A contingent claim analysis of preferential rate mortgages

2.3 Prepayment, Default and Mortgage Insurance Options The early mortgage OPM started off with a one-state variable model, either modeling a default-free mortgage or a non-prepayable mo

CHAPTER ONE:

1 INTRODUCTION

In a country where citizens own more real estate assets than stocks, the valuation of Singapore real estate assets and its related financing instruments becomes a critical research agenda for the country A government study (Singapore Department of Statistics, 2003) of Singapore household wealth and liabilities as of end of 2001 reveals that 48.2 percent of the households’ assets are in property and 72 percent of the liabilities are in mortgage loans The key concern here, as a result of this survey, is that any change

in policies or economic shocks in the real estate market will have deep and wide ranging effects on the general economy since households are highly exposed to the real estate sector

A recent change of housing and financing policies (Economic Review Committee, 2002)

in Singapore, as part of the economic restructuring programme, had sparked off a wave of loan wars in the mortgage lending sector In view of the variety and complexity of mortgages available on the market, there is considerable interest in the valuation and analysis of these financing instruments Preferential rate mortgages1 (PRMs), being the most prevalent type of mortgage in Singapore, lacks such a framework for the benefit of the mortgagor and mortgagee to facilitate comparison and pricing In addition, as a result

1 Preferential Rate Mortgages (PRMs) are adjustable rate mortgages (ARMs) that provide below-market interest rate over a specified promotional term of three to five years PRMs are quite similar in structure to US market’s ARM with teaser rates Their main difference lies in the fact the teaser rates exist only for the first year whereas the promotional

of the economic restructuring, the priority of charges on mortgages has been reversed which in turn boosts the incentives to develop a secondary market for mortgages In the light of such dynamic changes, we develop a mortgage theoretic option pricing model to accommodate the pricing, comparison and analysis of PRMs We believe that the development of such a valuation model will also contribute to the development of mortgage-backed securities in Singapore

1.1 Background

Economic Restructuring in Singapore

In Dec 2001, the government had initiated an economic review committee (ERC) against the backdrop of a significantly changed regional and world environment to fundamentally review Singapore’s development strategy, and formulate strategies to upgrade, transform and revitalize the economy (Economic Review Committee, 2002) Seven Sub-Committees and numerous working groups were formed to study issues such as taxation, wages, the Central Provident Fund (CPF) and land; promoting entrepreneurship and internationalization of Singapore companies; upgrading and growing the manufacturing sector; developing the services sector; growing domestic enterprises; developing our human capital; and helping Singaporeans to respond to changes and take advantage of new opportunities The key concerns for our study, arising from the restructuring initiatives, are the changes to the real estate financing policies These include the imposition of a CPF valuation withdrawal limit on bank mortgages, transfer of future

public housing market rate loans from the public housing authority to commercial mortgage lenders, changes to the CPF charge on a private property which is mortgaged, and changes to the cash downpayment requirement on mortgages

Mortgage Loan War

As part of the economic restructuring initiatives, the government had slowly devolved some non-core Housing Development Board (HDB) functions to the private sector which includes the transfer of future public housing market rate loans to commercial mortgage lenders This means that from 1st Jan 2003, the housing board will no longer grant loans

at market interest rate to buyers of new and resale HDB flats and these buyers must take

up their mortgage with an approved bank or finance company In addition, current HDB flat owners can choose to refinance their loans with these banks and finance companies

The move had opened up a previously untapped segment of the market to the private sector that is worth $3.2 billion As anticipated, the move had resulted in an eruption of mortgage loan war in Dec 2002 which involves not only banks, but also finance companies and even insurance companies The ferocity in the pursuit for market share in the housing loan market prompted lenders to aggressively slash their first and second year mortgage rates down to as low as 1.5% In addition, lenders are also coming up with various novel and competitive PRM packages to beat the competition In view of the variety and complexity of PRMs available on the market, potential borrowers are often perplexed by the choice of mortgage package Thus, we believe that a platform for assessment and comparison of PRMs is essential not only for the benefit of mortgagees

but also mortgagors This study attempts to fulfill this need by providing a valuation framework for PRMs

Securitization Potential of Mortgages in Singapore

A recent Singapore Department of Statistics survey reveals that the amount of outstanding mortgage held by households as of end of 2001, comprising of both HDB and private sector mortgages, totaled $105.79 billion (Singapore Department of Statistics, 2003) These mortgages take the form of either adjustable-rate mortgages (ARMs) or preferential rate mortgages (PRMs) The Monetary Authority of Singapore’s (MAS) recommendations in 1998 to deepen the debt markets have led to increased interest in the securitization of this mortgage pool However the lack of institutional and regulatory framework for MBS and the complexity of the mortgage instrument have slowed down efforts to develop the mortgage-backed securities Against the backdrop of a high liquidity and low interest rate environment, securitization efforts had been drawn into a standstill situation

One of the major impediments for securitization is that banks do not assume priority charge over the mortgaged property ahead of other institutions In the case of foreclosure, banks having the second charge may not be able to recover the full loan amount as the Central Provident Fund, a statutory board in charge of compulsory savings for a retirement fund in Singapore, had the first charge over the foreclosed proceeds This has deterred securitization efforts as MBS investors would not want to assume such

foreclosure risk in the event of default (Nang et al, 2003) A recent change in policies

resulting in the reversal of the charge position had renewed interest in securitization of mortgages As real estate securitization in Singapore is still in its infancy, the potential for the development of a mortgage secondary market is now immense Therefore, we believe the development of a mortgage valuation framework that thoroughly analyzes the intricate features of PRM could increase the pace of securitization and contribute toward

a secondary market for PRMs

1.2 Justification of Research

PRMs may soon dominate the mortgage portfolio in Singapore

Mortgages in Singapore take the form of either PRMs or ARMs in which most ARMs are provided by the public housing authority With the cessation of HDB granting market rate loans, future buyers of new and resale HDB flats who are not eligible for concessionary loans will have to take up PRMs with the private sector In the current low interest rate environment, many current HDB loan mortgagors are also refinancing their mortgages with private financing institutions In the light of such changes, PRMs’ share of the market will continue to expand and may soon become the dominant type of mortgage in the market Policymakers, mortgage lenders, borrowers and potential mortgage investors would hence be interested in understanding the pricing, risks and returns of these mortgages Yet, no robust valuation and analytical framework exist for PRMs This provides the impetus to develop such a framework

Valuation of PRMs is necessary for the structuring of mortgage-backed securities

The idea of real estate securitization in Singapore was mooted by the MAS in 1998, as part of a plan to develop the domestic capital markets However the development of mortgage securitization is slow due to the high liquidity and low interest environment as well as the lack of regulatory and institutional framework

With the reversal of the charge position on mortgages, the incentives for lenders to securitize their mortgages to raise liquidity have consequently increased Prepayment and default risks are key concerns in securitized mortgages and the financial community would have a strong interest in understanding such risks before the structuring of mortgage-backed securities In this study, we foresee the need of an analytical framework which provides a comprehensive understanding of the default and prepayment risk in mortgages

Inability of current valuation models to handle the complexity of PRMs

The traditional method of valuing assets is to do a net present value (NPV) calculation of the asset’s cashflows The net present value, however, is based on some implicit assumptions that are overlooked in the pricing of mortgages The orthodox theory of valuing assets has not recognized the interplay of irreversibility, uncertainty and the choice of timing that will determine the optimal default, prepayment and continuation decisions of mortgagors

The reason is that a borrower with an opportunity to prepay is holding a “prepayment option” analogous to a financial call option-he has the right but not the obligation to buy out future mortgage payments at some future time of its choosing When a mortgagor prepays a mortgage, he makes an irreversible decision and gives up future opportunities

to terminate the mortgage The lost opportunities are opportunity cost that must be part of the cost of prepaying the mortgage The NPV calculations fail to capture such rights and opportunities Similarly, mortgagors also hold a “default option” analogous to a financial put option in which the borrower has the right to sell out the possession of the house in exchange for the abandonment of mortgage payments Thus mortgages are usually modeled as a fixed income security with two embedded options, namely prepayment option and default option However, it is more complicated when preferential rate mortgages come into the scenario given that the mortgage has features of both fixed and variable cashflows In this study, we further develop the theory of mortgage valuation under uncertainty, emphasizing the option-like characteristics of mortgage termination behavior

Increasing need for mortgage lenders to evaluate risk in their mortgage portfolio

Since the Asian Financial Crisis, lenders are increasing aware of the vulnerability of their banking system to external economic forces Using data from MAS, Development Bank

of Singapore (DBS) market research team found that as of May 2004, property related lending accounts for at least 46 percent of the lending by banks in Singapore (Koh,

2004) Given that the real estate collateral and mortgages form a significant portion of the bank’s asset and liabilities, lenders recognize the increasing need to understand, evaluate and benchmark the risk in their mortgage portfolio To do so, as an initial starting point, a structural model is necessary not only to identify the sources of risk but also to generate the entire risk distribution Using option-pricing ideologies, we attempt not only to reveal the sources of risk but also to produce such risk distribution in our model We believe that such information would be of interest to lending institutions, the central bank and the mortgage investing community

1.3 Objectives of Research

1 To create a theoretical option pricing model to price PRMs

2 To provide a framework whereby different PRMs can be evaluated using Option Pricing theory

3 To provide a risk-evaluation framework for PRMs

4 To examine how different economic and mortgage contractual factors affect the mortgage value and risk distribution

1.4 Organization of Thesis

The following chapters of this thesis are organized as follows:

A review of the literature related to theoretical mortgage OPMs follows in the next

chapter Chapter 3 presents the research methodology, viz the subject of study, techniques

used and the various research innovations The simulation results of our model are then presented in Chapter 4 This thesis shall be concluded and recommendations will be made

in the final chapter

CHAPTER TWO

2 LITERATURE REVIEW

This section reviews the theoretical works on the pricing of mortgages as derivative assets, often termed the option-pricing approach to mortgage valuation (Kau and Keenan, 1995) The options approach recognizes the value of the right to prepay or default in a mortgage, following the seminal work on pricing options by Black and Scholes (1973) and Merton (1973) Default can be regarded as a European compound put option where the borrower has the right to turn over the possession of the house in exchange for the abandonment of payments The option is European because such default rationally occurs when a payment is due, and compound because there is a succession of payments over the life of the mortgage Similarly, the right to prepay can be considered an American-style call option, in which the borrower has the right to buy all future obligations remaining under the mortgage at a price equal to the loan’s outstanding balance

Default and prepayment have value in that they both have exercise value and time value The default and prepayment options both have exercise value as the borrower receives a premium payoff when they exercise the relevant option They have time value as the borrower is able to postpone termination of the mortgage by at least one period to see if the termination will be more optimal Note that the two options are mutually competing

in nature- that is if default occurs, the value of prepayment becomes zero; if prepayment occurs, the value of prepayment becomes zero The two options will compete, so that the

value of default is less in the presence of prepayment than in its absence, just as the value

of prepayment is less in the presence of default than in its absence

2.1 Pricing Mortgages as Contingent Claims

Traditional valuation in the world of certainty is straightforward: that is to do a present value calculation In the world of uncertainty, valuation of assets with early termination features becomes challenging as we do not know whether and when termination occurs Black and Scholes (1973), in the breakthrough paper present a valuation methodology for pricing derivative assets with early exercise features in a stochastic environment The beauty of this methodology lies in the case where the derivative asset value can be calculated without any reference to individual risk attitudes and, indeed, without any reference whatever to the mean value of the underlying asset price movements that govern the derivative’s riskiness In addition, the paper provides a form of parsimonious modeling where all the usual forces of demand and supply are present, but are entered into the model via the value of asset itself, from which the value of derivative naturally follows The underlying asset volatility, however, does matter, to dictate the value of the derivative Thus, the greater the volatility of a stock, the greater the value of a call option

Turning to mortgages, the two commonly identified sources of uncertainty are the house price and term structure The most popular choice for modeling term structure is the one-state variable Cox, Ingersoll, and Ross (1985a) mean-reverting process where we assume:

r

r r dz dt

r

Here r represents the spot rate, γ the mean reversion coefficient, θ the trend rate, σr the volatility parameter and dz the Wiener process The term structure is assumed to revert r

towards a trend rate θ (at a rate dictated byγ ) and it always avoids the negative interest rate condition As interest rate is not a directly tradable asset, in order to achieve risk-neutral pricing, we assume either that the Local Expectations Hypothesis holds (an assumption about intertemporal risk attitudes, see Cox, Ingersoll and Ross,1981; 1985b)

or that any such premium has been absorbed into the term structure parameters γ and θ (see Cox, Ingersoll and Ross, 1979) A number of articles that work with this process include Dunn and McConnell (1981a, 1981b), Schwartz and Torous (1992), Titman and Torous (1989), Buist and Yang (1998), and the various articles of Kau et al

Brennan and Schwartz (1985), as well as Schwartz and Torous (1989a, 1989b, 1991), instead work with a two-state term structure process where they consider the combination

of a spot rate and long rate The two state term structure clearly outperforms the single state structure as it provides more degree of freedom in describing the actual term structure However, the distinct disadvantage is that the two state term structure process model requires many more calculation to achieve a solution Buser, Hendershott, and Sanders (1990) compare the one-state and two-state term structure process and found the one state form adequate Litterman and Scheinkman (1991), on the hand, find the one-state interest process to be deficient when estimating the term structure In the presence

of default, the computational work of backward solving a three state-variable partial differential equation is rather technologically prohibitive and as a result, many authors still prefer the single term structure model rather than the two-state term structure In our

study, with the assistance of Monte Carlo simulation procedures, we chose a two-state variable term stochastic process to enhance the characterization of the term structure process

On the other hand, house price is commonly specified as a lognormal stochastic process

σ is the volatility parameter and dz is the Wiener process Papers that work with this H

specification of real estate asset includes Cunningham and Hendershott (1984), Epperson

et al (1985), Kau et al (1987, 1990a, 1990b, 1992, 1993a, 1993b 1993c), Schwartz and

Torous (1992), and Titman and Torous (1989) Here, the expected rate of return to housing comprises of expected capital appreciation and a rental or service flow:

s H

on some other economic factors As a consequence of such criticism, Kim (1991) proposes that an alternative method of formulating the service flow which seems more appropriate to the actual economic character of real estate assets Kim (1991) argues that

the service flow is interest rate dependent and that service flow should be decomposed into the difference between the required rate of return and expected capital gains The latter can be easily estimated on the basis of past experience while the former can be estimated from a consumption-based asset pricing model In our study, we adopted Kim(1991) specification of service flows to avoid the service flow criticism.

With the specification of the underlying sources of risk, we can value a mortgage

pH r H

r

X t

X r r

X s r H H

X

r

Hσσθ

γ

∂

∂

∂ +

∂

∂ +

−

∂

∂ +

X r H

2

12

dz

t

dz r( ) H( )=

Here, the fundamental PDE is common to all derivative assets driven by the two sources

of uncertainty, house price and term structure To solve the PDE, we need to further specify the terminal and boundary conditions The specific nature of the mortgage contract enters here through the specification of terminal and boundary conditions The

PDE can be solved in various ways depending on the type of mortgage contract and the modeler’s assumptions We shall review the various numerical methods to solve the PDE

in the section 2.7

2.2 Development of Mortgage Theoretic Option Pricing Models

The early developments in mortgage theoretic option pricing models started with Findlay and Capozza (1977); Asay (1978) and Dunn and McConnell (1981a, 1981b) Thereafter, research that has utilized option pricing theory to price mortgage securities has taken several directions The resulting models have differed from each other according to the inclusion or omission of diverse state variables (short-and long-term interest rates and house price) and prepayment and default options In this section, we shall survey the various developments in the literature For a review of the option-theoretic mortgage pricing models and discussion of related issues in modeling, see Kau and Keenan (1995)

2.3 Prepayment, Default and Mortgage Insurance Options

The early mortgage OPM started off with a one-state variable model, either modeling a default-free mortgage or a non-prepayable mortgage When default is excluded, the stochastic economic environment is solely described by the term structure A number of papers concentrate on prepayment risk and term structure modeling and ignore default and the related house price process They include Brennan-Schwartz (1985); Buser and Hendershott (1984); Buser, Hendershott, and Sanders (1985, 1990); Dale-Johnson and Langetieg (1986); Dietrich et al (1983); Dunn and McConnell (1981a, 1981b); McConnell and Singh (1993, 1994); Schwartz and Torous (1989a, 1989b, 1991); and Van

Drunen and McConnell (1988) Many of these papers include some forms of suboptimal prepayment, which, in the case of MBS, account for default

Similarly, a number of researchers analyzed a non-prepayable mortgage, concentrating on the modeling of house price and the related default risk Cunningham and Hendershott (1984); Riddiough and Thompson (1993) and Epperson et al (1985) price mortgage default options while ruling out the possibility of prepayment The subsequent expansion

of the literature was dominated by the integration of prepayment and default in a competing risk framework Examples include Foster and Van Order (1984, 1985), Titmann and Torous (1989), Kau et al (1987, 1990a, 1990b, 1992, 1993a, 1995), and Schwartz and Torous (1992)

Besides default and prepayment option, a number of researchers began to recognize the

existence of a third option, mortgage insurance, in the mortgage valuation framework

This additional feature has been considered by Chinloy (1992); and Cunningham and Hendershott (1984) in which they consider mortgage insurance in isolation from default and prepayment option Recent developments in the literature like Kau et al (1990a,

1992, 1993b); Kau, Keenan, and Muller (1993a); Kau, Keenan and Kim (2001); Kau and Keenan (1999); and Schwartz and Torous (1992) began to integrate mortgage insurance

as part of the overall mortgage valuation framework The treatment of mortgage insurance can be done in two ways: as an upfront lump sum payment or as part of the contract rate that determines monthly payments Most researchers adopted the former approach

2.4 The Expansion of State Variables

The initial works of Cunningham and Hendershott (1984); Epperson et al(1985); Dunn and McConnell (1981a, 1981b); Schwartz and Torous (1989a, 1989b) and Dietrich et al (1983) started off with a one-variable model solely capturing either default or prepayment risk in a mortgage

The one-state variable model was extended to a two-state variable model in two different ways Firstly, some researchers included long term interest rate as another variable in the model to price default-free prepayable mortgages Examples include Brennan and Schwartz (1982, 1983, 1985), Buser, Hendershott, and Sanders (1990), Schwartz and Torous (1989a, 1989b, 1991), and McConnell and Singh (1993, 1994) The second way is

to include house price in the model to capture default behavior Examples include Foster and Van Order (1984, 1985), Titmann and Torous (1989), Kau et al (1987, 1990a, 1990b,

1992, 1993b), Kau and Kim (1994), and Schwartz and Torous (1992) Recently, there has been some interest in building three state variable models like Chatterjee, Edmister and Hatfield (1995); Buist and Yang (1998) and Brunson, Kau and Keenan (2001) The three state variable models adopt two different forms in the literature: the first being the combination of long tem interest rate, spot rate and house price (Chatterjee, Edmister and Hatfield, 1995 and Brunson, Kau and Keenan, 2001) while the second is the combination

of income, house price and interest rate (Buist and Yang, 1998) Our study attempts to extend the literature by providing the first four-state variable model which includes income, house price and a two-state term structure process

2.5 Types of Mortgage Contract

The most common type of residential mortgages in US is the fixed rate mortgages which have a fixed contract rate and a fixed monthly payment From an option-theoretic viewpoint, several papers like Buser and Hendershott (1984), Epperson et al (1985), Kau

et al (1992,1995), and Pozdena and Iben (1984) explicitly considerd such fixed rate

mortgages Adjustable-rate mortgages (ARMs), on the other hand, are mortgages that have a contract rate that is adjusted periodically to reflect prevailing interest rates and as

a result, subject borrowers to the variability in payments

Most ARMs in US would have features like lifetime caps, lifetime floors, periodic caps, periodic floors and teaser rates For a comprehensive review of ARM features, see Schwartz and Torous (1991) Theoretical models explicitly analyzing ARMs and other variable rate contracts includes Buser, Hendershott, and Sanders (1985); Cox, Ingersoll, and Ross (1980); Findlay and Capozza (1977); Kau et al (1985, 1990b, 1993b); and Schwartz and Torous (1991) A point worth noting is that the problem of path-dependency is often associated with ARMs Unlike fixed-rate mortgages, ARMs have terms that depend on past interest rates This causes much difficulty for backward pricing methods as such historical information is not available in the pricing method for which the past is not yet known Nonetheless, this problem can be overcome in two ways The first way is to use a forward pricing method and the second way is to include past values

of the current state variables as additional state variable as suggested by Kau et al (1985) This device employed by Kau et al (1985, 1990b) for ARMs exploits the first-degree

homogeneity of a mortgage contract in loan size to reduce the required augmentation of the state space to a single dimension

Other works have also considered less common forms of residential mortgages like graduated payment mortgages (GPMs) and price level adjusted mortgages (PLAMs) From the theoretical viewpoint, Buser and Hendershott (1984) have considered GPMs and Kim (1987) analyzes PLAMs Another less common form of residential mortgages, which is unique to Singapore is the preferential rate mortgages (PRMs) PRMs are essentially ARMs that are served with a promotional rate within a promotional term of three to six years The promotional rate may come in two forms, either as a below-the-market fixed contract rate or as a contract rate that is pegged at a fixed discount below the prevailing market rate Some PRMs have a combination of both types within the promotional term The first and only serious attempt to consider PRMs from an option-theoretic perspective is Ong and Tan (2000) The paper uses a combination of prepayment options and discounted cash-flow method to value the incremental value of a PRM over and above an ARM Our study seeks to improve on Ong and Tan (2000) version of PRMs by considering additional features like the inclusion of default option, the competing risk nature between default and prepayment, transaction costs, suboptimal termination, suboptimal nontermination, cash flow induced terminations and mortgage underwriting regulations

2.6 Other Developments

Commercial vs Residential Mortgages

A whole literature has been developed on the pricing of residential mortgages and the mortgage-backed securities In contrast, there has been little work done on commercial mortgages Commerical and residential mortgages differ not only in type of collateral they represent; they often differ in their contractual features Commercial mortgages are often characterized by initial lockout periods, balloon payments and prepayment penalties during the lifetime of the mortgage Kau et al (1987, 1990a); Riddiough and Thompson (1993) and Titman and Torous (1989) have considered commercial mortgages from the option-theoretic point of view As for commercial mortgage-backed securities, only Kau

et al (1987, 1990a) have examined its pricing methodology

Pass-through Mortgage-Backed Securities

Theoretical OPMs have often associated not only with mortgages but also with MBS and their exotic derivative assets Papers valuing MBS include those of Brennan and Schwartz (1985), Dunn and McConnell (1981a, 1981b), Kau et al (1987, 1990a), Archer and Ling (1993) and Schwartz and Torous (1989a, 1989b, 1992) As most pass-through securities in US are backed by guarantees provided by mortgage insurance agencies, default will manifest to the lender like prepayment in which such terminations are still influenced by default-related factors like house price As a result, models that value MBS without considering default-related factors like house price are wrong in principle CMOs are tranches of securities formed from the pools of MBS, which sets a priority of claims

in case of mortgage termination CMOs theoretical models have been considered by

Dale-Johnson and Langetieg (1986) and McConnell and Singh (1993, 1994) Besides MBS and CMO, option-pricing models have also been used to price interest-only ( IO) and principal-only ( PO) stripped MBS An IO mortgage strips is a claim on the interest portion of an MBS pool whereas a PO strip is a claim to the principal portion Interest-only and principal-only mortgage strips are considered by Maris and Yang (1996), Schwartz and Torous (1989b) and McConnell and Singh (1994)

Delinquency and Reinstatement Options

Recent academic studies have only viewed default and delinquency as separate economic events There has been some interest recently to analyze delinquency in OPMs Ambrose and Buttimer (2000) attempt to separate the traditional default option into two components: the right to stop making payments and the right to give up property via foreclosure Traditional OPMs combine these components into the term ‘default’ In addition, the paper extends the current literature by documenting the impact of various loss mitigation programs, including forbearance and anti-deficiency judgments, as well as the value of credit on borrower default behavior

Game Theory and Option Pricing Theory

Traditional default and prepayment options have always been analyzed in a non-strategic setting There has been interest recently to integrate game theory and option pricing theory in mortgage pricing Jones and Nickerson (2002) is the probably the first to differ from traditional models by analyzing the option to default in a strategic settings, in which the borrower and lender play a well-defined game in a economy with complete

information and complete contingent claims market The paper also enlarges the based literature on mortgage termination to include endogenous response of supply credit

option-to both the strategic timing of default or prepayment of borrower, and simultaneously option-to the characteristics of the collateral offered by the borrower, in a perfect equilibrium

2.7 Numerical methods in mortgage modeling

The initial work of Black and Scholes (1973) establishes a closed-form solution to the option pricing problem However, under complex contract terms, such solution does not exist Mortgages are among the most complex contracts ever devised, so to value them,

we must resort to numerical solution techniques In the following section, we review the various numerical techniques used by theoretical mortgage researchers

Tree/Lattice approach

Lattice methods are founded on the discretization of the risk-nuetral processes followed

by the underlying state variables Thereafter, a backward induction process is used to solve for the option price The common and popular methods are binomial and trinomial trees The binomial method was introduced by Cox et al (1979) and Rendleman and Bartter (1979) which is based on the random walk approximation to the Brownian motion Bartter and Rendleman (1979), Pozdena and Iben (1984), Hall (1985), Chen and Ling (1989), Giliberto and Ling (1989) and Follain, Scott and Yang (1992) adopted binomial tree in pricing mortgages while Leung and Sirmans (1990) adopted a more complex trinomial tree approach in doing valuation calculations

Most lattice approaches are developed on the assumption that the underlying processes are lognormal distributed and the state variables are uncorrelated To relax such assumptions, Hilliard, Kau and Slawson (1998) suggested a bivariate binomial option pricing technique to price mortgages Following the introduction, a number of works like Ambrose and Buttimer (2000) and Kelly and Slawson (2001) adopted the technique to price mortgages

Finite Difference Schemes

An alternative technique is the finite difference method Option valuation under finite difference methods is done by backward induction in time as with lattices There are generally three classes of finite difference methods: explicit, implicit and Crank-Nicolson types Finite difference methods, except for explicit methods, are unconditionally convergent and stable but it is difficult to extend them to path-dependent claims or high-dimensional options For dimensions higher than three, Monte Carlo simulation is preferred

Stanton (1995) and Jones and Nickerson (2002) solved the fundamental differential equation using the Crank-Nicholson discrete approximations for the partial derivatives Buser, Hendershott and Sanders (1985) and Schwartz and Torous (1992), on the other hand chose the implicit finite difference method as the fundamental numerical solver The commonly used finite difference method, the explicit finite difference method, is adopted by Chatterjee, Edmister and Hatfield (1995) and various articles of Kau et al

Monte Carlo Approach

Monte Carlo simulation is suitable for path-dependent options and can be extended to price options that depend on multiple state variables, stochastic volatility and Poisson jump process Its major drawback is that it is a computationally intensive method as it usually requires many simulations to achieve convergence and stability With the aim of solving this problem, variance reduction techniques, such as antithetic variables and control variates, have been developed In pricing mortgages, the Monte Carlo method works only if the decision to terminate the mortgage is independent of the valuation processs Such an approach was considered by various authors such as Schwartz and Torous (1989a, 1989b, 1991) and Riddiough and Thompson (1993)

Schwartz and Torous (1989a) integrate an empirical prepayment function in a Monte Carlo settings to value default-free mortgage-backed securities Other works of Schwartz and Torous (1989b, 1991) adopted a similar approach Riddiough and Thompson (1993) uses Monte Carlo simulations along with empirically estimated default frequencies to produce mortgage price estimates as well as default premiums and loss severity

If the decision to terminate is dependent on the valuation process, the generic Monte Carlo approach is unable to capture the rational termination behaviour of borrowers To

do so, one would then have to work with backward pricing techniques

Hybrid Approach

To avoid the limitations of both forward and backward pricing methods, a number of researcher resort to utilizing hybrid methods involving the combination of both forward and backward pricing methodologies

McConnell and Singh (1993, 1994) employ a two step procedure to solve the implicit dynamic programming problem The first step use an implicit finite difference backward solution procedure to determine the prepayment boundary and the second step involves using Monte Carlo Simulation to value the CMO tranches Buist and Yang (1998), along

a similar light, uses an explicit finite difference method to determine the prepayment hurdle rates and then uses such rates in a Monte Carlo simulation procedure to value fixed rate mortgages

From our review of the various numerical techniques in mortgage pricing, we observe that most of the techniques suffer from one or more drawbacks such as the inability to extend to multiple stochastic factors, the curse of dimensionality, the inability to integrate rational decision-making and the inability to handle path-dependency This provides us with the impetus to search for a technique that can overcome these drawbacks Our search ends with advanced Monte Carlo techniques which offer promising results in handling securities of extreme complexity

2.8 Review of Monte Carlo Simulation Techniques for American Derivatives

Monte Carlo was first introduced in finance by Boyle (1977) Simulation is becoming a popular method to price securities due to its generality in the types of assets it can handle

In addition, it avoids the problem of dimensionality and path-dependency Its only drawback is its inability to handle the free-boundary aspect of American options Simulation techniques are “forward” algorithms in which all the calculation is done forward in time By contrast, pricing algorithms for assets with early exercise are generally backwards in nature The problem of using simulation to price American options stems from the difficulty to use a forward-based approach to solve a problem which requires a backward procedure to capture the rational decision-making process of option holders

Tilley (1993) is the first who attempt to price American options using Monte Carlo techniques He presents an algorithm in which, at each date, simulated paths are ordered

by asset prices and bundled into groups Then, for each group, an optimal exercise decision is taken According to Broadie and Glasserman (1997a), there are three problems with this algorithm Firstly, there are no convergence results for the algorithm Secondly, there is no stated or obvious way to generalize the algorithm to several state variables Thirdly, his algorithm requires all the simulated paths to be stored at one time for sorting and bundling purposes This storage requirement could be a significant computational problem if the number of simulated paths is large

Barraquand and Martineau (1995) propose to reduce the dimensionality of the valuation problem, grouping the simulated values into a set of “bins.” The transition probabilities between bins is determined by simulation and the option valuation is performed using each bin as a decision unit However, the algorithm does not prove itself to be a convergent method Raymar and Zwecher (1997) extend Barraquand and Martineau approach by basing the exercise decision on a partition of two state variables, rather than one This improves the approximation but such an extension does not solve the convergence problem and will leads to an exponential increase in work requirement

Broadie and Glasserman (1997a) develop an algorithm that creates point estimates and error bounds for American option prices It generates two estimates, one biased high and one biased low, both asymptotically unbiased and converging to the true price In this algorithm, the combination of the two estimates yields a confidence interval for the true price Though the algorithm can be easily extended to multiple state variable models, its computation time is exponential in the number of exercise opportunities Broadie et al (1997) further developed several enhancements to improve the efficiency of the two estimates

To price high-dimensional options with large number of exercise opportunities, Broadie and Glasserman (1997b) introduced another algorithm in which they named Stochastic Mesh method The method provides both upper and lower bounds, confidence intervals for the true price, and it is asymptotically converging The computational time is linear in the number of state variables, linear in the number of exercise opportunities, and

quadratic in the number of points in the mesh Such features make it viable to price dimensional American options

high-Ibañez and Zapatero (1998) devise a simulation technique for computing the optimal exercise frontier as the fixed point of an algorithm To compute the frontier, the values of all parameters but one are fixed and their algorithm is used to converge to the value of the remaining parameter in the optimal exercise frontier The authors further illustrated the algorithm to price put and call options on the maximum of two securities, assuming they are exercisable at a finite number of dates For a survey of Monte Carlo techniques, see Boyle et al (1997) and Broadie and Glasserman (1998) A comparison of various Monte Carlo methods for pricing American options can also be found in Laprise et al (2001)

The most recent development is by Longstaff and Schwartz (2001) which proposes a simulation technique that combines the use of least square regressions with Monte Carlo technique They name this technique Least-Square Monte Carlo (LSM) In this technique, the Monte Carlo function first generates the asset prices movement and provides the cross sectional information for regression In the least square function, they use a set of basis functions based on the asset prices to estimate the expected continuation values Comparing the fitted continuation values with the immediate exercise values, the algorithm identifies the optimal stopping rule This procedure is repeated recursively going back in time The price of the American option is computed via the average of the discounted cash flows Following their seminal work, various authors started to carry out detailed analysis of the algorithm

Moreno and Navas (2001) analyses the robustness of LSM for pricing American options The authors analyze the impact of different basis functions on option prices Numerical results for American put options provide evidence that this approach is very robust to the choice of different alternative polynomials and few basis functions are required to achieve convergence Clément et al (2002) and Egloff and Min-Oo (2002) prove the almost sure convergence of the complete least square Monte Carlo algorithm for a fixed finite set of basis functions and also establish a type of central limit theorem for the rate

of convergence of the Monte Carlo method, thus proving its normalized error is asymptotically Gaussian

A detailed analysis of LSM is also carried recently by Stentoft (2003) The author, based

on the analysis of various specifications of the cross-sectional regressions, suggested that

a modified version using ordinary monomials outperforms the Laguerre polynomial specification in terms of computational time In addition, the author proved that the LSM method is computationally more efficient than existing numerical methods in cases of high dimensional American options On the other hand, Rasmussen (2003) suggested two improvements to the technique to improve its stability and accuracy The first improvement is to use control variates on the discounted payoffs and the second is to disperse the initial state variables from which the asset paths are generated

As the technique is fairly recent, the application to specific examples is rather limited Gamba (2003) illustrated the use of LSM to value a wide set of capital budgeting

problems with many embedded real options dependent on many state variables Andreatta and Corradin (2003) proposed to use LSM to price a surrender option inherent in Italian life insurance policy As far as our literature search reveals, none has applied the LSM to mortgage pricing Our study is the first to apply the LSM to price embedded options in mortgages

2.9 Summary of Literature Review

Our review begins with the foundations of pricing mortgages as derivative assets and tracks the development of these theoretical models over time Our literature review has tracked the developments in terms of type of options, state variables, mortgage contracts, securitization models and type of property These developments have provided the basis

of developing our theoretic option pricing model Our review then explores the various types of numerical methods in mortgage modeling and observes that most type of numerical methods suffer from one or more drawbacks such as inability to extend to multiple stochastic factors, the curse of dimensionality, the inability to integrate rational decision-making and the inability to handle path-dependency We then reviewed the various Monte Carlo option pricing techniques in search of a suitable technique to resolve the mentioned drawbacks From our review of the literature, we discovered Longstaff and Schwartz (2001)’s LSM technique as a suitable candidate in pricing complex structure mortgages By employing this technique, our study has contributed to the literature by providing a more flexible and robust methodology to price mortgages In addition, our

study extends the mortgage theoretic literature by providing the first four-state variable model

below-Following the seminal works of Black and Scholes (1973) and Merton(1973), the theoretic approach to pricing mortgages offers a new insight into opportunities of prepay and default The early works of Dunn and McConnell (1981), Cunningham and Hendershott (1984), Brennan and Schwartz (1985) and Foster and Van Order (1984,1985) develops the theoretical foundation for rational prepayment and default Subsequent

option-works of Schwartz and Torous (1992), Ambrose, Buttimer and Capone (1997), Kau, Keenan and Kim (1993c), Kau and Slawson (2002) contributed additional features such

as transaction costs, suboptimal termination and sub-optimal non-termination to the existing literature Essentially, most models in the literature rely on the use of traditional dynamic programming techniques to solve the Bellman’s equation implicit in most American-style mortgage options However, they often suffer from two drawbacks Firstly, the traditional dynamic programming techniques are limited by the so-called curse of dimensionality; in other words, the more stochastic factors, the higher the memory and time requirements of the approach For complex mortgage options, the number of state variables can be substantial and the computational requirements become prohibitive Secondly, the backward solving method assumes path-independence and fails

to capture the path-dependent nature of borrower decisions (Buist and Yang, 1998) As for valuing hybrid mortgages, the path-dependent mortgage rate structure poses significant difficulty for such methods This provided the impetus to use a forward-pricing method, the LSM technique of Longstaff and Schwartz in this paper, to overcome the problem of dimensionality and path dependency

The term structure of interest rates significantly affects mortgage values, yet most research restricts the characterization of term structure to a single factor stochastic process One-factor term interest rate models, though simple, are more suitable for modeling short-term interest rates than long-term interest rates However in mortgage pricing, the underlying mortgage rates normally exhibit long-term interest rate characteristics and one-factor models may fail to adequately reflect the entire term

structure Furthermore, the mortgage rates are often used for both the risk neutral discounting process and also forward generation of trajectories for mortgage balances This paper posits a two-factor term structure process of Schaefer and Schwartz (1984) in which the above-mentioned functions are dichotomized by the employment of a risk-free rate and a mortgage rate

time-The ruthless default conditions suggested in most mortgage pricing literature (Kau et al, 1995) ignore other incentives of households not to exercise their default options Even when the default option is deep in-the-money, the borrower may not exercise the option

to avoid a bad credit rating and to preserve future access to credit To weaken such ruthless default assumptions, we follow in the spirit of Buist and Yang (1998) to append

an income process and mortgage underwriting constraints in the valuation framework

We posit that default will only arise when normal ruthless default conditions are accompanied by ability-to-pay constraints If declines in household income reach a point

at which the borrower cannot afford the mortgage expenses, the borrower must consider default or prepayment If ability-to-pay problems arise and the house price is greater than the mortgage value, the borrower will then engage in a pre-foreclosure sale of the property in which the borrower sells off the house to reduce his housing expenses in face

of impeding foreclosure Otherwise, the borrower will simply default on his property In addition, we endogenize mortgage underwriting constraints to bridge the gap between theoretical and observed prepayment behavior

Given a reduction in mortgage rates, a lender’s imposition of maximum loan-to-value (LTV) constrains the borrower’s choice to refinance In particular, declines in house price will violate the LTV limits, lowering prepayment rates Maximum mortgage payment-to-income ratios also constrain prepayment in refinancing situations Hence, a refinancing will occur when three conditions hold simultaneously: (1) the value of the underlying callable mortgage is sufficiently greater than the outstanding balance given that interest rates have fallen; (2) the house price is greater than the outstanding mortgage balance divided by the required loan-to-value ratio, and (3) the household income is higher than the new payments divided by the maximum allowable payment-to-income ratio The mortgage-theoretic OPM literature has mostly ignored conditions (2) and (3)

Even with the above specifications, the strictly rational model still might not adequately reflect the observed termination behavior as the model lacks certain reality constructs In reality, borrowers may not default, prepay and continue the mortgage as OPMs have indicated We argue that the existence of frictions in real-world environment has contributed to this disparity between presumed theoretical predictions and empirical observed results The paper supports such agenda by providing a frictions-adjustable OPM which allows the incorporation of transaction costs and suboptimal termination while maintaining financial optimality as the mainstream decision-making principle Furthermore, the formulation of frictions and incorporation of income statistics allow for borrower heterogeneity in the model which again attempts to draw our model closer to empirical observations

Lending institutions, being significantly exposed to booms and busts of the real estate cycle, are interested in understanding not only the sources of credit risk but also the entire risk distribution of their mortgage portfolio Given that distribution of credit risk is both disperse and skewed, the option value, being the unconditional expected discounted severities, do not adequately reveal the risk profile of mortgages Using OPMs, we can derive the market value of default, prepayment and the mortgage One may think of the value of an option as the product of the probability of exercising the option and the (discounted) payoff of such exercise in a very simplistic way From such relationship, we postulate that, besides the option value of default and prepayment, we could extract another two types of information from mortgage OPMs namely: mortgage termination probabilities and its associated severities Severities in this case refer to the expected loss

of the lender at the time of termination For the reason we mentioned earlier, more insight can be gained by going beyond the standard option value calculation, to determine termination probabilities and the distribution of severities that average up to the market cost of the lender’s liability This involves extracting probability distributions of default and prepayment within specified credit range from our Least Square Monte Carlo Option Pricing Model (LSM-OPM) We further note that in order to understand the overall risk

of the lender in mortgage operations; lenders should understand not only the underlying default risk but also the reinvestment risk associated with prepayment This paper serves

to fulfill these purposes by providing a theoretical framework to generate disaggregated risk statistics of mortgages such as default probabilities, prepayment probabilities, default probability distribution across time, prepayment probability distribution across time, default severities and prepayment severities Lastly, we provide an extensive sensitivity

analysis of these statistics with varying economic and contractual parameters to illustrate the flexibility and adaptability of our model in the risk evaluation of mortgages

The purpose of this paper is to introduce several innovations to the theoretical mortgage OPMs Firstly, the paper is the first to present the formulation for the market value of a PRM in a competing risk framework Secondly, the paper adopts a least square Monte Carlo approach to price mortgages and clearly refrains from following the usual backward pricing approach, to avoid the problems of dimensionality and path dependency Thirdly, the framework incorporates a household income process and a two-factor term structure process into the model to negate the ruthless default conditions and also to enhance the characterization of long term mortgage rate processes Fourthly, the paper provides a framework that allows for borrower heterogeneity and modeling of various frictions like transaction costs, suboptimal termination and suboptimal nontermination The fifth innovation is that our model allows for the endogenization of underwriting guidelines which is useful for evaluating policy changes related to the housing and mortgage markets Lastly, we illustrate a simple factorization of the value of termination into probability and severity to gain more insight into the risk profile of mortgages

3.2 The Valuation Model

The Economic Environment

To describe the mortgage termination behavior, one must first specify both the economic environment and the mortgage contract We assume the economic environment to be modeled as a stochastic economy (Longstaff and Schwartz, 2001), where an underlying complete probability space of (Ω,F,P)in a finite time horizon [ ]0,T with Ω, being the set of possible realizations of the economy and has typical ω representing a sample path,

W being the sample space where {ω∈W}, F being the sigma field of distinguishable events and P , being the probability measure defined on these events We define{Ft :t∈[ ]0,T } to be the augmented filtration generated by the four stochastic processes {(Y,r,s,H)∈ℜ4}for the securities and assume FT =F We introduce the notation )X(ω,t to be the sample value of X obtained during the sample path ωat time t

Formally, we assume that the household income Y, follows a jump diffusion process where we append a Poisson process to represent the arrival of unemployment events, ones which occur beyond the typical movements of income:

),(),()

(),(

),(

t dq t dz dt

t Y

t dY

y Y

Y Y

Y

µω

small interval of time dt, there is probability (1−λY)dt that nothing happens, so that

dq is independent of other Wiener processes in this model and in particular, independent

of general movements in the market This is in line with Merton (1976)’s assumption that the jump component represents nonsystematic risk

The term structure is assumed to follow a two factor mean-reverting stochastic process of Schaefer and Schwartz (1984):

),())

,((),

),())

,((),

where r is the instantaneously riskfree rate, s is the spread between the riskfree rate and

the long term rate (mortgage rate in our case), m, i.e s=m-r,θ represents the steady-state value, γ is the speed of adjustment and σ is the volatility parameter of the corresponding term structure

The house price H, on the other hand, is assumed to follow geometric Brownian motion (standard lognormal process) in the form:

),()

(),(

),(

t dz dt

s t

H

t dH

H H

µω

with µ and σH representing the instantaneous total expected return and proportionate volatility The return to using the house is the service flow, s; we assume the service flow

is interest rate dependent2 (Kim, 1991) and:

where the relation (5) states that the required rate of return on housing µHconsists of rental dividends and capital gains Ε[dH / H] Using the same past information that rational agents would use to form their anticipations of the capital gain process, the expected capital gains can be estimated using such information Without any loss in generality, the required rate of return can be estimated from a consumption-based asset-pricing model of:

)()

where the return3 of the housing asset depends on the market portfolio4, M, as well as

fluctuations in mortgage rate spread Here, αMis the anticipated return on the market portfolio, m is the mortgage rate and βM and βc are ordinary multiple regression coefficients of the housing asset with the market portfolio and the mortgage rate We introduce eqns.(5) and (6) to capture the influences of changes in term structure on the housing stochastic process Lastly, the correlations of the Wiener processes,

H s