields. More generally, macroeconomic news that conveys higher inflation and/or economic growth has a positive influence on mortgage and other interest rates. Finally, this study finds several announcements including durable goods orders, new home sales, personal consumption, non-farm payroll, trade balance and Treasury budget to have a significant influence on mortgage spreads.     

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

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.