Pricing and Hedging of Contingent Credit Lines
• 1 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

## Contributor Notes

Contingent credit lines (CCLs) are widely used in bank lending and also play an important role in the functioning of short-term capital markets. Yet, their pricing and hedging has not received much attention in the finance literature. Using a financial engineering approach, the paper analyzes the structure of simple CCLs, examines methods for their pricing, and discusses the problems faced in hedging CCL portfolios.

## Abstract

Contingent credit lines (CCLs) are widely used in bank lending and also play an important role in the functioning of short-term capital markets. Yet, their pricing and hedging has not received much attention in the finance literature. Using a financial engineering approach, the paper analyzes the structure of simple CCLs, examines methods for their pricing, and discusses the problems faced in hedging CCL portfolios.

where is the indicator function for the event that the firm is not downgraded before the caplet expires. The term B(t, ti+1)is the risky discount at time t on $1 to be paid at time ti+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 ti+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: $\begin{array}{ccc}{\left(cap\right)}_{t}=\sum _{i=1}^{n}{\left(cpl\right)}_{t}^{i}.& \phantom{\rule{7.0em}{0ex}}& \left(10\right)\end{array}$ 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 ti. 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, …, tn on whether to tap the CCL or not. If the CCL is tapped at the date ti, (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 ti+1 – ti = δ.14 At the next date ti+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 ti, 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 Lt.15 The firm buying the CCL has a credit risk represented by the spread ct, and the firm has raised$N in the commercial paper market as of time t.

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

$\begin{array}{ccc}c\left(t,{t}_{i},{t}_{i+1}\right)=f\left(t,{t}_{i},{t}_{i+1}\right)-F\left(t,{t}_{i},{t}_{i+1}\right).& \phantom{\rule{7.0em}{0ex}}& \left(11\right)\end{array}$

We model the dynamics for F (t, ti, ti+1), f (t, ti, ti+1) and c(t, ti, ti+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:

$\begin{array}{ccc}F\left(t,{t}_{i},{t}_{i+1}\right)={E}_{t}^{{P}^{{t}_{i+1}}}\left[{L}_{{t}_{i}}\right]& \phantom{\rule{7.0em}{0ex}}& \left(12\right)\end{array}$

where ${P}^{{t}_{i+1}}$ represents the ti+1-risk-neutral forward measure. This measure is obtained by a normalization using the time ti+1 default-free pure discount bond price P (t, ti+1) and yields a F (t, ti, ti+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 Ft≡F (t, t1, t2) and ft≡ f (t, t1, t2).18 It is well known that under the t2-forward measure the Ft process has no drift:

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

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

One approach is to use Schönbucher’s extension of the Forward LIBOR model. To this end, define $\stackrel{˜}{P}$ 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 ft has martingale dynamics:

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, (t) can be written as

where ωQ(t) is the Brownian motion under the spot-martingale measure, and ${\alpha }_{{t}_{2}}\left(t\right)$ can be recursively expressed as

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

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

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

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 Ft and ft 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 Ft, ft, c(t), and H(t) under a single measure as

Note that the dynamics for Ft 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 ${u}_{{t}_{0}}$:

where B(t0, t2) is the risky discount factor and 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 (ct) will be greater than the nego-tiated spread in the CCL contract $\left({s}_{{t}_{0}}\right)$, 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 σf, ${s}_{{t}_{0}}$ 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: σF = 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 σf 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 σf, 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 σf and then declines, since eventually high values for σf increase the probability of the knock-out threshold being crossed. Figure 4 looks at the effect of σf and c* on the CCL price, keeping ${s}_{{t}_{0}}$ constant. The relationship between σf and the CCL price is similar to the one seen in Figure 3. What is interesting here is that the σf 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 σf. The positive relationship between c* and the CCL price is also in line with our intuition. c* is the knock-out threshold; if ct 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 σf. 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 σf, the price curve flattens out after c* has reached a certain level. ### B. Method 2 An alternative way of proceeding in modeling the dynamics of the forward rates is to assume that when a firm defaults the recovery rate is bounded away from zero, and normalize using the price of the risky discount bond, B(t, ti+1). Under this forward measure ${\stackrel{˜}{P}}^{{t}_{i+1}}$, the forward rate f(t, ti, ti+1) is a martingale without drift. Hence, using this approach, only the drift of the risk-free forward rate F(t, ti, ti+1) under the forward measure ${\stackrel{˜}{P}}^{{t}_{i+1}}$ has to be specified. Using the same notation as in Method 1, the real-world dynamics of Ft, ft and ct are specified as where the $d{\omega }_{t}^{\left(1\right)}$ and $d{\omega }_{t}^{\left(2\right)}$ are standard, possibly correlated, Wiener processes. Under the forward measure ${\stackrel{˜}{P}}^{{t}_{2}}$ obtained through normalization using B(t, t2), the forward rate ft is a martingale without drift where $d{\stackrel{˜}{\omega }}_{t}^{\left(2\right)}$ is a standard Wiener process and volatility is assumed to be constant. To specify the drift of the risk-free forward rate Ft, we proceed as follows. We assume that Ft follows a mean-reversion model: $\begin{array}{ccc}d{F}_{t}=\beta \left(\mu -{F}_{t}\right)\mathit{dt}+{\sigma }^{F}{F}_{t}d{\omega }_{t}^{\left(1\right)},& \phantom{\rule{7.0em}{0ex}}& \left(29\right)\end{array}$ where µ is the long-term average rate and β is the speed of reversion. Under the B(t, t2) normalization, $E\left[d{\omega }_{t}^{\left(1\right)}\right]={\lambda }_{t}\mathit{dt}\ne 0$. By adding and subtracting the term λt σF Ft dt in (29), we get where $d{\stackrel{˜}{\omega }}_{t}^{\left(1\right)}$ is a Wiener process, such that and $E\left[d{\stackrel{˜}{\omega }}_{t}^{\left(1\right)}\right]=0$. We assume λt is a constant and calibrate it by making use of the fact that the caplet price can be calculated in two ways. Black’s formula gives a closed-form analytical solution.21 The other is a λt-dependent simulation based method that makes use of the interest rate dynamics specified in (30). We choose λt to equalize the two calculations of the caplet price.22 Suppose t0 (t0 < t1 < t2) is the contract initiation time and the caplet contract applies to the future period [t1, t2]. The payoff at time t2 of the caplet with a fixed strike rate of K is where N is the notional amount decided when entering the contract, δ is the days adjustment factor, and ${L}_{{t}_{1}}$ is the LIBOR rate observed at time t1. Black’s formula for a (at-the-money) caplet price with strike rate ${F}_{{t}_{0}}$ is (see, for example, Hull (2003)) where, P (t0, t2) is the time t0 value of the relevant default-free bond, the Φ(·) is the standard normal distribution, and σcpl is the average realized annual caplet volatility. The simulated caplet price is the expected payoff: We simulate Ft using the Euler discretization, The paths are simulated from time t to time t1 to obtain $F\left(t,{t}_{1},{t}_{2}\right)={E}_{t}^{{P}_{{t}_{2}}}\left[{L}_{{t}_{1}}\right]$. By simulating p (say, 3000) paths of Ft, we get the expected payoff of the caplet. The desired λ* is then the λ that makes (32) equal to (33). Having calculated λ*, the dynamics of the forward rates are given by We assume that the Wiener processes driving the two forward rates, ft and Ft, have an instantaneous correlation ρ: The Cholesky decomposition of the covariance matrix of allows us to express the dynamics of the forward rate processes in terms of two independent Wiener processes ${W}_{t}^{\left(1\right)}$ and ${W}_{t}^{\left(2\right)}$ where (see, for example, Brigo and Mercurio (2001)) $\begin{array}{l}\begin{array}{l}\begin{array}{lll}d{\stackrel{˜}{\omega }}_{t}^{\left(1\right)}=& d{W}_{t}^{\left(1\right)}& \phantom{\rule{7.0em}{0ex}}\end{array}\\ \begin{array}{lll}d{\stackrel{˜}{\omega }}_{t}^{\left(2\right)}=& \rho d{W}_{t}^{\left(1\right)}+\sqrt{1-{\rho }^{2}}d{W}_{t}^{\left(2\right)}.& \phantom{\rule{7.0em}{0ex}}\end{array}\end{array}\left(37\right)\end{array}$ The dynamics of ft and Ft can be written in terms of ${W}_{t}^{\left(1\right)}$ and ${W}_{t}^{\left(2\right)}$ as Since ct = ftFt, the simulated paths of ft and Ft, also yield a corresponding path for the credit spread. One can interpret $d{W}_{t}^{\left(1\right)}$ as a shock to the macroeconomic environment that affects all firms and $d{W}_{t}^{\left(2\right)}$ as an “idiosyncratic” shock that only affects the specific company. The simulations show the effects of σf, ρ, ${s}_{{t}_{0}}$ and c* on the price of the contingent credit line. Again, we assume that the CCL is for a notional sum of$1 and the values for the other parameters are as follows: σF = σcpl = 15%, β = 0.05, µ= 6:5%, t1 = 1, t2 = 2, ${F}_{{t}_{0}}=6.7%$, ${f}_{{t}_{0}}=8.3%$, ${c}_{{t}_{0}}=1.6%$. Figures 611 show the expected payoff at time t2 under different parameter specifications. Notice that the graphs in Figures 68 are similar to those in Figures 3-5, except that they are shifted up by a few basis points. In this alternative approach we also examine the sensitivity of the CCL price to the correlation parameter ρ.

Figure 9 shows how the CCL price varies with σf and ρ when ${s}_{{t}_{0}}$ and c* are held constant at the specified levels. For a fixed ρ, the graph of the CCL price against σf first increases and then decreases. Initially, as volatility σf increases it is more likely that ct will exceed ${s}_{{t}_{0}}$ and the CCL will be drawn. However, as volatility keeps increasing, eventually the knock-out effect dominates and the CCL price falls at still higher volatilities.

The effect of varying ρ while keeping σf constant is more complex. A higher correlation ρ between ft and Ft implies that ct is less volatile and the chance of ct exceeding ${s}_{{t}_{0}}$ and the CCL being drawn, is low. Hence, the CCL price is negatively related to the correlation ρ. For low levels of σf, this holds true. But for relatively high levels of σf, the story is different. A high ρ now makes it less likely that ct will exceed c*, therefore attenuating the knock-out effect. Thus, at high σf volatility, the CCL price and the correlation ρ can be positively related.

Figure 10 plots the CCL price against ρ and ${s}_{{t}_{0}}$, for given values of σf and c*. As before, it is clear that the relationship with ${s}_{{t}_{0}}$ is negative. With respect to ρ, the curve is relatively flat; an examination of cuts for given values of ${s}_{{t}_{0}}$, shows that for high values of ρ, the CCL price is a decreasing function of ρ. This can be explained as follows: the higher the ρ, the lower is the volatility of ct and hence the lower is the probability of drawing on the CCL. For low values of ρ, the graph is almost flat.

Figure 11 illustrates the effect of varying ρ and c* on the CCL price, for given values of σf and ${s}_{{t}_{0}}$: As expected, for given ρ, the CCL price increases with c*. The effect of ρ on the CCL price, however, depends on how large c* is relative to ${s}_{{t}_{0}}$. When c* is relatively large compared to ${s}_{{t}_{0}}$ (c* = 0.08; ${s}_{{t}_{0}}=0.025$), the relationship between ρ and the CCL price is clearly negative—the higher the ρ, the less volatile is ct and hence the lower is the payoff. When c* is small compared to ${s}_{{t}_{0}}$ (c* = 0.04, ${s}_{{t}_{0}}=0.025$), the range of spreads for which the CCL has positive value is small, and this is true irrespective of ρ: In this case for low values of ρ, the knock-out effect is larger; as ρ increases, ct becomes less volatile, lowering the probability of hitting c*. Hence, the curve is almost flat at low values of ρ, and has a very small positive slope at high values of ρ.

The above simulations show that the nature of the relationship between the CCL price and σf, ${s}_{{t}_{0}}$ and c* is reasonably clear. However, the relationship between the CCL and the correlation parameter ρ can be quite different depending on the values taken by the other parameters—Figure 12 plots the CCL price against ρ for different values of σf. The implication for the CCL issuer is that, depending on the market environment and the CCL characteristics, if the assumption for ρ is off the mark, the CCL could be dramatically mispriced.

## VI. Hedging Issues

We have shown that it may be possible to replicate a basic CCL by a cap that gets knocked out if the credit spread exceeds a pre-specified upper-limit c* during the life of the contract. The question then arises: can the bank issuing the CCL hedge it by following a procedure similar to that used in constructing market hedges for reverse knock-out options?

There are two major difficulties. First, even under reasonably normal conditions reverse knock-out options are quite difficult to hedge and this is especially so close to their time of expiration. A reverse knock-out option becomes void when it is in the money. When the credit spread ct increases and becomes larger than ${s}_{{t}_{0}}$ the value of the option increases. But as the credit risk keeps increasing and approaches the c* barrier, the value of the option decreases because it may get knocked-out. Thus, the delta of a reverse knock-out is first positive, and then it becomes negative. In fact, as its expiration time approaches this delta pattern becomes more pronounced, and close to expiration the delta can sharply turn from positive to negative for the option holder. Of course, the opposite is true for the option writer. Given such difficulties in hedging reverse knock-out options, some market makers are often forced to treat reverse knock-out option books the way insurers treat their insurance portfolios. They consider them as unhedgeable and use the principle of diversification to reduce the risk.

Second, a reverse knock out on a credit spread ct could be even more difficult to hedge because constructing the hedge may require the buying and selling of a complex portfolio of default swaps. This is difficult for vanilla instruments and is likely to be even more difficult for credit spread knock-outs. Default swaps are expensive and the market may not be very liquid. Also, the cap that we use in this paper will require not only default swaps, but preferably, forward markets in these instruments. Forward markets in default swaps, even if they exist, are definitely not liquid yet.

Hence, the bank may have no recourse but to treat a portfolio of CCL’s as an unhedgeable portfolio of risks and manage the risk through diversification. This is essentially how banks manage their loan portfolios and CCL’s are no different in this sense.

Another issue is that although a CCL can be designed to reduce credit risk, it is also meant to be a temporary solution to a market risk faced by borrowers in the CP market. If the CCL contains ratings provisions (where the knock-out depends on a downgrading of the borrower) and other covenants, then it cannot be considered a pure credit instrument. Even on the credit risk aspect, ratings may not correctly represent the true credit risk associated with the borrower and the bank may have to hedge its exposure with a credit default swap.

As mentioned earlier, a large proportion of bank lending is done through CCLs. Yet, hedging CCLs and altering such loan portfolios is difficult through the secondary market. This stems from two factors. First, due to the nature of CCLs and the option to prepay the loans drawn, funding obligations are continual. Second, the secondary market is illiquid due to the fact that CCLs cannot be sold without the consent of the borrower. Often the borrowers are large corporations that have a good relationship with the bank and the contractual arrangements may prevent the sale of the drawn portion of the loan in the secondary market.

Given the non-existence of a secondary market, an alternative for hedging CCLs could be provided by the credit default swap (CDS) market. The problem in this market has been the cost of these instruments. CDS rates are normally higher than the price of the CCL, and would make the fashioning of a hedge too costly under most conditions. Also, the purchase of a CDS on a particular corporation by a bank issuing a CCL to the same corporation would be a negative signal on the underlying credit and may hurt the corporation. Hence, this alternative for hedging is currently not much in use.

## VII. Concluding Remarks

In this paper we have examined the replication and pricing of a CCL facility to back up commercial paper issuance. It is shown that the CCL can be replicated by a cap written on the credit spread of the company, where the underlying caplets are reverse knock-out options. In general, pricing CCLs is difficult compared to pricing a simple option, because an option is exercised in full, or not exercised at all, but cannot be “partly exercised” as in the case of a credit line. Also, at present, as hedging can be costly, banks treat a portfolio of CCLs as being an unhedgeable portfolio of risks, and manage it through diversification.

The CCL structure considered in the paper can be extended in at least two ways. First, one could add a term-out option. Second, for longer-term CCLs one could model explicitly the pre-payment of loans drawn under a credit line.

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Salih Neftci is a Professor of Finance in the Graduate College of CUNY, New York; Elena Loukoianova is an Economist in the IMF’s African Department, and Sunil Sharma is the Director of the IMF-Singapore Regional Training Institute in Singapore. The authors would like to thank Ralph Chami, Connel Fullenkamp, Miguel Messmacher, Andre Santos and Felix Vardy for discussions and Yinqiu Lu for superb research assistance. The usual disclaimer applies.

See the recent survey by Ergungor (2002). An early exception is Thakor, Hong and Greebaum (1981) who utilize an option pricing approach to obtain the value of loan commitments and assess the sensitivity of these values to changes in interest rates.

Few credit lines, however, carry all three types of fees. Most of them usually have two types of fees: a usage fee combined with an upfront or annual fee.

See, for example, Shockley and Thakor (1997).

It should be pointed out that large, highly visible clients do not benefit much from such signals, because they are also monitored by rating agencies, the financial press, and bank analysts. Smaller and less visible clients benefit much more from the signals sent to the market by the approval of credit lines. The same is true for banks of different size. Large clients tend to use big highly reputable banks for large credit lines, since big banks are able to signal more reliable information to the market. Large lines of credit are usually underwritten by a syndicate of lenders, with a big financial institution being the lead manager.

See IFR November 2001.

In some cases, the contract may specify that the credit line can be opened only under special conditions or for special purposes.

Reliable empirical data on what proportion of the amount committed under CCLs is actually drawn is not available. However, casual evidence suggests that in a substantial majority of cases CCLs are not drawn at all.

For a discussion on partial takedown of CCLs see Thakor, Hong and Greenbaum (1981).

In this paper we use the terms cap and caplet to discuss the structure of the CCL contract. In fact, these are also spread options and we could formulate CCLs as a basket of spread options.

When a firm taps its CCL, the bank acquires a loan at a pre-determined spread ${s}_{{t}_{0}}$ on its balance sheet. The bank’s loan credit portfolio is affected and this raises questions regarding the management of CCL portfolios that are beyond the scope of the current paper.

Thus, in this characterization there is no pre-payment that takes place.

Normally, LIBOR is the funding cost for a AA-rated firm. But the LIBOR market model makes the assumption of zero credit risk.

See, Schönbucher (2000) and (2004).

See, for example, Rebonato (2002).

Note that the risky forward rate dynamics does not include a separate jump component to account for default by the underlying credit. This is a convenient approximation that can be justified in our set up because: (i) back stop facilities for highly rated clients have very small default probabilities; (ii) as in Calomiris (1989), CCLs are viewed as liquidity enhancers and not as tools for default protection; (iii) in very short periods of time the probability of credit deterioration from AAA to full default is likely to be very small; (iv) the existence of a MAC clause limits the credit exposure, and in the case of a big jump in credit spreads the option knocks out.

The spot martingale measure is also the t1-forward measure t0. See Musiela and Rutkowski (1998) for further details.

The discount factor for the CCL payoff is obtained under the assumption that the yield curve is flat. Since CCLs are generally short-term facilities, or are “reset” frequently, this is a reasonable assumption.

Note that here we are modeling the risk-free forward rate drift instead of calculating it explicitly under the risky bond normalization. However, this drift is then calibrated to arbitrage-free bond prices.

Pricing and Hedging of Contingent Credit Lines
Author: Ms. Elena Loukoianova, Salih N. Neftci, and Mr. Sunil Sharma