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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.

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

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 “

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 [

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 [

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

There exist few articles (e.g., works by Cossin and Lu [

Another contribution by Cossin and Lu [

There also exist mortgage prepayment decision models based on Poisson regression approach for mortgage loans (see, e.g., Schwartz and Torous [

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 [

Works involving Markov switched regimes and CIR dynamics appear in [

On the other hand numerical methods are proposed in [

Finally, we refer to [

We assume throughout the paper that the interest rate

It is known that if

Assume a loan with a fixed coupon defined by the interest rate

Therefore the loan value

The present value of the cash flows discounted at the (instantaneous) risky rate

For a perpetual loan the maturity

The margin

If an additional commercial margin

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 (

From definition (

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

As discussed above, the prepayment exercise results in a payoff

We will denote by

Denote for

Then

Then the price of the prepayment option is

We start with the first item: it is possible to obtain a general solution of (

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. [

Further results derived for the situation of a perpetual (standard) American put options [

The result builds heavily on the fact that the discounted payoff of the standard situation of an American put

on

the solution candidate

the function

The theorem also says that the borrower exercises his option on

We now show that

The requirement 2 implies that in the continuation region the price is the solution of the following PDE:

The following inequality holds:

Recall that

Note that Theorem

Two approaches can be considered; first, it is enough to find a zero of the following function

A different convenient procedure to find the critical

We consider a perpetual loan (

In order to find the initial contractual margin, we use (

At inception, the present value of cash flows is at par, so

The value

We illustrate here the dependence of

Loan value as a function of the intensity. The loan value is decreasing when there is a degradation of the credit quality (i.e.,

Prepayment option price

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

We assume the economic state of the market is described by a finite state Markov chain

Assuming the process

Assume a loan has a fixed coupon defined by the constant interest rate

The loan value

The PVRP

The crucial information here is that the coefficients

Note that (cf. Section

By differentiation with respect to

We have therefore analytic formulas for the PVRP

When all liquidity parameters

The margin

We will also need to introduce for any

It is useful for the following to introduce a PDE formulation for

Having defined the dynamics (

When the dynamics involve different coefficients of the CIR process for different regimes (cf. also Remark

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:

For any N-tuple

Similar arguments as in the proof of Theorem

To this end we use a similar technique as in Theorem 10.4.1 [

Denote by

The Lemma

Under the hypothesis of the Theorem

The nontrivial part of this lemma comes from the fact that if for fixed

From (

We now analyse the situation when

A last situation is when

Recalling the properties of

Several remarks are in order at this point.

When only one regime is present; that is,

When

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

The search for the candidate

If the optimization of

The numerical solution of the partial differential equation (

To avoid working with an infinite domain, a well-known approach is to define an artificial boundary

We consider a perpetual loan with a nominal amount

In order to find the initial contractual margin, we use (

The optimal value

We illustrate here the dependence of

In the state

The loan value will be equal to the nominal if the intensity decreases until the exercise region

Loan value as a function of the intensity. (a) Regime

The price of the prepayment option

When the liquidity parameters corresponding to given regimes are very different, it may happen that the optimization of

Further examination of Theorem

The boundary value set in (

It is interesting to know when such a situation can occur and how can one interpret it. Let us take a two-regime case (

We consider the same situation as in Section

The couple (

We illustrate here the dependence of

Loan value as a function of the intensity. (a) Regime

The price of the prepayment option

To be accepted as true price, the numerical solution

In the state

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.