## _wp0613 - References ## Source details **Canonical URL:** [_wp0613 - References](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp0613.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp0613.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp0613.pdf.json) --- ### I. Introduction - Contingent credit lines (CCLs) are widely used by banks for commercial and industrial lending and play an important role in short-term capital markets. - A U.S. Federal Reserve survey shows that over three-quarters of bank lending is done using commitment contracts; as of June 2005, the outstanding (unused) CCLs of U.S. firms were close to $1.72 trillion, more than double that in 1990. - Pricing and hedging of CCLs has received limited attention; recent advances in financial modeling and new liquid option contracts permit more detailed analysis. - This paper uses a financial engineering approach to examine the structure of simple CCLs and develops a method for their pricing. - Typical credit line contract characteristics: - Specifies a maximum amount (the commitment) that a bank is committed to lend over a given period; the client can draw any amount up to the maximum. - Specifies an interest rate on the drawdown, either fixed or as a spread over a reference rate such as LIBOR. - Specifies fees: an upfront commitment fee, an annual fee on the total committed amount, and a usage fee on the undrawn portion. - Contains an escape clause or material adverse change (MAC) clause allowing the bank to deny credit if the client’s financial condition changes. - Company motivations for using lines of credit: - Backstop facilities to provide flexibility and avoid borrowing during temporary spikes in commercial paper (CP) rates. - Signal of ability to pay for specific transactions, reducing credit risk on short-term borrowing. - Reputation effects: opening a credit line with a reputable bank reduces information asymmetries. - Renewal dynamics: - Renewal of credit lines validates credit-worthiness; termination can signal deteriorating financial health. - Clients have incentives to roll over existing credit lines with the same banks. - Organization of paper: - Section II discusses the market for CCLs. - Section III models a canonical credit line. - Section IV provides a replication strategy. - Section V proposes two pricing methods and uses Monte Carlo simulations to examine parameter effects. - Last section concludes. ### II. Market practice and fees - Short-term capital market financing types for investment grade borrowers: - Syndicated loan market for relatively large issuers. - Commercial paper (CP) market for high grade issuers to finance daily operations. - CCL facilities or CP backstop loans associated with CP markets and other operations (focus of paper). - Market size and fee levels: - In 2001, total volume of CP backstop facilities in the United States was around $290 billion. - Participation fees: banks in syndicates typically paid 1 basis point for every $10 million of the loan they underwrite. - Undrawn or facility fees: during 2001 facility fees were in the 8-13 basis point range depending on borrower creditworthiness. - Drawn fees (all-in drawn fee over LIBOR): - During 2001 this fee averaged 73 basis points and in 2002 the average was 53 basis points. - 1-year LIBOR averaged 3.8 percent in 2001 and 2.2 percent in 2002. - Term-out fees and maturity choices: - Before 2000 most CCLs were multi-year (up to five years); after corporate scandals, 60 percent of CCLs were marketed as one year (364 day) facilities during 2002-2003. - Term-out fees were 12.5 basis points in 2001 and increased to 25 basis points in 2002. - Some CCLs embed currency options allowing draws in more than one currency; such features are more complex but not necessarily harder to price (ignored in this paper). - Syndication and regulatory capital treatment: - Large CCLs are sold through syndications consisting of banks with close borrower relationships. - From the banks’ perspective, as long as the facility is not drawn, a commitment with original maturity of less than one year carries a zero risk-capital weight. - For commitments with original maturity greater than one year, banks are required to hold half the capital required for regular loans. ### III. Canonical CCL model and mechanics - Objective: transfer the risk associated with a low probability event A that may negatively affect a firm’s funding opportunities (e.g., temporary closing of the CP market or a temporary spike in CP rates). - Canonical contract features: - Contract written at time t0 for n periods with maturity T = tn. - Maximum loan size $N; model assumes full line is drawn when used. - Unused portion subject to a facility fee denoted by u_t0, interpreted here as the CCL premium paid up front at contract initiation. - Drawn portion subject to a drawn-fee s_t0 expressed as a spread over LIBOR; fees are fixed at negotiation. - Bermudan-type access: CCL can be tapped only at predetermined times t1, t2, ..., t_{n-1}. - When drawn at t_i, borrower pays all-in interest r_{t_i} = L_{t_i} + s_t0. - Borrower may prepay drawn amount if market credit spread falls below s_t0. - Modeling simplifications and notes: - Assumes full drawdown when facility is used (partial takedowns can be accommodated). - Bermudan access dates simplify modeling and replication when t_i − t_{i-1} is small. - Participation and term-out fees are ignored. - Casual evidence suggests a substantial majority of CCLs are not drawn at all. ### IV. Event B (downgrade) and CCL trigger mechanics - Event B (downgrade) voids the CCL if it occurs at some time t with 0 < t < T. - Trigger condition for downgrade: c_t > c. - Feasibility requirement: c > s_t0. - All-in interest for short-term borrowing at time t_i: - r_{t_i} = L_{t_i} + c_{t_i} - Borrower draws on CCL at t_i only if market credit spread exceeds all-in drawn fee: - c_{t_i} > s_t0 for t0 < t_i - Loan is repaid when market spread c_{t_i} goes below s_t0. - If c_{t_i} exceeds c the borrower is downgraded and the CCL becomes void. - CCLs function as liquidity and credit enhancement; draws are temporary and not intended to result in large permanent bank lending. ### V. Replicating portfolio and valuation structure - Replication instruments: - A cap (collection of caplets) written on the credit spread c_t with cap rate s_t0; caplet payoff at t_{i+1}: - Max[N (c_{t_i} s_t0); 0] - A reverse knock-out (barrier) option that knocks out upon the credit event {c_t > c for t0 ≤ t ≤ T}. - Replicating portfolio = cap on c_t + barrier option that knocks out the cap if {c_t > c} occurs during life of the contract. - Caplet fair market value at time t (when not knocked out): - (cpl)_i^t = B(t; t_{i+1}) E_t[ N  Max(c_{t_i} s_t0; 0) 1_{c_t < c; t0 ≤ t ≤ t_i} ] - Cap value at time t: (cap)_t = sum_{i=1}^n (cpl)_i^t - Heuristic exercise/rollover: - If CCL tapped at t_i, corporation pays s_t0 over L_{t_i} for period t_{i+1} − t_i = . - At t_{i+1} firm chooses to repay CCL and roll or borrow in CP market if c_{t_{i+1}} < s_t0. - CCL viewed as a sequence of options on floating-rate loan contracts struck at spread s_t0, each exercised if c_{t_i} > s_t0 and knocked out if c_{t_i} > c. ### VI. Pricing methods and model dynamics - Corporate credit spread (forward) defined as: - c(t; t_i; t_{i+1}) = f(t; t_i; t_{i+1}) F(t; t_i; t_{i+1}) - Method 1 (Schönbucher extension of forward-LIBOR model): - Forward LIBOR under t_{i+1}-forward measure: F(t; t_i; t_{i+1}) = E_{P_{t_{i+1}}}[ L_{t_i} ] - Under t_2-forward measure (notation F_t and f_t): - dF_t = F_t _F_t d! (no drift) - df_t = μ(f_t; (t)) dt + f_t _f_t d!(t) (unknown drift under t_2-forward) - Using t_2-survival measure (~P_{t_2}) makes f_t a martingale: - df_t = f_t _f_t d!(t) under ~P_{t_2} - Change-of-measure introduces known drift terms; Wiener processes connected via recursively defined _t^2(t) and _D^t_2(t). - Dynamics under t_2-survival measure summarized (as given in text): - dF_t = F_t _F_t d!(t) _D^t_2(t) dt - df_t = f_t _f_t d!(t) - dc(t) = F_t _F_t _D^t_2(t) dt + [ f_t _f_t F_t _F_t ] d!(t) - dH(t) = [ F_{t1} + F_t (1 + H(t)) _D^t_2(t) H(t) _H_t ] dt + H(t) _H_t d!(t) - Single-caplet CCL price expressed as discounted expectation under survival measure: - (CCL)_{t_0} = B(t_0; t_2) N  E[ Max(c_{t1} s_t0; 0) 1_{c_t < c; t0 ≤ t ≤ t1} ] - Method 2 (risky bond normalization / forward measure ~P_{t_{i+1}}): - Real-world dynamics: - dF_t = a(F_t; t) dt + _F_t F_t d!^{(1)}_t - df_t = b(F_t; f_t; t) dt + _f_t f_t d!^{(2)}_t - dc_t = df_t dF_t - Under B(t; t_2) normalization, f_t is a martingale without drift: - df_t = _f f_t d~!^{(2)}_t - Risk-free forward rate F_t assumed mean-reverting: - dF_t = ( F_t) dt + _F F_t d!^{(1)}_t - Drift calibration: introduce constant _t, calibrated by matching Black's closed-form caplet price with simulated caplet price. - Simulated dynamics after calibration: - df_t = _f f_t [  dW^{(1)}_t + sqrt(1 ^2) dW^{(2)}_t ] - dF_t = [ ( F_t) +  _F F_t ] dt + _F F_t dW^{(1)}_t - Instantaneous correlation: d~!^{(1)}_t d~!^{(2)}_t =  dt ### VII. Simulation setup and key parameter specifications - Notional sum for simulations: $1 - Common parameter values used in Method 1 simulations: - _F = 15%; t_1 = 1; t_2 = 2 - Initial values: F_{t0} = 6:7%, f_{t0} = 8:3%; c_{t0} = 1:6% - Example cap/knock-out parameter: c = 0:055 (illustrated in Figure 3) - Common parameter values used in Method 2 simulations: - _F = _cpl = 15%; = 0:05;  = 6:5%; t_1 = 1; t_2 = 2 - Initial values: F_{t0} = 6:7%, f_{t0} = 8:3%; c_{t0} = 1:6% - Correlation parameter:  (varied in Figures 6ñ12) - Caplet volatility input: _cpl - Caplet pricing formulas referenced: Black's caplet price (at-the-money) using P(t_0; t_2), N, , F_{t0}, and _cpl. - Simulation methodology: - Euler discretization for forward-rate SDEs. - Monte Carlo trajectories for c(t) to compute discounted expected payoff of replicating portfolio. - Units of reported prices: basis points. ### VIII. Simulation findings and sensitivities - General relationships: - CCL payoff/price increases with higher probability that c_t > s_t0. - Higher c_t also raises probability of breaching knock-out barrier c, which dampens CCL payoff. - Method 1 (Figures 3–5) qualitative findings: - CCL payoff is a decreasing function of s_t0 for fixed _f and c. - For fixed s_t0, CCL price initially rises with volatility _f and then declines as high _f increases knock-out probability. - The _f corresponding to peak CCL price increases with c. - Higher c (knock-out threshold) increases CCL price (lower knock-out probability). - Example parameter points shown in figures: - c = 0:055 (Figure 3) - s_t0 = 0:025 (Figure 4) - _f = 0:025 (Figure 5) - Method 2 (Figures 6–12) qualitative findings: - Graphs similar to Method 1 but shifted up by a few basis points. - Sensitivity to correlation : - For fixed  and increasing _f, CCL price first increases then decreases (knock-out effect). - Higher  tends to reduce volatility of c_t and thus generally lowers CCL price, but for high _f higher  can attenuate knock-out probability and potentially make CCL price positively related to . - Relationship between CCL price and  depends on relative sizes of c and s_t0: - If c is large relative to s_t0 (e.g., c = 0:08; s_t0 = 0:025) relationship between  and CCL price is clearly negative. - If c is small relative to s_t0 (e.g., c = 0:04; s_t0 = 0:025) curves are almost flat; small positive slope at high . - Figure 12 illustrates potential large pricing errors if  is misspecified; _f series shown: 0.10, 0.15, 0.20, 0.30, 0.40, 0.50. - Prices in figures reported in basis points. ### IX. Hedging issues and practical constraints - Two major hedging difficulties: 1. Reverse knock-out options are hard to hedge, especially near expiration: delta can change sign (positive to negative) as credit spread approaches barrier; option writer faces rapidly changing hedging needs. 2. Hedging a credit-spread knock-out may require trading complex portfolios of credit default swaps (CDS), including forward default-swap exposures; CDS markets may be illiquid and CDS contracts are relatively expensive. - Practical consequences: - Market makers may treat reverse knock-out option books as unhedgeable and manage via diversification (similar to insurers). - Secondary market limitations: - Funding obligations are continual due to prepayment options on drawn loans. - CCLs cannot be sold without borrower consent; secondary market illiquidity. - Hedging via CDS is costly and may send negative signals about borrower credit (purchase of CDS by the lender could harm the borrower), hence not widely used. - Banks typically manage CCL portfolios as unhedgeable risks and rely on diversification and loan-portfolio management techniques. ### X. Conclusions and extensions - CCLs can be replicated by a cap on the firm's credit spread composed of caplets that are reverse knock-out options. - Pricing CCLs is more complex than pricing standard options because credit lines can be partially exercised and include knock-out features. - Hedging CCLs is costly and difficult; banks often treat CCL portfolios as unhedgeable and rely on diversification. - Possible extensions to the CCL structure: - Add a term-out option. - Model explicitly the pre-payment of loans drawn under the credit line for longer-term CCLs. *Source: Excerpt from _wp0613 - References (PDF content provided).* ### References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 ### _wp0613 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 ### I. Introduction - Contingent credit lines (CCLs) are widely used by banks for commercial and industrial lending and play an important role in short-term capital markets. - A U.S. Federal Reserve survey 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. - Pricing and hedging of CCLs has received limited attention in the finance literature; recent advances in financial modeling and new liquid option contracts permit more detailed analysis. - This paper uses a financial engineering approach to examine the structure of simple CCLs and develops a method for their pricing. - Typical credit line contract characteristics: - Specifies a maximum amount (the commitment) that a bank is committed to lend over a given period; the client can draw any amount up to the maximum. - Specifies an interest rate on the drawdown, either fixed or as a spread over a reference rate such as LIBOR. - Specifies fees: an upfront commitment fee, an annual fee on the total committed amount, and a usage fee on the undrawn portion. - Contains an escape clause or material adverse change (MAC) clause allowing the bank to deny credit if the client’s financial condition changes. - Company motivations for using lines of credit: - Backstop facilities to provide flexibility and avoid borrowing during temporary spikes in commercial paper (CP) rates. - Signal of ability to pay for specific transactions, reducing credit risk on short-term borrowing. - Reputation effects: opening a credit line with a reputable bank reduces information asymmetries. - Renewal dynamics: - Renewal of credit lines validates credit-worthiness; termination can signal deteriorating financial health. - Clients have incentives to roll over existing credit lines with the same banks. - Relevant literature and precedents: - Mozebach (1999): banks often have considerable access to corporate information; granting a line signals firm viability and produces a positive market reaction on the protection buyer’s stock. - Calomiris (1989): bank loan commitments backing commercial paper provide insurance against systemic liquidity risk; CP back-stops function as a source of funds during extreme CP market volatility. - Early option-pricing approach to loan commitments: Thakor, Hong and Greenbaum (1981). - Paper organization: - Section II discusses the market for CCLs. - Section III models a canonical credit line. - Section IV provides a replication strategy. - Section V proposes two pricing methods and uses Monte Carlo simulations to examine parameter effects. - Last section concludes. ### II. Market Practice - Short-term capital market financing types for investment grade borrowers: - Syndicated loan market for relatively large issuers. - Commercial paper (CP) market for high grade issuers to finance daily operations. - CCL facilities or CP backstop loans associated with CP markets and other operations (focus of paper). - Market size and fees: - In 2001, total volume of CP backstop facilities in the United States was around $290 billion. - Participation fees: banks in syndicates typically paid 1 basis point for every $10 million of the loan they underwrite. - Undrawn or facility fees: during 2001 facility fees were in the 8-13 basis point range depending on borrower creditworthiness. - Drawn fees (major component): expressed as an all-in drawn fee (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 points. - 1-year LIBOR averaged 3.8 percent in 2001 and 2.2 percent in 2002. - Term-out fees: before 2000 most CCLs were multi-year (up to five years); after corporate scandals, 60 percent of CCLs were marketed as one year (364 day) facilities during 2002-2003. - Term-out fees were 12.5 basis points in 2001 and increased to 25 basis points in 2002. - Some CCLs embed currency options allowing draws in more than one currency; these are more complex but not necessarily harder to price (ignored in this paper). - Syndication and regulatory capital treatment: - Large CCLs are sold through syndications consisting of banks with close borrower relationships. - From the banks’ perspective, 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 preference for maturities 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 - Objective: transfer the risk associated with a low probability event A that may negatively affect a firm’s funding opportunities (e.g., temporary closing of the CP market or a temporary spike in CP rates). - CCL contract features (canonical model): - Contract written at time t0 for n periods (months or years) with maturity T = tn. - Maximum loan size $N; borrower can draw any amount up to $N. The model assumes the full line is drawn when used to simplify replication. - Unused portion subject to a facility fee denoted by u_t0, interpreted here as the CCL premium paid up front at contract initiation (different from market practice where u_t0 is paid as an annual fee over the contract). - Drawn portion subject to a drawn-fee s_t0 expressed as a spread over LIBOR; fees have subscript t0 as they are decided at negotiation and remain constant. - Contract is Bermudan-type: the CCL can be tapped only at predetermined times t1, t2, ..., t_{n-1} (simplification; in practice access may be at any time). - When the credit line is drawn at time t_i (i = 1,2,..., n-1), the borrower pays the annual all-in interest cost r_{t_i} = L_{t_i} + s_t0. - The borrower has the option to prepay the amount drawn; if the firm’s credit spread in the CP market falls below s_t0, the firm will choose to repay the bank. - Modeling simplifications and notes: - Assumes full drawdown when facility is used (can be relaxed for partial takedowns). - Bermudan access dates simplify modeling and replication when t_i - t_{i-1} is small. - Participation and term-out fees are ignored for simplicity. - Reliable empirical data on proportion of amount committed under CCLs actually drawn is not available; casual evidence suggests a substantial majority of CCLs are not drawn at all. *Source: Excerpt from _wp0613 - References . . . . . . . . . . . . . . . . . . . . . . . . . . 24* ### 7. If an eventBoccurs at some datet,t ### _wp0613 - 7. If an eventBoccurs at some datet,t ### CCL structure and mechanics - Event B (downgrade) voids the CCL if it occurs at some time t with 0 < t < T. - Trigger condition for downgrade: c_t > c. - Feasibility requirement: c > s_t0. - All-in interest for short-term borrowing at time t_i: - r_ti = L_ti + c_ti - Firm draws on CCL at t_i only if market credit spread exceeds all-in drawn fee: - c_ti > s_t0 for t0 < t_i - Loan is repaid when market spread c_ti goes below s_t0, ensuring the temporary nature of CCL draws. - If c_ti exceeds c the borrower is downgraded and the CCL becomes void. - CCLs function as liquidity and credit enhancement; draws are meant to be temporary and the facility is not intended to result in large permanent bank lending. ### Replicating portfolio for a CCL - Replication instruments: - A cap (collection of caplets) written on the credit spread c_t with cap rate s_t0; caplet payoff at t_i+1: - Max[N (c_ti s_t0); 0] - A reverse knock-out (barrier) option that knocks out upon the credit event: - c_t > c for t0 ≤ t ≤ T - Replicating portfolio = cap on c_t + barrier option that knocks out the cap if {c_t > c} occurs during life of the contract. - Caplet fair market value at time t (when not knocked out) includes indicator that borrower is not downgraded before caplet expiry: - (cpl)_i^t = B(t; t_i+1) E_t[ N  Max(c_ti s_t0; 0) 1_{c_t < c; t0 ≤ t ≤ t_i} ] - Cap value at time t: (cap)_t = sum_{i=1}^n (cpl)_i^t - Heuristic exercise/rollover description: - If CCL tapped at t_i, corporation pays s_t0 over L_ti for period t_{i+1} − t_i = . - At t_{i+1} firm chooses to repay CCL and roll or borrow in CP market if c_{t_{i+1}} < s_t0. - CCL can be seen as a sequence of options on floating-rate loan contracts struck at spread s_t0, each option exercised if c_ti > s_t0 and knocked out if c_ti > c. ### Pricing: modeling dynamics and valuation - Corporate credit spread (forward) defined as: - c(t; t_i; t_{i+1}) = f(t; t_i; t_{i+1}) F(t; t_i; t_{i+1}) - Two modeling methods presented. - Method 1 (Schönbucher extension of forward-LIBOR model): - Forward LIBOR under t_{i+1}-forward measure: F(t; t_i; t_{i+1}) = E_{P_{t_{i+1}}}[ L_{t_i} ] - Under t_2-forward measure (notation simplification F_t and f_t): - dF_t = F_t _F_t d! (no drift) - df_t = μ(f_t; (t)) dt + f_t _f_t d!(t) (unknown drift under t_2-forward) - Using t_2-survival measure (~P_{t_2}) makes f_t a martingale: - df_t = f_t _f_t d!(t) under ~P_{t_2} - Change-of-measure relationships introduce known drift terms; Wiener processes connected via recursively defined _t^2(t) and _D^t_2(t) (see text). - Dynamics under single measure (t_2-survival measure) summarized: - dF_t = F_t _F_t d!(t) _D^t_2(t) dt - df_t = f_t _f_t d!(t) - dc(t) = F_t _F_t _D^t_2(t) dt + [ f_t _f_t F_t _F_t ] d!(t) - dH(t) = [ F_{t1} + F_t (1 + H(t)) _D^t_2(t) H(t) _H_t ] dt + H(t) _H_t d!(t) - CCL price (single-caplet case) expressed as discounted expectation under survival measure: - (CCL)_{t_0} = B(t_0; t_2) N  E[ Max(c_{t1} s_t0; 0) 1_{c_t < c; t0 ≤ t ≤ t1} ] - Method 2 (risky bond normalization / forward measure ~P_{t_{i+1}}): - Real-world dynamics specified: - dF_t = a(F_t; t) dt + _F_t F_t d!^{(1)}_t - df_t = b(F_t; f_t; t) dt + _f_t f_t d!^{(2)}_t - dc_t = df_t dF_t - Under B(t; t_2) normalization, f_t is a martingale without drift: - df_t = _f f_t d~!^{(2)}_t - Risk-free forward rate F_t assumed mean-reverting: - dF_t = ( F_t) dt + _F F_t d!^{(1)}_t - Drift calibration: introduce constant _t, calibrated by matching Black's closed-form caplet price with simulated caplet price. - Simulated dynamics after calibration: - df_t = _f f_t [  dW^{(1)}_t + sqrt(1 ^2) dW^{(2)}_t ] - dF_t = [ ( F_t) +  _F F_t ] dt + _F F_t dW^{(1)}_t - Instantaneous correlation between driver Wiener processes: d~!^{(1)}_t d~!^{(2)}_t =  dt ### Simulation setup and key parameter specifications (as used in the paper) - Notional sum for simulations: $1 - Common parameter values used in Method 1 simulations: - _F = 15%; t_1 = 1; t_2 = 2 - Initial values: F_t0 = 6:7%, f_t0 = 8:3%; c_t0 = 1:6% - Example cap/knock-out parameter: c = 0:055 (illustrated in Figure 3) - Common parameter values used in Method 2 simulations: - _F = _cpl = 15%; = 0:05;  = 6:5%; t_1 = 1; t_2 = 2 - Initial values: F_t0 = 6:7%, f_t0 = 8:3%; c_t0 = 1:6% - Correlation parameter:  (varied in Figures 6ñ12) - Caplet volatility input: _cpl - Caplet pricing formulas referenced: - Black's caplet price (at-the-money) using P(t_0; t_2), N, , F_t0, and _cpl (see text for explicit formula). - Simulation methodology: - Euler discretization for forward-rate SDEs. - Monte Carlo trajectories for c(t) to compute discounted expected payoff of replicating portfolio. ### Simulation findings and sensitivity results (qualitative and numeric parameter points preserved) - General relationships: - CCL payoff/price increases with higher probability that c_t > s_t0. - Higher c_t also raises probability of breaching knock-out barrier c, which dampens CCL payoff. - From Method 1 (Figures 3–5): - CCL payoff is a decreasing function of s_t0 for fixed _f and c. - For fixed s_t0, CCL price initially rises with volatility _f and then declines as high _f increases knock-out probability. - The _f corresponding to peak CCL price increases with c. - Higher c (knock-out threshold) increases CCL price (lower knock-out probability). - Example parameter points shown in figures: - c = 0:055 (Figure 3) - s_t0 = 0:025 (Figure 4) - _f = 0:025 (Figure 5) - From Method 2 (Figures 6–12): - Graphs similar to Method 1 but shifted up by a few basis points. - Sensitivity to correlation : - For fixed  and increasing _f, CCL price first increases then decreases (knock-out effect). - Higher  tends to reduce volatility of c_t and thus generally lowers CCL price (negative relationship), but: - For high _f, higher  can attenuate knock-out probability and potentially make CCL price positively related to . - Relationship between CCL price and  depends on relative sizes of c and s_t0: - If c is large relative to s_t0 (e.g., c = 0:08; s_t0 = 0:025) relationship between  and CCL price is clearly negative. - If c is small relative to s_t0 (e.g., c = 0:04; s_t0 = 0:025) curves are almost flat; small positive slope at high . - Figure 12 illustrates potential large pricing errors if  is misspecified; plotted CCL price vs  for multiple _f values: - _f series shown: 0.10, 0.15, 0.20, 0.30, 0.40, 0.50 (as labeled in figure caption) - Units of reported prices: basis points (figures display CCL price in basis points). ### Hedging issues and practical constraints - Two major hedging difficulties: 1. Reverse knock-out options are hard to hedge, especially near expiration: delta can change sign (positive to negative) as credit spread approaches barrier; option writer faces rapidly changing hedging needs. 2. Hedging a credit-spread knock-out may require trading complex portfolios of credit default swaps (CDS), including forward default-swap exposures; CDS markets may be illiquid and CDS contracts are relatively expensive. - Practical consequences: - Market makers may treat reverse knock-out option books as unhedgeable and manage via diversification (similar to insurers). - Secondary market limitations: - Funding obligations are continual due to prepayment options on drawn loans. - CCLs cannot be sold without borrower consent; secondary market illiquidity. - Hedging via CDS is costly and may send negative signals about borrower credit (purchase of CDS by the lender could harm the borrower), hence not widely used. - Banks typically manage CCL portfolios as unhedgeable risks and rely on diversification and loan-portfolio management techniques. ### Conclusions and extensions - CCLs can be replicated by a cap on the firm's credit spread composed of caplets that are reverse knock-out options. - Pricing CCLs is more complex than pricing standard options because credit lines can be partially exercised and include knock-out features. - Hedging CCLs is costly and difficult; banks often treat CCL portfolios as unhedgeable and rely on diversification. - Possible extensions to the CCL structure: - Add a term-out option. - Model explicitly the pre-payment of loans drawn under the credit line for longer-term CCLs. *Source: _wp0613 - 7. If an eventBoccurs at some datet,t (PDF content provided).* ### References ### _wp0613 - References ### Bibliographic entries - Black, Fischer, 1976, ìThe Pricing of Commodity Contracts," Journal of Financial Eco- nomics, Vol. 3 (March), pp. 167-79. - Brigo, Damiano and Fabio Mercurio, 2001,Interest Rate Models: Theory and Practice, (Berlin: Springer). - Calomiris, Charles W, 1989, ìThe Motivations for Loan Commitments Backing Commer- cial Paper,îJournal of Banking and Finance, 13, 1989, pp. 271-277. - Ergungor, O. 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