1. ADVANCES IN MORTGAGE
VALUATION: AN
OPTION-THEORETIC APPROACH
A thesis submitted to the University of Manchester
for the degree of Doctor of Philosophy
in the Faculty of Engineering and Physical Sciences
2006
Nicholas J. Sharp
School of Mathematics
6. 7 References 177
A Fixed-rate mortgage valuation pseudocode 186
B Analytic approximation derivation 214
B.1 Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 214
B.2 Derivation of the general solution for any month . . . . . . . . . . . . 215
C Bridging solutions 218
Word count 49849 (main text only)
6
7. List of Tables
3.1 Contract specifications and other parameters which are fixed, all based on
parameters used in the literature. . . . . . . . . . . . . . . . . . . . . . 84
3.2 Comparison of equilibrium setting contract rates for σr = 5%, σH = 5%
calculated using the finite-difference approach (FD) and the perturbation
approach (Pert). The computation times for the two methods are also
shown. r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . . 88
3.3 Comparison of equilibrium setting contract rates for σr = 5%, σH = 10%
calculated using the finite-difference approach (FD) and the perturbation
approach (Pert). The computation times for the two methods are also
shown. r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . . 89
3.4 Comparison of equilibrium setting contract rates for σr = 10%, σH = 5%
calculated using the finite-difference approach (FD) and the perturbation
approach (Pert). The computation times for the two methods are also
shown. r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . . 90
3.5 Comparison of equilibrium setting contract rates for σr = 10%, σH = 10%
calculated using the finite-difference approach (FD) and the perturbation
approach (Pert). The computation times for the two methods are also
shown. r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . . 91
3.6 Comparison of mortgage component values for σr = 5%, σH = 5%, calcu-
lated using the ‘exact’ contract rate and the contract rate found using the
perturbation method, for different contract specifications. The loan is for
25 years, r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . 93
7
8. 3.7 Comparison of mortgage component values for σr = 5%, σH = 10%, calcu-
lated using the ‘exact’ contract rate and the contract rate found using the
perturbation method, for different contract specifications. The loan is for
25 years, r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . 94
3.8 Comparison of mortgage component values for σr = 10%, σH = 5%, calcu-
lated using the ‘exact’ contract rate and the contract rate found using the
perturbation method, for different contract specifications. The loan is for
25 years, r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . . . 95
3.9 Comparison of mortgage component values for σr = 10%, σH = 10%, cal-
culated using the ‘exact’ contract rate and the contract rate found using
the perturbation method, for different contract specifications. The loan is
for 25 years, r(0) = spot interest rate (%), ξ = arrangement fee (%). . . . 96
4.1 Comparison of equilibrium contract rates and mortgage component values
for σr = 5%, σH = 5%, for different prepayment assumptions. The loan is
for 15 years, r(0) = spot interest rate (%). . . . . . . . . . . . . . . . . . 138
4.2 As in figure 4.1 except that σr = 5%, σH = 10%. . . . . . . . . . . . . . 138
4.3 As in figure 4.1 except that σr = 10%, σH = 5%. . . . . . . . . . . . . . 139
4.4 As in figure 4.1 except that σr = 10%, σH = 10%. . . . . . . . . . . . . . 139
5.1 Error in value of payments (for a FRM) published in Kau et al. (1993).
The analytic value of payments is calculated using equation (5.25). θ = 0.1,
κ = 0.25, n = 180, initial house price $100000. LTV = ratio of loan
to initial value of house, r(0) = initial interest rate and contract rate are
shown as percentages. . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8
9. 5.2 Component values for the ARM calculated using the improved auxiliary-
variable approach. Results without parentheses are for a 1.5% teaser; results
with parentheses are without teasers. All results are to par value for a 15-
year loan: spot interest rate r(0) = 8%, long-term mean θ = 10%, speed
of reversion κ = 25%, correlation coefficient ρ = 0, service flow δ = 8.5%,
interest-rate volatility σr = 10%, house-price volatility σH = 15%, points
ξ = 1.5%, insurance coverage φ = 25%, and a 90% loan-to-value ratio.
Initial margin was set at 100 basis point. Fixed-rate component values
given for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.3 Component values for the ARM calculated using the new valuation method-
ology. Parameter details identical to table 5.2. . . . . . . . . . . . . . . . 170
9
10. List of Figures
1.1 An illustration of the creation of a generic MBS showing the movement of
cash. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1 A graph of index(r) against interest rate r. For each line style, the long-
term mean of the short-term interest rate θ, is 0.1, 0.2 and 0.3 from the
bottom to the top, with σr = 0.1 and κ = 0.25. . . . . . . . . . . . . . . 46
4.1 An illustration of the state space for a Parisian option. . . . . . . . . . . 107
4.2 Valuation of the Parisian up-and-out call option with E = 10, ¯S = 12,
¯T = 0.1, T = 1, σ = 0.2 and r = 0.05. . . . . . . . . . . . . . . . . . . . 113
4.3 Parisian up-and-out call option at three different barrier times with E = 10,
¯S = 12, ¯T = 0.1, T = 1, σ = 0.2 and r = 0.05. . . . . . . . . . . . . . . . 114
4.4 The delta of the Parisian up-and-out call with E = 10, ¯S = 12, ¯T = 0.1,
T = 1, σ = 0.2 and r = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . 114
4.5 The delta of the ParAsian up-and-out call with E = 10, ¯S = 12, ¯τ = 0.1,
T = 1, σ = 0.2 and r = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . 117
4.6 Comparison of the Parisian and ParAsian options with E = 10, ¯S = 12,
¯T = 0.1, T = 1, σ = 0.2 and r = 0.05. . . . . . . . . . . . . . . . . . . . 117
4.7 An illustration of the effect of waiting to prepay on the value of the mort-
gage, modelled using a consecutive occupation-time derivative. . . . . . . 121
4.8 An illustration of the general solution space at any time step for a FRM
mortgage with the new prepayment model. . . . . . . . . . . . . . . . . 122
10
11. 4.9 An illustration of the finite grid in the house price H and interest rate r
dimensions, the approximate location taken as the free boundary position
is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.10 Mortgage value at origination V (H = 120000, r, τ1 = T1, ¯τ = ¯T) against
interest rate r for eight different decision times. For each line style, the
decision time ¯T is zero (this corresponds to the original prepayment as-
sumption), T/8, T/4, T/2, 3T/4, T, 5T/4 and 3T/2 from the bottom to
the top. For the case when κ = 0.25, θ = 0.1, δ = 0.085, σH = 0.1,
σr = 0.1, ρ = 0, c = 0.111805, ratio of loan to initial value of house = 0.9,
H(0) = $100000, r(0) = 0.1 and ξ = 0.015 for a 15 year loan. . . . . . . . 131
4.11 Mortgage value at origination V (H, r, τ1 = T1, ¯τ = ¯T) when the decision
time is zero, ¯T = 0. The other parameters are identical to those stated in
figure 4.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.12 Mortgage value at origination V (H, r, τ1 = T1, ¯τ = ¯T) when the decision
time tends to infinity ¯T → ∞. The other parameters are identical to those
stated in figure 4.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.13 Prepayment value at origination C(H = 120000, r, τ1 = T1, ¯τ = ¯T) against
interest rate r for four different decision times. For each line style, the de-
cision time ¯T is zero (this corresponds to the original prepayment assump-
tion), T/2, T and 3T/2 from the top to the bottom. The other parameters
are identical to those stated in figure 4.10. . . . . . . . . . . . . . . . . . 135
4.14 Mortgage ‘value’ at origination V (H = 100000, r, τ1 = T1, ¯τ) against interest
rate r at selected times until prepayment ¯τ (equal intervals). The decision
time is 1.5 months ¯T = 3T/2, other parameters are identical to those stated
in figure 4.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.1 An illustration of the unknown data point V (x, y) surrounded by its nearest
grid points, at which the value of V is known. . . . . . . . . . . . . . . . 160
11
12. 5.2 A graph of contract rate c(i, r) against interest rate r. Shown is the initial
contract rate c(0, r) (solid line), the contract rate after the first adjust-
ment date c(1, r) (thick dashed line), the contract rate after the second
adjustment date c(2, r) (thiner dashed line) and the contract rate after the
final adjustment date c(14, r) (smallest dashed line). For the case when
r(0) = 0.08, κ = 0.25, σr = 0.1, margin = 0.019, teaser = 0.015, y = 0.01,
l = 0.05, 15 year loan. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.3 A graph of contract rate c(i, r) against adjustment date i. For each line
style, interest rates are 0, 0.016, 0.032, 0.048, 0.064, 0.08, 0.096, 0.112 and
0.12 from the bottom to the top. For the case when r(0) = 0.08, κ = 0.25,
σr = 0.1, margin = 0.019, teaser = 0.015, y = 0.01, l = 0.05, 15 year loan. 164
C.1 An illustration of the solution space for the default option in the final month
n. The thick line represents the position of the required bridging solution. 219
12
13. Abstract
This thesis improves on existing theoretical work on the pricing of mortgages as
derivative assets, generally termed the option-pricing approach to mortgage valuation.
In order that mortgage valuation is realistic and consequently not trivial, the future
must be uncertain; therefore, the problems considered in this thesis operate within a
stochastic economic environment.
A highly accurate numerical scheme is presented, to tackle the partial differential
equations that arise in fixed-rate mortgage valuation, and further a novel (analytic)
singular perturbation approach is also developed. The analytic approximations pro-
duced result in a significant increase in the efficiency of solution. A new prepayment
model is also developed, which improves the modelling of the borrower’s decision pro-
cess by incorporating occupation-time derivatives in the valuation framework. This
simulates a delay in prepayment by the borrower, thus increasing the value of the
mortgage to the lender. Empirical work supports this theory, and the new model
should have positive implications for accurate mortgage-backed security pricing. For
the more complex problem of adjustable-rate mortgage valuation, improvements are
made to an existing approach by employing a superior numerical technique, and then
a new drastically more efficient valuation methodology is developed.
13
14. Declaration
No portion of the work referred to in this thesis has been
submitted in support of an application for another degree
or qualification of this or any other university or other
institution of learning.
14
15. Copyright
Copyright in text of this thesis rests with the Author. Copies (by any process)
either in full, or of extracts, may be made only in accordance with instructions given
by the Author and lodged in the John Rylands University Library of Manchester.
Details may be obtained from the Librarian. This page must form part of any such
copies made. Further copies (by any process) of copies made in accordance with such
instructions may not be made without the permission (in writing) of the Author.
The ownership of any intellectual property rights which may be described in this
thesis is vested in The University of Manchester, subject to any prior agreement to
the contrary, and may not be made available for use by third parties without the
written permission of the University, which will prescribe the terms and conditions
of any such agreement.
Further information on the conditions under which disclosures and exploitation
may take place is available from the Head of the School of Mathematics.
15
16. Acknowledgements
I would like to thank my supervisors, Peter Duck and David Newton, for their advice,
ideas and guidance throughout the last three years. In particular, to David for
introducing me to this topic, helping me with suggestions and motivation and to
Peter for his patience and encouragement with technical aspects of my work. I thank
my parents for their continual encouragement and for their faith in me. I would
like to thank my colleague and great friend Paul Johnson for, among other things,
his insights on some numerical aspects of my work. And I acknowledge all of my
close friends for keeping me sane, especially Claire, for being there when I needed
her. Finally, I give thanks to the EPSRC for their generous funding, without which
I would not have been able to attend conferences in Boston, Cambridge and Tokyo.
16
18. The Author
A Mancunian born and bred, Nicholas Sharp received his BSc in Mathematics from
the University of Manchester in 2000. He continued his studies in the School of
Mathematics to pursue his PhD in Mathematical Finance under the joint supervision
of Prof. David P. Newton and Prof. Peter W. Duck.
Nicholas Sharp started his PhD programme in September 2003. Since then, one
paper (an adapted version of chapter 3 of this thesis) has been accepted at the 4th
World Congress of the Bachelier Finance Society 2006 in Tokyo, Japan. This paper
as also been accepted for publication in the Journal of Real Estate Finance and
Economics.
His principal research interests lie in the field of option-theoretic mortgage valua-
tion. He is currently improving theoretical models through advancements in the nu-
merical methods employed in pricing mortgage contracts and through the utilisation
of exotic option pricing techniques to improve the modelling of borrower behaviour.
18
19. Chapter 1
Introduction
A mortgage (literally meaning a dead pledge) is a type of financial contract which
falls under the fixed-income product umbrella. It is a legal document by which a real
estate asset is pledged as security for the repayment of a loan; the pledge is cancelled
when the debt is paid in full. This type of debt instrument can be treated as a
derivative security. The mortgage derives its value from the evolution of the global
economy, via the underlying house price and the term structure of interest rates. A
mortgage is a prime example of a financial product that can be modelled and then
valued using option-pricing theory.
The lender, who issues the contract, would like to know the value of the future
cashflows that will be received as the result of the borrower making the scheduled
monthly payments. The value of the mortgage is not simply the time value of these
payments, since the borrower may terminate the contract prior to maturity, thus
terminating the projected cashflows. The valuation of mortgages involves the bor-
rower’s two options embedded in the contract to minimise the market value of the
loan. The borrower has the option to prepay the remainder of the outstanding bal-
ance owed if interest rates are financially favourable; this is an American call option
which spans the whole mortgage. The borrower also has the option to default on
the mortgage when a monthly payment falls due; this amounts to a series of linked
monthly European options.
As the mortgage is a contract between two parties, it is assumed that neither
19
20. CHAPTER 1. INTRODUCTION 20
would enter into an agreement unless it was fair at the onset. This means that the
value of the mortgage to the lender at origination (when the contract begins) must
be equal to the amount lent to the borrower. If this is in fact the case then it can
be said that the contract is in equilibrium at origination. The mortgage value at
origination will permit contractual arbitrage unless this is the case.
To give an idea of the amount of money outstanding on residential mortgages,
£1 trillion was owed by British borrowers alone by the second quarter of 2006. This
figure is dwarfed by the collective worth of the unmortgaged property across the UK,
which stands at £3.6 trillion (according to figures from the Bank of England).
A related financial derivative is a mortgage-backed security. This product derives
its value from sets of mortgages where cash flows have been combined (securitised),
to form a more desirable debt instrument. If mortgage-backed securities are to be
priced accurately, then it is vital that mortgage loans themselves are valued just as
accurately. In 2003, the daily mortgage-backed security trading volume exceeded
$200 billion in the United States of America. For price calculations to be useful,
these must be timely and there remains a need for more rapid valuation.
This thesis introduces some novel techniques to value several different types of
mortgage loans accurately and efficiently. Also, a new termination model of prepay-
ment is introduced with a view to eventually improving the modelling and pricing
of mortgage-backed securities. Mortgage valuation is a complex derivative problem
which involves many intricate subtleties; these make the framework required to price
a mortgage very appealing from a mathematical point of view.
1.1 Types of mortgage considered
If a bank agrees to grant a loan to a borrower who uses the money for the specific
purpose of building or purchasing a house, the borrower typically pays a lower interest
rate than for standard consumer credit, because the home offers security to the bank.
If the loan cannot be repaid (this is a situation when the borrower exercises his
option to default on a monthly payment), the house can be sold and the proceeds
21. CHAPTER 1. INTRODUCTION 21
can be used for repayment. This mechanism is formally agreed in the terms of the
mortgage that the borrower receives from the bank. A mortgage is the legal claim the
bank holds, allowing the bank to satisfy the debt through foreclosure and sale of the
property, if necessary; the related loan is called a mortgage loan. There are various
types of mortgage loans. The interest rate the borrower has to pay can be fixed
or floating according to a specific index (Libor, for example); typically such loans
have a maturity of up to 30 years (in the US). Various types of fixed-rate mortgage
(abbreviated to FRM hereafter) are available. The constant-payment mortgage is
the most common type in the United States. In the United Kingdom, variable-rate
mortgages have been the historical norm, but in the past two decades especially,
FRMs have been offered. For a constant-payment mortgage, the monthly payment is
constant over the life of the loan.
All the information, for example, terminology, formulae, conditions etc, that is
necessary for the reader’s understanding of later chapters, is described in chapter 2.
1.1.1 Fixed-rate mortgage
This thesis will first concentrate on the typical case of constant-payment mortgages,
or fixed-repayment mortgages as they are known in the United Kingdom, with a
known initial maturity and a fixed contract rate. These mortgages are the dominant
collateral in the mortgage securitisation market. Also, since the mortgage market
is the largest component of the outstanding US bond market debt, it is important
that efficient models of the possible cash flows from these types of mortgages can
be realised.1
This thesis initially attempts to simplify the highly complex problem
that is the contingent claims mortgage valuation model. In chapter 3 an improved
FRM valuation methodology is introduced. An approximate analytic (singular per-
turbation) approach is used in a huge simplification of the valuation problem so that
it is reduced to calculating a few simple equations. As a benchmark, rather than
1
According to The Bond Market Association, as of June 2006, mortgage-related bond market
debt exceeded all other types of bond market debt (including municipal, treasury, corporate, federal
agency securities, money markets and asset backed).
22. CHAPTER 1. INTRODUCTION 22
using the finite-difference techniques already available in the literature, an improved
technique was first developed and this was employed as the benchmark with which to
compare the singular perturbation approach, to test the latter technique for efficiency
and accuracy.
Although the term fixed-repayment mortgage might suggest that future cash flows
are also fixed, this is not the case. In fact the real cash flows that originate from
a mortgage loan are not fixed at all. The reason is that borrowers generally have
the right to prepay the outstanding balance before the maturity of the loans. The
prepayment feature has similarities with the callability of a more usual bond.
In financial terms, the prepayment right can be viewed as an American call option.
Exercise is only considered here when it makes sense from a purely financial viewpoint.
No attempt is made to model any form of exogenous termination of the mortgage.
Further, only one form of endogenous termination is considered, which occurs when
the value of the mortgage to the bank is equal to the total debt the borrower has
to pay if they decide to prepay their mortgage. This is called financially rational
prepayment.
The idea of optimality for a borrower can be complicated and causes much debate
as to how to model correctly this idea in mortgage valuation. In reality a borrower
choses to prepay based on individual circumstances. For example, the borrower may
chose to prepay for any of the following reasons:
• The borrower comes into money and is risk-averse; as a result the money is
used to pay off the mortgage early;
• The borrower moves house and pays off the mortgage with the proceeds from
the sale;
• The house is catastrophically lost (fire, earthquake, severe flood, etc), falls down
and the insurance payment goes to the lender;
• Interest rates fall and the borrower finds a better deal from another lender; this
is known as refinancing.
23. CHAPTER 1. INTRODUCTION 23
For the first three reasons above, prepayment could be modelled in this way en-
dogenously using some type of hazard process, see section 1.2.1 for details regarding
this modelling approach. The final reason, which depends on the movement of the
underlying interest rate, is ideally suited to be modelled using an option-theoretic
approach. To model the prepayment decision within an endogenously driven frame-
work the usual assumption is that the borrower pays off the mortgage when interest
rates decrease sufficiently.
It is clear that prepayment significantly affects the value of a mortgage. The
second significant piece of work in this thesis, chapter 4, concentrates on improving
the model of a borrower’s option to prepay. It has been suggested that borrowers do
not actually choose to prepay when it is financially rational to do so, rather, that there
is a time lag between the arrival of the information to them and the actual decision
being made to prepay the mortgage (see section 4.1 for a discussion of why borrowers
would wait to prepay). A new model of the borrower’s decision process regarding
prepayment is offered, in which occupation-time derivatives are incorporated into
the mortgage termination framework, that allows for more flexibility when modelling
prepayment by the borrower, as a time lag from when it is initially financially optimal
is permitted.
This improvement in model flexibility will hopefully have implications in the im-
provement of mortgage-backed security pricing. Empirical research suggests that
conventional option-pricing mortgage valuation models do not contain the necessary
features to price mortgage-backed securities accurately. It is hoped that the contribu-
tions made by improving the way prepayment is modelled will allow mortgage-backed
securities to be valued accurately within an option-theoretic framework.
1.1.2 Adjustable-rate mortgage
The final contribution in this thesis, chapter 5, moves on to the valuation of adjustable-
rate mortgages (abbreviated to ARMs hereafter). The more contractually and math-
ematically complex problem of ARMs poses some interesting modelling and valuation
24. CHAPTER 1. INTRODUCTION 24
problems; for a review of the literature in this field, see section 5.2. In summary, the
difficulty in modelling an ARM occurs as the common solution technique using back-
wards valuation, of the asset valuation partial differential equation (abbreviated to
PDE hereafter), is in the opposing temporal direction to the propagation of informa-
tion about the varying contract rate. Innovative solution techniques must be used to
overcome this difficulty. The work on this topic first utilises the auxiliary contract-
rate variable approach of Kau et al. (1993) and then improves the numerical scheme
employed by these authors, and further introduces a new technique to circumvent
the problem with the opposing direction of the contract rate information and the
solution scheme. This new technique removes the need for the auxiliary variable,
simultaneously overcoming the problems with accuracy and solution efficiency that
are inherent in the approach of Kau et al. (1993).
1.2 Mortgages as derivative assets
In the past two decades, theoretical pricing models of mortgages as derivative assets
have been accepted by the financial community as tools to improve the understanding
of markets themselves. It is very rare that economic reasoning, applied to understand
the workings of markets, leads to tools that have practical consequences. This thesis
examines and extends the option-pricing approach to mortgage valuation.
Although applying financial mathematics to price options is a relatively recent de-
velopment (dating back to the early 1970’s), the foundations of option-based pricing
models were laid down far earlier in economic research. At the turn of the twen-
tieth century, the French mathematician Louis Bachelier (1900) was the pioneer of
the random walk of financial market prices, Brownian motion and martingales. His
innovations predated the famous work by Einstein (1906) on Brownian motion for
physical processes. It is usually suggested that financial mathematics borrows theo-
ries from leading physicists, but in this case at least, finance arrived at a theory first.
In recent times the work of Merton (1973), and Black and Scholes (1973) on option-
pricing theory produced closed-form solutions to the problem of valuing a European
25. CHAPTER 1. INTRODUCTION 25
call option on an underlying asset for short-run scenarios, in which the interest rate
may be regarded as constant.
Moving to mortgages, two sources of uncertainty are present: term structure risk
and default risk. As house price, the source of default risk, is itself a traded asset
(if not a standardised one, with frequent trading), the analogy between a mortgage
on a house and an option on a stock is quite close. Default by a borrower acts in a
manner similar to a put option, since by defaulting the borrower returns the asset.
As interest rates are not a directly traded asset, an equilibrium model can be used
to value interest-dependent contracts. This means attitudes toward interest-rate risk
as well as the trend of interest movements enter into the valuation of mortgages,
the corresponding elements for house prices are not a consideration. If, as typically
assumed in the literature, the term structure is captured by a single variable, the
spot rate, the result is a single market price of risk. The commonly assumed local
expectations hypothesis concerning the term structure is nothing but the requirement
that this market price of risk disappear (Cox et al., 1981).
Unfortunately it is not the case, unlike for Black and Scholes (1973), that closed-
form solutions exist for complex contracts such as mortgages. To value these, nu-
merical solution techniques must be generally employed. Nevertheless, good analytic
approximations of mortgage valuation can be found - see chapter 3 for details of
an approximation method involving a novel perturbation approach to value FRMs.
The approach in chapter 3 simplifies the complexities of mortgage valuation, by first
appealing to the assumption of mathematically small volatilities for house-price and
interest rate and then using these small parameters in an asymptotic analysis of the
asset valuation PDE. This results in some especially simple analytic formulae which
can be used specifically to determine equilibrium contract rates. This technique not
only reduces the complexity of the problem but also vastly increases the efficiency of
obtaining a solution to the valuation problem. This is abundantly evident when com-
paring the computation time required to achieve a solution using the perturbation
approach with that required using a full numerical solution scheme.
With mortgages there are three ascending levels of assets: the underlying physical
26. CHAPTER 1. INTRODUCTION 26
real estate asset, the house; the contracts financing this real estate, which makes
up the primary mortgage market; and mortgage-backed securities (MBS) that arise
from pooling such mortgages, which make up the secondary mortgage market. Since
MBS are pools of individual mortgages, analysing their performance depends on
understanding the behaviour of the constituent mortgages.
The main difficulties in pricing MBS are that borrowers:
1. do not all act the same, so how can this be modelled within a single framework?
2. do not exercise their option to prepay when this appears to be financially opti-
mal, so how should this be modelled?
The work in chapter 4 addresses the second of these difficulties by contributing a new
prepayment model. By incorporating occupation-time derivatives into the option-
theoretic framework for mortgages, time lagged prepayment can be included. The
next section introduces MBS more formally and describes past research on pricing
these derivatives.
When the ultimate aim is to value a MBS, it is still necessary to value the under-
lying mortgage with precision and with great care to model the subtleties that affect
the security. There are two strands in the valuation literature, the split depends
on whether an econometric (reduced-form) valuation model is used or whether an
option-theoretic rational (structural) model is employed.
1.2.1 Reduced-form models
Reduced-form models are such that borrower decisions are related to a set of predic-
tors via a functional form chosen by the modeller. There are no theoretical restrictions
on this functional form, speeding development time and providing a great deal of flex-
ibility to match the historical data record closely. Unfortunately, there is no way to
determine how the estimated parameters should change in response to changes in
the economic environment, and this results in reduced-form models performing badly
out-of-sample (see section 1.2.4 for more details).
27. CHAPTER 1. INTRODUCTION 27
1.2.2 Structural models
Structural models are used exclusively throughout this thesis. The reason will be ex-
plained after the following survey of the techniques used in the literature. Structural
models can produce informative forecasts in economic environments unlike those seen
in the past, since mortgage terminations are the result of optimising behaviour by
agents in the model. The structural methodology links option exercise events to the
underlying fundamentals faced by the borrower.
1.2.3 Mortgage-backed securities
A mortgage-backed security is a security based on a pool of underlying mortgages.
Investors then buy a piece of this pool and in return receive a fraction of the sum of
all the interest and principal payments. MBS are usually based on mortgages that are
guaranteed by a government agency for payment of principal and a guarantee of timely
payment. The analysis of MBS concentrates on the nature of the underlying payment
stream, particularly the prepayments of principal prior to maturity. By buying into
this pool of mortgages, the investor gets a stake in the housing-loan market, but
with less of the prepayment risk (if payments are arranged to be on equal terms to
all bond holders). Since there are so many individual mortgages in each security, it
would seem only necessary to model the average behaviour, but what is the behaviour
of the average borrower? Recent work introduces borrower heterogeneity to model
different types of borrower actions, see section 1.2.4 for details.
An illustration of the creation of a generic MBS is shown in figure 1.1, which
shows the flow of the mortgage loan cash flows as they are securitised by the MBS
issuer, then sold to investors, as a more desirable debt instrument (where the investors
are not subjected to the prepayment risk in exchange for a lower return). A more
specific example of a MBS is a Collateralized Mortgage Obligations (CMO). These
securities are based on a MBS but in which there has been further pooling and/or
splitting so as to create securities, with different maturities for example. A typical
CMO might receive interest and principal only over a certain future time frame. MBS
28. CHAPTER 1. INTRODUCTION 28
Interest
Scheduled principal repayment
Prepayments
Interest
Scheduled principal repayment
Prepayments
Interest
Scheduled principal repayment
Prepayments
Interest
Scheduled principal repayment
Prepayments
Interest
Scheduled principal repayment
Prepayments
Interest
Scheduled principal repayment
Prepayments
Interest
Scheduled principal repayment
Prepayments
Interest
Scheduled principal repayment
Prepayments
Interest
Scheduled principal repayment
Prepayments
Interest
Scheduled principal repayment
Prepayments
$ $ $ $
Rule for distribution of cash flow
Pro rata basis
Pooled monthly cash flow:
Interest
Scheduled principal repayment
Monthly cash flow Investors
Pool
Loans
Prepayments
Figure 1.1: An illustration of the creation of a generic MBS showing the movement of cash.
29. CHAPTER 1. INTRODUCTION 29
can be stripped into principal and interest components. Principal Only (PO) MBS
receive only the principal payments and become worth more as prepayment increases.
Interest Only (IO) MBS receive only the interest payments. The latter can be very
risky since high levels of prepayment mean many fewer interest payments.
1.2.4 Previous work
Confirmation that a contingent claims approach to pricing MBS is useful is given by
Dunn and McConnell (1981a). This early work was based on a general equilibrium
theory of the term structure of interest rates under uncertainty, modelled a default-
free fixed-rate version of a MBS, where the interest-rate process is modelled using a
mean reverting stationary Markov process; suboptimal prepayment is modelled using
a Poisson-driven process. In their follow up work, Dunn and McConnell (1981b)
compare the value of an amortising callable bond, with MBS specification, to three
types of default-free bonds to show the impact of call, amortisation and prepayment
features on pricing, returns and risks of MBS. They conclude that amortisation and
prepayment features increase the price of a MBS and the callability feature decreases
its price. The effect of all three features is to reduce the interest-rate risk, and
consequently the expected return of a MBS relative to other securities which do
not have these features. Dunn and McConnell (1981b) also express the need for an
empirical study to determine if the prices generated by their model were consistent
with observed market prices.
In Brennan and Schwartz (1985) three arbitrage-based models for MBS are con-
trasted. Both the interest-rate uncertainty and the call policy used are said to have
an effect on pricing. A two-factor term structure model consisting of the short rate
and the consol rate (the yield on a bond of infinite maturity) is used.
Stepping away from optimal call policy models, Schwartz and Torous (1989) offer
an empirically based, reduced-form model. Here, maximum-likelihood techniques
are used to estimate a prepayment function from recent (at the time of publication)
price information. A proportional-hazards model is then used to make prepayment
30. CHAPTER 1. INTRODUCTION 30
decisions. This prepayment function is integrated into the Brennan and Schwartz
(1985) two-factor model for valuing default-free interest-dependent claims. Monte
Carlo simulation methods are used in the solution of the problem. A comparison
with an optimal, value-minimising call policy model is given. In conclusion, they
claim optimal call policy models cannot explain the fact that borrowers prepay their
loans when the prevailing refinancing rate exceeds their loan’s contract rate, and
conversely other borrowers do not prepay even when the contract rate on their loan
exceeds the prevailing refinancing rate. Also, the use of an estimated prepayment
function produces mortgage prices which are consistent with traded MBS prices.
Although the model results are closer to actual prices used in practice than those
produced using optimal, value-minimising call conditions, these reduced-form models
are notorious for performing badly out-of-sample; see Downing et al. (2005) for a
discussion of this problem. Regardless of their shortcomings reduced-form models
can provide an understanding of the empirical performance of existing contracts and
their pricing dynamics.
The later work of Schwartz and Torous (1992) is the first to introduce the pos-
sibility of default in the valuation of MBS. The borrower’s conditional probability
of prepayment is given by a prepayment function, while the borrower’s conditional
probability of default is given by a default function. Although a MBS is default free,
as it is guaranteed by the government, the cash flows to the security are not. There-
fore, the value of the security is affected by default. In the work of Schwartz and
Torous (1993), still with the inclusion of a default possibility, a Poisson regression
technique is developed, instead of a likelihood method, to estimate the parameters of
a proportional-hazards model for prepayment and default decisions. This is simply
an alternative way to find the parameters for the termination functions from real
data.
Stanton (1995) comments on the shortcomings of reduced-form models and their
inability to perform in different economic environments; instead, a rational mortgage
prepayment model is proposed. Akin to the work of Dunn and McConnell (1981a,b),
Stanton’s model estimates heterogeneity in the transaction costs faced by mortgage
31. CHAPTER 1. INTRODUCTION 31
holders, and borrowers make prepayment decisions at discrete intervals. Burnout
dependence is produced by letting expected prepayment rates depend on cumulative
historical prepayment levels. This allows prices to exceed par by more than the
transaction costs. The model gives a simple rational representation for prepayment,
and MBS value is given as a weighted sum of the market values of the underlying
mortgages.
So far, the discussion of the literature for MBS valuation has focused on what type
of model to use, be it structural or reduced-form, but Chen and Yang (1995) explore
deeper and discuss which actual interest-rate process should be used in pricing MBS.
Four processes are used to compare MBS prices, and they conclude that mortgage
pricing models are less sensitive to the underlying interest-rate process than a simple
coupon bond, and that this is due to the prepayment feature with mortgages.
In the work of Kariya et al. (2002), borrowers are allowed to act differently within
mortgage pools. A framework is provided whereby a short term interest rate is used
for discounting and a mortgage rate is used as an incentive factor for refinancing; a
second prepayment incentive factor based upon rising property values is also used.
This falls into the non-option methodology for pricing.
The more recent work of Kau and Slawson (2002) incorporates frictions into a
theoretical options-pricing model for mortgages. Here the model is still a rational
model of mortgage valuation, where prepayment and default are financial decisions
but the effect of borrower characteristics is introduced without destroying the options
theoretic framework. Three categories of friction are allowed for, including fixed and
variable transaction costs, sub-optimal termination and sub-optimal non-termination.
The adaptability and flexibility of an option-theoretic model is illustrated. The ability
to include borrower heterogeneity is shown not to require the loss of optimality.
A full spectrum of refinancing behaviour is modelled using a notion of refinancing
efficiency by Kalotay et al. (2004). They focus on understanding the market value
of the mortgage, rather than trying to predict future cashflows. Two separate yield
curves are used, one for discounting mortgage cashflows and the other for MBS cash-
flows. They give the following reasons why option-theoretic models, at present, tend
32. CHAPTER 1. INTRODUCTION 32
not to be used for prepayment modelling: most homeowners do not exercise their
options optimally; and option-based models are not able to explain observed MBS
prices. The authors show that in fact, a ‘rigorously constructed’ option-pricing model
does explain MBS prices well. MBS are said to be priced well with the assumption
that most homeowners exercise their refinancing option near-optimally. As mortgages
are not always refinanced using an optimal strategy (sold at par by the borrower to
the lender), the authors account for borrower heterogeneity by breaking the mort-
gage pool into buckets and assume that each bucket represents different refinancing
behaviour to price MBS well.
The empirical test of Downing et al. (2005) as to the importance of a second
factor confirmed that including the house price as a factor in mortgage valuation and
MBS pricing (to capture the effect of default by the borrower) is necessary. This adds
weight to the research of Schwartz and Torous (1992) and that of Kau and Slawson
(2002), which both include the possibility that the borrower will default.
In the recent research of Longstaff (2005), a multi-factor term structure approach
is used to incorporate borrower credit into the analysis. Results show that optimal
refinancing strategy can delay prepayment relative to conventional models, and that
mortgage values can exceed par by much more than the cost of refinancing. The
notion that a borrower’s financial situation affects the rate at which he can refinance,
including credit worthiness, is introduced. The borrower’s optimal refinancing strat-
egy involves considering the life of loan affects of refinancing. If the borrower’s credit
is poor he will have to refinance at a premium rate; this is modelled by adding a
credit spread to the prepayment cost. A Poisson process is used to add in the chance
of exogenous refinancing reasons. Borrowers then find it optimal to delay prepay-
ment beyond the point at which conventional models imply the mortgage should be
prepaid.
Other recent models include Dierker et al. (2005) and Dunn and Spatt (2005). As
in the two previously discussed articles, option exercise is modelled by endogenous
decisions made by borrowers to minimise the present value of their current mortgage
position.
33. CHAPTER 1. INTRODUCTION 33
Both reduced-form and structural models have the same goal, which is to account
realistically for all the embedded options in mortgage contracts. Kalotay et al. (2004)
comment that there is evidence for the use of both types of model in practice. As
mentioned previously, reduced-form models contain a great deal of flexibility to match
historical data closely. However, there is no guarantee that a functional form which
works well in-sample will perform as well out-of-sample. Also, there is no way to
determine how the estimated parameters should change in response to a change in
the economic environment. Although with basic structural models it is true that it is
difficult to give prepayment predictions that match observed prepayment behaviour,
and impossible to allow prices to exceed par by more than the transaction costs,
Kalotay et al. (2004) and Longstaff (2005) show that they are flexible enough that
mortgage features such as friction, borrower heterogeneity and many other subtleties
can be included. These make it possible to achieve realistic mortgage values within a
rational framework (Kalotay et al., 2004 and Longstaff, 2005). Evidence has also been
given for the importance of house price as a necessary factor. It is a crucial factor
in capturing the information about the default behaviour of the borrower. Although
MBS are generally guaranteed against default, default affects the cashflows from the
underlying mortgages themselves, as changes in house value affect the borrower’s
decision to default, the cashflows to the MBS will also be affected indirectly by house
price changes.
The next section details the two state variables (house price and interest rate)
which are used in this thesis as the sources of uncertainty in the economic environment
for which all problems will be set. The modelling will be of the rational option-
theoretic variety for the reasons discussed above and the processes chosen for the
house price and interest rate variables are discussed next.
1.3 Underlying state variables
It is possible to treat a mortgage as a derivative asset which exists within a stochastic
economic environment. The uncertainty which comes with this modelling setup could
34. CHAPTER 1. INTRODUCTION 34
affect risk preferences. Option theory can be used to show that the role of prefer-
ences is actually quite limited when applied to derivative assets. Valuation can be
performed as if the world were risk neutral (with some risk adjustments), so that the
value of a derivative asset is simply the expected present value of its future payoffs;
see Cox et al. (1985a) for a discussion of risk-neutralised pricing.
A mortgage derives its value from two state variables. Possibly the most obvious
(from the borrower’s point of view) is the price of the underlying real estate asset,
the house. The term structure of interest rates is the other state variable. This could
be considered the most relevant to the lender, as it will ultimately determine the
value of the payments made by the borrower, but it is the interaction between these
two factors that must be considered simultaneously to determine the value of the
mortgage.
The assumption that underpins the whole option-theoretic approach to mortgages
is that even though mortgages depend on the real economy through the house price
and term structure, mortgages themselves are not necessary to determine this un-
derlying economy. As a derivative asset is one that is not necessary to describe the
underlying real economy, it is a redundant asset and its value depends entirely on the
variables that do determine the underlying economy.
The choice of the processes in this thesis for the two state variables that model
the economic environment is consistent with recent literature (Kau et al. 1995; Kau
and Slawson 2002; Azevedo-Pereira et al. 2002, 2003).
1.3.1 House price
Merton (1973) lognormal diffusion process
The house-price process, equation (1.1) below, models house price behaviour as a
lognormal diffusion process; see Merton (1973) for more details.
In the contingent claim framework, let the true process describing the underlying
estate asset, the house price H, be
dH = (µ − δ)Hdt + σHHdXH, (1.1)
35. CHAPTER 1. INTRODUCTION 35
where:
µ is the instantaneous average rate of house-price appreciation,
δ is the ‘dividend-type’ per unit service flow provided by the house,
σH is the house-price volatility,
XH is the standardised Wiener process for house price.
The house-price appreciation µ, is analogous to the drift term for the more stan-
dard stock-price model. The service flow δ is analogous to a dividend on a stock as
the borrower benefits from the underlying asset (the borrower is allowed to live in
the real estate asset during the life of the mortgage contract). The borrower benefits
from the asset, therefore the price must drop by this amount otherwise arbitrage
would occur.
1.3.2 The term structure of interest rates
CIR (1985) mean reverting square root process
The term structure of interest rates is modelled using the single factor Cox et al.
(1985b) mean-reverting square root process. The single factor r is taken to be the
spot rate of interest.
Within the contingent claim framework let the true process describing the term
structure of interest rates, the spot rate r, be
dr = κ(θ − r)dt + σr
√
rdXr, (1.2)
where:
κ is the speed of adjustment in the mean reverting process,
θ is the long-term mean of the short-term interest rate r,
σr is the interest-rate volatility,
Xr is the standardised Wiener process for interest rate.
36. CHAPTER 1. INTRODUCTION 36
1.3.3 Correlation
The stochastic elements of the house-price process (1.1) and the spot interest-rate
process (1.2) which involve the standardised Wiener processes, XH for house price
and Xr for interest rate respectively, are correlated according to
dXHdXr = ρdt, (1.3)
where ρ is the instantaneous correlation coefficient between the two Wiener processes.
1.3.4 Risk adjustment
For house price and term structure to have any value, preferences, technology and
supply and demand considerations are incorporated into the price. The only other
factor that influences a derivative asset’s price is its market price of risk, which is
included in the risk adjustment of that variable. As the real estate asset underly-
ing a mortgage is itself a traded asset (Kau et al., 1993, 1995; Kau and Slawson,
2002; Azevedo-Pereira et al. 2002, 2003), the risk adjustment involves no external
parameters whatsoever. As the term structure follows the Cox et al. (1985b) process,
the market price of risk can be regarded as having been absorbed into the estima-
tion of reversion κ and long-term average θ parameters of the interest rate-process.
As a result, the local expectations hypothesis requires that this market price of risk
also disappear (Cox et al. 1981). This means that with risk adjustments taken care
of, it is possible to proceed with the expected present value calculation. Using risk
neutrality arguments the instantaneous average rate of house-price appreciation (the
drift term) can be taken as the interest rate (as the market price of risk for the house
price is taken as zero, which is explained above), Therefore µ = r in the process for
the house price, equation (1.1). These risk adjustment reasons are standard in the
literature.
37. CHAPTER 1. INTRODUCTION 37
1.4 Derivation of the asset valuation PDE
This section demonstrates a derivation of the asset valuation PDE using standard
hedged portfolio arguments.
The PDE for the valuation of any asset F = F(H, r, t) whose value is a function
only of house price H, interest rate r, and time t, can be found as follows. House
price is described by the stochastic differential equation (1.1) and stochastic interest
rate follows equation (1.2). Using Itˆo’s lemma (see Itˆo, 1951, for the details) on the
function F(H, r, t), it can be shown that,
dF =
∂F
∂t
dt +
∂F
∂H
dH +
∂F
∂r
dr +
1
2
∂2
F
∂H2
dH2
+ 2
∂2
F
∂H∂r
dHdr +
∂2
F
∂r2
dr2
+ · · · (1.4)
From stochastic calculus; dt2
→ 0, dX2
→ dt, and dXdt = o(dt), as dt → 0; then
from equation (1.1)
dH2
→ σ2
HH2
dX2
H → σ2
HH2
dt; (1.5)
and from equation (1.2) note that,
dr2
→ σ2
r rdX2
r → σ2
r rdt; (1.6)
and finally from equation (1.1), (1.2) and (1.3)
dHdr → σHσrH
√
rdXHdXr = ρσHσrH
√
rdt. (1.7)
Thus, Itˆo’s lemma for the two stochastic variables governed by (1.1) and (1.2) is,
dF =
∂F
∂t
dt +
∂F
∂H
dH +
∂F
∂r
dr +
1
2
σ2
HH2 ∂2
F
∂H2
+ 2ρσHσrH
√
r
∂2
F
∂H∂r
+ σ2
r r
∂2
F
∂r2
dt.
(1.8)
Now construct a portfolio Π, long one asset F1(H, r, t) with maturity T1, short ∆2 of
an asset F2(H, r, t) with maturity T2, and short ∆1 of the underlying asset H. Thus,
Π = F1 − ∆2F2 − ∆1H. (1.9)
The change in this portfolio over a time dt is,
dΠ = dF1 − ∆2dF2 − ∆1dH, (1.10)
38. CHAPTER 1. INTRODUCTION 38
where ∆1 and ∆2 are constant during this time. The effect of the service flow δ is
to cause the price of the underlying asset H to drop in value by δH over a time
dt. Therefore, the portfolio must change by an amount −δH∆1dt during this time.
Thus, the correct change in the value of the portfolio over a time dt is
dΠ = dF1 − ∆2dF2 − ∆1(dH + δHdt). (1.11)
With a careful choice of
∆2 =
∂F1/∂r
∂F2/∂r
(1.12)
and
∆1 =
∂F1
∂H
− ∆2
∂F2
∂H
(1.13)
the risk from the portfolio can be eliminated, i.e. the random components of the dH
and dr terms vanish, and dΠ becomes
dΠ =
∂F1
∂t
dt +
1
2
σ2
HH2 ∂2
F1
∂H2
+ 2ρσHσrH
√
r
∂2
F1
∂H∂r
+ σ2
r r
∂2
F1
∂r2
dt − δH
∂F1
dH
dt
−
∂F1/∂r
∂F2/∂r
∂F2
∂t
dt +
1
2
σ2
HH2 ∂2
F2
∂H2
+ 2ρσHσrH
√
r
∂2
F2
∂H∂r
+ σ2
r r
∂2
F2
∂r2
dt − δH
∂F2
dH
dt
= r F1 −
∂F1/∂r
∂F2/∂r
F2 −
∂F1
∂H
H +
∂F1/∂r
∂F2/∂r
∂F2
∂H
H dt. (1.14)
Here arbitrage arguments have been used to set the return on the portfolio equal to
rΠdt, since the growth of the portfolio in a time step dt is equal to the risk-free growth
rate of the portfolio, as the portfolio is now completely deterministic. Dividing by dt
and separating the F1 and F2 terms leads to,
1
∂F1/∂r
∂F1
∂t
+
1
2
σ2
HH2 ∂2
F1
∂H2
+ ρσHσrH
√
r
∂2
F1
∂H∂r
1
2
σ2
r r
∂2
F1
∂r2
+ (r − δ)H
∂F1
∂H
− rF1
=
1
∂F2/∂r
∂F2
∂t
+
1
2
σ2
HH2 ∂2
F2
∂H2
+ρσHσrH
√
r
∂2
F2
∂H∂r
1
2
σ2
r r
∂2
F2
∂r2
+(r−δ)H
∂F2
∂H
−rF2 .
(1.15)
Although this is one equation in two unknowns, the left-hand side is a function of T1
but not of T2 and the right-hand side is a function of T2 but not of T1. The only way
for this to be possible is for both sides to be independent of maturity date. Thus,
39. CHAPTER 1. INTRODUCTION 39
removing the subscript from F,
1
∂F/∂r
∂F
∂t
+
1
2
σ2
HH2 ∂2
F
∂H2
+ ρσHσrH
√
r
∂2
F
∂H∂r
+
1
2
σ2
r r
∂2
F
∂r2
+ (r − δ)H
∂F
∂H
− rF = a(H, r, t), (1.16)
is obtained for some function a(H, r, t). It is convenient to write a(H, r, t) = −κ(θ−r)
(this is a standard procedure in the literature, see Kau et al. 1993, 1995; Azevedo-
Pereira et al. 2002, 2003), which leads to the asset valuation PDE for F(H, r, t),
1
2
H2
σ2
H
∂2
F
∂H2
+ ρH
√
rσHσr
∂2
F
∂H∂r
+
1
2
rσ2
r
∂2
F
∂r2
+κ(θ − r)
∂F
∂r
+ (r − δ)H
∂F
∂H
+
∂F
∂t
− rF = 0. (1.17)
This PDE will be used extensively in this thesis to value fixed-rate (chapter 3) and
adjustable-rate mortgages (chapter 5), and a modified version will be used to value
a fixed-rate mortgage containing a new prepayment assumption (chapter 4).
40. Chapter 2
Foundations of mortgage valuation
It is first necessary to describe some concepts and ideas that underlie the complex
problem of mortgage valuation. This chapter acts as a reference source and will be
referred to where necessary later in this thesis, which will remove excess formulation
in later discussions of the improvements to be implemented.
2.1 Formulae
Initially, the problem of valuing a FRM is explored. This type of loan is repaid by
a series of equal monthly payments, made on pre-determined, equally-spaced dates.
The monthly payment MP and the outstanding balance following each payment
OB(i) are calculated using standard annuity formulae, which will be given in the
next section.
2.1.1 Value of monthly payments
The asset valuation PDE (1.17) is solved using a backward valuation procedure (see
section 2.5 for details). It is necessary to start the process from the known information
at maturity, referring to these known cashflows at the final moment of the contract,
rather than using the more common actuarial procedure of referring all the cashflows
at the origination of the loan.
To define the value of each monthly payment it is necessary to recognise that the
40
41. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 41
future value of the outstanding debt in the terminal period of the contract must be
equal to the future value of all the payments, when this value is also referred to the
terminal moment of the contract. Consequently
OB(0) 1 +
c
12
n
= MP 1 +
c
12
n 1 − 1 + c
12
−n
c
12
,
which upon slightly simplifying yields
OB(0) 1 +
c
12
n
= MP
1 + c
12
n
− 1
c
12
,
and then making MP the subject of this equation
MP =
OB(0) 1 + c
12
n c
12
1 + c
12
n
− 1
, (2.1)
gives the formula for the value of the monthly payments, where OB(0) is the amount
initially loaned to the borrower, c is the fixed yearly contract rate, and n is the life
of the mortgage in months.
2.1.2 Value of the outstanding balance
Immediately after the ith
monthly payment has been made, the outstanding balance
OB(i) the borrower still has to repay can be expressed in the following way
OB(i) = OB(0) − MP
1 − 1 + c
12
−i
c
12
1 +
c
12
i
.
Making the substitution for MP from equation (2.1) yields
OB(i) = OB(0) −
OB(0) 1 + c
12
n c
12
1 + c
12
n
− 1
1 − 1 + c
12
−i
c
12
1 +
c
12
i
,
and then simplifying gives
OB(i) =
OB(0) 1 + c
12
n
− 1 + c
12
i
1 + c
12
n
− 1
, (2.2)
which is the formula for the value of the outstanding balance OB(i) after the ith
monthly payment has been made.
For an ARM the formulae to calculate the monthly payment and the outstanding
balance are given by equation (5.3) and (5.4), respectively. As the contract rate can
42. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 42
change each year the two formulae are also functions of the current year as well as the
current month. The only change in the formulae is that a year index is introduced so
the specific contract rate for the present year can be used in the calculation. Therefore
the derivation of the formulae for an ARM is omitted.
2.2 Equilibrium condition
As mentioned in chapter 1, the mortgage contract would not be agreed originally by
the two counter parties unless it was fair. This means that at origination the contract
must be in financial equilibrium, which is the case if the value of the mortgage to the
bank is equal to the amount lent to the borrower. A generalised equilibrium condition
for a generic mortgage loan (where the type of mortgage is irrelevant) is as follows
V (t = 0; c) + I(t = 0; c) = (1 − fee)loan. (2.3)
The bank’s position in the contract is V = A − D − C, i.e. the scheduled payments
minus the sum of the value of the borrower’s options to terminate the mortgage (D is
the value of the default option and C is the value of the prepayment option), plus any
insurance I the bank may have against the borrower defaulting on a payment. The
borrower’s position is the amount lent by the bank, which will be some percentage
of the initial house value, minus an arrangement fee (for a UK contract) or the
points (for a US loan) charged as a percentage of the loan amount. The equilibrium
constraint (2.3) is to avoid contractual arbitrage. The specific equilibrium condition
for a UK and US FRM is given by equation (3.22), and by equation (5.14) for a US
ARM.
The difference between the conditions for a UK contract and a US contract is
only in the terminology used for the fee paid by the borrower when the contract is
set up. The fixed-rate and the adjustable-rate conditions vary according to the free
parameter c which is used to balance the equilibrium condition. This is discussed in
the next section, as well as the method used to calculate the free parameter c that
will provide a contract in equilibrium at origination.
43. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 43
2.2.1 Newton method
All mortgage contracts discussed in this thesis require an equilibrium condition to be
set prior to the contract commencing, and so it is necessary to find the value of the
free parameter c (which is the contract rate for FRM valuation and the margin for
ARM valuation) which balances the relevant equilibrium condition.
The free variable can be found easily using an iterative process following Newton’s
method. Let f(c) be a function of c only, where c is the free variable used to balance
the equilibrium condition and f(c) is given by rearranging equation (2.3) to form
f(c) = V (t = 0; c) + I(t = 0; c) − (1 − fee)loan, (2.4)
which must be zero to satisfy the equilibrium condition. An initial estimate for the
value of c is made, let this estimate be c0. Then the values of the mortgage components
involved in the equilibrium condition are calculated with the initial estimate c0 used as
the value of the free parameter. Next, a tolerance to which the absolute value of f(c)
must be less than is specified; once f(c) is less than this tolerance the iterative process
is terminated. For FRMs, c at this point is the equilibrium setting contract rate, and
for ARMs, c at this point is the equilibrium setting margin. An estimate is required
for the initial increment change in c0; call this increment ∆0 (which is specified). The
next potential equilibrium setting free parameter c is given by c1 = c0 + ∆0. Given
this information it is then possible to calculate f(c1) and check if its absolute value is
less than the tolerance. If the absolute value of f(c1) and any further f(ci) is greater
than the tolerance the new increment for the change in c is calculated as follows,
∆i+1 = −
∆if(ci)
f(ci) − f(ci−1)
; where i ≥ 1. (2.5)
2.3 Interest-rate index
This section discusses interest-rate indices, how they drive the contract rate for
ARMs, and how the particular index used in chapter 5, on ARM valuation, is calcu-
lated.
44. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 44
Traditionally, in the UK, the variable rate has been adjustable at the discretion of
the lender, but in recent years mortgages that charge a rate tied to a specific interest
rate (such as LIBOR or the Bank of England repurchase rate) have become popular.
In the USA the adjustable rate can be based on any rate. Currently, the Federal
National Mortgage Association (FNMA), or Fannie Mae for short, offers a two-step
purchase programme which specifies that the new rate be calculated by adding 250
basis points to a weekly average of the 10-year constant maturity Treasury yield.
Fannie Mae also limit any increase in the mortgage rate to no more than 600 basis
points over the initial mortgage rate.
When valuing a theoretical ARM, it is necessary to decide how the contract rate
will change during the life of the mortgage. Usually a contract rate will change
according to an index. The precise details of how the index is derived may vary,
but it will depend on a specific interest rate. The model used in chapter 5 for the
valuation of an ARM uses an index which depends on the current interest rate plus
a margin - this is just one way to model an index. Another example is illustrated in
Stanton and Wallace (1995), where an index is used which lags behind shifts in the
term structure; this is discussed in section 5.2.1.
2.3.1 Calculation of the index
The index that is used in section 5.3 as part of the adjustment rule (see equation
(5.1)), is the mortgage-equivalent rate or yield, index(r), for a 1-year, default-free
pure discount bond (as used by Kau et al., 1993). Given the assumption of the Local
Expectations Hypothesis (see Cox et al., 1981) and that the interest-rate process is
the single-factor spot interest rate, there exists a closed-form solution for the pure
discount bond yield. Cox et al. (1985b) give full details, but the solution is sum-
marised next. The mortgage equivalent conversion takes into account the monthly
compounding. From Cox et al. (1985b), the PDE for the price of a discount bond
P(r, t), in the absence of the market price of risk, is
1
2
σ2
r r
∂2
P
∂r2
+ κ(θ − r)
∂P
∂r
+
∂P
∂t
− rP = 0, (2.6)
45. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 45
with the terminal (expiration) condition P(r, t = T) = 1.
The bond price takes the form
P(r, t) = A(t)e−B(t)r
, (2.7)
where
A(t) =
2γe(γ+κ)(T−t)/2
(γ + κ)(eγ(T−t) − 1) + 2γ
2κθ/σ2
r
, (2.8)
B(t) =
2(eγ(T−t)
− 1)
(γ + κ)(eγ(T−t) − 1) + 2γ
, (2.9)
γ = κ2 + 2σ2
r . (2.10)
The yield-to-maturity, R(r, t) is defined by e−(T−t)R(r,t)
= P(r, t). Therefore,
R(r, t) = [rB(t) − logA(t)]/(T − t). (2.11)
Equation (2.11) gives the pure discount bond yield. The mortgage-equivalent yield
index(r) is given by equating the yield on the the principal amount at the con-
tinuously compounded rate of R(r, t), to the yield on the principal amount at the
mortgage equivalent monthly compounded rate index(r), i.e.
P(r, t)eR(r,t)
= P(r, t) 1 +
index(r)
12
12
, (2.12)
and so
index(r) = 12[eR(r,t)/12
− 1]. (2.13)
Equation (2.13) gives the mortgage equivalent yield on a 1-year, default-free, pure
discount bond. This is used to calculate the index, given the current interest rate,
which is added to the margin when calculating the new contract rate at an adjustment
date for the ARM. Figure 2.1 shows the profile of the index as the interest rate
changes, for various different values of the long-term mean of the short-term interest
rate θ. Notice that initially when the interest rate is less than the long-term mean,
the index slightly leads the underlying interest rate, i.e. index(r) > r for r < θ. Once
the interest rate is above the long-term mean, the index lags behind the underlying
interest rate, i.e. index(r) < r for r > θ.
46. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 46
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
PSfragreplacements
Interest Rate r
index(r)
Figure 2.1: A graph of index(r) against interest rate r. For each line style, the long-term
mean of the short-term interest rate θ, is 0.1, 0.2 and 0.3 from the bottom to the top, with
σr = 0.1 and κ = 0.25.
2.4 Numerical methods
Most problems that arise in financial mathematics cannot be solved analytically.
Instead, numerical methods must be employed to obtain their solution. There are
several numerical methods that can be used to approximate the value of such deriva-
tive securities. Popular methods include the Monte Carlo method, lattice methods
(binomial or trinomial trees), quadrature and finite-difference methods. This study
will solely involve the implementation of the latter, although a few brief details, of
each method, is given below.
The Monte Carlo approach is a forward method, in that the solution starts from
the initiation of the option at time t = 0. Random sample paths are generated
according to which stochastic process is used to model the underlying asset. The
sample paths are then discounted at a specified interest rate to find the implied
option value. Boyle (1977) was first to develop a Monte Carlo simulation method for
solving option-valuation problems. The main drawback with this numerical method
is that it is a complex matter to value options which have early exercise features.
As simulated paths are generated forward in time, it is difficult to decide when it is
47. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 47
optimal to exercise the option. As mortgage valuation involves finding the optimal
time for the borrower to prepay, this effectively rules out the use of the Monte Carlo
method.
Moving to backward methods, the lattice approach and the finite-difference ap-
proach can both readily handle early exercise features. Lattice, or tree methods were
developed independently by Cox et al. (1979) and Rendlemann and Bartter (1979).
The theory behind the method is that at each discrete moment in time, the asset
price can either move up to a new level, down to a new level, or, in the case of the
trinomial lattice, move to a third level. As the value of an option is known at expiry
(the payoff) this value can be used to evaluate the option price at T − δt. This is
performed recursively so that ultimately the value of the option at t = 0 can be cal-
culated. Early exercise features are no problem, as valuation is performed backwards
in time the option value can be compared to the value of the option price if early
exercise is taken. The lattice approach suffers as the tree itself is not very flexible, it
is difficult to align nodes with important asset prices, such as a barrier or the exercise
price. Also, computations, are rather inefficient, as only a single option price is found
from each calculation, unlike the finite-difference approach which produces a range
of option values for each calculation.
The first author to employ numerical integration or quadrature techniques to
option pricing was Parkinson (1977). The more recent work of Andricopoulos et al.
(2003, 2004), for a single underlying, and Andricopoulos et al. (2006) for multi-
asset and complex path-dependent options, contain considerable improvements in
convergence and accuracy over previous quadrature benchmark methods. The main
difficulty with this numerical method is that the mathematics required to formulate
the integrand itself is often very difficult, even if it is actually possible to then perform
the integration. As such, the exact integrand required for assets following processes
other than the lognormal diffusion processes is still in the early stages of development.
The PDEs that arise in Mathematical Finance can often be best solved directly
using finite-difference methods. This type of numerical method is ideally suited to
optimal-stopping problems and it was for this type of problem that Brennan and
48. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 48
Schwartz (1977) first recognised the utility of this method to price American put
options. Solution via a backward method makes it simple to track the free bound-
ary, which determines when it is optimal to exercise the option. Finite-difference
methods are extremely flexible and allow the inclusion of complex path-dependent
option features, such as occupation-time derivatives, to cause very little problem to
the financial engineer attempting to value a problem of this nature. See section 4.2
for a detailed description of the method for pricing these types of derivatives.
2.4.1 Finite-difference methods
The framework for pricing options is built around the Black-Scholes (1973) equation.
Although this backward parabolic equation can be solved analytically (in simple
cases), it can be very efficiently solved by use of finite-difference methods. These can
be adapted to handle with ease many problems based on the Black-Scholes equation,
with mortgages treated as derivatives being no exception.
Finite-difference methods provide the user with an intuitive feel for the problem
and how the solution is produced. The underlying problem is converted from one
which exists over a continuous domain to a problem that can be described on a finite
domain. The derivatives in the partial differential equation are discretised to form
linear difference equations and the state space for the problem is replaced with a
mesh on which the problem is defined.
Brennan and Schwartz (1977), used an explicit method to obtain the price for
an American put option. There are other (improved) variations of finite-difference
methods that exist and are used today in more complicated financial problems. Other
methods can give better accuracy and can be used in a wider variety of situations.
Next the derivation of the most basic difference equations is shown.
From standard calculus, the following approximations are valid for the derivative
of a function u(x, t). A forward derivative approximation is
∂u(x, t)
∂x
=
u(x + ∆x, t) − u(x, t)
∆x
+ O(∆x), (2.14)
49. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 49
a backward derivative approximation is
∂u(x, t)
∂x
=
u(x, t) − u(x − ∆x, t)
∆x
+ O(∆x), (2.15)
and a central derivative approximation is
∂u(x, t)
∂x
=
u(x + ∆x, t) − u(x − ∆x, t)
2∆x
+ O(∆x2
). (2.16)
The difference equations are obtained by removing the error terms indicated by the
‘O’ notation. The order of the error for each of the equations is easily seen by
considering the Taylor series expansion of u about x,
u(x + ∆x, t) = u(x, t) + ∆x
∂u
∂x
+
∆x2
2
∂2
u
∂x2
+ . . . =
∞
n=0
∆xn
n!
∂n
u
∂xn
, (2.17)
and
u(x − ∆x, t) = u(x, t) − ∆x
∂u
∂x
+
∆x2
2
∂2
u
∂x2
+ . . . =
∞
n=0
(−1)n ∆xn
n!
∂n
u
∂xn
, (2.18)
where ∂nu
∂xn denotes the n-th order derivative of u with respect to x. Equation (2.17)
leads to the forward derivative approximation (2.14), whilst (2.18) leads to the back-
ward derivative equation (2.15), and both approximations have an error O(∆x). The
central derivative equation (2.16) is obtained by subtracting equation (2.17) from
equation (2.18), and has error O(∆x2
).
The terms O(∆x) and O(∆x2
) indicate the truncation error of the difference
equations. If a better approximation is required, the computational mesh (on which
the approximated solution is calculated), can be made finer (by making ∆x smaller)
or information can be added by including higher-order neighbouring terms, which
will involve additional mesh points.
In order to simplify the notation, when convenient, the discretisation points will
be labelled with appropriate indices. With uk
i ≡ u(xi, tk) where xi = i∆x, tk = k∆t,
for example equation (2.16) becomes
∂u
∂x
≈
uk
i+1 − uk
i−1
2∆x
. (2.19)
50. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 50
2.5 Numerical solution
Throughout the following study of various mortgage pricing techniques, the Crank-
Nicolson finite-difference method will be employed. Finite-difference algorithms re-
place derivatives with difference equations and approximate the solution of the PDE
by a set of algebraic equations. For convenience, the analysis will involve the following
transformation:
τ = T − t. (2.20)
This transforms the governing PDE (1.17) to a forward parabolic equation in τ. In
the physical world, parabolic equations are generally solved forward in time, starting
from an initial condition. The Crank-Nicolson finite-difference method will be used
as convergence for this method is superior to the more basic explicit and implicit
methods (which of both have convergence at the rate O(∆τ, ∆H2
, ∆r2
)). The Crank-
Nicolson method converges at the rate O(∆τ2
, ∆H2
, ∆r2
) and unlike the explicit
method, there is no stability constraint.
Section 3.5 provides a full exposition of how to space the finite-difference grid,
and how the mortgage valuation PDE (1.17) is discretised according to the Crank-
Nicolson finite-difference scheme follows.
2.5.1 Derivative approximations
The valuation PDE (1.17) is discretised following a Crank-Nicolson finite-difference
scheme to ensure second-order accuracy in underlying house price, interest rate and
time. The time derivative is approximated as
∂F(H, r, τ + 1
2
∆τ)
∂τ
≈
Fk+1
i,j − Fk
i,j
∆τ
. (2.21)
The spatial derivatives for house price H are approximated by
∂F(H, r, τ + 1
2
∆τ)
∂H
≈
(Fk+1
i+1,j − Fk+1
i−1,j + Fk
i+1,j − Fk
i−1,j)
4∆H
, (2.22)
∂2
F(H, r, τ + 1
2
∆τ)
∂H2
≈
(Fk+1
i+1,j − 2Fk+1
i,j + Fk+1
i−1,j + Fk
i+1,j − 2Fk
i,j + Fk
i−1,j)
2(∆H)2
. (2.23)
51. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 51
The spatial derivatives for interest rate r are approximated by
∂F(H, r, τ + 1
2
∆τ)
∂r
≈
(Fk+1
i,j+1 − Fk+1
i,j−1 + Fk
i,j+1 − Fk
i,j−1)
4∆r
, (2.24)
∂2
F(H, r, τ + 1
2
∆τ)
∂r2
≈
(Fk+1
i,j+1 − 2Fk+1
i,j + Fk+1
i,j−1 + Fk
i,j+1 − 2Fk
i,j + Fk
i,j−1)
2(∆r)2
. (2.25)
The cross-spatial derivative is approximated by
∂2
F(H, r, τ + 1
2
∆τ)
∂H∂r
≈
1
8∆H∆r
(Fk+1
i+1,j+1 − Fk+1
i−1,j+1 − Fk+1
i+1,j−1 + Fk+1
i−1,j−1
+Fk
i+1,j+1 − Fk
i−1,j+1 − Fk
i+1,j−1 + Fk
i−1,j−1). (2.26)
Finally, the asset F(H, r, τ) is approximated by
F H, r, τ +
1
2
∆τ ≈
Fk+1
i,j + Fk,l
i,j
2
. (2.27)
Overall the error in the approximate solution F k
i,j is of second-order accuracy in ∆H,
∆r and ∆τ.
2.5.2 Discrete representation
Upon substituting the derivative approximations from section 2.5.1 into the governing
PDE (1.17) and rearranging, the problem of solving the PDE reduces to solving the
following set of simultaneous linear equations for F k+1
i,j
ai,jFk+1
i,j−1 + bi,jFk+1
i,j + ci,jFk+1
i,j+1 + di,jFk+1
i−1,j + ei,jFk+1
i+1,j
+fi,j[Fk+1
i+1,j+1 − Fk+1
i,j+1 − Fk+1
i+1,j−1 + Fk+1
i−1,j−1] =
−ai,jFk
i,j−1 −
2
∆τ
+ bi,j Fk
i,j − ci,jFk
i,j+1 − di,jFk
i−1,j
−ei,jFk
i+1,j − fi,j[Fk
i+1,j+1 − Fk
i,j+1 − Fk
i+1,j−1 + Fk
i−1,j−1], (2.28)
where
ai,j =
rjσ2
r
4(∆r)2
−
κ(θ − rj)
4∆r
, (2.29)
bi,j = −
H2
j σ2
H
2(∆H)2
−
rjσ2
r
2(∆r)2
−
1
∆τ
−
rj
2
, (2.30)
52. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 52
ci,j =
rjσ2
r
4(∆r)2
+
κ(θ − rj)
4∆r
, (2.31)
di,j =
H2
i σ2
H
4(∆H)2
−
(rj − δ)Hi
4∆H
, (2.32)
ei,j =
H2
i σ2
H
4(∆H)2
+
(rj − δ)Hi
4∆H
, (2.33)
fi,j =
ρσHσrHi
√
rj
8∆H∆r
. (2.34)
Next, the solution of this set of algebraic equations is considered.
2.5.3 Solution of the difference equations
At any time step during the valuation of the mortgage, the value of the asset F(H, r, τ)
must be calculated for all house-price and interest-rate values. Moving to the discrete
representation of the problem, this means that F k+1
i,j must be found all for i and j. It
is not easy to directly solve the system of equations (2.28), since the two-dimensional
matrix problem produced is particularly complicated. For the results produced in
chapter 3, for the case of the solution of a two-factor UK FRM, a general LU solver
standard library package is employed (see Wilmott et al., 1993, for more on LU
decomposition). The default D, insurance I and coinsurance CI components are
calculated using the general LU solver standard library package. The coefficients
ai,j, bi,j, etc, are the input for the package; the output is the value of the particular
component at the present time step, for further details of the actual implementation
and for full details of the solution for the two-factor UK FRM valuation (using the
Crank-Nicolson finite-difference method), see the pseudocode given in Appendix A.
The value of the remaining payments is dependent only on the interest rate and time,
and this is discussed in section 2.6.1. This implies that the matrix problem produced
to calculate Ak+1
j using the Crank-Nicolson method is tridiagonal at each time step,
which can readily be solved using Gaussian elimination (see Smith, 1978). The value
of the mortgage to the lender V has the added complication of a free boundary, which
53. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 53
determines when it is optimal for the borrower to prepay. This component is valued
by considering the problem in its linear complementarity form (see Wilmott et al.,
1993) and then solving the resulting constrained-matrix problem using the projected
successive over-relaxation method (PSOR); see section 3.6.1 for the details.
An alternative method to the general LU solver library package is used in chapters
4 and 5 to solve for the default and insurance components. The successive over-
relaxation method (SOR) is used. The only difference between this method and the
PSOR method (Wilmott et al. 1993), as described generally in section 3.6.1, is that
equation (3.39) is simplified to
xk+1
i = xk
i + ω(yk+1
i − xk
i ), (2.35)
as the test to ensure that xk+1
i ≥ ci is not required since there is no free boundary
constraint for these components. It could be thought that this alternative iterative
technique would be less computationally efficient than the direct solution technique,
using the library package, but this is not the case, as discussed in section 4.4.
The solution for the value of the mortgage component with the new prepayment
model is discussed in chapter 4. The modelling details require that a more sophisti-
cated solver be used, as described in section 4.4.1.
Finally throughout this thesis the prepayment component is valued by rearranging
the relation V = A−D −C, so that once V , A and D are calculated the prepayment
component C can be inferred.
2.6 Boundary conditions
The valuation of the different mortgage models, discussed in chapters 3, 4 and 5,
require the following boundary conditions to close each problem.
2.6.1 The value of the remaining payments
The value of the remaining payments A(r, t) is dependent only on the term structure
of interest rates and time. Since A is independent of house price the valuation PDE
54. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 54
(1.17) for A reduces to
1
2
rσ2
r
∂2
A
∂r2
+ κ(θ − r)
∂A
∂r
+
∂A
∂t
− rA = 0. (2.36)
Condition at r = 0
Setting the interest rate to zero, r = 0, directly into equation (2.36) leads to,
κθ
∂A
∂r
+
∂A
∂t
= 0, (2.37)
which serves as a boundary condition for A(r = 0, t).
Condition as r → ∞
In the limit of large interest rates any expected future payment is worthless. In
accordance with Azevedo-Pereira et al. (2002), then
lim
r→∞
A(r, t) → 0. (2.38)
However, it is more computationally convenient to impose the corresponding Neu-
mann boundary condition, namely
lim
r→∞
∂A(r, t)
∂r
= 0. (2.39)
This is a ‘softer’ condition than equation (2.38), and enables a smaller domain trun-
cation rmax to be used.
The differing modelling details, between the contracts considered in this thesis,
make it more appropriate to discuss the payment-date conditions, which complete the
boundary conditions for the value of the remaining payments A(r, t), at the relevant
points in chapters 3, 4 and 5.
2.6.2 The value of the other mortgage components
Here, unless stated otherwise, F(H, r, t), represents any of the components: V , D,
C, I and CI (the coinsurance is only relevant to the UK FRM considered in chapter
3). For these remaining mortgage components, consider the boundary conditions to
be imposed at the extremes of the grid.
Condition at H = 0, r = 0
55. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 55
An intuitive condition can be derived if H and r are simply set equal to zero in
equation (1.17); the asset pricing PDE becomes,
κθ
∂F(0, 0, t)
∂r
+
∂F(0, 0, t)
∂t
= 0. (2.40)
Condition as H → ∞, r = 0
If H tends to infinity and r = 0 in equation (1.17); the asset pricing PDE reduces to,
lim
H→∞
∂F(H, 0, t)
∂H
→ 0. (2.41)
Condition as H → ∞, r → ∞ and at H = 0, r → ∞
In the limit of large interest rate any asset is worthless, therefore,
lim
r→∞
F(H, r, t) → 0. (2.42)
For the above condition the following equivalent Neumann boundary condition can
be used if it is numerically expedient to do so, namely
lim
r→∞
∂F(H, r, t)
∂r
→ 0. (2.43)
Condition along H = 0
If the house price becomes zero, the borrower will default and the mortgage is now
worth the same as the house, and so
V (0, r, t) = 0. (2.44)
Prepayment at this point is worthless, thus
C(0, r, t) = 0. (2.45)
The option to default is now equal to the value of the remaining payments. Since
D = A − C − V , then
D(0, r, t) = A(r, t). (2.46)
56. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 56
The value of I and CI is given by equation (2.47) with F(0, r, t) replaced by either
I or CI,
1
2
rσ2
r
∂2
F
∂r2
+ κ(θ − r)
∂F
∂r
+
∂F
∂t
− rF = 0, (2.47)
which is a degenerate form of equation (1.17) with H = 0.
Condition when r = 0
Substituting r = 0 directly into equation (1.17) gives
1
2
H2
σ2
H
∂2
F
∂H2
+ κθ
∂F
∂r
− δH
∂F
∂H
+
∂F
∂t
= 0. (2.48)
Condition along H → ∞
As H → ∞ the value of the default option tends to zero. Since there is no value in
default, the insurance and the coinsurance have no value, and therefore:
lim
H→∞
D(H, r, t) = 0, (2.49)
lim
H→∞
I(H, r, t) = 0, (2.50)
lim
H→∞
CI(H, r, t) = 0. (2.51)
The value of the mortgage V is constant as H tends to infinity, implying ∂V/∂H → 0.
V is then determined by a degenerate form of equation (1.17) with ∂V/∂H and
∂2
V/∂H2
both set to zero, namely
1
2
rσ2
r
∂2
V
∂r2
+ κ(θ − r)
∂V
∂r
+
∂V
∂t
− rV = 0. (2.52)
Since the value of the mortgage is the difference between the value of the remaining
future payments and the borrower’s joint option to terminate the mortgage, the
prepayment option value at this extreme is given by
lim
H→∞
C(H, r, t) = A(r, t) − lim
H→∞
V (H, r, t). (2.53)
57. CHAPTER 2. FOUNDATIONS OF MORTGAGE VALUATION 57
Condition as r → ∞
Since, in the limit of infinite interest rate, any asset is worthless, the following is
taken as the boundary condition for F as r tends to infinity,
lim
r→∞
F(H, r, t) = 0. (2.54)
Alternatively, if necessary, it is possible to use the corresponding Neumann condition
(similar to the limit when r → ∞) for the corners of the grid (see equation (2.43)).
Again, due to the differing modelling details between the contracts considered,
it is more appropriate to discuss the payment-date conditions, which complete the
boundary conditions for the mortgage components V , D, C, I and CI, at the relevant
points in chapters 3, 4 and 5.
2.6.3 Default boundary
The option to default is serial-European in nature, since it can only be exercised (if
the borrower chooses to do so) on the payment date in any particular month. Also,
default cannot occur if the option to prepay is exercised. Therefore, default is only
rational outside the prepayment region and the default boundary is described fully
by the payment-date conditions.
2.6.4 Prepayment boundary
The prepayment boundary is discussed at the relevant point in each chapter. Section
3.6.1 explains the prepayment boundary condition when prepayment occurs ratio-
nally, as an effort by borrower’s to minimise the cost of the mortgage to themselves.
This assumption is used again in chapter 5 on valuing ARMs. Chapter 4 considers
an alternative prepayment assumption, as an attempt to improve FRM valuation, as
actual borrowers tend to wait for a time after theory says it is optimal to prepay,
a time lag before the borrowers prepay is introduced. This is explained in detail in
section 4.3.1.
58. Chapter 3
An improved fixed-rate mortgage
valuation methodology with
interacting prepayment and default
options
The work in this chapter draws extensively on that presented in Sharp et al. (2006).
3.1 Introduction
This chapter considers in detail a realistic mortgage valuation model (including the
potential for early prepayment and the risk of default), based on stochastic house-
price and interest-rate models. As well as the development of a highly accurate nu-
merical scheme to tackle the resulting partial differential equations, this chapter also
exploits singular perturbation theory (a mathematically rigorous procedure, based
on the idea of the smallness of the volatilities), whereby mortgage valuation can be
quite accurately approximated by very simple closed-form solutions. Determination
of equilibrium contract rates, previously requiring many computational hours (using
the highly accurate numerical scheme) is reduced to just a few seconds, rendering this
a highly useful portfolio management tool; these approximations compare favourably
58
59. CHAPTER 3. IMPROVED FIXED-RATE MORTGAGE VALUATION 59
with the full numerical solutions. The method is of wide applicability in US or other
mortgage markets and is demonstrated for UK FRMs, including insurance and coin-
surance.
Contingent claims analysis leads to the modelling of many derivative securities
as PDEs. The need to model increasingly sophisticated products realistically has
resulted in the development of extremely complex valuation frameworks, whose com-
putation often prove excessively time-consuming. Collin-Dufresne and Harding (1999)
note the utility of reduced calculation times for mortgage values as a useful tool for
portfolio management and develop a closed-form formula for the value of a fixed-
rate residential mortgage dependent only on a single state variable. However, until
now, for models using two state variables including both default and prepayment, no
closed-form solutions of any kind have been available.
Several numerical procedures based on the explicit finite-difference method have
been published for the solution of a contingent claims valuation model aimed at
valuing mortgage-related products, including the work of Kau et al. (1992, 1995), on
US mortgages and Azevedo-Pereira et al. (2000, 2002, 2003), on UK mortgages, which
use two state variables. Brunson et al. (2001), describe a three-state variable model
with a two factor term structure and a one factor property process. They argue that
two-state variable models lack flexibility, failing to reflect the evolution of the whole
term structure by using just one factor to represent the term structure and therefore
these misprice the mortgage value. Conversely, empirical work by Chatterjee et al.
(1998) indicates that a two variable model (short rate and building value) is the most
efficient, in terms of pricing accuracy, of all the alternative mortgage valuation models
that are available.
The problem addressed in this chapter will follow a two-state variable model, and
will also establish a new technique for extremely rapid valuation, using UK FRMs as
the practical example. However, the technique is more broadly applicable to US and
foreign mortgages; for example, it would be directly applicable to the US mortgage-
backed securities model of Downing et al. (2005). The model falls into the structural
category, where default and prepayment are treated as the exercise of options held
