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Valuation of the Prepayment Option of a PerpetualCorporate Loan
Timothée Papin, Gabriel Turinici
To cite this version:Timothée Papin, Gabriel Turinici. Valuation of the Prepayment Option of a Perpetual CorporateLoan. Abstract and Applied Analysis, Hindawi Publishing Corporation, 2013, 2013, pp.960789.�10.1155/2013/960789�. �hal-00653041v4�
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 960789, 13 pageshttp://dx.doi.org/10.1155/2013/960789
Research Article
Valuation of the Prepayment Option ofa Perpetual Corporate Loan
Timothee Papin1 and Gabriel Turinici2
1 BNP Paribas CIB Resource Portfolio Management and CEREMADE, Universite Paris Dauphine, 75016 Paris, France2CEREMADE, Universite Paris Dauphine, 75016 Paris, France
Correspondence should be addressed to Gabriel Turinici; [email protected]
Received 13 December 2012; Accepted 2 February 2013
Academic Editor: Dragos-Patru Covei
Copyright © 2013 T. Papin and G. Turinici. his is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
We investigate in this paper a perpetual prepayment option related to a corporate loan.he default intensity of the irm is supposedto follow a CIR process. We assume that the contractual margin of the loan is deined by the credit quality of the borrower andthe liquidity cost that relects the funding cost of the bank. Two frameworks are discussed: irstly a loan margin without liquiditycost and secondly a multiregime framework with a liquidity cost dependent on the regime. he prepayment option needs speciicattention as the payof itself is an implicit function of the parameters of the problem and of the dynamics. In the unique regimecase, we establish quasianalytic formulas for the payof of the option; in both cases we give a veriication result that allows forthe computation of the price of the option. Numerical results that implement the indings are also presented and are completelyconsistent with the theory; it is seen that when liquidity parameters are very diferent (i.e., when a liquidity crisis occurs) in thehigh liquidity cost regime, the exercise domain may entirely disappear, meaning that it is not optimal for the borrower to prepayduring such a liquidity crisis. he method allows for quantiication and interpretation of these indings.
1. Introduction
When a irm needs money, it can turn to its bank which lendsit against, for example, periodic payments in a form of a loan.In almost every loan contract, the borrower has the optionto prepay a portion or all the nominal at any time withoutpenalties.
We assume in this model that the riskless interest rate,denoted by �, is constant and known. he liquidity costdynamics will be described later. he interest rate of the loanis the sum of the constant interest rate, a margin deinedaccording to the credit quality of the borrower, and a liquiditycost that relects the funding costs of the lender, the bank.
In order to decide whether the exercise of the option isworthwhile, the borrower (the irm) compares the actualizedvalue of the remaining payments with the nominal value topay. If the remaining payments exceed the nominal value,then it is optimal for the borrower to reinance his debt at alower rate.
When the borrower is subject to default, the computationof the actualization is less straightforward. It starts with
considering all possible scenarios of evolution for the defaultintensity in a risk-neutral framework and computing theaverage value of the remaining payments (including the inalpayment of the principal if applicable); this quantity will becalled “PVRP” (denoted by �) and is the present value of theremaining payments, that is, the cash amount equivalent, bothfor borrower and lender in this model of the set of remainingpayments. he PVRP is compared with the nominal: if thePVRP value is larger than the nominal, then the borrowershould prepay, otherwise not. Recall that at the initial timethe payments correspond to a rate, the sum of the interest rateand a contractual margin �0, which is precisely making thetwo quantities equal. Note that in order to compute the priceof the embedded prepayment option, the lender also uses thePVRP as it will be seen below.
For a bank, the prepayment option is essentially a rein-vestment risk, that is, the risk that the borrower decides torepay earlier his/her loan and that the bank cannot reinvestits excess of cash in a new loan. So the longer the maturityof the loan, the riskier the prepayment option. herefore,it is interesting to study long-term loans that are set for
2 Abstract and Applied Analysis
more than three years and can run for more than twentyyears. he valuation problem of the prepayment option canbe modeled as an American embedded option on a riskydebt owned by the borrower. AsMonte-Carlo simulations areslow to converge and the binomial tree techniques are timeconsuming for long-term loans (cf. works by Cossin and Lu[1]), we decided to focus, in this paper, on the prepaymentoption for perpetual loan.
When valuing inancial products with long maturity,the robustness with respect to shocks and other exogenousvariabilities is important. Among problems that have to betreated is the liquidity and its variability. Liquidity is crucialfor the stability of the inancial system. Past events like theAsian crisis of 1997 [2]; the Russian inancial crisis of 1998 [3];the defaults of hedge funds and investment irms like LTCM,Enron, Worldcom, and Lehman Brothers defaults, sovereigndebts crisis of 2010-11, and so on prove that banks holdsigniicant liquidity risk in their balance sheets. A liquiditycrisis can have a severe impact on bank’s funding costs,its market access (reputation risk), and short-term fundingcapabilities.
Following the state of the economic environment, theliquidity can be deined by distinct states. Between two crises,investors are conident and banks ind it easier to launchtheir long-term reinancing programs through regular bondsissuances. hus the liquidity market is stable. Unfortunately,during crisis, liquidity becomes scarce, pushing the liquiditycurve to very high levels which can only decrease if coni-dence returns to themarket.he transition between these twodistinct behaviors is rarely smooth but rather sudden.
In order to model the presence of distinct liquiditybehaviors, we will simulate the liquidity cost by a continuoustime Markov chain that can have a discrete set of possiblevalues, one for each regime that is encountered in the liquidityevolution.
From a technical point of view, this paper faces severalnonstandard conditions: although the goal is to value aperpetual American option, the payof of the option is highlynonstandard (is dependent on the PVRP). As a consequence,the characterization of the exercise region is not standard andtechnical conditions have to be met. Furthermore, our focushere is on a speciic type of dynamics (of CIR type) with evenmore speciic interest on the situation when several regimesare present.
he balance of the paper is as follows: in the remainderof this section (Section 1.1) we review the related existingliterature; in Section 2, we consider that the liquidity costis negligible and that the borrower credit risk is deinedby his/her default intensity (called in the following simply“intensity”) which follows a CIR stochastic process. In thissituation, we are able to obtain a quasianalytic formula forthe PVRP. In Section 3 we explore the situation when theliquidity cost, deined as the cost of the lender to accessthe cash on the market, has several distinct regimes that wemodel by a Markov chain. We write the pricing formulas andtheoretically support an algorithm to identify the boundaryof the exercise region; numerical examples and concludingremarks close the paper.
1.1. Related Literature. here exist few articles (e.g., works byCossin and Lu [1]) on the loan prepayment option but a closesubject, the prepayment option in a ixed-rate mortgage loan,has been widely covered in several papers by Hilliard et al.[4] and more recent works by Chen et al. [5]. To approximatethe PDE satisied by the prepayment option, they deine twostate variables (interest rate and house price). heir approachis based on a bivariate binomial option pricing techniquewitha stochastic interest rate and a stochastic house value.
Another contribution by Cossin and Lu [1] applies thebinomial tree technique (but of course it is time consumingfor long-term loans due to the nature of binomial trees) tocorporate loans. hey consider a prepayment option with a1-year loan with a quarterly step, but it is diicult to have anaccurate assessment of the option price for a 10-year loan.
here also exist mortgage prepayment decision modelsbased on Poisson regression approach for mortgage loans(see, e.g., Schwartz and Torous [6]). Unfortunately, thevolume and history of data are very weak in the corporateloan market.
Due to the form of their approach, these papers did nothave to consider the geometry of the exercise region becauseit is explicitly given by the numerical algorithm.his is not thecase for us and requires that particular care be taken whenstating the optimality of the solution. Furthermore, to thebest of our knowledge, none of these approaches explored thecircumstance when several regimes exist.
he analysis of Markov-modulated regimes has beeninvestigated in the literature when the underlying(s) followthe Black-Scholes dynamics with drit and volatility havingMarkov jumps; several works are of interest in this area: Guoand Zhang [7] have derived the closed-form solutions forvanilla American put; Guo analyses in [8] Russian (i.e., per-petual look-back) options and is able to derive explicit solu-tions for the optimal stopping time; in [9] Xu andWu analysethe situation of a two-asset perpetual American option wherethe payof function is a homogeneous function of degree one;Mamon and Rodrigo [10] ind explicit solutions to vanillaEuropean options. Buington and Elliott [11] study Europeanand American options and obtain equations for the price.A distinct approach (Wiener-Hopf factorization) is used byJobert and Rogers [12] to derive very good approximationsof the option prices for, among others, American puts. Othercontributions include [13, 14].
Works involving Markov switched regimes and CIRdynamics appear in [15] where the bond valuation problem isconsidered (but not in the form of an American option; theirapproach will be relevant to the computation of the payof ofour American option although in their model only the meanreverting level is subject to Markov jumps) and in [16] wherethe term structure of the interest rates is analysed.
On the other hand numerical methods are proposed in[17] where it is found that a ixed point policy iterationcoupled with a direct control formulation seems to performbest.
Finally, we refer to [18] for theoretical results concerningthe pricing of American options in general.
Abstract and Applied Analysis 3
2. Perpetual Prepayment Option witha Stochastic Intensity CIR Model
We assume throughout the paper that the interest rate � isconstant. herefore, the price of the prepayment option onlydepends on the intensity evolution over time. We model theintensity dynamics by a Cox-Ingersoll-Ross process (see [19–21] for theoretical and numerical aspects of CIR processesand the situations where the CIR process has been used ininance):
��� = � (� − ��) �� + �√�����, �, �, � > 0, �0 = �0.(1)
It is known that if 2�� ≥ �2, then CIR process ensures anintensity strictly positive. Fortunately, as it will be seen in thefollowing, the PVRP is given by an analytic formula.
2.1. Analytical Formulas for the PVRP. Assume a loan witha ixed coupon deined by the interest rate � and an initialcontractual margin �0. Here �0 does not take into accountany commercial margin (see Remark 1). Let �(�, �, �) be thepresent value of the remaining payments at time � of acorporate loan with initial contractual margin �0 (dependingon �0), intensity at time �, ��, following the risk-neutralequation (1) with �� = �; the loan has nominal amount� andcontractual maturity �. Here the assignment �� = � meansthat the dynamics of �� start at time � from the numericalvalue �. All random variables will be conditional by this event(see, e.g., (3)).
herefore the loan value LV(�, �, �) is equal to the presentvalue of the remaining payments �(�, �, �)minus the prepay-ment option value �(�, �, �):
LV (�, �, �) = � (�, �, �) − � (�, �, �) . (2)
he present value of the cash lows discounted at the(instantaneous) risky rate � + �� is denoted by �. heininitesimal cash low at time � is �(� + �0) and the inalpayment of the principal�. hen
� (�, �, �)= E [� ⋅ (� + �0) ∫�
��−∫�� (�+��)���� + ��−∫�� �+���� | ��= �] .
(3)
For a perpetual loan the maturity � = +∞. Since �� isalways positive, � + �� > 0, and thus the last term tends tozero when � → ∞. A second remark is that since �, �, and� are independent of time, � is independent of the startingtime �:
� (�, �) = E [� ⋅ (� + �0) ∫+∞
��−∫�� �+������ | �� = �] (4)
= E [� ⋅ (� + �0) ∫+∞
0�−∫�0 �+������ | �0 = �]
=: � (�) ,(5)
where the last equality is a deinition. For a CIR stochasticprocess, we obtain (see [19, 21])
� (�) = � ⋅ (� + �0) ∫+∞
0�−��� (0, �, �) ��, (6)
where for general �, � we use the notation� (�, �, �) = E [�−∫�� ���� | �� = �] . (7)
Note that�(�, �, �) is a familiar quantity, and analytic formulasare available for (7) (see Lando [22, page 292]). he intensityis following a CIR dynamic; therefore, for general �, �
� (�, �, �) = � (�, �) �−�(�,�)� (8)
with
� (�, �) = ( 2ℎ �(�+ℎ)((�−�)/2)2ℎ + (� + ℎ) (�( �−�)ℎ − 1))
2��/�2 ,
� (�, �) = 2 (�( �−�)ℎ − 1)2ℎ + (� + ℎ) (�( �−�)ℎ − 1) ,
where ℎ = √�2 + 2�2,
(9)
where � and � are the parameters of the CIR process of theintensity in (1). Obviously�(0, �, �) is monotonic with respectto �; thus the same holds for �.
hemargin �0 is the solution of the following equilibriumequation:
� (�0) = � (10)
which can be interpreted as the fact that the present valueof the cash lows (according to the probability of survival) isequal to the nominal�:
�0 = 1∫+∞0 �−��� (0, �, �0) �� − �. (11)
Note that we assume no additional commercial margin.
Remark 1. If an additional commercial margin �0 is consid-ered, then �0 is irst computed as above and then replaced
by �0 = �0 + �0 in (6). Equations (10) and (11) will not be
veriied as such but will still hold with some �0 instead of �0;for instance, we will have
�0 = 1∫+∞0 �−���(0, �, �0)�� − �.
(12)
With these changes all results in the paper are valid, exceptthat when computing for operational purposes once the priceof the prepayment option is computed for all �, one will use� = �0 as price relevant to practice.
4 Abstract and Applied Analysis
Remark 2. Some banks allow (per year) a certain percentageof the prepaid amount without penalty and the rest with apenalty. his circumstance could be incorporated into themodel by changing the deinition of the payof by subtractingthe penalty. his will impact formula (15).
From deinition (7) of �(�, �, �) it follows that �(�, �, �) <1; thus�−��� (0, �, �0) < �−�� (13)
and as a consequence
∫+∞
0�−��� (0, �, �0) �� < ∫+∞
0�−���� = 1� (14)
which implies that �0 > 0.2.2. Valuation of the Prepayment Option. he valuationproblem of the prepayment option can be modeled as anAmerican call option on a risky debt owned by the borrower.Here the prepayment option allows borrower to buy back andreinance his/her debt according to the current contractualmargin at any time during the life of the option. As theperpetual loan, the option value will be assumed independentof the time �.
As discussed above, the prepayment exercise results ina payof (�(�, �, �) − �)+ for the borrower. he option istherefore an American call option on the risky asset �(�, �, ��)and the principal� (the amount to be reimbursed) being thestrike. Otherwise we can see it as an American option on therisky �� with payof
� (�, �) := (� (�, �) − �)+ (15)
or, for our perpetual option,
� (�) := (� (�) − �)+. (16)
We will denote by A the characteristic operator (cf. [23,Chapter 7.5]) of the CIR process, that is, the operator that actson any �2 class function V by
(AV) (�) = � (� − �) ��V (�) + 12�2����V (�) . (17)
Denote for �, � ∈ R and � ≥ 0 by �(�, �, �) the solutionto the conluent hypergeometric diferential (also known asthe Kummer) equation [24]:
���� (�) + (� − �) �� (�) − �� (�) = 0 (18)
that increases at most polynomially at ininity and is inite(not null) at the origin. Recall also that this function isproportional to the conluent hypergeometric function of thesecond kind �(�, �, �) (also known as the Kummer functionof the second kind, Tricomi function, or Gordon function);for �, � > 0 the function �(�, �, �) is given by the formula
� (�, �, �) = 1Γ (�) ∫+∞
0�−����−1(1 + �)�−�−1��. (19)
When � ≤ 0, one uses other representations (see the citedreferences; for instance, one can use a direct computation orthe recurrence formula �(�, �, �) = (2� − � + � − 2)�(� +1, �, �) − (� + 1)(� − � + 2)�(� + 2, �, �)) it is known that�(�, �, �) behaves as �−� at ininity. Also introduce for � ≥ 0� (�) = ��((�−ℎ)/�2)�(�2−2��)/�2
× �(−−��2 − �2ℎ + �2� + �ℎ��2ℎ , 2 − 2���2 , 2ℎ�2 �) ,(20)
where ℎ = √�2 + 2�2.
heorem 3. (1) Introduce for Λ > 0 the family of functions:�Λ(�) such that�Λ (�) = � (�) ∀� ∈ [0, Λ] , (21)
(A�Λ) (�) − (� + �) �Λ (�) = 0, ∀� > Λ, (22)
lim�→Λ
�Λ (�) = � (Λ) , (23)
lim�→∞
�Λ (�) = 0. (24)
hen
�Λ (�) = {{{{{� (�) ∀� ∈ [0, Λ]� (Λ)� (Λ)� (�) ∀� ≥ Λ. (25)
(2) Suppose now a Λ∗ ∈ ]0, �0 ∧ �0[ exists such that��Λ∗ (�)��
���������= (Λ∗)+= ��(�)��
���������= (Λ∗)−. (26)
hen the price of the prepayment option is �(�) = �Λ∗(�).Proof. We start with the irst item: it is possible to obtain ageneral solution of (22) in an analytic form. We recall that�(�) = �(�, �, �) is the solution of the Kummer equation(18). A cumbersome but straightforward computation showsthat the general solution vanishing at ininity of the PDE (22)is�(�); thus
�Λ (�) = �Λ�(�) ∀� > Λ (27)
with some �Λ > 0 to be determined. Now use the boundaryconditions. If � = Λ, by continuity �(Λ) = �Λ(Λ) = �Λ�(Λ).hus, �Λ = �(Λ)/�(Λ). Division by� is legitimate becauseby deinition,�(�) > 0 for all � > 0.
We now continue with the second part of the theorem.he valuation problem of an American option goes throughseveral steps: irst one introduces the admissible trading andconsumption strategies (cf. [25, Chapter 5]); then one realizesusing results in the cited reference (also see [21, 26]) thatthe price �(�) of the prepayment option involves computinga stopping time associated to the payof. Denote by T theensemble of (positive) stopping times; we conclude that
� (�) = sup�∈T
E (�−∫�0 �+����� (��) | �0 = �) . (28)
Abstract and Applied Analysis 5
Further results derived for the situation of a perpetual(standard) American put options [18, 27] show that thestopping time has a simple structure: a critical level exists thatsplits the positive axis into two regions: the exercise region(to the let) where it is optimal to exercise and where theprice equals the payof and a continuation region (to theright) where the price satisies a partial diferential equationsimilar to Black-Scholes equation. We refer to [28] for howto adapt the theoretical arguments for the situation when thedynamics is not Black-Scholes-like but a CIR process.
he result builds heavily on the fact that the discountedpayof of the standard situation of an American put �−��(� −�)− is a submartingale. For us the discounted payof is
�−∫�0 �+����� (��) = �−∫�0 �+����(� (��) − �)+, (29)
and checking this condition requires here more carefulexamination which is the object of Lemma 4. It is nowpossible to apply heorem 10.4.1 [23, Section 10.4, page 227](see also [28] for speciic treatment of the CIR process) whichwill show that �(�) is the true option price if the followingconditions are satisied:
(1) on ]0, Λ∗[we have �(�) = �(�) = (�(�)−�)+ and therelation (35) holds;
(2) the solution candidate �(�) satisies the relation(A�) (�) − (� + �) � (�) = 0, ∀� > Λ∗; (30)
(3) the function �(�) is �1 everywhere, continuous at
the origin and �2 on each subinterval ]0, Λ∗[ and]Λ∗,∞[.he theorem also says that the borrower exercises his
option on the exercise region [0, Λ∗]while on the continuationregion ]Λ∗,∞[ the borrower keeps the option because it isworth more nonexercised.
We now show that �Λ∗ veriies all conditions above whichwill allow to conclude that � = �Λ∗ . he requirement 1 isproved in Lemma 4; the requirement 3 amounts to asking thatthe optimal frontier value Λ∗ is chosen such that
��Λ∗ (�)�����������= (Λ∗)+
= �� (�)�����������= (Λ∗)−
. (31)
he requirement 2 implies that in the continuation regionthe price is the solution of the following PDE:
(A�) (�) − (� + �) � (�) = 0, ∀� > Λ∗. (32)
For this PDE we need boundary conditions.he condition at� = Λ∗ is
� (�)|�=Λ∗ = � (�)�����=Λ∗ . (33)
When � = +∞, the default intensity is ininite; thus the timeto failure is zero, and thus the borrower has failed; in this casethe option is worthless; that is,
lim�→∞
� (�) = 0. (34)
hese conditions give exactly the deinition of �Λ∗ .
Lemma 4. he following inequality holds:
(A�) (�) − (� + �) � (�) < 0, ∀� < �0 ∧ �0. (35)
Proof. Recall that �(�) = (�(�) − �)+; deinition (5) of �implies (cf. [23, Section 8.2 and exercise 9.12, page 203]) that� is solution of the following PDE:
(A�) (�) − (� + �) � (�) + (� + �0)� = 0, ∀� > 0. (36)
For � < �0 we have �(�) > � = �(�0); thus(A(� (⋅) − �)+) (�) − (� + �) (� (�) − �)+
= (A (� (⋅) − �)) (�) − (� + �) (� (�) − �)= (A�) (�) − (� + �) � (�) + (� + �)�= − (� + �0)� + (� + �)�= (� − �0)� < 0 ∀� < �0 ∧ �0.
(37)
Note that heorem 3 is only a suicient result (a so-called “veriication” result); under the assumption that a Λ∗
fulilling the hypotheses of the theorem exists the question ishow to ind it.
Two approaches can be considered; irst, it is enoughto ind a zero of the following function Λ �→ Υ(Λ) :=((��Λ(�)/��)|�=Λ+ − (��(�)/��)|�=Λ−) (the last equalityis a deinition). Of course (��(�)/��)|�=�0+� = 0 and
(���0+�(�)/��)|�=�0+�< 0; thus Υ(�0 + �) < 0 for any � > 0
and hence Υ(�0) ≤ 0. hus it is natural not to look for Λ∗
outside the interval [0, �0]. he theorem asks furthermore to
restrict the search to the interval [0, �0 ∧ �0].A diferent convenient procedure to ind the critical Λ∗
is to consider the dependence Λ �→ �Λ(�0). Let us considerthe stopping time �Λ that stops upon entering the domain[0, Λ]. We remark that by a Feynman-Kac formula (cf. [23,page 203]),
�Λ (�) = E (�−∫�Λ0 �+����� (��Λ) | �0 = �) . (38)
From (28) �(�) ≥ �Λ(�) for any �; thus Λ∗ is the value that
maximizes (with respect to Λ) the function Λ �→ �Λ(�0). Tocomply with the theorem, the maximization is performed in
the interval [0, �0 ∧ �0].2.3. Numerical Application. We consider a perpetual loan(� = +∞) with a nominal amount � = 1 and the borrowerdefault intensity �� follows a CIR dynamics with parameters:
initial intensity �0 = 300 bps, volatility � = 0.05, averageintensity � = 200 bps, and reversion coeicient � = 0.5. Weassume a constant interest rate � = 300 bps; that is, � = 3%.Recall that a basis point, denoted by “1 bps”, equals 10−4.
In order to ind the initial contractual margin, we use (11)and ind �0 = 208 bps.
6 Abstract and Applied Analysis
00
0.005
0.005
0.01
0.01
0.015
0.015
0.02
0.02 0.025
0.025
0.03Λ
Figure 1: We illustrate here the dependence of �Λ(�0) as a functionofΛ; this allows for inding the optimal valueΛ∗ that maximizes theoption price. For the numerical example described here we obtainΛ∗ = 123 bps.
At inception, the present value of cash lows is at par,
so �(�0) = 1. he prepayment option price is �(+∞, �0) =0.0232; that is, �(�0) = 2.32% ⋅ �. herefore the loan value
equals �(�0) − �(�0) = 0,9768.he valueΛ∗ = 123 bps is obtained bymaximizing�Λ(�0)
as indicated in the remarks above; the dependence of �Λ(�0)with respect to Λ is illustrated in Figure 1. he loan value willbe equal to par if the intensity decreases until the exerciseregion (� < Λ∗) (see Figure 2).he continuation and exerciseregions are depicted in Figure 3. We postpone to Section 3.5the description of the numerical method to solve (22).
3. Perpetual Prepayment Option witha Switching Regime
In this second part, the perpetual prepayment option is stillan option on the credit risk, intensity, and also the liquiditycost. he liquidity cost is deined as the speciic cost of abank to access the cash on the market. his cost will bemodeled with a switching regime with a Markov chain ofinite states of the economy.he interest rate � is still assumedconstant. herefore, the assessment of the loan value andits prepayment option is an �-dimensional problem. heintensity is still deined by a Cox-Ingersoll-Ross process with2�� ≥ �2:
��� = � (� − ��) �� + �√�����, �0 = �0. (39)
3.1. heoretical Regime Switching Framework. We assume theeconomic state of the market is described by a inite stateMarkov chain X = {��, � ≥ 0}. he state space X can betaken to be, without loss of generality, the set of unit vectors� = {�1, �2, . . . , ��}, �� = (0, . . . , 0, 1, 0, . . . , 0)� ∈ R
�. Here �is the transposition operator.
1
0.98
0.96
0.94
0.92
0.9
0.88
0.860 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
�
Figure 2: Loan value as a function of the intensity.he loan value isdecreasing when there is a degradation of the credit quality (i.e., �increases) and converges to 0.0.06
0.05
0.04
0.03
0.02
0.01
0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1�
PricePayof
Figure 3: Prepayment option price �(�) (solid line) and payof �(�)(dashed line) as a function of the intensity �. Two regions appear:the continuation region � > Λ∗ and the exercise region � ≤ Λ∗.
Assuming the process�� is homogeneous in time and has
a rate matrix �, then if �� = E[��] ∈ R�,
����� = ���,�� = �0 + ∫�
0����� + ��,
(40)
where M = {��, � ≥ 0} is a martingale with respect to theiltration generated by X. In diferential form
��� = ����� + ���, �0 = �0. (41)
Abstract and Applied Analysis 7
We assume the instantaneous liquidity cost of the bankdepends on the state X of the economy, so that
�� = ⟨l, ��⟩ . (42)
Denote by ��,� the entry on the line � and the column � of the� × � matrix � with ��,� ≥ 0 for � = � and ∑�
�=1 ��,� = 0 forany �.3.2. Analytical Formulas for the PVRP. Assume a loan has aixed coupon deined by the constant interest rate � and aninitial contractual margin �0 calculated at the inception for apar value of the loan. Let �(�, �, ��, ��) be the present value ofthe remaining payments at time � of a corporate loan, where�� is the intensity at time �, � is the contractual maturity, �is the nominal amount, and �� is the state of the economy attime �.
he loan value LV(�, �, �) is still equal to the present valueof the remaining payments �(�, �, �) minus the prepaymentoption value �(�, �, �):
LV (�, �, �) = � (�, �, �) − � (�, �, �) . (43)
he PVRP � is the present value of the cash lowsdiscounted at the risky rate, where the risky rate at time � isthe constant risk-free rate � plus the liquidity cost �� and theintensity ��. Similar to the discussion in the Section 2.1, � isnot depending on time when � = +∞ (perpetual loan). Sowe denote that,
� (�,�) := � (� + �0)× E [∫+∞
0�−∫�0 �+��+������ | �0 = �,�0 = �] .
(44)
We consider that there is no correlation between the creditrisk, that is, the intensity ��, of the borrower and the cost toaccess the cash on the market, that is, the liquidity cost ��, ofthe lender. herefore, we have
� (�,�) = � (� + �0) ∫+∞
0�−��E [�−∫�0 ���� | �0 = �]× E [�−∫�0 ���� | �0 = �]��.
(45)
Remark 5. he crucial information here is that the coei-cients �, �, and � of the CIR process are not depending onthe regime �; thus we can separate the CIR dynamics andthe Markov dynamics at this level. A diferent approach canextend this result by using the properties of the PVRP asexplained in the next section.
Note that (cf. Section 2.1, equation (7))
E [�−∫�0 ���� | �0 = �] = � (0, �, �) (46)
and �(0, �, �) is evaluated using (8)–(11). In order to compute
E [�−∫�0 ���� | �0 = �] , (47)
let ��(�) be deined by
�� (�) = E [�−∫�0 ���� | �0 = ��] . (48)
Let � be the time of the irst jump from �0 = ⟨X, ��⟩ tosome other state. We know (cf. Lando [22] paragraph 7.7,page 211) that � is a random variable following an exponentialdistribution of parameter �� with
�� = ∑� = �
��,�. (49)
We also know that conditional to the fact that a jump hasoccurred at time �, the probability that the jump is from state�� to state �� is ��,�, where
��,� = ��,��� . (50)
hus,
�� (�) = P (� > �) �−���+ P (� ≤ �) �−���× ∑
� = �P (�� = ��)E [�−∫�� ���� | �� = ⟨X, ��⟩]
= �−(��+��)� + �� ∫�
0�−(��+��)� ∑
� = ���,��� (� − �) ��.
(51)
hen,
�(��+��)��� (�) = 1 + �� ∫�
0�(��+��)(�−�) ∑
� = ���,��� (� − �) ��
= 1 + �� ∫�
0�(��+��)� ∑
� = ���,��� (�) ��.
(52)
By diferentiation with respect to �,��� [�(��+��)��� (�)] = ���(��+��)� ∑
� = ���,��� (�) . (53)
hen
��� (�)�� + (�� + ��) �� (�) = �� ∑� = �
��,��� (�) . (54)
hus,
��� (�)�� = [[∑� = �
����,��� (�)]]− (�� + ��) �� (�) . (55)
Denote by �(�) the vector (�1(�), �2(�), . . . , ��(�))� and intro-duce the� × �matrix �
��,� = {����,� if � = �− (�� + ��) if � = �. (56)
8 Abstract and Applied Analysis
From (55) we obtain
�� (�)�� = �� (�) thus � (�) = ���� (0) (57)
with the initial condition
� (0) = (�� (0))��=1 = (1, 1, . . . , 1)� ∈ R�. (58)
Wehave therefore analytic formulas for the PVRP �(�,�).We refer the reader to [15] for similar considerations onrelated CIR switched dynamics.
Remark 6. When all liquidity parameters �� are equal (tosome quantity �), then � = � − � ⋅ ��, and then we obtain
(ater some computations) that ��(�) = �−��; thus the payofis equal to that of one-regime dynamics with interest rate� + �, which is consistent with intuitive image we may have.Another limiting case is when the switching is very fast (seealso Remark 10 item 5 for further details).
he margin �0 is set to satisfy the equilibrium equation:
� (�0, �0) = �. (59)
Similar arguments to that in previous section show that �0 >min��� > 0. See Remark 1 for the situationwhen an additionalcommercial margin is to be considered.
We will also need to introduce for any � = 1, . . . , � the
value Λ0� such that
� (Λ0�, ��) = �. (60)
Of course, Λ0�0 = �0. Recall that �(�, ��) is decreasing with
respect to �; when �(0, ��) < �, there is no solution to (59)
and we will choose by convention Λ0� = 0.
3.3. Further Properties of the PVRP �. It is useful for thefollowing to introduce a PDE formulation for �. To ease
the notations, we introduce the operator AR that acts on
functions V(�, �) as follows:(AR
V) (�, ��) = (AV) (�, ��) − (� + �� + �) V (�, ��)+ �∑
� = 1��,� (V (�, ��) − V (�, ��)) . (61)
Having deined the dynamics (39) and (41), one can usean adapted version of the Feynman-Kac formula in order toconclude that PVRP deined by (44) satisies the equation
(AR�) + (� + �0)� = 0. (62)
Remark 7. When the dynamics involve diferent coeicientsof the CIR process for diferent regimes (cf. also Remark 5),(62) changes in that it will involve, for �(⋅, ��), the operator
A� (V) (�) = �� (�� − �) ��V (�) + 12�2�����V (�) (63)
instead ofA.
3.4. Valuation of the Prepayment Option. hevaluation prob-lemof the prepayment option can bemodeled as anAmericancall option on a risky debt owned by the borrowerwith payof:
� (�,�) = (� (�,�) − �)+. (64)
Here the prepayment option allows borrower to buy back andreinance his/her debt according to the current contractualmargin at any time during the life of the option.
heorem8. For any N-tupleΛ = (Λ �)��=1 ∈ (R+)� introducethe function �Λ(�, �) such that
�Λ (�, ��) = � (�, ��) ∀� ∈ [0, Λ �] , (65)
(AR�Λ) (�, ��) = 0, ∀� > Λ �, � = 1, . . . , �, (66)
lim�→Λ �
�Λ (�, ��) = � (Λ �, ��) , if Λ � > 0, � = 1, . . . , �,(67)
lim�→∞
�Λ (�, ��) = 0, � = 1, . . . , �. (68)
Suppose a Λ∗ ∈ ∏��=1[0, (�0 − ��)+ ∧ Λ0
�] exists such that forall � = 1, . . . , �:
�Λ∗ (�, �) ≥ � (�,�) ∀�,�, (69)
��Λ∗ (�, ��)������������= (Λ∗�)
+= �� (�, ��)��
����������= (Λ∗�)−
if Λ∗� > 0,
(70)
�∑� = 1
��,� (�Λ∗ (�, ��) − � (�, ��)) + � (� + �� − �0) ≤ 0
∀� ∈ ]min�
Λ∗�, Λ∗
�[ .(71)
hen � = �Λ∗ .Proof. Similar arguments as in the proof ofheorem 3 lead toconsider the American option price in the form
� (�,�) = sup�∈T
E[�−∫�0 �+��+����� (��, ��) | �0 = �, �0=�] .(72)
We note that for Λ ∈ (R∗+)� if �Λ is the stopping time that
stops upon exiting the domain � > Λ � when� = ��, then�Λ (�, �)
= E [�−∫�Λ0 �+��+����� (��Λ , ��Λ) | �0 = �, �0 = �] .(73)
Remark that for Λ ∈ (R∗+)� the stopping time �Λ is inite a.e.
hus for any Λ ∈ (R∗+)� we have � ≥ �Λ; when Λ has some
null coordinates, the continuity (ensured among others by theboundary condition (65)) shows that we still have � ≥ �Λ. In
Abstract and Applied Analysis 9
particular for Λ∗ we obtain � ≥ �Λ∗ ; all that remains to beproved is the reverse inequality; that is, � ≤ �Λ∗ .
To this end we use a similar technique as in heorem10.4.1 [23, Section 10.4, page 227] (see also [7] for similarconsiderations). First one can invoke the same arguments asin the cited reference (cf. Appendix D for technicalities) andwork as if �Λ∗ is �2 (not only �1 as the hypothesis ensures).
Denote by �Λ∗ the set {(�, ��) | � ∈ [0, Λ∗�], � =1, . . . , �} (which will be the exercise region), and denote by�Λ∗ its complementary with respect toR+ ×� (which will be
the continuation region).
Lemma 9 shows that AR�Λ∗ is nonpositive everywhere(and is null on �Λ∗). he Ito formula shows that
�(�−∫�0 �+��+�����Λ∗ (��, ��))= �−∫�0 �+��+���� (AR�Λ∗) (��, ��) �� + � (martingale) .
(74)
Taking averages and integrating from 0 to some stopping time�, it follows fromAR�Λ∗ ≤ 0 that
�Λ∗ (�, �) ≥ E [�−∫�0 �+��+�����Λ∗ (��, ��) | �0 = �, �0 = �]≥ E [�−∫�0 �+��+����� (��, ��) | �0 = �, �0 = �] .
(75)
Since this is true for any stopping time �, the conclusionfollows.
Lemma9. Under the hypothesis of theheorem 8 the followinginequality holds (strongly except for the values (�, �) =(Λ∗
�, ��) and everywhere in a weak sense):
(AR�Λ∗) (�,�) ≤ 0, ∀� > 0, ∀�. (76)
Proof. he nontrivial part of this lemma comes from the factthat if for ixed � we have for � in a neighborhood of some�1: �Λ∗(�, ��) = �(�, ��), this does not necessarily imply(AR�Λ∗)(�1, ��) = (AR�)(�1, ��) because AR depends onother values �Λ∗(�, ��) with � = �.
From (66) the conclusion is trivially veriied for � = ��for any � ∈ ]Λ∗
�,∞[.We now analyse the situation when � < min�Λ∗
�; this
means in particular that 0 ≤ � < min�Λ∗� ≤ Λ0
ℓ for any ℓ; thusΛ0ℓ > 0. Note that Λ∗
� < Λ0� implies �(Λ∗
�, ��) ≥ �(Λ0�, ��) = �
for any � = 1, . . . , �; thus �(�, ��) = �(�, ��) − � for any� ∈ [0, Λ∗�] and any �. Furthermore since � < min�Λ∗
�, we
have�Λ∗(�, ��) = �(�, ��) = �(�, ��)−� for any �. Fix� = ��;then
(AR�Λ∗) (�, ��) = (AR�) (�, ��) = (AR (� − �)) (�, ��)= (AR�) (�, ��) −A
R (�)= − (� + �0)� − (� + �� + �)�= � (�� + � − �0)≤ � (�� + Λ∗
� − �0) ≤ 0,(77)
the last inequality being true by hypothesis.A last situation is when � ∈ ]min�Λ∗
�, Λ∗�[; then�Λ∗(�, ��) = �(�, ��) but some terms �Λ∗(�, ��) for � = �
may difer from �(�, ��). More involved arguments areinvoked in this case. his point is speciic to the fact that thepayof � itself has a complex structure and as such was notemphasized in previous works (e.g., [7], etc.).
Recalling the properties of �, one obtains (and since�Λ∗(�, ��) = �(�, ��)) the following:(AR�Λ∗) (�, ��) = (A�) (�, ��) − (� + �� + �) � (�, ��)
+ �∑� = 1
��,� (�Λ∗ (�, ��) − � (�, ��))= (AR�) (�, ��)
+ �∑� = 1
��,� (�Λ∗ (�, ��) − � (�, ��))= (AR�) (�, ��) −A
R (�)+ �∑
� = 1��,� (�Λ∗ (�, ��) − � (�, ��))
= − � (� + �0) + (� + �� + �)�+ �∑
� = 1��,� (�Λ∗ (�, ��) − � (�, ��)) ≤ 0,
(78)
where for the last inequality we use hypothesis (71). Finally,
since we proved that (AR�Λ∗)(�, �) ≤ 0 strongly except for
the values (�, �) = (Λ∗�, ��) and since �Λ∗ is of �1 class, we
obtain the conclusion (the weak formulation only uses theirst derivative of �Λ∗).Remark 10. Several remarks are in order at this point.
(1) When only one regime is present; that is, � = 1,the hypothesis of the theorem is identical to that ofheorem 3 since (71) is automatically satisied.
(2) When � > 1 checking (71) does not involve anycomputation of derivatives and is straightforward.
10 Abstract and Applied Analysis
(3) As mentioned in the previous section, the theoremis a veriication result, that is, it only gives suicientconditions for a candidate to be the option price.Two possible partial converse results are possible: theirst one to prove that the optimal price is indeed anelement of the family �Λ. he second converse resultis to prove that supposing � = �Λ∗ , then Λ∗ ∈∏�
�=1[0, (�0 − ��)+ ∧ Λ0�], and (69)–(71) are satisied.
(4) he search for the candidate Λ∗ can be done eitherby looking for a zero of the function Λ �→ Υ(Λ) :=((��Λ∗(�, ��)/��)|�= (Λ∗�)
+−(��(�, ��)/��)|�= (Λ∗�)−)��=1
or by maximizing on ∏��=1]0, (�0 − ��) ∧ Λ0
�[ the
function Λ �→ �Λ(�0, �0).(5) If the optimization of �Λ(�0, �0) is diicult to per-
form, one can use a continuation argument withrespect to the coupling matrix �. Denote by Λ∗(�)the optimal value of Λ∗ as function of �. When � =0, each Λ∗
� is found as in Section 2 (the problemseparates into � independent, that is, no coupled,valuation problems, each of which requiring to solvea one-dimensional optimization) and we constructthus Λ∗(0). When considering �� with � → ∞ atthe limit, the optimal Λ ∗(∞�) has all entries equalto Λ∗
mean, where Λ∗mean is the optimal value for one-
regime (� = 1) dynamics with riskless interest rate �being replaced by �+(∑�
�=1 ��/��)/(∑��=1 1/��). Hav-
ing established the two extremal points, the candidateΛ∗(�) is searchedwithin the�-dimensional segment[Λ∗(0), Λ∗(∞�)].3.5. Numerical Application. he numerical solution of thepartial diferential equation (66) is required. We use a initediferencemethod.he irst derivative is approximated by theinite diference formula
����Λ (�, �) = �Λ (� + ��,�) − �Λ (� − ��,�)2�� + � (��2) ,(79)
while the second derivative is approximated by
�2��2�Λ (�, �)= �Λ (� + ��,�) − 2�Λ (� + ��,�) + �Λ (� − ��,�)��2
+ � (��2) .(80)
To avoid working with an ininite domain, a well-knownapproach is to deine an artiicial boundary �max. hen aboundary condition is imposed on �max which leads to a
numerical problem in the inite domain ∪��=1[Λ∗
�, �max].In this numerical application, �max = 400 bps. We dis-cretize [Λ∗, �max] with a grid such that �� = 1 bps. Twoapproaches have been considered for imposing a boundaryvalue at �max: either consider that �Λ(�max, ��) = 0, ∀� =
0.03 0.030.02 0.020.01 0.010
0.025
0.02
0.015
0.01
0.005
0
Λ 1Λ 2
Figure 4: We illustrate here the dependence of �Λ(�0, �0) as afunction of Λ; this allows for inding the optimal (Λ∗
1 = 122 bps,Λ∗2 = 64 bps) that maximizes the option price.
1, . . . , � (homogeneous Dirichlet boundary condition) orthat (�/��)�Λ(�max, ��) = 0, ∀� = 1, . . . , � (homogeneousNeumann boundary condition). Both are correct in thelimit �max → ∞. We tested the precision of the resultsby comparing with numerical results obtained on a muchlarger grid (10 times larger) while using the same ��. heNeumann boundary condition gives much better results forthe situations we considered and as such was always chosen(see also Figure 6).
We consider a perpetual loanwith a nominal amount� =1 and the borrower default intensity �� follows CIR dynamics
with parameters: initial intensity �0 = 300 bps, volatility � =0.05, average intensity � = 200 bps, and reversion coeicient� = 0.5. We assume a constant interest rate � = 1% and aliquidity cost deined by a Markov chain of two states �1 =150 bps and �2 = 200 bps. For � = 2 the rate � matrix iscompletely deined by �1 = 1/3, �2 = 1.
In order to ind the initial contractual margin, we use (11)and ind �0 = 331 bps in the state 1. he contractual margintakes into account the credit risk (default intensity) and the
liquidity cost. We have thus Λ01 = �0; we obtain then Λ0
2 =260 bps.he optimal value Λ∗ is obtained by maximizing�Λ(�0, �0) and turns out to be (Λ∗
1, Λ∗2) = (122 bps, 64 bps)
(see Figure 4). To be accepted, this numerical solution hasto verify all conditions of heorem 8. Hypotheses (69) and(71) are satisied (see Figure 6) and hypothesis (71) is accepted
ater calculation. Moreover Λ∗1 ≤ (�0 − �1) ∧ Λ0
1 and theanalogous holds for Λ∗
2.In the state �0 = 1, the present value of cash lows
is at par, so �(�0, �0) = 1. he prepayment option price
is �(�0, �0) = 0.0240. herefore the loan value equals�(�0, �0) − �(�0, �0) = 0.9760.he loan value will be equal to the nominal if the intensity
decreases until the exercise region � ≤ Λ∗ (see Figure 5). hecontinuation and exercise regions are depicted in Figure 6.
Abstract and Applied Analysis 11
1.05
1
0.95
0.9
0.850 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
�(a)
1
0.95
0.9
0.850 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
�(b)
Figure 5: Loan value as a function of the intensity. (a) Regime� = �1; (b) regime � = �2. he loan value is decreasing whenthere is a degradation of the credit quality (i.e., when � increases)and converges to 0.
3.6. Regimes inWhich Is Never Optimal to Exercise. When theliquidity parameters corresponding to given regimes are very
diferent, it may happen that the optimization of �Λ(�0, �0)overΛ gives an optimumvalueΛ∗with somenull coordinatesΛ �� , � = 1, . . . .hismay hint to the fact that in this situation itis never optimal to exercise during the regimes ��� , � = 1, . . . .his is not surprising in itself (remember that this is thecase of an American call option) but needs more care when
being dealt with. Of course when in addition Λ0�� = 0, the
payof being null, it is intuitive that the option should not beexercised.
Remark 11. Further examination of heorem 3 calls for thefollowing remarks.
(1) he boundary value set in (65) for some regime�� with Λ∗� = 0 deserves an interpretation. he
boundary value does not serve to enforce continuityof � �→ �Λ(�) because there is no exercise region in
this regime. Moreover when 2�� ≥ �2, the intensity�� does not touch 0; thus the stopping time �Λ∗ isininite in the regime �� (thus the boundary valuein 0 is never used and thus need not be enforced);from a mathematical point of view it is known thatno boundary conditions are required at points wherethe leading order diferential operator is degenerate.
(2) It is interesting to know when such a situation canoccur and how can one interpret it. Let us take a two-regime case (� = 2): �1 a “normal” regime and �2 the“crisis” regime (�2 ≥ �1); when the agent contemplatesprepayment, the more severe the crisis (i.e., larger�2 − �1), the less he/she is likely to prepay duringthe crisis when the cash is expensive (high liquidity
0.010 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1�
Price
Payof
0.06
0.04
0.02
0
(a)
0.010 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1�
Price
Payof
0.050.040.030.020.010
(b)
Figure 6: he price of the prepayment option �Λ∗ (�) (solid line)and the payof �(�) (dashed line) as a function of the intensity �. (a)Regime � = �1; (b) regime � = �2. For each regime two regionsappear: the continuation region � > Λ∗
� and the exercise region � ≤Λ∗� .
cost). We will most likely see that for �1 = �2 someexercise regions exist while starting from some large�2 the exercise region will disappear in regime �2. hisis completely consistent with the numerical resultsreported in this paper.
3.7. Numerical Application. We consider the same situationas in Section 3.7 except that �1 = 50 bps and �2 = 250 bps.In order to ind the initial contractual margin, we use (11)and ind �0 = 305 bps in the state 1. he contractual margintakes into account the credit risk (default intensity) and the
liquidity cost. As before Λ01 = �0 but here we obtain Λ0
2 =221 bps.he couple (Λ∗
1 = 121 bps, Λ∗2 = 0) (see Figure 7)
maximizes �Λ(�0, �0). here does not exist an exerciseboundary in the state 2. he loan value will equal the par ifthe intensity decreases until the exercise region � ≤ Λ∗ (seeFigure 8).he continuation and exercise regions are depictedin Figure 9.
To be accepted as true price, the numerical solution �Λ∗has to verify all hypotheses and conditions of heorem 8.In the regime � = �1, hypotheses (69) and (70) areveriied numerically (see also Figure 9) and hypothesis (71)
is accepted ater calculation. MoreoverΛ∗� ≤ (�0−��)∧Λ0
� for� = 1, 2.
12 Abstract and Applied Analysis
0.025
0.02
0.015
0.01
0.005
00.03 0.030.02 0.020.01 0.010 0Λ 1Λ 2
Figure 7: We illustrate here the dependence of �Λ(�0, �0) as afunction of the exercise boundary Λ; this allows to ind the optimal(Λ∗
1 = 121 bps, Λ∗2 = 0) that maximizes the option price.
1
0.95
0.9
0.85 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1�
(a)
1
0.95
0.9
0.80
0.85
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1�
(b)
Figure 8: Loan value as a function of the intensity. (a) Regime� = �1; (b) regime � = �2. he loan value is decreasing whenthere is a degradation of the credit quality (i.e., when � increases)and converges to 0.
In the state � = �1, the present value of cash lows is at
par, so �(�0, �0) = � = 1. he prepayment option price is�(�0) = 0.0245. herefore the loan value LV equals �(�0) −�(�0) = 0.9755.4. Concluding Remarks
We proved in this paper two suicient theoretical resultsconcerning the prepayment option of corporate loans. Inour model the interest rate is constant, the default intensityfollows a CIR process, and the liquidity cost follows a discretespace Markov jump process. he theoretical results wereimplemented numerically and show that the prepayment
0.06
0.04
0.02
00.010 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
�
Price
Payof
(a)
0.050.040.030.020.0100.010 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
�
Price
Payof
�: 0.0221�: 6.039� − 05
(b)
Figure 9: he price of the prepayment option �Λ∗ (�) (solid line)and the payof �(�) (dashed line) as function of the intensity �.(a) regime � = �1; (b) regime � = �2. Two regions appear: thecontinuation region � > Λ∗
1 and the exercise region � ≤ Λ∗1. For
the second regime there is no exercise region.
option cost is not negligible and should be taken into accountin the asset liability management of the bank. Moreover it isseen that when liquidity parameters are very diferent (i.e.,when a liquidity crisis occur) in the high liquidity cost regime,the exercise domainmay entirely disappear, meaning that it isnot optimal for the borrower to prepay during such a liquiditycrisis.
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