Valuation of the Prepayment Option of a Perpetual Corporate Loan

Abstract

We investigate in this paper a perpetual prepayment option related to a corporate loan. The default intensity of the firm is supposed to follow a CIR process. We assume that the contractual margin of the loan is defined by the credit quality of the borrower and the liquidity cost that reflects the funding cost of the bank. Two frameworks are discussed: firstly a loan margin without liquidity cost and secondly a multiregime framework with a liquidity cost dependent on the regime. The prepayment option needs specific attention as the payoff itself is an implicit function of the parameters of the problem and of the dynamics. In the unique regime case, we establish quasianalytic formulas for the payoff of the option; in both cases we give a verification result that allows for the computation of the price of the option. Numerical results that implement the findings are also presented and are completely consistent with the theory; it is seen that when liquidity parameters are very different (i.e., when a liquidity crisis occurs) 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 for quantification and interpretation of these findings.

1. Introduction

When a firm needs money, it can turn to its bank which lends it against, for example, periodic payments in a form of a loan. In almost every loan contract, the borrower has the option to prepay a portion or all the nominal at any time without penalties.

We assume in this model that the riskless interest rate, denoted by 𝑟, is constant and known. The liquidity cost dynamics will be described later. The interest rate of the loan is the sum of the constant interest rate, a margin defined according to the credit quality of the borrower, and a liquidity cost that reflects the funding costs of the lender, the bank.

In order to decide whether the exercise of the option is worthwhile, the borrower (the firm) compares the actualized value of the remaining payments with the nominal value to pay. If the remaining payments exceed the nominal value, then it is optimal for the borrower to refinance his debt at a lower rate.

When the borrower is subject to default, the computation of the actualization is less straightforward. It starts with considering all possible scenarios of evolution for the default intensity in a risk-neutral framework and computing the average value of the remaining payments (including the final payment of the principal if applicable); this quantity will be called “PVRP” (denoted by 𝜉) and is the present value of the remaining payments, that is, the cash amount equivalent, both for borrower and lender in this model of the set of remaining payments. The PVRP is compared with the nominal: if the PVRP 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 𝜌₀, 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 PVRP as it will be seen below.

For a bank, the prepayment option is essentially a reinvestment risk, that is, the risk that the borrower decides to repay earlier his/her loan and that the bank cannot reinvest its excess of cash in a new loan. So the longer 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 modeled as an American embedded option on a risky debt owned by the borrower. As Monte-Carlo simulations are slow to converge and the binomial tree techniques are time consuming for long-term loans (cf. works by Cossin and Lu [1]), we decided to focus, in this paper, on the prepayment option for perpetual loan.

When valuing financial 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 crucial for the stability of the financial system. Past events like the Asian crisis of 1997 [2]; the Russian financial crisis of 1998 [3]; the defaults of hedge funds and investment firms like LTCM, Enron, Worldcom, and Lehman Brothers defaults, sovereign debts crisis of 2010-11, and so on prove that banks hold significant liquidity risk in their balance sheets. A liquidity crisis can have a severe impact on 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 defined by distinct states. Between two crises, investors are confident and banks find it easier to launch their long-term refinancing programs through regular bonds issuances. Thus the liquidity market is stable. Unfortunately, during crisis, liquidity becomes scarce, pushing the liquidity curve to very high levels which can only decrease if confidence returns to the market. The transition 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 nonstandard conditions: although the goal is to value a perpetual American option, the payoff of the option is highly nonstandard (is dependent on the PVRP). As a consequence, the characterization of the exercise region is not standard and technical conditions have to be met. Furthermore, our focus here is on a specific type of dynamics (of CIR type) with even more specific interest on the situation when several regimes are present.

The balance of the paper is as follows: in the remainder of this section (Section 1.1) we review the related existing literature; in Section 2, we consider that the liquidity cost is negligible and that the borrower credit risk is defined by his/her default intensity (called in the following simply “intensity”) which follows a CIR stochastic process. In this situation, we are able to obtain a quasianalytic formula for the PVRP. In Section 3 we explore the situation when the liquidity cost, defined 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 exercise region; numerical examples and concluding remarks close the paper.

1.1. Related Literature

There exist few articles (e.g., works by Cossin and Lu [1]) on the loan prepayment option but a close subject, the prepayment option in a fixed-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 approximate the PDE satisfied by the prepayment option, they define 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 Cossin and Lu [1] 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 difficult to have an accurate assessment of the option price for a 10-year loan.

There also exist mortgage prepayment decision models based on Poisson regression approach for mortgage loans (see, e.g., Schwartz and Torous [6]). 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 exercise 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 [7] have derived the closed-form solutions for vanilla American put; Guo analyses in [8] Russian (i.e., perpetual look-back) options and is able to derive explicit solutions for the optimal stopping time; in [9] Xu and Wu analyse the situation of a two-asset perpetual American option where the payoff function is a homogeneous function of degree one; Mamon and Rodrigo [10] find explicit solutions to vanilla European options. Buffington and Elliott [11] study European and American options and obtain equations for the price. A distinct approach (Wiener-Hopf factorization) is used by Jobert and Rogers [12] to derive very good approximations of the option prices for, among others, American puts. Other contributions include [13, 14].

Works involving Markov switched regimes and CIR dynamics appear in [15] 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 payoff of our American option although in their model only the mean reverting level is subject to Markov jumps) and in [16] where the term structure of the interest rates is analysed.

On the other hand numerical methods are proposed in [17] where it is found that a fixed point policy iteration coupled with a direct control formulation seems to perform best.

Finally, we refer to [18] 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 [19–21] for theoretical and numerical aspects of CIR processes and the situations where the CIR process has been used in finance): 𝑑𝜆_𝑠=𝛾𝜃−𝜆_𝑠𝑑𝑠+𝜎𝜆_𝑠𝑑𝑊_𝑠,𝛾,𝜃,𝜎>0,𝜆₀=𝜆₀.(1)

It is known that if 2𝛾𝜃≥𝜎², 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 fixed coupon defined by the interest rate 𝑟 and an initial contractual margin 𝜌₀. Here 𝜌₀ does not take into account any commercial margin (see Remark 1). Let 𝜉(𝑡,𝑇,𝜆) be the present value of the remaining payments at time 𝑡 of a corporate loan with initial contractual margin 𝜌₀ (depending on 𝜆₀), intensity at time 𝑡, 𝜆_𝑡, following the risk-neutral equation (1) with 𝜆_𝑡=𝜆; the loan has nominal amount 𝐾 and contractual maturity 𝑇. Here the assignment 𝜆_𝑡=𝜆 means that the dynamics of 𝜆_𝑡 start at time 𝑡 from the numerical value 𝜆. All random variables will be conditional by this event (see, e.g., (3)).

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): LV(𝑡,𝑇,𝜆)=𝜉(𝑡,𝑇,𝜆)−𝑃(𝑡,𝑇,𝜆).(2)

The present value of the cash flows discounted at the (instantaneous) risky rate 𝑟+𝜆_𝑡 is denoted by 𝜉. The infinitesimal cash flow at time 𝑡 is 𝐾(𝑟+𝜌₀) and the final payment of the principal 𝐾. Then 𝜉(𝑡,𝑇,𝜆)=𝔼𝐾⋅𝑟+𝜌₀^𝑇_𝑡𝑒^{−∫∼𝑡𝑡(𝑟+𝜆𝑢)𝑑𝑢}𝑑^∼𝑡+𝐾𝑒^{−∫𝑇𝑡𝑟+𝜆𝑢𝑑𝑢}∣𝜆_𝑡=𝜆.(3)

For a perpetual loan the maturity 𝑇=+∞. Since 𝜆_𝑡 is always positive, 𝑟+𝜆_𝑡>0, and thus the last term tends to zero when 𝑇→∞. A second remark is that since 𝛾,𝜃, and 𝜎 are independent of time, 𝜉 is independent of the starting time 𝑡: 𝜉(𝑡,𝜆)=𝔼𝐾⋅𝑟+𝜌_0𝑡^+∞𝑒^{−∫∼𝑡𝑡𝑟+𝜆𝑢𝑑𝑢}𝑑^∼𝑡∣𝜆_𝑡=𝜆(4)=𝔼𝐾⋅𝑟+𝜌₀₀^+∞𝑒^{−∫∼𝑡0𝑟+𝜆𝑢𝑑𝑢}𝑑^∼𝑡∣𝜆₀(𝜆),=𝜆=∶𝜉(5) where the last equality is a definition. For a CIR stochastic process, we obtain (see [19, 21]) 𝜉(𝜆)=𝐾⋅𝑟+𝜌₀₀^+∞𝑒^{−𝑟∼𝑡}𝐵0,^∼𝑡,𝜆𝑑^∼𝑡,(6) where for general 𝑡,^∼𝑡 we use the notation 𝐵𝑡,^∼𝑡,𝜆=𝔼𝑒^{−∫∼𝑡𝑡𝜆𝑢𝑑𝑢}∣𝜆_𝑡=𝜆.(7) Note that 𝐵(𝑡,^∼𝑡,𝜆) is a familiar quantity, and analytic formulas are available for (7) (see Lando [22, page 292]). The intensity is following a CIR dynamic; therefore, for general 𝑡,^∼𝑡𝐵𝑡,^∼𝑡,𝜆=𝛼𝑡,^∼𝑡𝑒^{−𝛽(𝑡,∼𝑡)𝜆}(8) with 𝛼𝑡,^∼𝑡=2ℎ𝑒^{(𝛾+ℎ)((∼𝑡−𝑡)/2)}2ℎ+𝛾+ℎ𝑒^{(∼𝑡−𝑡)ℎ}−1^{2𝛾𝜃/𝜎2},𝛽𝑡,^∼𝑡=2𝑒^{(∼𝑡−𝑡)ℎ}−12ℎ+𝛾+ℎ𝑒^{(∼𝑡−𝑡)ℎ},−1whereℎ=𝛾²+2𝜎²,(9) where 𝛾 and 𝜎 are the parameters of the CIR process of the intensity in (1). Obviously 𝐵(0,𝑡,𝜆) is monotonic with respect to 𝜆; thus the same holds for 𝜉.

The margin 𝜌₀ is the solution of the following equilibrium equation: 𝜉𝜆₀=𝐾(10) which can be interpreted as the fact that the present value of the cash flows (according to the probability of survival) is equal to the nominal 𝐾: 𝜌₀=1∫₀^+∞𝑒^{−𝑟∼𝑡}𝐵0,^∼𝑡,𝜆₀𝑑^∼𝑡−𝑟.(11) Note that we assume no additional commercial margin.

Remark 1. If an additional commercial margin 𝜇₀ is considered, then 𝜌₀ is first computed as above and then replaced by 𝜌₀=𝜌₀+𝜇₀ in (6). Equations (10) and (11) will not be verified as such but will still hold with some 𝜆₀ instead of 𝜆₀; for instance, we will have 𝜌₀=1∫₀^+∞𝑒^{−𝑟∼𝑡}𝐵0,^∼𝑡,𝜆₀𝑑^∼𝑡−𝑟.(12) 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=𝜆₀ as price relevant to practice.

Remark 2. Some banks allow (per year) a certain percentage of the prepaid amount without penalty and the rest with a penalty. This circumstance could be incorporated into the model by changing the definition of the payoff by subtracting the penalty. This will impact formula (15).

From definition (7) of 𝐵(𝑡,^∼𝑡,𝜆) it follows that 𝐵(𝑡,^∼𝑡,𝜆)<1; thus 𝑒^{−𝑟∼𝑡}𝐵0,^∼𝑡,𝜆₀<𝑒^{−𝑟∼𝑡}(13) and as a consequence ₀^+∞𝑒^{−𝑟∼𝑡}𝐵0,^∼𝑡,𝜆₀𝑑^∼𝑡<₀^+∞𝑒^{−𝑟∼𝑡}𝑑^∼𝑡=1𝑟(14) which implies that 𝜌₀>0.

2.2. Valuation of the Prepayment Option

The valuation problem of the prepayment option can be modeled as an American call option on a risky debt owned by the borrower. Here the prepayment option allows borrower to buy back and refinance 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 payoff (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 payoff 𝜒(𝑡,𝜆)∶=(𝜉(𝑡,𝜆)−𝐾)⁺(15) or, for our perpetual option, 𝜒(𝜆)∶=(𝜉(𝜆)−𝐾)⁺.(16)

We will denote by 𝒜 the characteristic operator (cf. [23, Chapter 7.5]) of the CIR process, that is, the operator that acts on any 𝐶² class function 𝑣 by (𝒜𝑣)(𝜆)=𝛾(𝜃−𝜆)𝜕_𝜆𝑣(𝜆)+12𝜎²𝜆𝜕_𝜆𝜆𝑣(𝜆).(17)

Denote for 𝑎,𝑏∈ℝ and 𝑥≥0 by 𝑈(𝑎,𝑏,𝑥) the solution to the confluent hypergeometric differential (also known as the Kummer) equation [24]: 𝑥𝑧^′′(𝑥)+(𝑏−𝑥)𝑧^′(𝑥)−𝑎𝑧(𝑥)=0(18) that increases at most polynomially at infinity and is finite (not null) at the origin. Recall also that this function is proportional to the confluent hypergeometric function of the second kind 𝑈(𝑎,𝑏,𝑥) (also known as the Kummer 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 𝑈()=1𝑎,𝑏,𝑥Γ(𝑎)₀^+∞𝑒^−𝑥𝑡𝑡^𝑎−1(1+𝑡)^{𝑏−𝑎−1}𝑑𝑡.(19) When 𝑎≤0, one uses other representations (see the cited references; for instance, one can use a direct computation or the recurrence formula 𝑈(𝑎,𝑏,𝑥)=(2𝑎−𝑏+𝑧−2)𝑈(𝑎+1,𝑏,𝑥)−(𝑎+1)(𝑎−𝑏+2)𝑈(𝑎+2,𝑏,𝑥)) it is known that 𝑈(𝑎,𝑏,𝑥) behaves as 𝑥^−𝑎 at infinity. Also introduce for 𝑥≥0𝑊(𝑥)=𝑒^{𝑥((𝛾−ℎ)/𝜎2)}𝑥^{(𝜎2−2𝛾𝜃)/𝜎2}×𝑈−−𝑟𝜎²−𝜎²ℎ+𝛾²𝜃+𝛾ℎ𝜃𝜎²ℎ,2−2𝛾𝜃𝜎²,2ℎ𝜎²𝑥,(20) where ℎ=𝛾²+2𝜎².

Theorem 3. (1) Introduce for Λ>0 the family of functions: 𝑃_Λ(𝜆) such that 𝑃_Λ(𝜆)=𝜒(𝜆)∀𝜆∈[0,Λ],(21)𝒜𝑃_Λ(𝜆)−(𝑟+𝜆)𝑃_Λ(𝜆)=0,∀𝜆>Λ,(22)lim_𝜆→Λ𝑃_Λ(𝜆)=𝜒(Λ),(23)lim_𝜆→∞𝑃_Λ(𝜆)=0.(24)
Then 𝑃_Λ(𝜆)=𝜒(𝜆)[]𝜒(Λ)∀𝜆∈0,Λ𝑊(Λ)𝑊(𝜆)∀𝜆≥Λ.(25)
(2) Suppose now a Λ^∗∈]0,𝜌₀∧𝜆₀[ exists such that 𝑑𝑃_Λ^∗(𝜆)𝑑𝜆_𝜆=(Λ^∗₎⁺=𝑑𝜒(𝜆)𝑑𝜆_𝜆=(Λ^∗₎⁻.(26)
Then the price of the prepayment option is 𝑃(𝜆)=𝑃_Λ^∗(𝜆).

Proof. We start with the first item: it is possible to obtain a general solution of (22) in an analytic form. We recall that 𝑧(𝑋)=𝑈(𝑎,𝑏,𝑋) is the solution of the Kummer equation (18). A cumbersome but straightforward computation shows that the general solution vanishing at infinity of the PDE (22) is 𝑊(𝜆); thus 𝑃_Λ(𝜆)=𝐶_Λ𝑊(𝜆)∀𝜆>Λ(27) with some 𝐶_Λ>0 to be determined. Now use the boundary conditions. If 𝜆=Λ, by continuity 𝜒(Λ)=𝑃_Λ(Λ)=𝐶_Λ𝑊(Λ). Thus, 𝐶_Λ=𝜒(Λ)/𝑊(Λ). Division by 𝑊 is legitimate because by definition, 𝑊(𝑥)>0 for all 𝑥>0.
We now continue with the second part of the theorem. The valuation problem of an American option goes through several steps: first one introduces the admissible trading and consumption strategies (cf. [25, Chapter 5]); then one realizes using results in the cited reference (also see [21, 26]) that the price 𝑃(𝜆) of the prepayment option involves computing a stopping time associated to the payoff. Denote by 𝒯 the ensemble of (positive) stopping times; we conclude that 𝑃(𝜆)=sup_𝜏∈𝒯𝔼𝑒^{−∫𝜏0𝑟+𝜆𝑢𝑑𝑢}𝜒𝜆_𝜏∣𝜆₀=𝜆.(28)
Further results derived for the situation of a perpetual (standard) American put options [18, 27] show that the stopping time has a simple structure: a critical level exists that splits the positive axis into two regions: the exercise region (to the left) where it is optimal to exercise and where the price equals the payoff and a continuation region (to the right) where the price satisfies a partial differential equation similar to Black-Scholes equation. We refer to [28] 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 payoff of the standard situation of an American put 𝑒^−𝑟𝑡(𝑆−𝐾)⁻ is a submartingale. For us the discounted payoff is 𝑒^{−∫𝑡0𝑟+𝜆𝑢𝑑𝑢}𝜒𝜆_𝑡=𝑒^{−∫𝑡0𝑟+𝜆𝑢𝑑𝑢}𝜉𝜆_𝑡−𝐾⁺,(29) and checking this condition requires here more careful examination which is the object of Lemma 4. It is now possible to apply Theorem 10.4.1 [23, Section 10.4, page 227] (see also [28] for specific treatment of the CIR process) which will show that 𝑃(𝜆) is the true option price if the following conditions are satisfied:(1) on ]0,Λ^∗[ we have 𝑃(𝜆)=𝜒(𝜆)=(𝜉(𝜆)−𝐾)⁺ and the relation (35) holds;(2) the solution candidate 𝑃(𝜆) satisfies the relation (𝒜𝑃)(𝜆)−(𝑟+𝜆)𝑃(𝜆)=0,∀𝜆>Λ^∗;(30)(3) the function 𝑃(𝜆) is 𝐶¹ everywhere, continuous at the origin and 𝐶² on each subinterval ]0,Λ^∗[ and ]Λ^∗,∞[.
The theorem also says that the borrower exercises his option on the exercise region [0,Λ^∗] while on the continuation region ]Λ^∗,∞[ the borrower keeps the option because it is worth more nonexercised.
We now show that 𝑃_Λ^∗ verifies all conditions above which will allow to conclude that 𝑃=𝑃_Λ^∗. The requirement 1 is proved in Lemma 4; the requirement 3 amounts to asking that the optimal frontier value Λ^∗ is chosen such that 𝑑𝑃_Λ^∗(𝜆)𝑑𝜆_𝜆=(Λ^∗₎⁺=𝑑𝜒(𝜆)𝑑𝜆_𝜆=(Λ^∗₎⁻.(31)
The requirement 2 implies that in the continuation region the price is the solution of the following PDE: (𝒜𝑃)(𝜆)−(𝑟+𝜆)𝑃(𝜆)=0,∀𝜆>Λ^∗.(32) For this PDE we need boundary conditions. The condition at 𝜆=Λ^∗ is 𝑃(𝜆)|_𝜆=Λ^∗=𝜒(𝜆)_𝜆=Λ∗.(33) When 𝜆=+∞, the default intensity is infinite; thus the time to failure is zero, and thus the borrower has failed; in this case the option is worthless; that is, lim_𝜆→∞𝑃(𝜆)=0.(34) These conditions give exactly the definition of 𝑃_Λ^∗.

Lemma 4. The following inequality holds: 𝒜𝜒(𝜆)−(𝑟+𝜆)𝜒(𝜆)<0,∀𝜆<𝜌₀∧𝜆₀.(35)

Proof. Recall that 𝜒(𝜆)=(𝜉(𝜆)−𝐾)⁺; definition (5) of 𝜉 implies (cf. [23, Section 8.2 and exercise 9.12, page 203]) that 𝜉 is solution of the following PDE: (𝒜𝜉)(𝜆)−(𝑟+𝜆)𝜉(𝜆)+𝑟+𝜌₀𝐾=0,∀𝜆>0.(36) For 𝜆<𝜆₀ we have 𝜉(𝜆)>𝐾=𝜉(𝜆₀); thus 𝒜(𝜉(⋅)−𝐾)⁺(𝜆)−(𝑟+𝜆)(𝜉(𝜆)−𝐾)⁺=(𝒜(𝜉(⋅)𝜆)−(𝜉(𝜆))=(𝜆)−()𝜉(𝜆)+()𝐾−𝐾))(𝑟+𝜆)(−𝐾𝒜𝜉)(𝑟+𝜆𝑟+𝜆=−𝑟+𝜌₀()𝐾=𝐾+𝑟+𝜆𝜆−𝜌₀𝐾<0∀𝜆<𝜌₀∧𝜆₀.(37)

Note that Theorem 3 is only a sufficient result (a so-called “verification” result); under the assumption that a Λ^∗ fulfilling the hypotheses of the theorem exists the question is how to find it.

Two approaches can be considered; first, it is enough to find a zero of the following function Λ↦Υ(Λ)∶=((𝜕𝑃_Λ(𝜆)/𝜕𝜆)|_𝜆=Λ⁺−(𝜕𝜒(𝜆)/𝜕𝜆)|_𝜆=Λ⁻) (the last equality is a definition). Of course (𝜕𝜒(𝜆)/𝜕𝜆)|_{𝜆=𝜆0+𝜖}=0 and (𝜕𝑃_𝜆0+𝜖(𝜆)/𝜕𝜆)|_{𝜆=𝜆0+𝜖}<0; thus Υ(𝜆₀+𝜖)<0 for any 𝜖>0 and hence Υ(𝜆₀)≤0. Thus it is natural not to look for Λ^∗ outside the interval [0,𝜆₀]. The theorem asks furthermore to restrict the search to the interval [0,𝜆₀∧𝜌₀].

A different convenient procedure to find the critical Λ^∗ is to consider the dependence Λ↦𝑃_Λ(𝜆₀). 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. [23, page 203]), 𝑃_Λ(𝜆)=𝔼𝑒^{−∫𝜏Λ0𝑟+𝜆𝑢𝑑𝑢}𝜒𝜆_𝜏Λ∣𝜆₀=𝜆.(38) From (28) 𝑃(𝜆)≥𝑃_Λ(𝜆) for any 𝜆; thus Λ^∗ is the value that maximizes (with respect to Λ) the function Λ↦𝑃_Λ(𝜆₀). To comply with the theorem, the maximization is performed in the interval [0,𝜆₀∧𝜌₀].

2.3. Numerical Application

We consider a perpetual loan (𝑇=+∞) with a nominal amount 𝐾=1 and the borrower default intensity 𝜆_𝑡 follows a CIR dynamics with parameters: initial intensity 𝜆₀=300bps, volatility 𝜎=0.05, average intensity 𝜃=200bps, and reversion coefficient 𝛾=0.5. We assume a constant interest rate 𝑟=300bps; that is, 𝑟=3%. Recall that a basis point, denoted by “1 bps”, equals 10⁻⁴.

In order to find the initial contractual margin, we use (11) and find 𝜌₀=208bps.

At inception, the present value of cash flows is at par, so 𝜉(𝜆₀)=1. The prepayment option price is 𝑃(+∞,𝜆₀)=0.0232; that is, 𝑃(𝜆₀)=2.32%⋅𝐾. Therefore the loan value equals 𝜉(𝜆₀)−𝑃(𝜆₀)=0,9768.

The value Λ^∗=123bps is obtained by maximizing 𝑃_Λ(𝜆₀) as indicated in the remarks above; the dependence of 𝑃_Λ(𝜆₀) with respect to Λ is illustrated in Figure 1. The loan value will be equal to par if the intensity decreases until the exercise region (𝜆<Λ^∗) (see Figure 2). The continuation and exercise regions are depicted in Figure 3. We postpone to Section 3.5 the description of the numerical method to solve (22).

Figure 1 

We illustrate here the dependence of 𝑃Λ(𝜆0) as a function of Λ; this allows for finding the optimal value Λ∗ that maximizes the option price. For the numerical example described here we obtain Λ∗=123bps.

Figure 2 

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.

Figure 3 

Prepayment option price 𝑃(𝜆) (solid line) and payoff 𝜒(𝜆) (dashed line) as a function of the intensity 𝜆. Two regions appear: the continuation region 𝜆>Λ∗ and the exercise region 𝜆≤Λ∗.

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, and also the liquidity cost. The liquidity cost is defined as the specific cost of a bank to access the cash on the market. This cost will be modeled with a switching regime with a Markov chain of finite states of the economy. The interest rate 𝑟 is still assumed constant. Therefore, the assessment of the loan value and its prepayment option is an 𝑁-dimensional problem. The intensity is still defined by a Cox-Ingersoll-Ross process with 2𝑘𝜃≥𝜎²: 𝑑𝜆_𝑡=𝛾𝜃−𝜆_𝑡𝑑𝑡+𝜎𝜆_𝑡𝑑𝑊_𝑡,𝜆₀=𝜆₀.(39)

3.1. Theoretical Regime Switching Framework

We assume the economic state of the market is described by a finite state Markov chain 𝐗={𝑋_𝑡,𝑡≥0}. The state space 𝐗 can be taken to be, without loss of generality, the set of unit vectors 𝐸={𝑒₁,𝑒₂,…,𝑒_𝑁},𝑒_𝑖=(0,…,0,1,0,…,0)^𝑇∈ℝ^𝑁. Here 𝑇 is the transposition operator.

Assuming the process 𝑋_𝑡 is homogeneous in time and has a rate matrix 𝐴, then if 𝑝_𝑡=𝔼[𝑋_𝑡]∈ℝ^𝑁, 𝑑𝑝_𝑡𝑑𝑡=𝐴𝑝_𝑡,𝑋_𝑡=𝑋₀+^𝑡₀𝐴𝑋_𝑢𝑑𝑢+𝑀_𝑡,(40) where 𝐌={𝑀_𝑡,𝑡≥0} is a martingale with respect to the filtration generated by 𝐗. In differential form 𝑑𝑋_𝑡=𝐴𝑋_𝑡𝑑𝑡+𝑑𝑀_𝑡,𝑋₀=𝑋₀.(41) We assume the instantaneous liquidity cost of the bank depends on the state 𝐗 of the economy, so that 𝑙_𝑡=𝐥,𝑋_𝑡.(42) 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=1𝑎_𝑘,𝑗=0 for any 𝑘.

3.2. Analytical Formulas for the PVRP

Assume a loan has a fixed coupon defined by the constant interest rate 𝑟 and an initial contractual margin 𝜌₀ 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): LV(𝑡,𝑇,𝜆)=𝜉(𝑡,𝑇,𝜆)−𝑃(𝑡,𝑇,𝜆).(43)

The PVRP 𝜉 is the present value of the cash flows 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 Section 2.1, 𝜉 is not depending on time when 𝑇=+∞ (perpetual loan). So we denote that, 𝜉(𝜆,𝑋)∶=𝐾𝑟+𝜌₀×𝔼₀^+∞𝑒^{−∫∼𝑡0𝑟+𝑙𝑢+𝜆𝑢𝑑𝑢}𝑑^∼𝑡∣𝜆₀=𝜆,𝑋₀=𝑋.(44) We consider that there is no correlation between the credit risk, that is, the intensity 𝜆_𝑡, of the borrower and the cost to access the cash on the market, that is, the liquidity cost 𝑙_𝑡, of the lender. Therefore, we have 𝜉(𝜆,𝑋)=𝐾𝑟+𝜌₀₀^+∞𝑒^{−𝑟∼𝑡}𝔼𝑒^{−∫∼𝑡0𝜆𝑢𝑑𝑢}∣𝜆₀𝑒=𝜆×𝔼^{−∫∼𝑡0𝑙𝑢𝑑𝑢}∣𝑋₀=𝑋𝑑^∼𝑡.(45)

Remark 5. The crucial information here is that the coefficients 𝛾, 𝜃, and 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 different approach can extend this result by using the properties of the PVRP as explained in the next section.

Note that (cf. Section 2.1, equation (7)) 𝔼𝑒^{−∫𝑡0𝜆𝑢𝑑𝑢}∣𝜆₀=𝜆=𝐵(0,𝑡,𝜆)(46) and 𝐵(0,𝑡,𝜆) is evaluated using (8)–(11). In order to compute 𝔼𝑒^{−∫∼𝑡0𝑙𝑢𝑑𝑢}∣𝑋₀=𝑋,(47) let 𝑓_𝑘(𝑡) be defined by 𝑓_𝑘(𝑡)=𝔼𝑒^{−∫𝑡0𝑙𝑢𝑑𝑢}∣𝑋₀=𝑒_𝑘.(48) Let 𝜏 be the time of the first jump from 𝑋₀=⟨𝐗,𝑒_𝑘⟩ to some other state. We know (cf. Lando [22] paragraph 7.7, page 211) that 𝜏 is a random variable following an exponential distribution of parameter 𝛼_𝑘 with 𝛼_𝑘=_𝑗≠𝑘𝑎_𝑘,𝑗.(49) 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 𝑝_𝑘,𝑗=𝑎_𝑘,𝑗𝛼_𝑘.(50) Thus, 𝑓_𝑘(𝑡)=ℙ(𝜏>𝑡)𝑒^{−𝑙𝑘𝑡}+ℙ(𝜏≤𝑡)𝑒^{−𝑙𝑘𝜏}×_𝑗≠𝑘ℙ𝑙_𝜏=𝑙_𝑗𝔼𝑒^{−∫𝑡𝜏𝑙𝑢𝑑𝑢}∣𝑋_𝜏=𝐗,𝑒_𝑗=𝑒^{−(𝑙𝑘+𝛼𝑘)𝑡}+𝛼_𝑘^𝑡₀𝑒^{−(𝑙𝑘+𝛼𝑘)𝜏}_𝑗≠𝑘𝑝_𝑘,𝑗𝑓_𝑗(𝑡−𝜏)𝑑𝜏.(51) Then, 𝑒^{(𝑙𝑘+𝛼𝑘)𝑡}𝑓_𝑘(𝑡)=1+𝛼_𝑘^𝑡₀𝑒^{(𝑙𝑘+𝛼𝑘)(𝑡−𝜏)}_𝑗≠𝑘𝑝_𝑘,𝑗𝑓_𝑗(𝑡−𝜏)𝑑𝜏=1+𝛼_𝑘^𝑡₀𝑒^{(𝑙𝑘+𝛼𝑘)𝑠}_𝑗≠𝑘𝑝_𝑘,𝑗𝑓_𝑗(𝑠)𝑑𝑠.(52)

By differentiation with respect to 𝑡, 𝑑𝑒𝑑𝑡^{(𝑙𝑘+𝛼𝑘)𝑡}𝑓_𝑘(𝑡)=𝛼_𝑘𝑒^{(𝑙𝑘+𝛼𝑘)𝑡}_𝑗≠𝑘𝑝_𝑘,𝑗𝑓_𝑗(𝑡).(53) Then 𝑑𝑓_𝑘(𝑡)+𝑙𝑑𝑡^𝑘+𝛼_𝑘𝑓_𝑘(𝑡)=𝛼_{𝑘𝑗≠𝑘}𝑝_𝑘,𝑗𝑓_𝑗(𝑡).(54) Thus, 𝑑𝑓_𝑘(𝑡)=𝑑𝑡_𝑗≠𝑘𝛼_𝑘𝑝_𝑘,𝑗𝑓_𝑗(𝑡)−𝑙_𝑘+𝛼_𝑘𝑓_𝑘(𝑡).(55) Denote by 𝐹(𝑡) the vector (𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑁(𝑡))^𝑇 and introduce the 𝑁×𝑁 matrix 𝐵𝐵_𝑖,𝑗=𝛼_𝑖𝑝_𝑖,𝑗−𝛼if𝑖≠𝑗_𝑖+𝑙_𝑖if𝑖=𝑗.(56) From (55) we obtain 𝑑𝐹(𝑡)(𝑡)(𝑡)𝑑𝑡=𝐵𝐹thus𝐹=𝑒^𝐵𝑡𝐹(0)(57) with the initial condition 𝐹(0)=𝑓_𝑘(0)^𝑁_𝑘=1=(1,1,…,1)^𝑇∈ℝ^𝑁.(58)

We have therefore analytic formulas for the PVRP 𝜉(𝜆,𝑋). We refer the reader to [15] for similar considerations on related CIR switched dynamics.

Remark 6. 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 payoff is equal to that of 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 10 item 5 for further details).

The margin 𝜌₀ is set to satisfy the equilibrium equation: 𝜉𝜆₀,𝑋₀=𝐾.(59) Similar arguments to that in previous section show that 𝜌₀>min_𝑘𝑙_𝑘>0. See Remark 1 for the situation when an additional commercial margin is to be considered.

We will also need to introduce for any 𝑘=1,…,𝑁 the value Λ⁰_𝑘 such that 𝜉Λ⁰_𝑘,𝑒_𝑘=𝐾.(60) Of course, Λ⁰_𝑋0=𝜆₀. Recall that 𝜉(𝜆,𝑒_𝑘) is decreasing with respect to 𝜆; when 𝜉(0,𝑒_𝑘)<𝐾, there is no solution to (59) and we will choose by convention Λ⁰_𝑘=0.

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: 𝒜^ℛ𝑣𝜆,𝑒_𝑘=(𝒜𝑣)𝜆,𝑒_𝑘−𝑟+𝑙_𝑘+𝜆𝑣𝜆,𝑒_𝑘+^𝑁_𝑗=1𝑎_𝑘,𝑗𝑣𝜆,𝑒_𝑗−𝑣𝜆,𝑒_𝑘.(61)

Having defined the dynamics (39) and (41), one can use an adapted version of the Feynman-Kac formula in order to conclude that PVRP defined by (44) satisfies the equation 𝒜^ℛ𝜉+𝑟+𝜌₀𝐾=0.(62)

Remark 7. When the dynamics involve different coefficients of the CIR process for different regimes (cf. also Remark 5), (62) changes in that it will involve, for 𝜉(⋅,𝑒_𝑘), the operator 𝒜_𝑘(𝑣)(𝜆)=𝛾_𝑘𝜃_𝑘−𝜆𝜕_𝜆𝑣(𝜆)+12𝜎²_𝑘𝜆𝜕_𝜆𝜆𝑣(𝜆)(63) instead of 𝒜.

3.4. Valuation of the Prepayment Option

The valuation problem of the prepayment option can be modeled as an American call option on a risky debt owned by the borrower with payoff: 𝜒(𝜆,𝑋)=(𝜉(𝜆,𝑋)−𝐾)⁺.(64) Here the prepayment option allows borrower to buy back and refinance his/her debt according to the current contractual margin at any time during the life of the option.

Theorem 8. For any N-tuple Λ=(Λ_𝑘)^𝑁_𝑘=1∈(ℝ₊)^𝑁 introduce the function 𝑃_Λ(𝜆,𝑋) such that 𝑃_Λ𝜆,𝑒_𝑘=𝜒𝜆,𝑒_𝑘∀𝜆∈0,Λ_𝑘,𝒜(65)_ℛ𝑃_Λ𝜆,𝑒_𝑘=0,∀𝜆>Λ_𝑘,𝑘=1,…,𝑁,(66)lim_{𝜆→Λ𝑘}𝑃_Λ𝜆,𝑒_𝑘=𝜒Λ_𝑘,𝑒_𝑘,𝑖𝑓Λ_𝑘>0,𝑘=1,…,𝑁,(67)lim_𝜆→∞𝑃_Λ𝜆,𝑒_𝑘=0,𝑘=1,…,𝑁.(68) Suppose a Λ^∗∈∏^𝑁_𝑘=1[0,(𝜌₀−𝑙_𝑘)⁺∧Λ⁰_𝑘] exists such that for all 𝑘=1,…,𝑁: 𝑃_Λ^∗(𝜆,𝑋)≥𝜒(𝜆,𝑋)∀𝜆,𝑋,(69)𝜕𝑃_Λ^∗𝜆,𝑒_𝑘𝜕𝜆_{𝜆=(Λ∗𝑘)+}=𝜕𝜒𝜆,𝑒_𝑘𝜕𝜆_{𝜆=(Λ∗𝑘)−}𝑖𝑓Λ^∗_𝑘>0,(70)_𝑁𝑗=1𝑎_𝑘,𝑗𝑃_Λ^∗𝜆,𝑒_𝑗−𝜒𝜆,𝑒_𝑗+𝐾𝜆+𝑙_𝑘−𝜌₀≤0∀𝜆∈min_𝑗Λ^∗_𝑗,Λ^∗_𝑘.(71) Then 𝑃=𝑃_Λ^∗.

Proof. Similar arguments as in the proof of Theorem 3 lead to consider the American option price in the form 𝑃(𝜆,𝑋)=sup_𝜏∈𝒯𝔼𝑒^{−∫𝜏0𝑟+𝑙𝑢+𝜆𝑢𝑑𝑢}𝜒𝜆_𝜏,𝑋_𝜏∣𝜆₀=𝜆,𝑋₀=𝑋.(72) 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 𝑃_Λ()𝑒𝜆,𝑋=𝔼^{−∫𝜏Λ0𝑟+𝑙𝑢+𝜆𝑢𝑑𝑢}𝜒𝜆_𝜏Λ,𝑋_𝜏Λ∣𝜆₀=𝜆,𝑋₀=𝑋.(73) Remark that for Λ∈(ℝ^∗₊)^𝑁 the stopping time 𝜏_Λ is finite 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 (65)) 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; that is, 𝑃≤𝑃_Λ^∗.
To this end we use a similar technique as in Theorem 10.4.1 [23, Section 10.4, page 227] (see also [7] for similar considerations). First one can invoke the same arguments as in the cited reference (cf. Appendix D for technicalities) and work as if 𝑃_Λ^∗ is 𝐶² (not only 𝐶¹ as the hypothesis ensures).
Denote by 𝐷_Λ^∗ the set {(𝜆,𝑒_𝑘)∣𝜆∈[0,Λ^∗_𝑘],𝑘=1,…,𝑁} (which will be the exercise region), and denote by 𝐶_Λ^∗ its complementary with respect to ℝ₊×𝐸 (which will be the continuation region).
The Lemma 9 shows that 𝒜^ℛ𝑃_Λ^∗ is nonpositive everywhere (and is null on 𝐶_Λ^∗). The Itô formula shows that 𝑑𝑒^{−∫𝑡0𝑟+𝑙𝑠+𝜆𝑠𝑑𝑠}𝑃_Λ^∗𝜆_𝑡,𝑋_𝑡=𝑒^{−∫𝑡0𝑟+𝑙𝑠+𝜆𝑠𝑑𝑠}𝒜^ℛ𝑃_Λ^∗𝜆_𝑡,𝑋_𝑡𝑑𝑡+𝑑martingale.(74) Taking averages and integrating from 0 to some stopping time 𝜏, it follows from 𝒜^ℛ𝑃_Λ^∗≤0 that 𝑃_Λ^∗(𝜆,𝑋)≥𝔼𝑒^{−∫𝜏0𝑟+𝑙𝑢+𝜆𝑢𝑑𝑢}𝑃_Λ^∗𝜆_𝜏,𝑋_𝜏∣𝜆₀=𝜆,𝑋₀𝑒=𝑋≥𝔼^{−∫𝜏0𝑟+𝑙𝑢+𝜆𝑢𝑑𝑢}𝜒𝜆_𝜏,𝑋_𝜏∣𝜆₀=𝜆,𝑋₀=𝑋.(75) Since this is true for any stopping time 𝜏, the conclusion follows.

Lemma 9. Under the hypothesis of the Theorem 8 the following inequality holds (strongly except for the values (𝜆,𝑋)=(Λ^∗_𝑗,𝑒_𝑘) and everywhere in a weak sense): 𝒜^ℛ𝑃_Λ^∗(𝜆,𝑋)≤0,∀𝜆>0,∀𝑋.(76)

Proof. The nontrivial part of this lemma comes from the fact that if for fixed 𝑘 we have for 𝜆 in a neighborhood of some 𝜆₁: 𝑃_Λ^∗(𝜆,𝑒_𝑘)=𝜒(𝜆,𝑒_𝑘), this does not necessarily imply (𝒜^ℛ𝑃_Λ^∗)(𝜆₁,𝑒_𝑘)=(𝒜^ℛ𝜒)(𝜆₁,𝑒_𝑘) because 𝒜^ℛ depends on other values 𝑃_Λ^∗(𝜆,𝑒_𝑗) with 𝑗≠𝑘.
From (66) the conclusion is trivially verified for 𝑋=𝑒_𝑘 for any 𝜆∈]Λ^∗_𝑘,∞[.
We now analyse the situation when 𝜆<min_𝑗Λ^∗_𝑗; this means in particular that 0≤𝜆<min_𝑗Λ^∗_𝑗≤Λ⁰_ℓ for any ℓ; thus Λ⁰_ℓ>0. Note that Λ^∗_𝑘<Λ⁰_𝑘 implies 𝜉(Λ^∗_𝑘,𝑒_𝑘)≥𝜉(Λ⁰_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_𝑗Λ^∗_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 𝒜^ℛ𝑃_Λ^∗𝜆,𝑒_𝑘=𝒜^ℛ𝜒𝜆,𝑒_𝑘=𝒜^ℛ(𝜉−𝐾)𝜆,𝑒_𝑘=𝒜^ℛ𝜉𝜆,𝑒_𝑘−𝒜^ℛ(𝐾)=−𝑟+𝜌₀𝐾−𝑟+𝑙_𝑘𝐾𝑙+𝜆=𝐾_𝑘+𝜆−𝜌₀≤𝐾𝑙_𝑘+Λ^∗_𝑘−𝜌₀≤0,(77) the last inequality being true by hypothesis.
A last situation is when 𝜆∈]min_𝑗Λ^∗_𝑗,Λ^∗_𝑘[; then 𝑃_Λ^∗(𝜆,𝑒_𝑘)=𝜒(𝜆,𝑒_𝑘) but some terms 𝑃_Λ^∗(𝜆,𝑒_𝑗) for 𝑗≠𝑘 may differ from 𝜒(𝜆,𝑒_𝑗). More involved arguments are invoked in this case. This point is specific to the fact that the payoff 𝜒 itself has a complex structure and as such was not emphasized in previous works (e.g., [7], etc.).
Recalling the properties of 𝜉, one obtains (and since 𝑃_Λ^∗(𝜆,𝑒_𝑘)=𝜒(𝜆,𝑒_𝑘)) the following: 𝒜^ℛ𝑃_Λ^∗𝜆,𝑒_𝑘=𝒜𝜒𝜆,𝑒_𝑘−𝑟+𝑙_𝑘+𝜆𝜒𝜆,𝑒_𝑘+^𝑁_𝑗=1𝑎_𝑘,𝑗𝑃_Λ^∗𝜆,𝑒_𝑗−𝜒𝜆,𝑒_𝑘=𝒜^ℛ𝜒𝜆,𝑒_𝑘+^𝑁_𝑗=1𝑎_𝑘,𝑗𝑃_Λ^∗𝜆,𝑒_𝑗−𝜒𝜆,𝑒_𝑗=𝒜^ℛ𝜉𝜆,𝑒_𝑘−𝒜^ℛ(𝐾)+^𝑁_𝑗=1𝑎_𝑘,𝑗𝑃_Λ^∗𝜆,𝑒_𝑗−𝜒𝜆,𝑒_𝑗=−𝐾𝑟+𝜌₀+𝑟+𝑙_𝑘𝐾++𝜆_𝑁𝑗=1𝑎_𝑘,𝑗𝑃_Λ^∗𝜆,𝑒_𝑗−𝜒𝜆,𝑒_𝑗≤0,(78) where for the last inequality we use hypothesis (71). Finally, since we proved that (𝒜^ℛ𝑃_Λ^∗)(𝜆,𝑋)≤0 strongly except for the values (𝜆,𝑋)=(Λ^∗_𝑗,𝑒_𝑘) and since 𝑃_Λ^∗ is of 𝐶¹ class, we obtain the conclusion (the weak formulation only uses the first derivative of 𝑃_Λ^∗).

Remark 10. Several remarks are in order at this point.(1) When only one regime is present; that is, 𝑁=1, the hypothesis of the theorem is identical to that of Theorem 3 since (71) is automatically satisfied.(2) When 𝑁>1 checking (71) does not involve any computation of derivatives and is straightforward.(3) As mentioned in the previous section, the theorem is a verification result, that is, it only gives sufficient conditions for a candidate to be the option price. Two possible partial converse results are possible: the first one to prove that the optimal price is indeed an element of the family 𝑃_Λ. The second converse result is to prove that supposing 𝑃=𝑃_Λ^∗, then Λ^∗∈∏^𝑁_𝑘=1[0,(𝜌₀−𝑙_𝑘)⁺∧Λ⁰_𝑘], and (69)–(71) are satisfied.(4) The search for the candidate Λ^∗ can be done either by looking for a zero of the function Λ↦Υ(Λ)∶=((𝜕𝑃_Λ^∗(𝜆,𝑒_𝑘)/𝜕𝜆)|_𝜆=(Λ^∗_𝑘)⁺−(𝜕𝜒(𝜆,𝑒_𝑘)/𝜕𝜆)|_𝜆=(Λ^∗_𝑘)⁻)^𝑁_𝑘=1 or by maximizing on ∏^𝑁_𝑘=1]0,(𝜌₀−𝑙_𝑘)∧Λ⁰_𝑘[ the function Λ↦𝑃_Λ(𝜆₀,𝑋₀).(5) If the optimization of 𝑃_Λ(𝜆₀,𝑋₀) is difficult 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, that is, 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 one-regime (𝑁=1) dynamics with riskless interest rate 𝑟 being replaced by 𝑟+(∑^𝑁_𝑘=1𝑙_𝑘/𝛼_𝑘)/(∑^𝑁_𝑘=11/𝛼_𝑘). Having established the two extremal points, the candidate Λ^∗(𝐴) is searched within the 𝑁-dimensional segment [Λ^∗(0),Λ^∗(∞𝐴)].

3.5. Numerical Application

The numerical solution of the partial differential equation (66) is required. We use a finite difference method. The first derivative is approximated by the finite difference formula 𝜕𝑃𝜕𝜆^Λ()=𝑃𝜆,𝑋^Λ(𝜆+𝛿𝜆,𝑋)−𝑃_Λ(𝜆−𝛿𝜆,𝑋)2𝛿𝜆+𝑂𝛿𝜆²,(79) while the second derivative is approximated by 𝜕²𝜕𝜆²𝑃_Λ()=𝑃𝜆,𝑋_Λ(𝜆+𝛿𝜆,𝑋)−2𝑃_Λ(𝜆+𝛿𝜆,𝑋)+𝑃_Λ(𝜆−𝛿𝜆,𝑋)𝛿𝜆²+𝑂𝛿𝜆².(80)

To avoid working with an infinite domain, a well-known approach is to define an artificial boundary 𝜆_max. Then a boundary condition is imposed on 𝜆_max which leads to a numerical problem in the finite domain ∪^𝑁_𝑘=1[Λ^∗_𝑘,𝜆_max]. In this numerical application, 𝜆_max=400bps. 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 (homogeneous Dirichlet boundary condition) or that (𝜕/𝜕𝜆)𝑃_Λ(𝜆_max,𝑒_𝑘)=0, ∀𝑘=1,…,𝑁 (homogeneous Neumann 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 the same 𝛿𝜆. The Neumann boundary condition gives much better results for the situations we considered and as such was always chosen (see also Figure 6).

We consider a perpetual loan with a nominal amount 𝐾=1 and the borrower default intensity 𝜆_𝑡 follows CIR dynamics with parameters: initial intensity 𝜆₀=300bps, volatility 𝜎=0.05, average intensity 𝜃=200bps, and reversion coefficient 𝛾=0.5. We assume a constant interest rate 𝑟=1% and a liquidity cost defined by a Markov chain of two states 𝑙₁=150bps and 𝑙₂=200bps. For 𝑁=2 the rate 𝐴 matrix is completely defined by 𝛼₁=1/3, 𝛼₂=1.

In order to find the initial contractual margin, we use (11) and find 𝜌₀=331bps in the state 1. The contractual margin takes into account the credit risk (default intensity) and the liquidity cost. We have thus Λ⁰₁=𝜆₀; we obtain then Λ⁰₂=260bps.

The optimal value Λ^∗ is obtained by maximizing 𝑃_Λ(𝜆₀,𝑋₀) and turns out to be (Λ^∗₁,Λ^∗₂)=(122bps,64bps) (see Figure 4). To be accepted, this numerical solution has to verify all conditions of Theorem 8. Hypotheses (69) and (71) are satisfied (see Figure 6) and hypothesis (71) is accepted after calculation. Moreover Λ^∗₁≤(𝜌₀−𝑙₁)∧Λ⁰₁ and the analogous holds for Λ^∗₂.

Figure 4 

We illustrate here the dependence of 𝑃Λ(𝜆0,𝑋0) as a function of Λ; this allows for finding the optimal (Λ∗1=122bps, Λ∗2=64bps) that maximizes the option price.

In the state 𝑋₀=1, the present value of cash flows is at par, so 𝜉(𝜆₀,𝑋₀)=1. The prepayment option price is 𝑃(𝜆₀,𝑋₀)=0.0240. Therefore the loan value equals 𝜉(𝜆₀,𝑋₀)−𝑃(𝜆₀,𝑋₀)=0.9760.

The loan value will be equal to the nominal if the intensity decreases until the exercise region 𝜆≤Λ^∗ (see Figure 5). The continuation and exercise regions are depicted in Figure 6.

(a)

(b)

Figure 5 

Loan value as a function of the intensity. (a) Regime 𝑋=𝑒1; (b) regime 𝑋=𝑒2. The loan value is decreasing when there is a degradation of the credit quality (i.e., when 𝜆 increases) and converges to 0.

(a)

(b)

Figure 6 

The price of the prepayment option 𝑃Λ∗(𝜆) (solid line) and the payoff 𝜒(𝜆) (dashed line) as a function of the intensity 𝜆. (a) Regime 𝑋=𝑒1; (b) regime 𝑋=𝑒2. For each regime two regions appear: the continuation region 𝜆>Λ∗𝑖 and the exercise region 𝜆≤Λ∗𝑖.

3.6. Regimes in Which Is Never Optimal to Exercise

When the liquidity parameters corresponding to given regimes are very different, it may happen that the optimization of 𝑃_Λ(𝜆₀,𝑋₀) 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 being dealt with. Of course when in addition Λ⁰_𝑘𝑖=0, the payoff being null, it is intuitive that the option should not be exercised.

Remark 11. Further examination of Theorem 3 calls for the following remarks.(1) The boundary value set in (65) 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. Moreover when 2𝛾𝜃≥𝜎², the intensity 𝜆_𝑢 does not touch 0; thus the stopping time 𝜏_Λ^∗ is infinite in the regime 𝑒_𝑘 (thus the boundary value in 0 is never used and thus need not be enforced); from a mathematical point of view it is known that no boundary conditions are required at points where the leading order differential operator is degenerate.(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): 𝑙₁ a “normal” regime and 𝑙₂ the “crisis” regime (𝑙₂≥𝑙₁); when the agent contemplates prepayment, the more severe the crisis (i.e., larger 𝑙₂−𝑙₁), the less he/she is likely to prepay during the crisis when the cash is expensive (high liquidity cost). We will most likely see that for 𝑙₁=𝑙₂ some exercise regions exist while starting from some large 𝑙₂ the exercise region will disappear in regime 𝑒₂. 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 𝑙₁=50bps and 𝑙₂=250bps. In order to find the initial contractual margin, we use (11) and find 𝜌₀=305bps in the state 1. The contractual margin takes into account the credit risk (default intensity) and the liquidity cost. As before Λ⁰₁=𝜆₀ but here we obtain Λ⁰₂=221bps.

The couple (Λ^∗₁=121bps, Λ^∗₂=0) (see Figure 7) maximizes 𝑃_Λ(𝜆₀,𝑋₀). There does not exist an exercise boundary in the state 2. The loan value will equal the par if the intensity decreases until the exercise region 𝜆≤Λ^∗ (see Figure 8). The continuation and exercise regions are depicted in Figure 9.

Figure 7 

We illustrate here the dependence of 𝑃Λ(𝜆0,𝑋0) as a function of the exercise boundary Λ; this allows to find the optimal (Λ∗1=121bps, Λ∗2=0) that maximizes the option price.

(a)

(b)

Figure 8 

Loan value as a function of the intensity. (a) Regime 𝑋=𝑒1; (b) regime 𝑋=𝑒2. The loan value is decreasing when there is a degradation of the credit quality (i.e., when 𝜆 increases) and converges to 0.

(a)

(b)

Figure 9 

The price of the prepayment option 𝑃Λ∗(𝜆) (solid line) and the payoff 𝜒(𝜆) (dashed line) as function of the intensity 𝜆. (a) regime 𝑋=𝑒1; (b) regime 𝑋=𝑒2. Two regions appear: the continuation region 𝜆>Λ∗1 and the exercise region 𝜆≤Λ∗1. For the second regime there is no exercise region.

To be accepted as true price, the numerical solution 𝑃_Λ^∗ has to verify all hypotheses and conditions of Theorem 8. In the regime 𝑋=𝑒₁, hypotheses (69) and (70) are verified numerically (see also Figure 9) and hypothesis (71) is accepted after calculation. Moreover Λ^∗_𝑘≤(𝜌₀−𝑙_𝑘)∧Λ⁰_𝑘 for 𝑘=1,2.

In the state 𝑋=𝑒₁, the present value of cash flows is at par, so 𝜉(𝜆₀,𝑋₀)=𝐾=1. The prepayment option price is 𝑃(𝜆₀)=0.0245. Therefore the loan value LV equals 𝜉(𝜆₀)−𝑃(𝜆₀)=0.9755.

4. Concluding Remarks

We proved in this paper two sufficient theoretical results concerning the prepayment option of corporate loans. In our model the interest rate is constant, the default intensity follows a CIR process, and the liquidity cost follows a discrete space Markov jump process. The theoretical results were implemented numerically and show that the prepayment option cost is not negligible and should be taken into account in the asset liability management of the bank. Moreover it is seen that when liquidity parameters are very different (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.