Valuation of the prepayment option of a perpetual corporate loan

. We investigate in this paper a perpetual prepayment option related to a corporate loan. The default intensity of the (cid:28)rm is supposed to follow a CIR process. Two frameworks are discussed: (cid:28)rst a constant interest rate and a secondly a multi-regime framework where the interest rate is augmented by a liquidity factor dependent on the regime. The prepayment option needs speci(cid:28)c attention as the payo(cid:27) itself is an implicit function of the parameters of the problem and of the dynamics. We establish in the unique regime case analytic formulas for the payo(cid:27) of the option; in both cases we give a veri(cid:28)cation result that allows to compute the price of the option. Numerical results that implement the (cid:28)ndings are also presented and are completely consistent with the theory; it is seen that when liquidity parameters are very di(cid:27)erent (i.e., when a liquidity crisis occur) in the high liquidity cost regime the exercise domain may entirely disappear meaning that it is not optimal for the borrower to prepay during such a liquidity crisis. The method allows to quantify and interpret these (cid:28)ndings.


Introduction. When a rm needs money it can turn to its bank which lends
it against e.g., periodic payments in a form of a loan. In almost every loan contract, the borrower has the free option to prepay a portion or all the nominal. Even if the technicalities are, as it will be seen in the following, dierent, the concept of this option is very close to the embedded option of a callable bond. When its credit spread has gone down the issuer of the bond can buy back his debt at a dened call price before the bond reaches its maturity date. It allows the issuer to renance its debt at a cheaper rate.
In order to decide whether the exercise of the option is worthwhile the borrower compares the remaining payments (actualized by the interest rate he can obtain at that time) with the nominal value. If the remaining payments exceed the nominal value then it is optimal for the borrower to renance his debt at a lower rate.
When the interest rates are not constant or borrower is subject to default the computation of the actualization is less straightforward. It starts with considering all possible scenarios of evolution for interest rate and default intensity in a risk-neutral framework and compute the average value of remaining payments (including the nal payment of the principal if applicable); this quantity will be called P V RP (denoted ξ) and is the present value of the remaining payments i.e., the cash amount equivalent, both for borrower and lender in this model of the set of remaining payments. The P V RP is compared with the nominal : if the P V RP value is larger than the nominal then the borrower should prepay, otherwise not. Recall that at the initial time the payments correspond to a rate, the sum of the interest rate and a contractual margin ρ 0 , which is precisely making the two quantities equal. Note that in order to compute the price of the embedded prepayment option the lender also uses the P V RP as it will be seen below.
For a bank, the prepayment option is essentially a reinvestment risk i.e., the risk that the borrower decides to repay earlier his/her loan and that the bank can not reinvest his/her excess of cash in a new loan. So the longest the maturity of the loan, the riskier the prepayment option. Therefore, it is interesting to study long-term loans that are set for more than three years and can run for more than twenty years.
The valuation problem of the prepayment option can be modelled as an American embedded option on a risky debt owned by the borrower. As Monte-Carlo simulations are slow to converge to assess accurately the continuation value of the option during the life of the loan and that the binomial tree techniques are time-consuming for longterm loans (cf. works by D. Cossin et al. [8]), we decided to focus, in this paper, on the prepayment option for perpetual loan.
When valuing nancial products with long maturity the robustness with respect to shocks and other exogenous variabilities is important. Among problems that have to be treated is the liquidity and its variability. Liquidity is the key of the stability of sheets. Even if liquidity problems have a very low probability to occur, a liquidity crisis can have a severe impact on a bank's funding costs, its market access (reputation risk) and short-term funding capabilities.
Following the state of the economic environment, the liquidity can be dened by distinct states. Between two crises, investors are condent and banks nd it easier to launch their long term renancing programs through regular bonds issuances. Thus the liquidity market is stable. Unfortunately, during crisis, liquidity become scarce, pushing the liquidity curve to very high levels which can only decrease if condence returns to the market. The transition is between these two distinct behaviors is rarely smooth but rather sudden. In order to model the presence of distinct liquidity behaviors we will simulate the liquidity cost by a continuous time Markov chain that can have a discrete set of possible values, one for each regime that is encountered in the liquidity evolution.
From a technical point of view this paper faces several non-standard conditions: although the goal is to value a perpetual American option the payo of the option is highly non-standard (is dependent on the P V RP ). As a consequence the characterization of the exercice region is not standard and technical conditions have to be met.
Furthermore our focus here is on a specic type of dynamics (of CIR type) with even more specic interest on the situation when several regimes are present.
The balance of the paper is as follows: in the remainder of this section (Sub-Section 1.1) we review the related existing literature; in Section 2, we consider that the liquidity cost is negligible and the borrower credit risk dened by his/her default intensity (called in the following simply intensity) which follows a CIR stochastic process. We are able to obtain in this situation a quasi-analytic formula for the prepayment option price. In Section 3 we explore the situation when the liquidity cost, dened as the cost of the lender to access the cash on the market, has several distinct regimes that we model by a Markov chain. We write the pricing formulas and theoretically support an algorithm to identify the boundary of the exercice region; nal numerical examples close the paper.
1.1. Related literature. There exist few articles (e.g., works by D. Cossin et al. [8]) on the loan prepayment option but a close subject, the prepayment option in a xed-rate mortgage loan, has been widely covered in several papers by J.E. Hilliard and J.B. Kau [11] and more recent works by Chen et al. [6]. To approximate the PDE satised by the prepayment option, they dene two state variables (interest rate and house price). Their approach is based on a bivariate binomial option pricing technique with a stochastic interest rate and a stochastic house value.
Another contribution by D. Cossin et al. [8] applies the binomial tree technique (but of course it is time-consuming for long-term loans due to the nature of binomial trees) to corporate loans. They consider a prepayment option with a 1 year loan with a quarterly step but it is dicult to have an accurate assessment of the option price for a 10 years loan.
There also exist mortgage prepayment decision models based on Poisson regression approach for mortgage loans. See, for example, E.S. Schwartz and W.N.
Torous [22]. Unfortunately, the volume and history of data are very weak in the corporate loan market.
Due to the form of their approach, these papers did not have to consider the geometry of the exercice region because it is explicitly given by the numerical algorithm. This is not the case for us and requires that particular care be taken when stating the optimality of the solution. Furthermore, to the best of our knowledge, none of these approaches explored the circumstance when several regimes exist.
The analysis of Markov-modulated regimes has been investigated in the literature when the underlying(s) follow the Black& Scholes dynamics with drift and volatility having Markov jumps; several works are of interest in this area: Guo and Zhang [26] have derived the closed-form solutions for vanilla American put; Guo analyses in [10] Russian (i.e., perpetual look-back) options and is able to derive explicit solutions for the optimal stopping time; in [24] Y. Xu and Y. Wu analyse the situation of a two-asset perpetual American option where the pay-o function is a homogeneous function of degree one; Mamon and Rodrigo [18] nd explicit solutions to vanilla European options. Bungton and Elliott [4] study European and American options and obtain equations for the price. A distinct approach (Hopf factorization) is used by Jobert and Rogers [14] to derive very good approximations of the option prices for, among others, American puts. Other contributions include [25,23] etc.
Works involving Markov switched regimes and CIR dynamics appears in [9] where the bond valuation problem is considered (but not in the form of an American option; their approach will be relevant to the computation of the payo of our American option although in their model only the mean reverting level is subject to Markov jumps) and in [27] where the term structure of the interest rates is analyzed.
On the other hand numerical methods are proposed in [12] where it is found that a xed point policy iteration coupled with a direct control formulation seems to perform best.
Finally, we refer to [13] for theoretical results concerning the pricing of American options in general.
2. Perpetual prepayment option with a stochastic intensity CIR model. We assume throughout the paper that the interest rate r is constant. Therefore, the price of the prepayment option only depends on the intensity evolution over time.
We model the intensity dynamics by a Cox-Ingersoll-Ross process (see [5,2,16] for theoretical and numerical aspects of CIR processes and the situations where the CIR 3 hal-00653041, version 2 -3 Dec 2012 process has been used in nance): It is known that if 2γθ ≥ σ 2 then CIR process ensures an intensity strictly positive.
Fortunately, as it will be seen in the following, the PVRP is given by an analytic formula.
2.1. Analytical formulas for the PVRP. Assume a loan with a xed coupon dened by the interest rate r and an initial contractual margin ρ 0 . Let ξ(t, T, λ) be, the present value of the remaining payments at time t of a corporate loan with initial contractual margin ρ 0 (depending on λ 0 ), intensity at time t, λ t , following the riskneutral equation (2.1) with λ t = λ, has nominal amount K and contractual maturity T .
Therefore the loan value LV (t, T, λ) is equal to the present value of the remaining payments ξ(t, T, λ) minus the prepayment option value P (t, T, λ).
The present value of the cash ows discounted at the (instantaneous) risky rate r + λ t , is denoted by ξ. The innitesimal cash ow at time t is K(r + ρ 0 ) and the nal payment of the principal K. Then: For a perpetual loan the maturity T = +∞. Since λ t is always positive r + λ t > 0 and thus the last term tend to zero when T → ∞. A second remark is that since γ, θ and σ independent of time, ξ is independent of the starting time t : +∞ 0 e − t 0 r+λudu dt λ 0 = λ =: ξ(λ), (2.5) where the last equality is a denition. For a CIR stochastic process, we obtain (see [5,16]), where for general t,t we use the notation: Note that B(t,t, λ) is a familiar quantity: it is formally the same formula as the price of a zero-coupon bond where the interest rates follow a CIR dynamics. Of course here the interest rate is constant and the intensity is following a CIR dynamics nevertheless the same formula applies for general t,t: Obviously B(0, t, λ) is monotonic with respect to λ, thus the same holds for ξ. The margin ρ 0 is the solution of the following equilibrium equation: which can be interpreted as the fact that the present value of the cash ows (according to the probability of survival) is equal to the nominal K: Note that we assume no additional commercial margin.
Remark 1. If an additional commercial margin µ 0 is considered then ρ 0 is rst computed as above and then replaced by ρ 0 = ρ 0 + µ 0 in Equation (2.6). Equations (2.10) and (2.11) will not be veried as such but will still hold with some λ 0 instead of λ 0 ; for instance we will have With these changes all results in the paper are valid, except that when computing for operational purposes once the price of the prepayment option is computed for all λ one will use λ = λ 0 as price relevant to practice. From denition (2.7) of B(t,t, λ) it follows that B(t,t, λ) < 1 thus e −rt B(0,t, λ 0 ) < e −rt and as consequence which implies that ρ 0 > 0.

2.2.
Valuation of the prepayment option. The valuation problem of the prepayment option can be modelled as an American call option on a risky debt owned by the borrower. Here the prepayment option allows borrower to buy back and renance his/her debt according to the current contractual margin at any time during the life of the option. As the perpetual loan, the option value will be assumed independent of the time t. As discussed above, the prepayment exercise results in a pay-o (ξ(t, T, λ) − K) + for the borrower. The option is therefore an American call option on the risky asset ξ(t, T, λ t ) and the principal K (the amount to be reimbursed) being the strike. Otherwise we can see it as an American option on the risky λ t with pay-o, 14) or, for our perpetual option: We will denote by A the characteristic operator (cf. [15,Chapter 7.5]) of the CIR process i.e. the operator that acts on any C 2 class function v by Denote for a, b ∈ R and x ≥ 0 by U (a, b, x) the solution to the conuent hypergeometric dierential (also known as the Kummer) equation [1]: that increase at most polynomially at innity and is nite (not null) at the origin.
Recall also that this function is proportional to the the conuent hypergeometric function of the second kind U (a, b, x) (also known as the Kummer's function of the second kind, Tricomi function, or Gordon function); for a, x > 0 the function U (a, b, x) is given by the formula: When a ≤ 0 one uses other representations (see the cited references; for instance one can use a direct computation or the recurrence formula U behaves as x −a at innity. Also introduce for x ≥ 0: Theorem 2. 1. Introduce for Λ > 0 the family of functions: P Λ (λ) such that: Then the price of the prepayment option is P (λ) = P Λ * (λ). Proof. We start with the rst item: it is possible to obtain a general solution of (2.21) in an analytic form. We recall that z(X) = U (a, b, X) is the solution of with some C Λ > 0 to be determined. Now use the boundary conditions. If λ = Λ by continuity We now continue with the second part of the theorem. The valuation problem of an American option goes through several steps: rst one introduces the admissible trading and consumptions strategies cf. [19,Chapter 5]; then one realizes using results in cited reference (also see [20,16]) that the price P (λ) of the prepayment option involves computing a stopping time associated to the pay-o. Denote by T the ensemble of (positive) stopping times; we conclude that: Further results derived for the situation of a perpetual (standard) American put options [13,3] show that the stopping time has a simple structure: a critical level exists that split the positive axis into two regions: to the left the exercice region where it is optimal to exercice and where the price equals the payo and a continuation region (to the right) where the price satises a partial dierential equation similar to Black-Scholes equation. We refer to [7] for how to adapt the theoretical arguments for the situation when the dynamics is not Black-Scholes like but a CIR process.
The result builds heavily on the fact that the discounted payo of the standard situation of an American put e −rt (S − K) − , is a submartingale. For us the discounted payo is (see also [7] for specic treatment of the CIR process) which will show that P (λ) is the true option price if the following conditions are satised:  We now show that P Λ * veries all conditions above which will allow to conclude that P = P Λ * . The requirement 1 is treated in Lemma 2.1; the requirement 3 amounts at asking that the optimal frontier value Λ * be chosen such that: The requirement 2 implies that in the continuation region the price is the solution of the following PDE: For this PDE we need boundary conditions. The condition at λ = Λ * is  (Aχ)(λ) − (r + λ)χ(λ) < 0, ∀λ < ρ 0 ∧ λ 0 .
Note that the Theorem 2 is only a sucient result (a so-called "verication" result) ; under the assumption that a Λ * fullling the hypotheses of the Theorem exist the question is how to nd it.
Two approaches can be considered; rst, it is enough to nd a zero of the following ∂λ λ=Λ − (the last equality is a denition).
Of course ∂χ(λ) ∂λ λ=λ0+ = 0 and The theorem asks furthermore to restrict the search to the interval [0, λ 0 ∧ ρ 0 ]. A dierent convenient procedure to nd the critical Λ * is to consider the dependence Λ → P Λ (λ 0 ). Let us consider the stopping time τ Λ that stops upon entering the domain [0, Λ]. We remark that by a Feynman-Kac formula (cf. [15, p 203 We illustrate here the dependence of P Λ (λ 0 ) as a function of Λ; this allows to nd the optimal value Λ * that maximize the option price. For the numerical example described here we obtain Λ * = 123 bps. Loan value as a function of the intensity. The loan value is decreasing when there is a degradation of the credit quality (i.e., λ increases) and converges to 0.
In order to nd the initial contractual margin we use equation (2.11) and nd ρ 0 = 208 bps.
The value Λ * = 123 bps is obtained by maximizing P Λ (λ 0 ) as indicated in the Remarks above; the dependence of P Λ (λ 0 ) with respect to Λ is illustrated in Figure 2   3. Perpetual prepayment option with a switching regime. In this second part, the perpetual prepayment option is still an option on the credit risk, intensity, but now also the liquidity cost. The liquidity cost is dened as the specic cost of a bank to access the cash on the market. This cost will be modelled with a switching regime with a Markov chain of nite states of the economy. We assume an interbank oered rate IBOR r to be constant. Therefore, the assessment of the loan value and its prepayment option is a N -dimensional problem. The intensity is still dened by a Cox-Ingersoll-Ross process with 2kθ ≥ σ 2 : dλ t = γ(θ − λ t )dt + σ λ t dW t , λ 0 = λ 0 . Assuming the process X t is homogeneous in time and has a rate matrix A, then and, where M = {M t , t ≥ 0} is a martingale with respect to the ltration generated by X.
In dierential form dX t = AX t dt + dM t , X 0 = X 0 .

(3.4)
We assume the instantaneous liquidity cost of the bank depends on the state X of the economy, so that l t = l, X t

(3.5)
Denote by a k,j the entry on the line k and the column j of the N × N matrix A with a k,j ≥ 0 for j = k and N j=1 a k,j = 0 for any k.
3.2. Analytical formulas for the PVRP. Assume a loan has a xed coupon dened by the interest rate r and an initial contractual margin ρ 0 calculated at the inception for a par value of the loan. Let ξ(t, T, λ t , X t ) be, the present value of the remaining payments at time t of a corporate loan where: λ t is the intensity at time t; T is the contractual maturity; K is the nominal amount and X t is the state of the economy at time t. The loan value LV (t, T, λ) is still equal to the present value of the remaining payments ξ(t, T, λ) minus the prepayment option value P (t, T, λ).
The PVRP ξ is the present value of the cash ows discounted at the risky rate, where the risky rate at time t is the constant risk-free rate r plus the liquidity cost l t and the intensity λ t . Similar to the discussion in the Subsection 2.1, ξ is not depending on time when T = +∞ (perpetual loan). So we denote, We consider that there is no correlation between the credit risk, i.e., the intensity λ t , of the borrower and the cost to access the cash on the market, i.e. the liquidity cost l t , of the lender. Therefore, we have, Remark 3. The crucial information here is that the coecients γ, θ, σ of the CIR process are not depending on the regime X thus we can separate the CIR dynamics and the Markov dynamics at this level. A dierent approach can extend this result by using the properties of the PVRP as explained in the next section.
Note that (cf. Subsection 2.1 equation (2.7)) E e − t 0 λudu λ 0 = λ = B(0, t, λ) (3.9) and B(0, t, λ) is evaluated using equations (2.8) -(2.11). In order to compute (3.10) Let τ , the time of the rst jump from X 0 =< X, e k > to some other state. We know (cf. Lando [17] paragraph 7.7 p 211) that τ is a random variable following an exponential distribution of parameter α k with, We also know that conditional to the fact that a jump has occurred at time τ the probability that the jump is from state e k to state e j is p k,j , where Then, By dierentiation with respect to t: T and introduce the N × N matrix B, From equation (3.13) we obtain, with the initial condition, We have therefore analytical formulas for the PVRP ξ(λ, X). We refer the reader to [9] for similar considerations on a related CIR switched dynamics.
Remark 4. When all liquidity parameters l k are equal (to some quantity l) then B = A − l · Id and then we obtain (after some computations) that f k (t) = e −lt thus the payo is equal to that of a one-regime dynamics with interest rate r + l, which is consistent with intuitive image we may have. Another limiting case is when the switching is very fast, see also Remark 7 item 6 for further details.
The margin ρ 0 is set to satisfy the equilibrium equation ξ(λ 0 , X 0 ) = K. (3.17) Similar arguments to that in previous section show that ρ 0 > min k l k > 0. See Remark 1 for the situation when a additional commercial margin is to be considered.
3.3. Further properties of the PVRP ξ. It is useful for the following to introduce a PDE formulation for ξ. To ease the notations we introduce the operator A R that acts on functions v(λ, X) as follows:  Remark 5. When the dynamics involves dierent coecients of the CIR process for dierent regimes (cf. also Remark 3) the Equation (3.20) changes in that it will involve, for ξ(·, e k ), the operator Here the prepayment option allows borrower to buy back and renance his/her debt according to the current contractual margin at any time during the life of the option.
Proof. Similar arguments as in the proof of Thm. 2 lead to consider the American option price in the form We note that for Λ ∈ (R * + ) N if τ Λ is the stopping time that stops upon exiting the domain λ > Λ k when X = e k then Remark that for Λ ∈ (R * + ) N the stopping time τ Λ is nite a.e. Thus for any Λ ∈ (R * + ) N we have P ≥ P Λ ; when Λ has some null coordinates the continuity (ensured among others by the boundary condition (3.23)) shows that we still have P ≥ P Λ . In particular for Λ * we obtain P ≥ P Λ * ; all that remains to be proved is the reverse inequality i.e. P ≤ P Λ * .
To this end we use a similar technique as in Thm. 10 [26] for similar considerations). First one can invoke the same arguments as in cited reference (cf. Appendix D for technicalities) and work as if P Λ * is C 2 (not only C 1 as the hypothesis ensures).
The Lemma 3.1 shows that A R P Λ * is non-positive everywhere (and is null on C Λ * ). The Îto formula shows that d e − t 0 r+ls+λsds P Λ * (λ t , X t )) = e − t 0 r+ls+λsds (A R P Λ * )(λ t , X t ))dt + d(martingale) (3.30) Taking averages and integrating from 0 to some stopping time τ it follows from Since this is true for any stopping time τ the conclusion follows. Proof. The non-trivial part of this lemma comes from the fact that if for xed k we have for λ in a neighborhood of some λ 1 : P Λ * (λ, e k ) = χ(λ, e k ) this does not necessarily imply (A R P Λ * )(λ 1 , e k ) = (A R χ)(λ 1 , e k ) because A R depends on other values P Λ * (λ, e j ) with j = k. From (3.24) the conclusion is trivially veried for X = e k for any λ ∈]Λ * k , ∞[. We now analyze the situation when λ < min j Λ * j ; this means in particular that k , e k ) = K for any k = 1, ..., N thus χ(λ, e k ) = ξ(λ, e k ) − K for any λ ∈ [0, Λ * k ] and any k. Furthermore since λ < min j Λ * j we have P Λ * (λ, e k ) = χ(λ, e k ) = ξ(λ, e k ) − K for any k. Fix X = e k ; then the last inequality being true by hypothesis.
A last situation is when λ ∈] min j Λ * j , Λ * k [; there P Λ * (λ, e k ) = χ(λ, e k ) but some terms P Λ * (λ, e j ) for j = k may dier from χ(λ, e j ). More involved arguments are invoked in this case. This point is specic to the fact that the payo χ itself has a complex structure and as such was not emphasized in previous works (e.g., [26], etc.).
2. when N > 1 checking (3.29) does not involve any computation of derivatives and is straightforward.
3. as mentioned in the previous section, the Theorem is a verication result i.e., only gives sucient conditions for a candidate to be the option price.
Two possible partial converse results are possible: a rst one to prove that the optimal price is indeed an element of the family P Λ . A second converse result is to prove that supposing P = P Λ * then Λ * ∈ N k=1 [0, (ρ 0 − l k ) + ∧ Λ 4. a more general verication result for dierent payo function χ can be proven, cf [21] for details. 5. the search for the candidate Λ * can be done either by looking for a zero of if the optimization of P Λ (λ 0 , X 0 ) is dicult to perform, one can use a continuation argument with respect to the coupling matrix A. Denote by Λ * (A) the optimal value of Λ * as function of A. When A = 0 each Λ * k is found as in Section 2 (the problem separates into N independent i.e., no coupled, valuation problems, each of which requiring to solve a one dimensional optimization) and we construct thus Λ * (0). When considering µA with µ → ∞ at the limit the optimal Λ * (∞A) has all entries equal to Λ * mean where Λ * mean is the optimal value for a one-regime (N = 1) dynamics with riskless interest rate r being replaced by r + 7. note that this continuation procedure above works even when the CIR parameters depend on k (cf. [21] for details).
3.5. Numerical Application. The numerical solution of the partial dierential equation (3.24) is required. We use a nite dierence method. The rst derivative is approximated by the nite dierence formula: while the second derivative is approximated by: To avoid working with an innite domain a well-known approach is to dene an articial boundary λ max . Then a boundary condition is imposed on λ max which leads to a numerical problem in the nite domain ∪ N k=1 [Λ * k , λ max ]. In this numerical application, λ max = 400 bps. We discretize [Λ * , λ max ] with a grid such that δλ = 1bps. Two approaches have been considered for imposing a boundary value at λ max : either consider that P Λ (λ max , e k ) = 0, ∀k = 1, ..., N (homogenous Dirichlet boundary condition) or that ∂ ∂λ P Λ (λ max , e k ) = 0, ∀k = 1, ..., N (homogeneous Neuman boundary condition). Both are correct in the limit λ max → ∞. We tested the precision of the results by comparing with numerical results obtained on a much larger grid (10 times larger) while using same δλ. The Neumann boundary condition gives much better results for the situations we considered and as such was always chosen (see also We consider a perpetual loan with a nominal amount K = 1 and the borrower default intensity λ t follows a CIR dynamics with parameters: initial intensity λ 0 = 300bps, volatility σ = 0.05, average intensity θ = 200bps, reversion coecient γ = 0.5. We assume a constant interest rate r = 1% and a liquidity cost dened by a Markov chain of two states l 1 = 150bps and l 2 = 200bps. For N = 2 the rate A matrix is completely dened by α 1 = 1/3, α 2 = 1. In order to nd the initial contractual margin we use equation (2.11) and nd ρ 0 = 331 bps in the state 1. The contractual margin takes into account the credit We illustrate here the dependence of P Λ (λ 0 , X 0 ) as a function of Λ; this allows to nd the optimal (Λ * 1 = 122bps, Λ * 2 = 64bps) that maximizes the option price. The optimal value Λ * is obtained by maximizing P Λ (λ 0 , X 0 ) and turns out to be (Λ * 1 , Λ * 2 ) = (122bps, 64bps), see In the state X 0 = 1, the present value of cash ows is at par, so ξ(λ 0 , X 0 ) = 1. The prepayment option price is P (λ 0 , X 0 ) = 0.0240. Therefore the loan value equals ξ(λ 0 , X 0 ) − P (λ 0 , X 0 ) = 0.9760.
The loan value will equal to the nominal if the intensity decreases until the exercise region λ ≤ Λ * see  3.6. Regimes when is never optimal to exercise. When the liquidity parameters corresponding to given regimes are very dierent it may happen that the optimization of P Λ (λ 0 , X 0 ) over Λ gives an optimum value Λ * with some null coordinates Λ ki , i = 1, .... This may hint to the fact that in this situation it is never optimal to exercise during the regimes e ki , i = 1, .... This is not surprising in itself (remember that this is the case of an American call option) but needs more care when dealing with. Of course when in addition Λ 0 ki = 0 the payo being null it is intuitive that the option should not be exercised.
Remark 8. Further examination of the Theorem 2 calls for the following remarks: 1. the boundary value set in eqn. (3.23) for some regime e k with Λ * k = 0 deserves an interpretation. The boundary value does not serve to enforce continuity of λ → P Λ (λ) because there is no exercise region in this regime thus any value will do. Moreover when 2γθ ≥ σ 2 the intensity λ u does not touch 0 thus the stopping time τ Λ * is innite in the regime e k (thus the boundary value in 0 can be set to any arbitrary number since it is never used). The real meaning of the value P Λ * (0, e k ) comes from arbitrage considerations: when one proves in the demonstration of the Theorem that P ≥ P Λ * one uses continuity of P Λ with respect to the parameter Λ; in order to still have this conclusion one has to set P Λ * (0, e k ) ≤ lim Λ∈(R * + ) N →Λ * P Λ (0, e k ) = χ(0, e k ). On the contrary, in order to have P ≤ P Λ * , since P ≥ χ is it required that P Λ * (0, e k ) ≥ P (0, e k ) ≥ χ(0, e k ). Thus only P Λ * (0, e k ) = χ(0, e k ) can prevent arbitrage.
2. it is interesting to know when such a situation can occur and how can one interpret it. Let us take a two-regime case (N = 2): l 1 a normal regime and l 2 the crisis regime (l 2 ≥ l 1 ); when the agent contemplates prepayment the more severe the crisis (i.e. larger l 2 −l 1 ) less he/she is likely to prepay during the crisis the cash is expensive (high liquidity cost). We will most likely see that for l 1 = l 2 some exercise region exists while starting from some large l 2 the exercise region will disappear in regime e 2 . This is completely consistent with the numerical results reported in this paper. 3.7. Numerical Application. We consider the same situation as in Section 3.7 except that l 1 = 50bps and l 2 = 250 bps. In order to nd the initial contractual margin we use equation (2.11) and nd ρ 0 = 305 bps in the state 1. The contractual margin We illustrate here the dependence of P Λ (λ 0 , X 0 ) as a function of the exercise boundary Λ; this allows to nd the optimal (Λ * 1 = 121bps, Λ * 2 = 0) that maximizes the option price.  Fig. 3.5. Loan value as a function of the intensity. Top: regime X = e 1 ; bottom: regime X = e 2 . The loan value is decreasing when there is a degradation of the credit quality (i.e. when λ increases) and converges to 0. takes into account the credit risk (default intensity) and the liquidity cost. As before Λ 0 1 = λ 0 but here we obtain Λ 0 2 = 221bps. The couple (Λ * 1 = 121bps, Λ * 2 = 0) (see Figure 3.4) maximizes P Λ (λ 0 , X 0 ). There does not exist a exercise boundary in the state 2. The loan value will equal the par if the intensity decreases until the exercise region λ ≤ Λ * see To be accepted as true price the numerical solution P Λ * has to verify all hypothesis and conditions of the Theorem 6. In the regime X = e 1 , the hypothesis (3.27) and (3.28) are veried numerically (see also Figure 3.6) and the hypothesis (3.29) is accepted after calculation. Moreover Λ * k ≤ (ρ 0 − l k ) ∧ Λ 0 k for k = 1, 2. In the state X = e 1 , the present value of cash ows is at par, so ξ(λ 0 , X 0 ) = K = 1. The prepayment option price is P (λ 0 ) = 0.0245. Therefore the loan value LV equals ξ(λ 0 ) − P (λ 0 ) = 0.9755.