an intensity-based approach to the valuation of mortgage contracts and computation of the endogenous...

August 21, 2006 20:22 WSPC-104-IJTAF SPI-J071 00387

International Journal of Theoretical and Applied FinanceVol. 9, No. 6 (2006) 889–914c© World Scientific Publishing Company

AN INTENSITY-BASED APPROACH TO THE VALUATION OFMORTGAGE CONTRACTS AND COMPUTATION OF THE

ENDOGENOUS MORTGAGE RATE

YEVGENY GONCHAROV

Department of Mathematics, Florida State University202A Love Bldg, Tallahassee, FL 32303, USA

[email protected]

Received 6 November 2004Accepted 3 November 2005

This paper gives a general mortgage model subject to sub-optimal prepayment behav-ior of borrowers. The proposed classification of approaches to the option-based andmortgage-rate-based (MRB) specification is validated by the fact that these twoapproaches require different analytical and numerical tools. With the option-based spec-ification of the prepayment intensity, our model is the first continuous-time model; thisgives advantage in the numerical treatment of the model. We propose a new approachbased on the MRB idea with the endogenous mortgage rate process. The mortgage rateprocess is defined and the existence of a solution is proven for the option-based speci-fication of the prepayment process. A numerical procedure for finding the endogenousmortgage rate is proposed.

Keywords: Mortgage; option-based approach; prepayment; intensity; mortgage rate.

1. Introduction

Mortgage markets, both primary and secondary, evolved considerably during thelast two decades. A large body of literature, both academic and institutional, isdevoted to the rational pricing of mortgage-related assets. Considerable progresshas been made with the understanding of relevant factors for mortgage pricing, butmore work is required for a fully satisfactory, complete understanding.

The prepayment option is what makes mortgage related securities complicatedassets to price. Mortgagors have the option to fully or partially prepay their loansprior to the maturity dates. This right has a dramatic effect on the valuation byintroducing cash flow uncertainty which depends on the mortgage holder’s view ofpossible future opportunities (e.g., the mortgage holder’s expectation of the futurebehavior of the yield curve) to refinance the loan.

A very important feature of a mortgage model is how one models the prepay-ment incentives of the borrower. This determines the borrower’s decision-makingimplied by the model. Due to the different characteristics of approaches to model

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890 Y. Goncharov

prepayment incentive measures (i.e., a borrower’s idea of profitability of prepay-ment), mortgage models in the literature can be classified into two groups. Thefirst group is a type of model which is closely related to the pricing of Americanoptions and therefore is called the option-based or option-theoretic approach. Theoption-based measure of prepayment incentive is endogenous. The central object ofthis approach is a borrower’s liability, which can be defined as the present value ofthe cash flow that the borrower will pay off to repay his/her loan plus various trans-action costs which are incurred in case of prepayment or default. Most option-basedmodels in the modern literature measure the mortgagor’s incentive to prepay as thedifference between this liability and outstanding principal. This assumption some-what simplifies the problem of mortgage modeling, because it removes the necessityof the mortgage rate modeling, although it (unrealistically) implies that borrowersrefinance their mortgages with loans at the risk-free interest rate.

The option-based approach has evolved from frictionless models with optimalprepayment behavior (e.g., [10, 27]), where the mortgage liability to the borrowerand the mortgage asset to the investor are not distinguished (or they differ due tofees only) and the borrowers terminate their mortgages if and only if it is finan-cially optimal, to models which recognize the importance of taking into account thesubstantial presence of transaction costs and non-optimal behavior (the borrowercan fail to prepay when it is financially profitable and he/she can prepay when itis not profitable to do so). Transaction costs were first incorporated in the modelas a part of refinancing threshold for optimal liability (e.g., [11]). All these men-tioned models assumed optimal prepayment plus “background” prepayment due torelocation, divorce, etc. Stanton [39] first acknowledged the fact that borrowers failto prepay even if it is optimal to do so. Stanton’s model, as opposed to the modelsabove, permits mortgage prices to be higher than the outstanding principal plusthe transaction costs (this phenomena is actually observed on market). Recently,Longstaff [32] proposed a model with an optimal borrower’s behavior, which, never-theless, is able to explain this phenomena. The key observation in this paper, whichseparates it from previous models in the literature, is that borrowers refinance theirloans with another loan which might be “expensive” due to credit rating constraints.Independently, a similar idea (but under prepayment sub-optimality assumption)was implemented in the model called “complete option-based” by Goncharov [15].

The second class of models are sometimes called “option-based” models too (e.g.,[6, 7]) though the name does not precisely (if we understand “option approach”in the sense of [33]) reflect the nature of the prepayment decision-making that isassumed in the models. To avoid confusion, we will call them mortgage-rate-basedmodels or MRB for short. In this type of model, the prepayment policy is based ona comparison of prevailing mortgage and contract rates.1 This comparison can bedone with a “naive” mortgage formula (see formula (2.1) below) such as a precise

1For example, the mortgage holder prepays as soon as the mortgage rate drops below a specifiedthreshold.

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Intensity-Based Approach 891

calculation of how much is saved on, say, monthly payments by refinancing (e.g.,[5–7], or it can be done through a direct comparison such as the difference or ratioof two mortgage rates (e.g., [24, 25, 36–38]). So called empirical models can be fitinto this category because the prepayment rate is regressed, in particular, on somelong term (e.g., 10-year Treasury yield) interest rate, which might be regarded asa proxy for the mortgage rate process. Early MRB models (e.g., [37]) recognizednon-optimality of the borrower’s behavior; later models incorporated transactioncosts (e.g., [25], where the authors assumed optimal prepayment however).

Looking at this (far from being complete) list of papers which addressed thequestion of prepayment modeling, we can see that there is still no consensus evenon the way the refinancing incentive should be modeled. It is almost universally rec-ognized that any prepayment model is doomed to be “wrong” to some degree — thatis, prepayment surprises (not explained by the prepayment model) are inevitable.The market responds to this “unaccounted” (by a prepayment model) risk withthe increased yield on mortgage securities. The option-adjusted spread (OAS) is apopular tool to account for it. Levin [31], instead of the “myopic” OAS, considereda prepayment model “imperfection” explicitly in his model. As a result, Levin intro-duced a new concept called prepayment-risk-and-option-adjusted spread (prOAS).Though the idea was formulated for an empirical approach to model the prepaymentrates, it is applicable to the option-based models as well.

This paper gives a theoretic treatment of the topic of mortgage modeling. Weconcentrate only on models with sub-optimal borrower behavior.2 The peculiarity ofthe problem (illiquidity of prepayment option, substantial transaction costs, prepay-ment modeling in general) is recognized in the academic literature. But mortgageresearch mostly consists of empirical tests of realizations of some particular models.We build a first general continuous-time mortgage model and propose a classifica-tion of approaches according to the way the refinancing incentive is modeled. Weshow that the two approaches (the option-based and MRB) in the literature leadto two distinct mathematical problems. The option-based mortgage price is definedas a solution to an integral equation. If the state variables are modeled as a diffu-sion, the option-based mortgage valuation problem is equivalent to the solution ofa semi-linear reaction-diffusion PDE (all current option-based models are discretefirst-order approximations to the underlying PDE). In the case of the MRB specifi-cation of the prepayment intensity, we propose to extend the approach by definingthe mortgage rate endogenously, thus, making a MRB model endogenous.

In Sec. 2 we introduce a general mortgage model. A conventional expectation(2.4) was widely used in some forms in the literature (e.g., as a PDE or various“tree” computations), while an alternative representation for mortgage price (2.5)was not well-known. Since it is resolved around a “fair” mortgage price (which isthe outstanding principal) it is computationally more efficient than the conventional

2The optimal behavior can be regarded as a limit of our model when the refinancing intensity goesto infinity.

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892 Y. Goncharov

formula (2.4) if a PDE approach3 is used (since time dependence around the interestrates values “of interest” is flattened). Presence of the prepayment incentive onlyin the discount factor of this representation gives a new idea to price complicatedcollateralized mortgage obligations. In Sec. 3 we define and consider the option-based and MRB approaches to model the prepayment intensity. In particular, weshow an example illustrating a fundamental difference between them. In Sec. 4 weprove that an option-based mortgage model is well defined, i.e., a solution to themortgage price equation exists and is unique. Diffusion state setting is consideredin Sec. 5. It is of particular interest for the option-based approach because theFeynman-Kac result is not straightforward in this case. In Sec. 6, with the help of thealternative representation from Sec. 2, we formulate the problem of the endogenousmortgage rate in the form of a fixed-point problem and prove the existence of asolution in the case of the option-based approach. For the MRB setting the problemseems to be a nontrivial new mathematical problem and is the subject of futureresearch. In Sec. 7 we propose a numerical procedure to solve the mortgage rateequation. We conclude the paper in Sec. 8.

2. General Mortgage Model

2.1. A basic mortgage

We consider the following contract: a borrower takes a loan of P0 dollars at someinitial time and assumes the obligation to pay scheduled coupons at rate ct ≥ 0continuously for a duration T of the contract. The loan is secured by the collateralof some specified real estate property, which obliges the borrower to make the pay-ments. For a mortgage originated at time s, interest on the principal is compoundedaccording to a contract rate ms

t , t ≥ s, where time dependence as the subscript t

reflects a possible change of the rate as specified by the contract and the superscripts shows the time when this mortgage rate schedule was contracted. When s = 0we omit the superscript, so that mt = m0

t . When we are dealing with fixed-ratemortgages (no dependence on the subscript) initiated at time t we simply write mt

(the maturity of the mortgage is implicit).A straightforward argument shows that, in the absence of prepayment, the out-

standing principal P (t) at time t equals the future (time-t) value of the originalprincipal P0 less the future (time-t) value of the cumulative coupon payments:

P (t) = P0eR t0 mθdθ −

∫ t

0

cseR t

smθdθds. (2.1)

The borrower has the right to settle his/her obligation during an interval specifiedby the contract4 and prepay the outstanding principal P (t) in a lump sum. If

3Or a “tree” approach as a variant of the finite difference numerical solution of the PDE.4Commercial mortgages, for example, often have a prepayment lockout period.

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the borrower prepays, then he is forced to pay concomitant transaction costs. Thetransaction cost process, Ft, is assumed to be a defined part of the mortgage model.

The coupon payment rate ct and mortgage rate mt can depend on time andthe current state of the economy. They can be deterministic functions and evenstochastic processes. But they are not independent. In the case of a fully amortizedmortgage, i.e., P (T ) = 0, Eq. (2.1) implies that rates ct and mt must satisfy

P0 =∫ T

0

cse−R

s0 mθdθds. (2.2)

Although some commercial mortgages may have a balloon payment at maturity, weassume that the mortgage is fully amortized. The extension is straightforward andcan be taken care of within the framework we present.

In the case of the most popular level-payment fixed-rate mortgage, both ct andmt are constants. If one fixes either rate, c or m, then the other rate can be foundfrom this equation. Other examples of fixed-rate mortgages are graduated-paymentmortgages (GPM) and growing-equity mortgage (GEM). For a GPM mortgage onechooses a mortgage rate m and then finds a (deterministic) function ct which hassome ability to reflect the borrower’s “preferences” about when and how paymentsshould increase. The payment schedule ct nevertheless must satisfy (2.2) for a fixedmaturity T. GEM mortgages with a fixed mortgage rate m start with the samepayment as a level-payment mortgage (thus avoiding negative amortization whichis common for a GPM mortgage), and are gradually increased with time. In thiscase ct is an increasing positive function (c0 = c) and Eq. (2.2) produces some newmaturity time T ′ that will be less than the maturity time of the correspondinglevel-payment mortgage.

An adjustable-rate mortgage is an example of a stochastic coupon payment ratect and mortgage rate mt. The contract mortgage rate mt is reset periodically inaccordance with some appropriately chosen reference rate; the coupon rate ct maybe found the same way as it is done in different kinds of fixed-rate mortgages. Thereference rate depends on the current state of economy and, therefore, is stochastic.

Another situation where we may want to consider a stochastic coupon rate ct isto facilitate curtailment, i.e., partial prepayment. In this case, the maturity time T

depends on the whole history of the process ct via (2.1). Full prepayment and cur-tailment have different impacts on mortgage securities. If full prepayment reducescoupon payments in a pool, but leaves the maturity time the same, then curtail-ment, on the contrary, shortens the maturity time while leaving coupon paymentsthe same. Curtailment has not received the attention it deserves (we can mention[4] on the topic). As reported in Hayre and Rajan [18], it amounts roughly tothe same portion of total prepayment in a pool as default (though both are stillrelatively low).

As can be seen, the outstanding principal P (t) depends on the choice of thecontracted mortgage rate mt and the contracted coupon payments ct. In whatfollows, this dependence will be suppressed for notational transparency.

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894 Y. Goncharov

2.2. Mortgage securities

The mortgage market has undergone significant structural changes since the 1980s.Innovations have occurred in terms of the design of new mortgage instrumentsand the development of products that use pools of mortgages as collateral for theissuance of a security. Such securities, called mortgage-backed securities (MBS),remove prepayment uncertainty to some extent but introduce a strong historydependence in the form of burnout.5 One simple (not model-based) way to model theburnout effect is purely empirical. For example, Schwartz and Torous [37] assumethat a pool consists of borrowers with homogeneous characteristics and thereforeare forced to take burnout (recall that burnout is a product of a pool’s heterogene-ity) into account as the explanatory variable ln[P (t)/P ∗(t)], where P (t) representsthe dollar amount of the pool outstanding at time t, while P ∗(t) is the pool’sprincipal that would prevail at t in the absence of prepayments but reflecting theamortization of the underlying mortgages. The adequacy of this approach is ques-tionable because the burnout effect is “non-Markovian”; it should depend on thewhole history path of prepayments. The modeling approach (e.g., [31, 39]) is basedon the observation that one particular mortgage does not have burnout by defini-tion.6 Therefore, using conditional independence of individual mortgages, one canincorporate burnout into a model by modeling the prepayment behavior for indi-vidual mortgages and constructing a pool’s prepayment behavior as a combinationof individual prepayments. In the first approach, due to homogeneity, the priceof a pool is the number of borrowers in the pool multiplied by the price of onemortgage. In the modeling approach, the price of a pool is the summation of theprices of all individual mortgages in the pool. In both cases, the price of a pool(and therefore the price of a mortgage pass-through security)7 is based on a “basicmortgage”.

The same conclusion is true for other mortgage-backed securities if one wantsto apply an endogenous refinancing incentive.8 Therefore, the valuation of an indi-vidual mortgage (a basic mortgage) is of fundamental importance and, keeping inmind possible generalizations to MBS, in the present paper we will deal only witha model for one mortgage with an emphasis on the way the mortgage prepaymentcan be defined.

5If the pool experienced a wave of prepayment then it is likely that most of those who used thisrefinancing opportunity are “fast,” financially astute borrowers, whereas most of those who remainare “slower” borrowers. Therefore, in the presence of another refinancing opportunity, the pool isexpected to be less active — the pool is “burned out”.6This idea stands behind active-passive decomposition (ADP) of Levin [31].7A holder of this security receives a fixed portion of cash flow from the pool.8MRB approach with the exogenously specified mortgage rate process does not need prices of indi-vidual mortgages in order to price, say, CMO’s, since the distribution of prepayment is determinedby the mortgage rate.

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2.3. Mathematical apparatus

We formalize our setup by introducing a completed filtered probability space(Ω,G, Gtt≥0,Q),9 where the σ-algebra Gt represents all observations availableto an investor at time t, Ω is a set of all possible outcomes and Q is the probabilityon G (⊇ ⋃

t≥0 Gt). The prepayment time, for which we use notation τ, is then apositive stopping time on this filtered probability space (i.e., at any arbitrary timet we can tell if prepayment “occurred” given information Gt).

We introduce information concerning only the timing of the prepayment as thefiltration Dt = σ(Iτ≤u|u ≤ t). Now given Dt and the original filtration Gt, weare interested in decomposing Gt into Dt and an additional filtration Ft. Formally,it can be defined as a solution of the equality Gt = Dt ∨ Ft.

10 We accept thiscomplementary filtration Ftt≥0 as given. In the financial interpretation, Ftt≥0

is assumed to model the flow of observations available to the lender prior to theprepayment time τ . Given only this information Ftt≥0, the lender cannot antic-ipate the prepayment since he/she does not have complete information about theborrower (such as intention to move, to divorce, etc.) and/or the borrower does notprepay as soon as it is profitable to do so (see, e.g., [18]), i.e., Q(τ ≤ |Ft) < 1 for allt ≥ 0. Thus, we assume that prepayment time τ is not an Ft-stopping time. Thissituation is the same as for default time in the reduced-form modeling of creditderivatives. The technique is standard and we refer the reader to Jeanblanc andRutkowski [21, 22] for detailed treatment of the topic.

Definition 2.1. The process Γt = −ln[1−Q(τ ≤ t|Ft)

]is called the hazard process

of the random time τ. Equivalently Q(τ > t|Ft) = e−Γt .

We assume that Γt is an increasing process. Since there is no particular convenientor special day for prepayment,11 Γt is assumed to be continuous. Indeed, we makethe slightly more restrictive assumption that the process Γt is an absolutely con-tinuous process, i.e., Γt =

∫ t

0 γθdθ for some process γt, called the intensity of therandom time τ. In this case, Definition 1 has certain similarities with the definitionof the intensity of a Poisson process, thereby giving us intuition behind the name“intensity” of γt.

12

9We make throughout the technical assumption that this filtered probability space satisfy the“usual hypothesis” and all filtrations we work with later are (Q, G)−complete.10It is clear that the equality does not define Ft uniquely. For example, if Gt = Dt, we may takethe trivial filtration, but also Ft = Gt. We interested in having Ft be the “minimal” such filtration.In most applications, Ft is the natural filtration of processes which define the interest rate andreal estate processes.11For example, for defaultable mortgages with discrete payments it could be reasonable to assumethat default times are distributed at due dates, in which case the due dates would be “special” fordefault. We work with continuous coupon cash flows.12If τ is the first jump of a Poisson process with an intensity function γ(t) then Q(τ > t) =

e−R t0 γ(θ)dθ . Etymology of the term “intensity” (or “hazard rate”) for γ(t) came from the fact that

for small t we have Q(t < τ ≤ t + t|τ > t) = γ(t)t + o(t) for almost all t ∈ R+.

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Throughout this paper we assume that all given processes are positive andFt-progressively measurable. The latter condition is purely technical and is neededto have Ft-adapted time-integrals of the processes. One simple (and sufficient formost applications) condition for a process to be Ft-progressively measurable is thatthis process is adapted and right- or left- continuous (see, e.g., [34]).

A process rt will represent a short-term interest rate, so that at any time t itis possible to invest one unit in a default-free deposit account and “roll-over” theproceeds until a later time s for a market value at that time of e

R st

rθdθ.

The Fundamental Theorem of Arbitrage Pricing (see [17]) states that the absenceof arbitrage implies that all securities are priced in terms of this short-rate processrt and an equivalent martingale measure. Therefore, it is convenient to assume thatthe probability Q in the introduced probability space is this martingale measureand, thus, all expectations in the paper will be taken under this martingale measureQ without reminder. The connection between real-world and martingale probabilitymeasures will be briefly discussed at the end of this section.

If we assume that a no arbitrage opportunity exists on the market, then thetime-t price of a security Mt that pays a coupon payment continuously with therate of ct up to time τ ∧ T and Zτ in a lump sum at time τ, if τ ≤ T (e.g., aswith a mortgage), equals the expected discounted value of the future cash flow,with expectation with respect to the martingale measure Q (see, e.g., [2] where thissecurity is implied to be a defaultable bond). In other words, we have

Mt = E[∫ τ∧T

t

cse−R

st

rθdθds + It<τ≤TZτe−R

τt

rθdθ

∣∣∣∣Gt

]. (2.3)

This is not a convenient formula for calculation because of its explicit dependence onthe stopping time τ. In the credit-risk literature (e.g., we can reformulate Proposi-tion 8.2.1 from [2]) it is known that the value of the security Mt admits the followingrepresentation

Mt = Iτ>tE[∫ T

t

(cs + Zsγs

)e−

R st(rθ+γθ)dθds

∣∣∣∣Ft

]. (2.4)

This result removes the explicit involvement of τ in (2.3) by employing its intensityprocess γt.

Note on the connection to illiquid securities. The expectation (2.3) has beenextensively studied in the literature on defaultable securities (see, for example,[2, 21, 22] on the subject). The timing of default τ is uncertain given the informationavailable to investors. But, nevertheless, the likelihood of default is influenced bythis information. This is reflected by the Ft-adapted intensity γt. Our idea is that asimilar set-up can be applied to securities with “liquidity risk.” Here under “liquidityrisk,” we understand the risk not to have the ability to execute the transaction whenone plans to do it, i.e., not the way it is understood in the academic literature (asask and bid spread) but the way practitioners call it. For example, if one wants

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to execute an illiquid option (i.e., based on available information he/she concludedthat it is profitable to do so), the investor may not be able to do it at the righttime or may miss the opportunity altogether. The mortgage prepayment option isan example of a very illiquid security for the mortgagor.

If Zt := P (t), the expectations (2.3) and (2.4) define the mortgage price. Weassume that the mortgage rate process mt is uniformly bounded. From (2.1) and(2.2) we can conclude that P (t) and

∫ t

0 csds are uniformly bounded too. Therefore,from (2.3), the mortgage price is uniformly bounded as well.

Note on discrete coupon payments. If one assumes discrete coupon payments,then the analogous result can follow from the same Proposition 8.2.1 in [2]) with adividend process specified as a step-function. This would be the framework, forexample, with the set-up of Kau et al. [27] if we extended their model to thesub-optimal prepayment case. It can be analyzed along the same lines in whatfollows.

With the help of integration by parts and after some manipulations we can get thefollowing alternative formula for a mortgage price:

Mt = P (t) + E[ ∫ T

t

(ms − rs)P (s)e−R s

t(rθ+γθ)dθds

∣∣∣∣Ft

]. (2.5)

An analogous formula for securities which retire at a given rate can be found inLevin [30]. The intensity γ in (2.5) can have an interpretation of “retirement rate”in a risk neutral world. Formula (2.5) “resolves” the mortgage price around “fair”value P (t). It gives a promising (computational and theoretical) alternative to theformula (2.4) which is (often implicitly) used currently in the literature.

Formula (2.5) shows that a mortgage can be viewed as a defaultable interest-rateswap on a notional amount P (t). The parties exchange interest payments calculatedaccording to the scheduled (usually fixed) rate mt and interest rate rt up to pos-sible “default” (i.e., prepayment in our case), the timing of which is driven be theintensity γt.

Note on applications of formulae (2.4) and (2.5). If Zt is the outstandingprincipal P (t) and τ is called the prepayment time, then the security Mt in (2.4)should be called a mortgage. But (2.4) is much more general. For example:

• Later in Sec. 4 we will use (2.4) to define the mortgagor’s liability.• Instead of representing the whole coupon payment, ct can be used for a speci-

fied part of the coupon payment, as in pricing interest only (IO) and principalonly (PO) strips for example. Specifically, the interest only part, i.e., the part ofthe payment which covers interest on the principal P (t), is mtP (t). The rest ofthe coupon payment ct−mtP (t) goes to pay off the principal itself. Therefore thevalues of an IO strip, for instance, are given by formula (2.4) with substitution of

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898 Y. Goncharov

its “ct” and “P (t)” processes with appropriate payments, namely, with mtP (t)and 0 respectively. Hence the value of an IO strip is given by:

IO(t) = Iτ>tE[∫ T

t

msP (s)e−R s

t(rθ+γθ)dθds

∣∣∣∣Ft

].

• The alternative representation (2.5) can be used to price complicated mortgage-backed securities such as CMO’s. For example, consider sequential CMO’s. Wereceive a cash flow from a pool of mortgages only up to the nth prepayment inthe pool. Then the price of the security is given by

Mt = P (t) + E[∫ T

t

(ms − rs)P (s)Q(τn > s|Fs)e−R s

trθdθds

∣∣∣∣Ft

], (2.6)

where τn is the time of the nth prepayment and Q(τn > t|Ft) can be calculatedvia probabilities of individual borrowers (see [15] for further details). In particular,CMOs’ prices can be computed using a PDE approach as opposed to simulationsof prepayment used in the literature now.

Note on the real world and martingale measures. In the literature on mort-gage valuation, authors often capture the stochastic nature of prepayments withan empirically estimated prepayment function of some state process such as theborrower’s prepayment incentive, loan-to-value ratio, interest rates, etc. The com-mon feature of this literature is that the prepayment function is assumed to be thesame under the real world and martingale measures. Jarrow et al. [20], in their workon default risk, argue that this equivalence is an example of an implicitly appliedassumption of conditional diversification. Briefly (and rephrasing the authors soour wording is in terms of mortgage prepayment rather then default), the notionof conditional diversifibility requires that conditioning on the evolution of the stateprocesses, the prepayment processes of borrowers are independent of each other.This captures the idea that once the systematic parts of prepayment risk have beenisolated, the residual parts represent idiosyncratic, or borrower-specific, shocks thatare uncorrelated across borrowers. Examples of such shocks include divorce, a newjob or a lost job, arrival of a new family member, etc. As the nonsystematic risk isnot priced, it may justify the practice of using an empirically estimated prepaymentfunction for valuation purposes.

We can add that the creation of mortgage-backed securities, namely, the practiceof pooling individual mortgages, can be seen as the “physical” implementation of theabove argument. As such, we may view the prepayment intensity γt as a model forprepayment. In general, we may state that “mortgage model” = “intensity model” =“prepayment model” in the “ideal” world with a “perfect” prepayment model. Theprepayment model risk (in the sense of [31]), however, will have an impact on anintensity model, which will make the intensities different from the prepayment ratesobserved on the market. Questions about how close the prepayment intensity is

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under the risk-neutral and physical probability measures deserve deep empiricalresearch (see [28, 31]).

3. Specification of Refinancing Incentive

As we can see from (2.5), γt determines the price of a mortgage. It governs the tim-ing of prepayment, the most specific characteristic of mortgage modeling. Thereforeits specification is a cornerstone of mortgage pricing. Let Πt be a scalar quantitywhich measures the refinancing incentive of the borrower. We assume that refi-nancing becomes profitable as soon as the refinancing incentive is higher than therefinancing transaction cost, i.e., Πt > Ft. The larger the refinancing incentive themore likely the borrower will refinance his/her mortgage. Therefore, to emphasizethis dependence, we single out variable Π in the notation of the prepayment inten-sity, i.e., the process γt = γt(Π) is a function of Π for fixed (t, ω) ∈ [0, T ]× D. Thefunction γt(·) measures how “fast” the borrower is expected to use a refinancingopportunity. Existing mortgage models mostly assume that γt(Π) is a constant asa function of Π up to a point where Π is less than transaction costs and a greaterconstant or merely growing linear function. The actual form of γt dependence on Πis borrower specific and, therefore, it is a (important!) matter of statistical research.Once we find γt(·), it should be stable with respect to changes in economical vari-ables and structural changes in market. These changes are reflected in the refinanc-ing incentive Π itself.13 According to the way this quantity is constructed we canclassify the models in the literature for valuation of mortgage contracts into oneof two categories: the option-based models and the empirical mortgage-rate-basedmodels (empirical MRB for short).

Note on the empirical comparison of the approaches. This classificationgives an idea of “targeted” mortgage research. If one wants to investigate whichapproach is better, i.e., gives more precise prepayment rates, then, ideally, he/sheneeds to test models with appropriate Πt specifications, but the same class of theprepayment intensity functions γt(·). Otherwise, the answer cannot be conclusive.For example, Stanton [39] reports that his prepayment model based on a one-factorinterest rate model performs as well as Schwartz and Torous [37] prepayment modelbased on a two-factor interest rate model. This does not mean that the option-based approach is better than the empirical MRB because the outcome might beattributed to the better family of the prepayment intensity functions. Indeed, a step-function (implied by the Stanton’s model) clearly fits the known form of empiricalprepayment intensity (e.g., [18]) better than a “linear” intensity implied in Schwartzand Torous [37].

13Given fixed Π, γt(Π) may still depend on time to explain seasonality and other explanatoryvariables such as unemployment rate, etc.

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900 Y. Goncharov

Since most considerations will be done for fixed-rate mortgages, we will assume thatthe mortgage rate is fixed at the origination of the mortgage contract, i.e., mt is afixed mortgage rate for a loan originated at time t.14

3.1. MRB approaches

A simple way to measure the prepayment incentive is to compare the contractmortgage m0 with an available-for-refinancing mortgage rate mt. On the web, forexample, one can find numerous calculators which can say if it is profitable torefinance (and how much one saves) on the basis of mortgage rates information. Theapproach is intuitively clear and assumes that the borrower considers refinancing toanother mortgage of the same type. As Hayre and Rajan [18] report, the fixed-rate30-year mortgage has retained its popularity as the refinancing vehicle of choicefor mortgagors with an existing 30-year loan. That gives a rationale for the MRBdefinition of the refinancing incentive.

The refinancing incentive Πt can be the difference m0 − mt (models by[24, 25, 37, 38] assume this specification), the rational expression m0/mt (as wasargued in [18, 36], it is able to capture the refinancing incentive better than thedifference), or it can be given by direct computation of how much the borrower willsave on coupon payments with refinancing [5–7]. Depending on the type (e.g., a 30-year or 15-year mortgage rate) of mortgage rates mt and m0 considered, the expres-sions above approximate/evaluate potential savings from refinancing a m0-fixed-ratemortgage with a (30-year or 15-year mortgage rate) mt-fixed-rate mortgage.

The mortgage rate can be defined exogenously (empirically). In this (empiricalMRB) case a common choice is to model the mortgage rate as the 10-year Treasurynote yield plus a constant. As reported in Boudoukh et al. [3], the 10-year Treasurynote yield has correlation of 0.98 with the mortgage rate. Belbase and Szakallas [1]found that Libor swap rates are better predictors than the Treasury yield (thecorrelation of 30-year mortgage rate vs. 10-year Libor swap rate and vs. 10-yearTreasury yield are 0.983 and 0.915, respectively, in their study).

We can extend this approach by defining the mortgage rate process endoge-nously (this is done in Sec. 6). After solving a mortgage rate equation we havethe endogenous prepayment intensity γt(Πt) (i.e., prepayment model). As aresult we get a model which is “immuned” to structural changes in the underlyingvariables.

MRB approaches allow an easy fit to prepayment data as opposed to option-based approaches. The mortgage rate data are readily available, thus the refinanc-ing incentive is available without “additional computation”. We can use this datatogether with prepayment data to calibrate the prepayment intensity function.

14That may not mean that the coupon payments ct are constant. It can change according tosome schedule. Examples of such mortgages are graduated payment mortgages and growing equitymortgages. We refer the reader to Sec. 2.1 or Fabozzi [12].

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3.2. “Traditional” option-based approach

The MRB approach is “naive:” the MRB refinancing incentive Πt(m0, mt) does notdepend on, for example, volatility of the interest rates.15 As is well known fromoption pricing theory (mortgages have embedded options!), the value of an optionis sensitive to changes in volatility of an underlying asset (see also [23]). We mayexpect that Πt decreases if the volatility increases (keeping the other factors thesame) since the probability of the option being (deeper) “in-the-money” is higher.This feature is fully incorporated in the following approach, which is based on theoption pricing idea.

Intuitively, the price of an American option can be found under an assumptionthat a holder of the option exercises it as soon as doing so is profitable, that is,when the benefit of exercising the option is higher than the benefit of keeping it. Inthe context of mortgage pricing, if a borrower prepays the mortgage, he/she mustpay the outstanding principal P (t) plus (perhaps substantial) transaction costs and,in return, he/she is liberated from his/her obligations to pay coupons, i.e., he/she“gets back” his/her liability Lt. This idea stands behind the option-based definitionof the refinancing incentive, where the refinancing incentive is Πt = Lt − P (t).Theoretically, from this point of view16 it is optimal to prepay if Πt is greater thanthe transaction costs.

The liability Lt itself can be defined with the help of (2.4). We note that thedistinction between the liability to the mortgagor and the price of the asset tothe investor is that at the time of prepayment a holder of the security receives theoutstanding principal P (t), but the mortgagor pays P (t) plus a transaction cost,which we assume to be uniformly bounded and denote by Ft. Therefore, the liabilityLt has the interpretation of the price of a security paying coupons ct continuouslyup to time τ ∨ T, and P (t) + Ft upon prepayment of the mortgage. So we just useformula (2.4) with P (t) + Ft instead of Zt

17:

Lt = E[∫ T

t

(cs + [P (s) + Fs]γs)e−R

st

[rθ+γθ ]dθds

∣∣∣∣Ft

]. (3.1)

The traditional option-based approach implies that γt = γt(Lt−P (t)) and, thus,(3.1) is not a formula but an equation. The existence of a solution is discussed inSecs. 4 and 5. Here we note that the liability is uniformly bounded (it can be shownthe same way as boundness of the mortgage price in Sec. 2.3).

15One can include this dependence (and others), but it would be an ad-hoc approach.16This option-based approach was up to the recent time the “only” one in the literature (see, e.g.,[8, 26, 39]) in certain sense. This is the reason for naming this approach “traditional.” But it doesnot take into account the fact that borrowers mostly prepay with other mortgages and, therefore,assume a new (cheaper) liability. See Goncharov [15, 16] for further details and new option-basedapproaches which utilize this feature.17We could also change the coupon rate in the mortgage pricing formula to account for servicingfee. This is a relatively small amount and is usually neglected.

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902 Y. Goncharov

Note on the liability specification. The liability Lt is not just the discountedvalue of the cash flow ct; it takes into account future prepayment opportunities.It is important to note the presence of γt and Ft in the definition of the liability,i.e., the borrower is aware of the possibilities to miss prepayment opportunities inthe future and takes into account the future transaction costs associated with theprepayment. This feature is sometimes overlooked in the option-based mortgagemodeling literature.

After we solve Eq. (3.1), the prepayment intensity γt is an Ft-adapted processes;we are in the framework of Sec. 2.3, and can find the price of the mortgage or relatedmortgage security.

The traditional option-based approach does not require one to model the mort-gage rate process. The knowledge of the contract rate ct (i.e., the mortgage rate mt

at the origination of the mortgage) is sufficient. However, things get more compli-cated when we calibrate the prepayment intensity function. The borrower liabilityis not available to us without additional computations and, therefore, the refinanc-ing incentive data are not “free” as they were in the case of the MRB approaches.Therefore, the prepayment intensity calibration procedure should be done togetherwith the liability valuation, and this is potentially an expensive procedure.

3.3. MRB vs. option-based specification of refinancing incentive

An option-based measure of the refinancing incentive is based on a comparison ofthe outstanding principal P (t) and the liability of the mortgagor. Whatever kindof a loan the mortgagor chooses to pay off, the price of the loan at the originationshould coincide with P (t). On the other hand, we have to keep in mind that theborrower probably refinances to another mortgage of the same type and, thus, pro-longs the time of repayment of the loan (that may be undesirable for the mortgagor).However, from the option-based approach point of view, the borrower is indifferentto the length of repayment time. This can lead to a situation when the option-basedand MRB approaches give contradictory refinancing advices to the mortgagor. Theparadox comes from the absence of shorter-term refinancing vehicle. The followingsimple example illustrates this point.

Example. Let us consider a 30-year mortgage. Assume that the risk-free interestrate is deterministic and equals the constant r1 from 0 to 15 years, then 0 from 15to 30 years and the constant r2 from 30 years on. The prepayment intensity γt isarbitrary, but bounded.

From the mortgage rate Eq. (6.1) (see below) we can conclude that the mortgagerate at time 0 is less than r1, i.e., m0 < r1 (the particular value of m0 dependson the prepayment intensity γt which we do not specify here), and m0 → +∞monotonically as r1 → +∞. From the same equation we can see that at timet = 15 the mortgage rate m15 is monotonically increasing to +∞ as r2 → +∞.

The liability L15 is higher than P (15) because the interest rate rt is zero for t ∈

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(15, 30). Moreover, Lt − P (t) can be made larger than any specified transactioncost by choosing r1 large enough. This means that at time t = 15 it is profitableto refinance the mortgage from the option-based point of view for that value ofr1. However, choosing r2 large enough we can have m15 > m0. Therefore, at timet = 15 refinancing to another 30-year mortgage is unprofitable from the MRBpoint of view. It is unlikely that the borrower will follow “option-based” advice torefinance to another mortgage with a higher rate.

This shows that the option-based measure of refinancing incentive may not beadequate. This measure neglects the borrower’s preferences and ability to pay a cer-tain cash flow. The outstanding principal P (t) is compared with the liability, whichis based on the behavior of economic variables over the period of time [t, T ], thusmaking the traditional option-based refinancing incentive measure “comparable”18

with refinancing the current mortgage into a (T −t)-year fixed-rate mortgage (whichis valid over the same period of time), the choice of which is not readily availableto a residential mortgagor. This effect may be negligible for new mortgages.

Another reason why the MRB choice of incentive measure may be more appro-priate than the option-based measure is that the typical residential borrower is notfinancially astute, as sophisticated financial models are not freely accessible for thegeneral public. A mortgagor calculates his/her amount of savings (true and ultimatemeasure of incentive to prepay his/her mortgage!) with refinancing, using currentlyavailable alternative mortgage rates and a straight-forward “simple” financial model(2.1) (like in [36], from which the authors get m0/mt as an appropriate incentivemeasure). Such “refinancing” calculators are very popular, easily accessible on theworld wide web and are often used as an advertisement for refinancing. This mayintroduce a bias in prepayment behavior of mortgagors which may work in favor ofthe MRB approach.

Nevertheless, more empirical research is needed to compare these approaches“side-to-side.” New option-based approaches (see, [16]) may address the drawbackspointed out above.

Note on close-to-maturity mortgages. The option-based approach can be well-suited for close-to-maturity mortgages since borrowers can take advantage of lowshort term interest rates, (the mortgage rates can still be high) to repay their mort-gages using their own savings or refinancing to balloon or adjustable-rate mortgages.

4. Prepayment Intensity for Option-Based Approaches

In this section we prove the existence and uniqueness of a solution to Eq. (2.4) inthe case when the prepayment intensity γt is a function of the t-time price of the

18Actually, it is more precise to say that the traditional option-based approach assumes that amortgage is refinanced by taking a loan from a bank at a risk-free rate. This is not realistic andcontributes to the overestimated transaction costs in option-based models (see [15, 16]).

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904 Y. Goncharov

underlying security Mt, i.e., γt = γt(Mt). This covers the option-based models formortgage securities, e.g., liability equation (3.1).

Lemma 4.1. Let the processes At(x) and Bt(x) depend on the parameter x. Assumethat they are uniformly Lipschitz continuous in x, i.e., there exist constants L1 andL2 so that for all t ≥ 0

|At(x) − At(y)| ≤ L1|x − y|, |Bt(x) − Bt(y)| ≤ L2|x − y|.Next, suppose At(x) is bounded by some finite constant L3. Then there exists asolution to equation

xt = J[x]t := E[∫ T

t

As(xs)e−R

st

Bθ(xθ)dθds

∣∣∣∣Ft

]. (4.1)

This solution is unique in the set of bounded Ft-adapted processes.

Proof. Let Dαb be a space of a bounded Ft-adapted processes with the norm

‖x‖α = sup(t,ω)∈[0,T ]×Ω

e−α(T−t)|xt(ω)|.

The theorem is equivalent to the problem of existence and uniqueness of the fixedpoint x ∈ Dα

b of the operator J[x], which is defined as the right-hand side of theEq. (4.1). Clearly, J maps Dα

b into itself. Moreover, the operator is a contraction inthat space for α large enough. Indeed, for bounded Ft-adapted processes xt and yt

we have

∣∣J[x]t − J[y]t∣∣ ≤ E

[∫ T

t

|As(xs) − As(ys)|e−R

st

Bθ(xθ)dθds

∣∣∣∣Ft

]

+E[∫ T

t

As(ys)∣∣e− R

st

Bθ(xθ) dθ − e−R

st

Bθ(yθ)dθ∣∣ ds

∣∣∣∣Ft

]

≤∫ T

t

L1‖x − y‖αeα(T−s)ds

+E[∫ T

t

As(ys)∫ s

t

L2‖x − y‖αe−α(T−θ)dθds

∣∣∣∣Ft

]

<L1 + L2L3(T − t)

αeα(T−t)‖x − y‖α.

Taking the norm ‖ · ‖α of this inequality we see that operator J is a contraction ifα > L2 + L1L3(T − t):

‖J[x] − J[y]‖α ≤ β‖x − y‖α,

where the constant β = [L2 +L1L3(T − t)]/α < 1. We thus conclude that the equa-tion xt = J[x]t has a unique solution in the set of bounded Ft-adapted processes.

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Theorem 4.1. Suppose the process γt(x) is uniformly Lipschitz continuous in theparameter x ∈ (0, M ′) uniformly in (t, ω) ∈ [0, T ] × D for sufficiently large M ′.Then there exists a unique solution to Eq. (2.4) in the set of bounded Ft-adaptedprocesses.

Proof. The result is a straightforward consequence of Lemma 4.1. The conditionsof Lemma 4.1 are easily verified if we take into account uniform boundness of theprocesses and the fact that Mt is a priory bounded from (2.3).

5. Diffusion State Process

To formulate the mortgage model in a continuous-time diffusion state process set-ting, which is popular in financial applications, we assume that all relevant factors inthe model (i.e., rt, ct, P (t) and γt) are deterministic functions of time t and a stateprocess Xt = (X1

t , . . . , Xnt ), for some n. Additionally, if the option-based approach

is under consideration, γt may aso depend on the underlying security Mt (it may bethe borrower’s liability Lt which is defined in Sec. 3.2 or the price of the mortgageMt itself — both are defined with the help of (2.4)). To avoid long expressions wewill use ft for f(t, Xt), so that instead of writing r(t, Xt) and γ(t, Xt, M(t, Xt)) weuse rt and γt(Mt).

The state process Xt is a diffusion process following the stochastic differentialequation

dXt = µ(t, Xt)dt + σ(t, Xt)dWt, Xs = x ∈ D (5.1)

with an m-dimensional Brownian motion Wt ∈ Rm and functions µ : [0, T ] × D →

Rn, σ : [0, T ]× D → R

n × Rm, where D is a domain (i.e., open and connected set)

in Rn. If the dependence of Xt on the initial conditions (s, x) will be important,

the notation Xs,xt will be used, i.e., in particular we have Xt,x

t = x. If the initialtime (and the initial position) will be fixed and clear from the context we skip thedependence on it and use notation Xx

t (or notation Xt).We assume the following conditions on the coefficients of the stochastic differ-

ential equation:

A The functions µ(t, x) and σ(t, x) are locally Lipschitz continuous in x ∈ D,

uniformly in t ∈ [0, T ].B The solution Xt of Eq. (5.1) neither explodes nor leaves D before T, i.e.,

Q[supt∈[s,T ] |Xs,x

t | < ∞ and Xs,xt ∈ D for all t ∈ [t, T ]

]= 1 for all (s, x) ∈

[0, T ) × D.

These assumptions guarantee that SDE (5.1) has a unique solution Xs,xt on the

interval [s, T ] and that this solution is continuous as a function of (t, s, x) (seeTheorem II.5.2 of [29]).

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906 Y. Goncharov

Finally, to state the equivalence of the martingale and PDE approaches weadditionally need these assumptions:

C The functions r, c, P and γ are Holder continuous in (t, x) ∈ [0, T ] × D; thefunction γ is uniformly Lipschitz continuous in M uniformly for all (t, x) ∈[0, T ]× D.

D σ(t, x) = 0 for all (t, x) ∈ [0, T ]× D.

Let operator A be the generator of the diffusion state process Xt, i.e.,

A :=12

∑i,j

(σσT )i,j∂2

∂xi∂xj+

∑i

µi∂

∂xi.

We can state the following Feynman-Kac representation result.

Theorem 5.1. Assume that Conditions A, B, C and D hold. Then the expectation

Mt(x) = E[∫ T

t

(cs + P (s)γs(Ms))e−R

st

[rθ+γθ(Mθ)]dθds

∣∣∣∣Xt = x

](5.2)

is a unique classical solution Mt (i.e., Mt ∈ C1,2([0, T ] × D)) of the followingbackward semi-linear parabolic PDE19

∂M

∂t+ AM − [r + γ(M)]M + c + Pγ(M) = 0,

MT (x) = 0, (t, x) ∈ (0, T ) × D. (5.3)

Proof. This result can be deduced from Theorem 1 of Heath and Schweizer [19]which is proved for a linear case but is easily generalized to our semi-linear set-up with the help of the fact that condition (A3) in their paper (which carriesmuch of the proof burden), modified to a semi-linear case, can be verified, say, byTheorem 7.4.9 of Friedman [13]. The nontrivial boundary condition Mt(x) ∈ C2+δ

(for x on boundary) in this theorem can be verified by Gilbarg and Trudinger [14],where we, in turn, can use a priori continuity of Mt(x) (see Lemma 5.1 below).Additionally, we use the fact that all conditions stated on a “set of growing closeddomains” in Theorem 1 of Heath and Schweizer [19] can be streamlined becauseHolder and Lipschitz continuity on D imply uniform Holder and Lipschitz continuityon a bounded closed subset of the domain D. The existence of such (growing to D)domains with sufficiently smooth boundaries is trivial.

Note on the technical conditions. There are many sets of sufficient conditionsfor equivalence of martingale and PDE approaches (Feynman-Kac type results).Note, that most of them address only a linear case, while the above conditions are

19We left out the dependence of M, r, c, γ, P on time t and the value of the state variables x fornotational simplicity and to underline the nonlinear dependence of γ on M (we will keep thispractice in what follows).

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applicable for a case when the intensity γ depends on the security we are pricing.This is the case for the option-based approaches. Also, the conditions above areconsiderably less restrictive (see, e.g., Appendix E of [9] for an overview of theconditions and references). See Heath and Schweizer [19] for illustrations of thispoint.

A solution of the mortgage equation (5.2) or, equivalently, (5.3) depends on thetime t, the initial condition x of the diffusion state process Xt (i.e., X0 = x) andthe contract mortgage rate m, i.e., we have the map (t, x, m) → M(t, x; m) which isdefined by solving (5.2) or (5.3). Later (and for the proof of the above theorem) wewill need continuity of Mt with respect to m. It is stated in the following lemma.

Lemma 5.1. Let M(t, x, m) be a solution of Eq. (5.2). Then the map (t, x, m) →M(t, x, m) is continuous for (t, x, m) ∈ [0, T ]× D × [0,∞).

Proof. First, for a function u(t, x, m) on [0, T ]×D × [0,∞) we define an operatorJ as follows:

J[u]t := E[∫ T

t

(cs + Psγs(us))e−R

st[rθ+γθ(uθ)]dθ ds

∣∣∣∣Xt = x

],

where the notation ut is used for u(t, Xt, m). Then the solution M(t, x, m) ofEq. (5.2) is a fixed point of the operator J, i.e. u = J[u].

The operator J maps C([0, T ] × D × [0,∞)) into itself. Indeed, from TheoremII.5.2 of [29] we can conclude that the conditions stated in the present sectionguarantee not only the existence of a solution Xs,x

t to the diffusion equation (5.1) forany initial conditions Xs,x

s = x ∈ D, but also continuity of the map (t, s, x) → Xs,xt

on [s, T ] × [0, T ] × D. This map is uniformly continuous on any compact subset of[s, T ]× [0, T ]×D. Therefore

∫ t

s[r(θ, Xs,x

θ ) + γ(θ, Xs,xθ , u(t, Xs,x

θ , m)]dθ and∫ T

s[ct +

γθ(uθ)Pθ] ds are continuous. Since the expression under the expectation in (5.2) isbounded (it follows from the boundness of

∫ T

tcsds, P (t) and Mt which we assumed

in Sec. 2.3), the function J[u](t, x, m) is continuous for (t, x, m) ∈ [0, T ]×D× [0,∞)from the dominated convergence theorem.

Next, the proof of the Lemma 4.1 can be repeated here to show that the operatorJ is a contraction on the space of continuous functions C([0, T ]×D × [0,∞)) withthe norm

‖ut‖α = sup(t,x,m)∈[0,T ]×D×[0,∞)

e−α(T−t)|u(t, x, m)|

for some α large enough. Therefore, the unique fixed point of the operator J (whichis guaranteed by Lemma 4.1 lies in the space of continuous functions C([0, T ]×D×[0,∞)).

6. Mortgage Rate

It is of a vital practical interest for Wall Street to define the mortgage rate endoge-nously, i.e., to find what mortgage rate is implied by the current (riskless) yield

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curve and the prepayment behavior of a representative mortgagor. Consider thecase of fixed-rate mortgages, i.e., mt is a mortgage rate of a fixed-rate mortgageoriginated at time t. Then this can be done through the postulate that at originationthe value of a mortgage should be equal to the initial principal. For example, from(2.5), using the arbitrage argument that M0 = P0, we can get that the mortgagerate mt of a fixed-rate default-free mortgage at time t is a weighted average of therisk-free interest rate over the life-time T of the contract, that is,20

mt =E

[∫ T+t

t rsP (s − t)e−R

s0 (rθ+γθ)dθds

∣∣Ft

]E

[∫ T+t

t P (s − t)e−R

s0 (rθ+γθ)dθds

∣∣Ft

] . (6.1)

Unfortunately, this is an equation rather than a formula because at least P (s) andthus the right-hand side of (6.1) depend on the mortgage rate too. The prepaymentintensity γt may also depend on the mortgage rate. This dependence can be quitedifferent for different approaches. The prepayment intensity γt can be modeled insuch a way that it depends only on the contract mortgage rate mt for all s ∈ [t, T +t](this is the case of the traditional option-based approach where γt depends on mt

implicitly through the mortgage price or the borrower’s liability) or it can depend onthe future mortgage rates ms as well (as with the MRB approaches). The latter case(MRB) greatly complicates the problem because Eq. (6.1) is a functional equationrather than just a nonlinear equation in one variable as in the former case (theoption-based approaches). For results in a discrete time setting see Pliska [35].

If γt depends on the price of the mortgage Mt (the reduced option-basedapproach in what follows21) then we have the following theorem on the existence ofa solution.

Theorem 6.1. Consider the diffusion-state process specification of Sec. 5 for γt =γ(t, Xt, Mt). Then a continuous solution m(x) of the mortgage rate equation existswith specification22:

m(x) =E

[∫ T

0rsP (s)e−

R s0 [rθ+γθ(M

m(x)t )]dθds

∣∣X0 = x]

E[∫ T

0 P (s)e−R s0 [rθ+γθ(M

m(x)t )]dθds

∣∣X0 = x] (6.2)

Proof. From Lemma 5.1 we have that M(t, x, m) is continuous for (t, x, m) ∈[0, T ] × D × [0,∞). From Theorem II.5.2 of Kunita [29] we can conclude that theconditions of Sec. 5 guarantee existence of a solution Xx

t to the diffusion equa-tion (5.1) of Sec. 5 for any initial conditions Xx

0 = x ∈ D. Additionally, the theo-rem states that a realization Xx

t is continuous as a function of (t, x) ∈ [0, T ] × D.

20Mortgages are mostly traded in pools. Therefore, if one assumes heterogeneity of borrowers, thedenominator and the numerator of Eq. (6.1) should be “averaged” over a possible “risk-neutral”distribution of borrowers’ prepayment processes γt.21This is equally applicable to the traditional option-based approach.22We use notation M

m(x)t for the mortgage price to remind the reader about the ulterior depen-

dence of the mortgage price on the mortgage rate m.

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Therefore M(t, x, m) and Xxt are uniformly continuous on any compact subset

of [0, T ] × D × [0,∞) and we have that∫ t

0[r(θ, Xx

θ ) + γ(θ, Xxθ , M

m(x)t )]dθ and∫ T

0 r(θ, X0,xθ )ds are continuous. The expression under the expectation in the denom-

inator of (6.1) is bounded by P0 and, thus, is uniformly integrable. It is easy toshow that the expression under the expectation in the numerator is uniformly inte-grable too:

∫ T

0

rsP (s)e−R s0 (rθ+γθ(M

m(x)t ))dθds ≤ P0

∫ T

0

rse−R s

0 rθdθds < P0.

Therefore the expectations in the denominator and numerator of the mortgage rateEq. (6.2) are continuous in (x, m) ∈ D× [0,∞). Now let us fix x ∈ D. The existenceof a solution of the mortgage rate equation follows from the fact that the right handside of (6.1) is uniformly bounded from above and positive for all m for a fixed x

and thus Eq. (6.1) has a fixed point. The continuity of this solution m(x) followsfrom shown continuity of the right hand side of (6.1) as a function of m and x.

7. Numerical Scheme for the Mortgage Rate Equation

In this section we assume that the only stochastic process in the model is theinstantaneous interest rate rt. We consider a time-homogeneous model. Thus themortgage rate process mt is merely a function of the interest rate rt, i.e., mt = m(rt).For the right hand side of the mortgage rate Eq. (6.1) we use notation A(m), wherem = m(r)|r ≥ 0 is the mortgage rate function. The mortgage rate equationitself can be written as m = A(m) in this case, i.e., a solution to the mortgagerate equation is a fixed point of the operator A. This equation can be written asa family of uncoupled scalar equations m(r) = A(m(r)), r ≥ 0, for the traditionalor reduced option-based approaches, but it is essentially a functional equation for(non-empirical) MRB approaches. The numerical scheme below is formulated forthe latter case, but it is equally applicable for the option-based approaches if onedesires to find the mortgage rate as a function of the interest rate.

The refinancing incentive at time t is assumed to be a function of the con-tract and current mortgage rates, i.e., Πt = Π(m0, mt). Since it is not likelythat a borrower will refinance to a mortgage with the higher mortgage rate wepostulate that if m0 ≤ m1 then Π(m0, m1) = 0. We find it natural to assumemonotonicity of the mortgage rate, i.e., m(r1) ≤ m(r2) if r1 ≤ r2. A model thatviolates this assumption would look “suspicious” since the fall in interest rateswould imply an increase in mortgage rates with all else being equal. Next, weassume that Π(m0, m2) ≤ Π(m0, m1) for m1 ≤ m2, i.e., the lower the available-for-refinancing mortgage rate the more likely borrowers refinance their mortgages(recall that the prepayment intensity is an increasing function of the refinancingincentive Πt). Combining these monotonicity assumptions on m(·) and Π(m0, ·) weget that the prepayment intensity of some particular borrower is a decreasing func-tion of the interest rate rt, just as one would intuitively expect. Finally, we assume

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910 Y. Goncharov

that the prepayment intensity is a constant γo23 for the refinancing incentive Πt

less than the transaction cost.From the above assumptions, it follows that if one wants to know, say, the value

of some mortgage-backed security for a specific interest rate r∗, then one needs toknow only the mortgage rates for the values of the interest rates not greater thanr∗, i.e., the set m(r)|r ≤ r∗. In particular, m(r∗) can be found from Eq. (6.1)knowing only the set m(r)|r < r∗. This is the idea behind the following numericalscheme.

Let M = ri | i = 0, 1, 2, . . . , N be a grid over the interest rate values and leti = ri − ri−1 be its step sizes. Our purpose is to find approximations of m(ri),i = 0, 1, 2, . . . , N for which we use mi. We do not specify a numerical method forfinding the expectations in the mortgage rate Eq. (6.1). If this numerical methodrequires the knowledge of m(r) for intermediate r’s in the interest rate grid, thenwe assume that they are defined with the help of some interpolation based on M.Assume that r0 is the lowest value (for computational purposes) of the interest rate(i.e., zero for most models). Therefore m0 can be found as a solution of (6.1) withthe constant γ0 instead of the process γt, i.e., in this case we expect exogenousprepayment only (e.g., sale of the house due to divorce or new job in another city).This solution can be found with the help of an iterative procedure. Let m0

0 be our

01

2

3

3

3

4

4

4

0

i

0

i

r

m(r)

r r r1 2 3

m

m

m

m

m

m

m

m

m

Fig. 1. Consecutive application of the iterative scheme (7.3).

23γo can be a function of the interest rate rt and/or time since the origination of the mortgage tomodel a burnout effect. It does not change the method below.

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initial guess, then the sequence mn0 , n = 1, 2, . . . is determined inductively:

mi+10 = A(mi

0) =E

[∫ T

0rsP (s; mi

0)e−R s

trθdθ−γosds

∣∣r0

]E

[∫ T

0P (s; mi

0)e−R s

trθdθ−γosds

∣∣r0

] , (7.3)

where mi0 enters the right hand side through the definition of P (t) only. Now we

expect that mi0 → m0 for i → ∞.

From our specification we can conclude that to find mn for n = 1, 2, . . . , N weneed to know only the values of mk, k = 0, . . . , n − 1. Indeed, for rk, k = n, . . . , N

(i.e., rk ≤ rn) we have m(rk) ≥ m(rn) and, thus, γt is just a constant γo for thesevalues of the interest rate. Therefore, we can find mn for n = 1, 2, . . . , N in order.If we know the first n values of mn then the next mortgage rate mn+1 can be foundwith the help of the iterative procedure

mi+1n+1 = A(mi

n+1; mk| k = 0, . . . , n)

=E

[∫ T

0rsP (s; mi

n+1)e−R

st[rθ+γ(mi

n+1;mk| k=0,...,n)]dθds∣∣rn+1

]E

[∫ T

0P (s; mi

n+1)e−R s

t[rθ+γ(mi

n+1;mk| k=0,...,n)]dθds∣∣rn+1

] .

Finding mn’s successively, we provide ourselves with a good initial guess forthis iterative procedure. From econometric considerations we expect the functionm(r) to be continuous for a reasonable model. Therefore, mn should not be “far”from mn−1. If the underlying model gives differentiability of m(r)24 then we canuse Euler’s rule to make a better guess m0

n+1 = mn + (mn − mn−1)(n+1/n).This guess reduces the number of iterations needed for a given precision.

Note on the interest rate grid. Suppose the expectations in the mortgage rateEq. (6.1) are evaluated on the grid R = ri| i = 0, 1, 2, . . . , M with the characteris-tic step size R. Assume that the mortgage rate function m(r) is twice differentiableand is approximated on the grid M = ri| i = 0, 1, 2, . . . , N with the characteris-tic step size M. If the method employed to find the expectations is of the secondorder then it is reasonable to take M = R. However, if the method is of the firstorder then we can calculate the mortgage rate function m(r) on the grid M ⊂ Rwith the characteristic step size M = O(

√R) interpolating the values of m(r)to R (for calculation of the expectations) with the help of a higher order spline.

8. Summary and Concluding Remarks

This paper developed a general mortgage model subject to sub-optimal prepay-ment risk by using an intensity-based approach. The model is flexible and can beapplied to all types of mortgages available on the market. We classified (parametric)

24As is well known, monotonic functions are differentiable almost everywhere.

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mortgage models into MRB (Mortgage-Rate-Based) and option-based groups dueto the way the borrower’s prepayment incentive is measured. This division is naturaland is validated by the fact that implementations of MRB and option-based modelsuse different analytical and numerical approaches. The proposed MRB approachwith the endogenously defined mortgage rate process is new and needs a furthertheoretical and empirical investigation.

The general model in the paper gives background for “conclusive” empiricalcomparison of option-based and MRB approaches. Rather than comparing mort-gage models available in the literature we propose to address the narrower problemof identifying the “right” prepayment incentive leaving the question of “implementa-tion” of models (the interest rate model, the set of borrowers’ prepayment functions,computation of expectations) for a separate investigation. This “targeted research”is important in view of the fact that both approaches found their application onthe Wall Street25 while leading to different results; in particular (see Sec. 3.3), theygive opposite prepayment prediction in some circumstances.

Our general model is not tied to a particular numerical procedure as are option-based models which recognize non-optimality of prepayment (e.g., [8, 26]). Knowl-edge of the real underlying process can give an edge to a researcher since he/shecan be numerically more efficient. For example, Stanton’s model,26 is in fact a vari-ant of a splitting-up, numerical method applied to our model (see [16]) and is ofthe first order of convergence in time. It can be easily “upgraded” to the secondorder (e.g., using predictor-corrector for PDE (5.3)); such numerical schemes canhave acceptable precision with significantly reduced number of time-steps. This isvery important for mortgage securities, because mortgage modeling (calibration, inparticular) is a computationally heavy problem.

We formulated the problem of the exogenously defined mortgage rate process.We proved the existence of a solution for the option-based mortgage models. Thequestion of the existence and uniqueness of a solution for MRB approaches staysopen (for a discrete set-up, see [35]). The proposed numerical procedure for themortgage rate equation can be used to find a fixed point solution of the mortgagerate equation.

Acknowledgments

This paper is partially based on my Ph.D. dissertation. I am grateful to my adviserStanley R. Pliska for introducing me to this topic, helping me with comments andsuggestions, and for his patience and encouragement.

25The empirical approaches (which can be considered as MRB as pointed out in the introduction)prevail though; the author spoke with only one practitioner who said his firm used the empiricaland the option-based models. However, more papers in the academic literature find the option-based models promising.26It is the most advanced model in the academic option-based literature in a certain sense.

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an intensity-based approach to the valuation of mortgage contracts and computation of the endogenous...

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