I. Introduction

Contingent credit lines (CCLs) are widely used by banks for commercial and industrial lending and also play an important role in the functioning of short-term capital markets. A recent survey by the U.S. Federal Reserve shows that over three-quarters of bank lending is done using commitment contracts, and as of June 2005, the outstanding (unused) CCLs of U.S. firms were close to $1.72 trillion, more than double that in 1990.² Yet, the pricing and hedging of CCLs has not received much attention in the finance literature.³ Recent advances in financial modeling and the emergence of new liquid option contracts have begun to permit a more detailed analysis of pricing and hedging of CCLs. Using a financial engineering approach, this paper examines the structure of simple CCLs and develops a method for their pricing.

A typical credit line contract has the following characteristics. First, it specifies a maximum amount that a financial institution (henceforth a bank) is committed to lending a client over a given period; this amount is called the commitment. Typically, the client has the right to draw any amount up to the maximum that is committed by the lender. Second, the contract specifies an interest rate that will apply to the amount borrowed or the drawdown. This is specified as a fixed interest rate or, more commonly, as a spread over some reference rate such as the London interbank offered rate (LIBOR). Third, the contract specifies the various fees charged by the lenders—an upfront commitment fee, an annual fee levied on the total amount committed, and a usage fee levied annually on the undrawn portion of the commitment.⁴ Fourth, contingent credit lines contain an escape clause, sometimes called a material adverse change (MAC) clause. Such a clause allows the bank to deny credit if the client’s financial condition changes in a substantive way—for example, if the borrower is downgraded.⁵

Companies use lines of credit for three general reasons. First, many credit lines are issued as backstop facilities that give flexibility to issuers in capital markets. For example, companies with unused credit lines will avoid borrowing during periods when commercial paper (CP) rates have temporarily spiked in the market due to unforeseen events. Second, having a credit line in place signals that the company has the ability to pay for specific transactions and hence reduces credit risk on short term borrowing. In the commercial paper market, for example, back-up credit lines provide “insurance” to the investors for their short-term unsecured lending. Third, opening a credit line with a highly reputable bank usually sends a positive signal to other financial market participants. It reduces information asymmetries between the company management and the market about the company’s financial condition (Fama (1985)). Normally, credit lines are rolled over after examining the financial health of the company. Renewal is important since it validates a companies credit-worthiness; termination, on the other hand, could be a negative signal about a clients financial health or business opportunities. Hence, clients have an incentive to roll over their existing credit lines with the same banks.⁶

Mozebach (1999) suggests that, in most cases, banks have considerable access to corporate information, and they are generally in the unique position of having little information asymmetry vis-à-vis the firms they lend to. Hence, the granting of a line of credit by a bank is a signal of a company’s financial viability, and helps reduce the information asymmetry for other market players. Mozebach’s paper also shows that there is a positive and significant market reaction on the protection buyer’s stock to the announcement that a credit line has been granted. In addition, the bank implicitly sends a signal about itself, since it enters into a contract to provide funds on demand in the future. Therefore, lines of credit are an important signaling device for both firms (protection buyers) and banks (protection sellers).

Calomiris (1989) provides the framework that is closest to our treatment of contingent credit lines. He discusses evidence that bank loan commitments backing commercial paper provide insurance against systemic liquidity risk. The main function of commercial paper back-stops is to be another source of funds during periods of extreme CP market volatility. They are infrequently used and become automatically void if the underlying credit changes significantly. This paper is an attempt to replicate and price such CCLs.

The rest of the paper is organized as follows. Section II discusses the market for CCLs. Section III deals with modeling a canonical credit line. Section IV provides a replication strategy for the basic credit line. Section V proposes two methods for pricing CCLs and, using Monte Carlo simulations, examines how the CCL price is affected by contract and market parameters. The last section concludes.

II. Market Practice

In short-term capital markets, there are at least three standard types of financing for investment grade borrowers. The first is the syndicated loan market for relatively large issuers. The second is the commercial paper (CP) market where high grade issuers can borrow to finance their daily operations. And third, there are CCL facilities or CP backstop loans that are mainly associated with CP markets and other operations.

In this paper, we focus on this third category. In 2001, total volume of CP backstop facilities in the United States was around $290 billion.⁷ Such credit lines are generally opened by prime borrowers and carry various types of fees. Before we formally model such contracts it is useful to describe current market practices. Reviewing some of these practices will motivate certain features of CCLs:

• Participation fees. Large contingent credit lines are issued by bank syndicates. The banks that participate in these syndicates are typically paid 1 basis point for every $10 million of the loan they underwrite. Obviously, this is a relatively small portion of the total cost of a CCL.

• Undrawn or facility fees. Once the CCL is opened, the borrower pays an annual fee even when the lines are not used. During the year 2001 the facility fees were in the 8-13 basis point range depending on the creditworthiness of the borrower.

• Drawn fees. This fee is the major component of the cost of a CCL. It is the spread over LIBOR on the portion of the CCL that is eventually drawn. The standard market practice is to express the price of the CCL as an all-in drawn fee, i.e., the total funding cost over LIBOR paid by a borrower who avails of the CCL. During 2001 this fee averaged 73 basis points and in 2002 the average was 53 basis points⁸. This is a significant amount given that 1-year LIBOR averaged 3.8 percent in 2001 and 2.2 percent in 2002.

• Term-out fees. Before the year 2000 most CCLs were multi-year facilities and went up to five years. Then, with the collapse of the tech bubble, the bankruptcy of companies like Enron and WorldCom, and the revelation of accounting improprieties at many U.S. corporations, the time-length of CCLs was adversely affected. During 2002-2003, 60 percent of CCLs were marketed as one year (364 day) facilities. However, to lengthen the term of the CCL, the buyer can pay a term-out fee to buy an option to increase the term by another year. Such term-out fees were 12.5 basis points in 2001, but increased to 25 basis points in 2002.

• Some of the CCLs have embedded currency options. The options give the borrower the right to draw the loan in more than one currency. Such options make the product more complex but not necessarily more difficult to price. In this paper we will ignore such options.

Large CCLs are sold through syndications. The syndicates often consist of banks with which the borrower has a close relationship and the CCL facility is routinely extended. From the point of view of the banks, as long as the facility is not drawn, a commitment with original maturity of less than one year carries a zero risk-capital weight. Hence, the preference for maturities of less than a year. For commitments with original maturity greater than one year banks are required to hold half the capital required for regular loans.

III. Modeling a CCL

To fix ideas, suppose a corporation would like to transfer the risk associated with a low probability event A that may negatively affect its funding opportunities. In our case, this event could be a temporary closing of the CP market to the firm or a temporary spike in CP rates due to some outside shock. If the event A occurs the firm may need to secure some alternative financing and it decides to hedge this risk through a contingent credit line contract.

A CCL is a contract between a bank and a borrower, signed at time t₀, that specifies the time T till which the contract is valid, the maximum loan amount ($N) that is available and the term of the loan. Any amount, up to the maximum available under the contract, can be drawn at an interest rate that is either fixed at the time the contract is written, or more generally specified as a spread, $s_{t_0}$, over some benchmark rate such as LIBOR, L_t. At the borrower’s discretion, this amount can be drawn at any time in the interval [t₀, T]. The contract also contains a material adverse clause that could lead to the contract being voided under certain circumstances, for example, a deterioration of the borrower’s credit rating.⁹

At regular times {t_i, i = 1, 2, …, n} the firm has to roll over a funding requirement of $N in the money market. It pays the LIBOR rate $L_{t_i}$ plus the credit spread $c_{t_i}$ at the settlement date t_i+1. The CCL contract designed as a CP backstop has the following structure.

1. The contract is written at time t₀ for n periods (say, months or years). The maturity date is denoted by T, with T = t_n.

2. The credit line has a maximum size $N. Normally, the borrower can draw any amount up to $N. We assume that when the facility is used the full line is drawn.¹⁰ This simplifies the replication process and can be relaxed by assuming other rules for drawing the line.¹¹

3. The unused portion of the CCL is subject to a facility fee denoted by $u_{t_0}$. In this paper, $u_{t_0}$ is interpreted as the CCL premium that is paid up front at contract initiation. This is different than market practice, where $u_{t_0}$ is paid over the contract period as an annual fee. In addition, the drawn portion is subject to a drawn-fee, $s_{t_0}$, expressed as a spread over LIBOR. These fees have subscript t₀ since they are decided when the contract is negotiated and remain constant throughout the contract period.

4. The CCL contract is of the Bermudan-type and the CCL can be tapped only at the predetermined times t₁, t₂, …, t_n−1. This is a simplification since in general the CCL lines can be accessed at any time during the life of the contract. However, if t_i − t_i−1 is small, this is an innocuous assumption that greatly simplifies the modeling and replication of the CCL.

5. When the credit line is drawn at time t_i, i = 1, 2,…, n−1, the borrower pays the annual all-in interest cost:

   $\begin{array}{ccc}{r}_{t_i}=L_{t_i}+s_{t_0}.& & (1)\end{array}$

   For simplicity, we ignore the participation and the term-out fees.

6. At times t_i, the borrower has the option to prepay the amount drawn from the CCL. Thus, if the firm’s credit spread in the CP market falls below $s_{t_0}$ the firm will choose to repay the bank.

7. If an event В occurs at some date t, t₀ < t < T, the CCL contract is automatically voided. For the CCL being considered, event В is defined to be a rating downgrade of the borrower. This is assumed to occur if the time t credit spread observed in the market, c_t, exceeds a level c*:

   $\begin{array}{ccc}{c}_t>c^{*},& & (2)\end{array}$
   where, obviously, we need to have

   $\begin{array}{ccc}{c}^{*}>s_{t_0\cdot }& & (3)\end{array}$

The structure described above captures many important aspects of CCLs. A simplified version is provided in Figure 1. CCLs are tools that essentially decrease a company’s cost of funding by lowering the credit risk for short-term lenders and by providing (partial) insurance against unforeseen market events. A large issue of CP is likely to be better received by the market if the investors know that the firm has access to a CCL and hence will be in a position to redeem its CP even under potentially adverse market conditions. It is worth noting that in offering a backstop loan or CCL facility, it is not the intention of the bank to lend (possibly) large sums. Rather, a CCL facility should be seen as a liquidity and credit enhancement, and not as a loan. And any loan that originates from drawing the CCL should be regarded as temporary in nature.

CCL Structure

Citation: IMF Working Papers 2006, 013; 10.5089/9781451862737.001.A001

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CCL Structure

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CCL Structure

Citation: IMF Working Papers 2006, 013; 10.5089/9781451862737.001.A001

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The CCL structure described above accomplishes this in the following way. If the corporation borrows from the short-term money markets at time t_i, the all-in interest paid will be:

where $L_{t_i}$ is again the appropriate LIBOR. Under these conditions, a firm with a CCL facility will draw on it at time t_i only if the credit spread in the market exceeds the all-in drawn fee on the credit line:

Once the market spread $c_{t_i}$ goes below $s_{t_0}$ the loan is repaid, ensuring the temporary nature of the instrument. In contrast, when $c_{t_i}$ exceeds the level c*, the borrower is downgraded and the CCL becomes void.

IV. Replicating Portfolio

In this section we show that the CCL contract can be replicated by using two well known instruments written on the credit spread c_t — a cap on the credit spread c_t with exercise dates t_i and a reverse knock-out option on c_t.

The first instrument is the caplet and its associated cap. A plain vanilla caplet is like a call option written on an interest rate, say x, such that at settlement time t_i+1 the caplet pays

where ĸ is the cap rate and the δ is the period of the loan expressed in years. In other words, if at the expiration date t_i, the interest rate $x_{t_i}$ exceeds the specified cap rate ĸ, the buyer of the caplet gets compensated for the difference.¹²

In our case, instead of the interest rate $x_{t_i}$ we assume that the caplet is written on the market credit spread $c_{t_i}$, and that the cap rate ĸ is the all-in drawn fee $s_{t_0}$. Thus the time t_i+1 payoff of the caplet is assumed to be,

Hence, if $c_{t_i}$ exceeds the pre-specified level $s_{t_0}$, the CP market becomes less attractive than the CCL and the credit line is used (fully).¹³ Putting together n such “caplets” with exercise dates t_i, i = 1, 2, …, n, in a single contract, we obtain a cap written on the credit spread $c_{t_i}$.

The second instrument we use in engineering a CCL is a reverse barrier option. In particular, we consider a knock-out option that is contingent on the “credit event”:

If this event occurs, the firm is downgraded and the CCL contract is voided and the bank is under no obligation to provide the loan.

The underlying risk is c_t, the credit spread at time t, t₀ ≤ t ≤ t_n = Т. The replicating portfolio for the CCL structure described above consists of a cap written on the credit spread c_t, combined with a barrier option that leads to the cap being knocked-out if the event $\left\{c_t>c_t^{*}\right\}$ occurs during the life of the contract. Since the CCL knocks-out when the cap is in-the-money, we are dealing with a reverse barrier option. The payoff structure is shown in Figure 2

Reverse Barrier Option

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Reverse Barrier Option

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Reverse Barrier Option

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We assume that at exercise times t_i of the caplets, the firm could get a loan of size $N from the bank. This loan has an interest cost made up of the floating rate $L_{t_i}$ and the all-in-fee, $s_{t_0}$. The fair market value at time t of the i^th (knock-out) caplet that expires at time t_i+1 can be written as:

where $1_{\{c_t<c^{*},t_0\le t\le t_i\}}$ is the indicator function for the event that the firm is not downgraded before the caplet expires. The term B(t, t_i+1)is the risky discount at time t on $1 to be paid at time t_i+1 and is calculated from the appropriate risky zero-coupon bonds. The valuation equation written above uses the discount bonds with default risk for normalizing securities prices—the t_i+1 risky forward-measure is used in evaluating the expectation. This normalization requires that during default the recovery rate is bounded away from zero with probability one. The cap, formed by the collection of caplets, has a value at time t of:

We can now complete the replication process. Suppose the client buys the knock-out cap from the bank at the all-in cost $s_{t_0}$ and, at the same time, obtains a floating rate loan in the money markets that has rates $L_{t_i}+c_{t_i}$ at reset dates t_i. This portfolio will be equivalent to a market loan backed by a contingent credit line written by the bank.

The contract can be described in heuristic terms as follows. The corporation has to make a decision at the dates t₁, …, t_n on whether to tap the CCL or not. If the CCL is tapped at the date t_i, (by assumption in the full amount) then the corporation pays the drawn fee $s_{t_0}$ over the $L_{t_i}$ rate observed at that time. The loan is for the period t_i+1 – t_i = δ.¹⁴ At the next date t_i+1, the corporation has two choices. The corporation may decide to pay back the CCL loan and draw it again (i.e. roll it over), or it may want to pay the bank and borrow in the CP market rather than access the CCL. The latter will be the case if $c_{t_{i+1}}<s_{t_0}$.

In this setup, the CCL amounts to a sequence of options on a series of floating rate loan contracts struck at the credit spread $s_{t_0}$. At t_i, the option to draw on the CCL will be exercised if the observed credit spread $c_{t_i}$ exceeds $s_{t_0}$. This option is “knocked out” if this excess is too much and the corporation is downgraded. The price of the CCL depends on the underlying $L_{t_i}$, $c_{t_i}$, their volatilities, and other parameters like $s_{t_0}$, c^*, and whether there are options to prepay or extend the maturity of the CCL. Having the prepayment and term-out privileges, would involve adding two more options to the replicating portfolio.

V. Pricing

The pricing of the CCL contract cannot use the standard approach for foreign exchange and equity options. Normally, a reverse knock-out option can be priced with reasonable accuracy by combining an American digital option with a regular knock-out option. However, in our case the standard approach to pricing knock-out options is not appropriate because we are dealing with interest rates and credit spreads. The standard Black-Scholes methodology has one factor and assumes constant interest rates. In our case there may be two factors and, depending on the term of the CCL contract, constant interest rates may not be a realistic assumption.

Suppose for simplicity that the cost of borrowing in the CP market for a firm with no default-risk is L_t.¹⁵ The firm buying the CCL has a credit risk represented by the spread c_t, and the firm has raised $N in the commercial paper market as of time t.

Let F (t, t_i, t_i+1) be the forward LIBOR rate at time t for a loan made at time t_i that matures at time t_i+1, where t < t_i < t_i+1. We assume that this rate applies to loans with no default risk. We use the notation f (t, t_i, t_i+1) for the forward rate that applies to loans made to a specific firm. Pricing of the CCL contract requires modeling the dynamics of: (i) n risk-free forward rates, F (t, t_i, t_i+1), i = 1, …, n−1 and (ii) n risky forward rates, f (t, t_i, t_i+1), that apply to loans made to a specific firm at times t_i and maturing at times t_i+1, i = 1, …, n-1. The resulting dynamics for c(t, t_i, t_i+1), the (forward) corporate credit spread, is derived from

We model the dynamics for F (t, t_i, t_i+1), f (t, t_i, t_i+1) and c(t, t_i, t_i+1) using two methods.

A. Method 1

The first method uses a framework provided by Schönbucher’s extension of the forward-LIBOR model.¹⁶ For the forward-LIBOR process we have:

$\begin{array}{ccc}F(t,t_i,t_{i+1})=E_t^{P^{t_{i+1}}}[L_{t_i}]& & (12)\end{array}$

where $P^{t_{i+1}}$ represents the t_i+1-risk-neutral forward measure. This measure is obtained by a normalization using the time t_i+1 default-free pure discount bond price P (t, t_i+1) and yields a F (t, t_i, t_i+1) process that has no drift.¹⁷

For expositional clarity, we consider the pricing of a single caplet in the CCL structure described in the previous section. Assuming that the caplet applies to the period [t₁, t₂], considerably simplifies the notation. Dropping the i subscripts, we redefine the forward rates as F_t≡F (t, t₁, t₂) and f_t≡ f (t, t₁, t₂).¹⁸ It is well known that under the t₂-forward measure the F_t process has no drift:

$\begin{array}{ccc}d{F}_t=F_t{\sigma }_t^{F}d\omega (t)& & (13)\end{array}$

where ${\sigma }_t^F$ is the volatility parameter for F_t. and ω(t) is the Brownian motion under the t₂-forward measure. However, under the t₂-forward measure the f_t process has an unknown drift that depends on the market prices for interest rate and credit risks, and it does not have martingale dynamics:

$\begin{array}{ccc}d{f}_t=\gamma \left(f_t,\lambda \left(t\right)\right)\mathit{dt}+f_t{\sigma }_t^{f}d\omega \left(t\right)& & (14)\end{array}$

where the λ(t) is a vector of market prices and ${\sigma }_t^f$ is the volatility parameter for f_t.

One approach is to use Schönbucher’s extension of the Forward LIBOR model. To this end, define $\tilde{P}$ as the t₂-survival measure. This measure is obtained by scaling the time t₂ state-price vector using the value of the time-t₂ maturity defaultable bond issued by the borrower. Schönbucher (2004) shows that when this measure is used f_t has martingale dynamics:

$\begin{array}{ccc}d{f}_t=f_t{\sigma }_t^{f}d\overline{\omega }\left(t\right)& & (15)\end{array}$

where $\overline{\omega }\left(t\right)$ is the Brownian motion under the t₂-survival measure.

These two measures can be connected by using a spot-martingale measure, which is obtained by normalizing asset prices using a properly defined savings account.¹⁹ After changing the probability under the spot-martingale measure to that under the t₂-forward measure, dω(t) can be written as

$\begin{array}{ccc}d\omega \left(t\right)=d{\omega }_Q\left(t\right)+{\alpha }_{t_2}\left(t\right)\mathit{dt},& & (16)\end{array}$

where ω_Q(t) is the Brownian motion under the spot-martingale measure, and ${\alpha }_{t_2}\left(t\right)$ can be recursively expressed as

$\begin{array}{ccc}{\alpha }_{t_2}\left(t\right)={\alpha }_{t_1}\left(t\right)+\frac{\delta F_t}{1+\delta F_t}{\sigma }_t^F.& & (17)\end{array}$

Here ${\alpha }_{t_1}$ is defined as minus the volatility of the default free bond P (t, t₁). Changing measure from spot-martingale measure to t₂-survival measure, we get

$\begin{array}{ccc}d\overline{\omega }\left(t\right)=d{\omega }_Q\left(t\right)+{\overline{\alpha }}_{t_2}\left(t\right)\mathit{dt},& & (18)\end{array}$

$\begin{array}{ccc}{\overline{\alpha }}_{t_2}\left(t\right)={\overline{\alpha }}_{t_1}\left(t\right)+\frac{\delta f_t}{1+\delta f_t}{\sigma }_t^f,& & (19)\end{array}$

where ${\overline{\alpha }}_{t_1}$ is minus the volatility of the defaultable bond B(t, t₁). These connections between the Wiener process increments are used in writing the dynamics of the underling processes under a single measure. This is needed since pseudo-random numbers need to be drawn from one probability distribution in the Monte Carlo approach.

Using (16) and (18), the Wiener processes under the t₂-forward measure and the t₂-survival measure are related by

$\begin{array}{ccc}d\overline{\omega }\left(t\right)=\mathit{d}\omega \left(t\right)+{\alpha }_{t_2}^D\left(t\right)\mathit{dt}& & (20)\end{array}$

where, ${\alpha }_{t_2}^D\left(t\right)$, defined as ${\overline{\alpha }}_{t_2}\left(t\right)-{\alpha }_{t_2}\left(t\right)$, has the recursion formula

$\begin{array}{ccc}{\alpha }_{t_2}^D\left(t\right)={\alpha }_{t_1}^D\left(t\right)+\frac{\delta H\left(t\right)}{1+\delta H\left(t\right)}{\sigma }_t^H.& & (21)\end{array}$

Here H(t) is interpreted as the hazard rate at time t. Using the equality in (20) we can now express the two martingale dynamics for F_t and f_t under one single measure. This introduces a drift term in one of the martingale dynamics, but this can be calculated since ${\alpha }_{t_2}^D\left(t\right)$ is known.

Using the t₂-survival measure, we write the dynamics of F_t, f_t, c(t), and H(t) under a single measure as

$\begin{array}{ccc}d{F}_t=F_t{\sigma }_t^F\left(d\overline{\omega }\left(t\right)-{\alpha }_{t_2}^D\left(t\right)\mathit{dt}\right),& & (22)\end{array}$

$\begin{array}{ccc}d{f}_t=f_t{\sigma }_t^{f}d\overline{\omega }\left(t\right),& & (23)\end{array}$

$\begin{array}{ccc}dc\left(t\right)=F_t{\sigma }_t^F{\alpha }_{t_2}^D\left(t\right)\mathit{dt}+\left[f_t{\sigma }_t^f-F_t{\sigma }_t^F\right]d\overline{\omega }\left(t\right),& & (24)\end{array}$

$\begin{array}{ccc}dH\left(t\right)=\frac{\delta F_t}{1+\delta F_t}\left[(1+\delta H(t)){\alpha }_{t_2}^D(t)-\delta H(t){\sigma }_t^H\right]dt+H(t){\sigma }_t^{H}d\overline{\omega }(t).& & (25)\end{array}$

Note that the dynamics for F_t now has a non-zero drift. Note also that this drift is known at time t. Discretizing these equations using an Euler scheme, we obtain Monte Carlo trajectories for c(t), which we use to price the CCL contract.

The CCL price is the discounted payoff from the replicating portfolio consisting of the caplet and the associated knock-out option. In the introduction, this price is also denoted by the symbol $u_{t_0}$:

$\begin{array}{ccc}{(CCL)}_{t_0}=B(t_0,t_2)N\delta E\left[\mathit{Max}(c_{t_1}-s_{t_0},0)1_{\{c_t<c^{*},t_0\le t\le t_1\}}\right]& & (26)\end{array}$

where B(t₀, t₂) is the risky discount factor and $1_{\{c_t<c^{*},t_0\le t\le t_1\}}$ is the indicator function representing the requirement that the firm is not downgraded before time t₁.

The pricing approach used above is built on Schönbucher’s extension of the Forward LIBOR Model. Our discussion of synthetically creating a CCL was based on the dynamics of the credit spread and did not directly model the event of default. Schönbucher (2004) provides the rationale for our procedure. Note that the synthetic CCL and its pricing is done using the survival measure. This eliminates the need to directly model the default event. The key to this is modeling the dynamics conditional on the fact that default has not occurred —that is, using a normalization that depends on the value of the defaultable bond, but under the condition that default has not yet occurred. Schönbucher(2004) shows how a model that is conditional on no default having occurred, can be used to price defaultable securities.

The structure of the replicating portfolio for the CCL makes two tendencies clear. The higher the probability that the borrower’s credit spread (c_t) will be greater than the nego-tiated spread in the CCL contract $(s_{t_0})$, the larger is the payoff to the CCL and hence the higher is its price. However, higher credit spreads also increase the probability of breaching the knock-out barrier c^*, and the increase in this probability tends to dampen the payoff to the CCL.

The simulations are designed to show the effects of σ^f, $s_{t_0}$ and c^* on the price of the contingent credit line. We assume that the CCL is for a notional sum of $1 and the values for the other parameters are as follows: σ^F = 15%, t₁ = 1, t₂ = 2. For the simulation exercise, we take the initial values to be $F_{t_0}=6.7\%$, $f_{t_0}=8.3\%$, $c_{t_0}=1.6\%$. Figures 3-5 show the expected payoff²⁰ at time t₂ under different parameter specifications. Figure 3 depicts the effect of σ^f and $s_{t_0}$ on the CCL price, holding c^* constant. It shows that the payoff is a decreasing function of $s_{t_0}$, for given levels of σ^f, and c^*. The higher is $s_{t_0}$, the lower the probability that the CCL will be drawn, and hence the lower is its price. For fixed c^*, the higher is $s_{t_0}\left(<c^{*}\right)$, the smaller is the range over which the CCL has positive value. For fixed $s_{t_0}$, the CCL price initially rises with volatility σ^f and then declines, since eventually high values for σ^f increase the probability of the knock-out threshold being crossed.

CCL price: c*= 0.055

Citation: IMF Working Papers 2006, 013; 10.5089/9781451862737.001.A001

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CCL price: c*= 0.055

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CCL price: c*= 0.055

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CCL Price: $s_{t_0}=0.025$

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CCL Price: $s_{t_0}=0.025$

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CCL Price: $s_{t_0}=0.025$

Citation: IMF Working Papers 2006, 013; 10.5089/9781451862737.001.A001

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CCL Price: σ^f = 0.025

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CCL Price: σ^f = 0.025

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CCL Price: σ^f = 0.025

Citation: IMF Working Papers 2006, 013; 10.5089/9781451862737.001.A001

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Figure 4 looks at the effect of σ^f and c^* on the CCL price, keeping $s_{t_0}$ constant. The relationship between σ^f and the CCL price is similar to the one seen in Figure 3. What is interesting here is that the σ^f corresponding to the peak CCL price for a given c^* increases with c^*. Intuitively, given $s_{t_0}$, for a higher c^* the “knock-out” effect begins to dominate at a higher σ^f. The positive relationship between c^* and the CCL price is also in line with our intuition. c^* is the knock-out threshold; if c_t exceeds c^*, the CCL terminates. The higher the c^*, the lower is the knock-out probability, and hence the higher the payoff.

Figure 5 shows how the CCL price varies with $s_{t_0}$ and c^*, for fixed values of σ^f. The results are intuitive and clear-cut. For fixed c^*, raising $s_{t_0}$ lowers the price of the CCL. For fixed $s_{t_0}$, raising c^* increases the CCL price. As one would expect, for given values of σ^f, the price curve flattens out after c^* has reached a certain level.