Walter J. Muller

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.

Pricing Default Risk in Mortgages

This paper examines the valuation of fixed-rate mortgages and the pricing of insurance against default on such mortgages. Both the mortgage and the insurance are treated as compound European put options. A put is the right, but not the obligation, to turn over an asset to another party for a specified payment, and being a European put indicates that this can only occur at a specified expiration date. The mortgage contract, and hence the insurance on it, fit into a European option framework because no rational borrower would ever choose to default until a payment is due. Mortgages are compound options in nature because at each payment data prior to the last one, the borrower either defaults or purchases a new option to default at the next payment date by making the scheduled payment.Since the current value of the mortgage is affected by options to default in the future, the problem is solved working backwards in time with the value of later options feeding into the earlier ones, so that the process builds on itself in a recursive fashion. Using familiar arguments from option-pricing theory, the value of any of the assets in the model is expressed as the solution to a partial differential equation, where the terms of the contract yield the appropriate terminal conditions. Standard numerical procedures are then used to produce the value of the mortgage and the insurance under various economic conditions.The simulations indicate that the prime determinants of the value of the assets considered are the volatility of the house price and the volatility of the spot interest rate. Sensitivity tests show that changing either of these parameters affects the results substantially more than any of the other parameters examined.The paper completely analyzes the default option and insurance against default on the mortgage. It is one part of a complete model of fixed-rate mortgages that would allow for both prepayment and default and treat the interaction of the two options. The general approach outlined in this paper can be used to develop such a model as well as to value any mortgage-related security. In light of the increasing variety and the complexity of such instruments in the market today, the presentation of our approach to these valuation problems is perhaps the most important contribution of the paper. Copyright American Real Estate and Urban Economics Association.

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 valuation and securitization of commercial and multifamily mortgages

Abstract This paper develops a pricing model capable of accurately valuing commercial mortgages and their mortgage backed securities (MBS). It is the first example in the general literature on mortgage pricing in which the valuation of an MBS is explicitly tied to that of the underlying mortgages, making possible a comparison of the performance of the loans and the securities they back. We have shown that while there are similarities between mortgages and their mortgage backed securities, they act in different ways. In general, it turns out that despite being the more passive asset, the mortgage backed security exhibits the more complicated behavior.

Determinacy of equilibrium in stationary economies with both finite and infinite lived consumers

Abstract We consider the determinacy of perfect foresight equilibrium near steady-state equilibria of stationary infinite-horizon economies. The work of T. J. Kehoe and D. K. Levine (Econometrica 53 (1985) , 433–453) is generalized by the inclusion of both infinite lived agents and overlapping generations of finite lived agents, and the inclusion of production and non-depreciating pure rents. We show that indeterminacy is possible in robust examples, and that the dimension of the indeterminacy can be as large as the number of goods per period. We also discuss extreme cases in which determinacy is guaranteed.

An Option-Based Pricing Model of Private Mortgage Insurance

This study uses option-pricing techniques to determine the impact of changes in the mortgage contract or in the economic environment on mortgage insurance values. We found that actual variations in insurance prices for changes in the loan-to-value ratio are substantially less than those called for in theory. This article demonstrates that option-based pricing models can play a useful role in providing firms with estimated insurance prices that reflect the economic environment and the terms of the mortgage contract.

Rational Pricing of Adjustable Rate Mortgages

In this paper we develop a general method for valuing adjustable rate mortgages and by producing a set of simulation results, we show that our approach can be implemented. While the simulations are of interest in their own right, we view the approach itself as the major contribution of the paper. Copyright American Real Estate and Urban Economics Association.

A note on bias resulting from imposing expedient conditions on mortgage valuation models

Abstract This study seeks to understand how mortage valuation behaves under the alternative simplifications of nonamortization, continual interest payments, continuously exercisable default, and a prepayment option that may be exercised only periodically. The goal is to compare the effect of these differing simplifications against exact results for the standard mortgage offered in the market place. The comparison provides researchers with evidence in support of making such expedient choices in modeling. Except for amortization, there is little loss in precision from choosing the less contractually accurate mortgage representations.