The earliest credit model that employed the option pricing theory can be credited to BSM. Black-Scholes, in the last section of their seminal option pricing paper, explicitly articulated that corporate liabilities can be viewed as a covered call: own the asset but short a call option. In the simplest set- ting where the company has only one zero-coupon debt, at the maturity of the debt, the debt holder either gets paid the face value of the debt—in such a case, the ownership of the company is transferred to the equity holder—
or takes control of the company—in such a case, the equity holder receives nothing. The debt holder of the company therefore is subject to default risk for he or she may not be able to receive the face value of his or her invest- ment. BSM effectively turned a risky debt evaluation into a covered call evaluation whereby the option pricing formulas can readily apply.
In BSM, the company balance sheet consists of issued equity with a market value at time tequal to E(t). On the liability side is debt with a face value of Kissued in the form of a zero-coupon bond which matures at time T. The market value of this debt at time tis denoted by D(t,T).
The value of the assets of the firm at time t is given by A(t).
At time T(the maturity of the debt), the market value of the issued equity of the company is the amount remaining after the debts have been paid out of the firm’s assets; that is,
This payoff is identical to that of a call option on the value of the firm’s assets struck at the face value of the debt. The payoff is graphed as a function of the asset value in Exhibit 8.1. The holders of the risky cor- porate debt get paid either the face value, K, under no default or take over the firm, A, under default. Hence the value of the debt on the maturity date is given by
E T( ) = max{A T( )–K,0}
(8.1)
(8.2)
The equations provide two interpretations. Equation (8.1) decom- poses the risky debt into the asset and a short call. This interpretation was first given by Black and Scholes that equity owners essentially own a call option of the company. If the company performs well, then the equity owners should call the company; or otherwise, the equity owners let the debt owners own the company. Equation (8.2) decomposes the risky debt into a risk-free debt and a short put. This interpretation explains the default risk of the corporate debt. The issuer (equity own- ers) can put the company back to the debt owner when the performance is bad.7 The default risk hence is the put option. These relationships are shown in Exhibit 8.1. Exhibits 8.1(a) and 8.1(b) explain the relationship between equity and risky debt and Exhibits 8.1(b) and 8.1(c) explain the relationship between risky and risk-free debts.
Note that the value of the equity and debt when added together must equal the assets of the firm at all times, i.e., A(t) = E(t) + D(t,T). Clearly, at maturity, this is true as we have
as required.
7Acovered callis a combination of a selling call option and owning the same face value of the shares, which might have to be delivered should the option expire in the money. If the option expires in the money, a net profit equal to the strike is made. If the option expires worthless, then the position is worth the stock price.
EXHIBIT 8.1 Payoff Diagrams at Maturity for Equity, Risky Debt, and Risk-Free Debt
D T T( , ) = min{A T( ),K}
A T( )–max{A T( )–K,0}
=
K–max{K A T– ( ),0}
=
E T( )+D T T( , ) = max{A T( )–K,0}+min{A T( ),K} A T( )
=
184 CREDIT DERIVATIVES: INSTRUMENTS, APPLICATIONS, AND PRICING
Since any corporate debt is a contingent claim on the firm’s future asset value at the time the debt matures, this is what we must model in order to capture the default. BSM assumed that the dynamics of the asset value follow a lognormal stochastic process of the form
(8.3)
whereris the instantaneous risk-free rate which is assumed constant, σ is the percentage volatility, and W(t) is the Wiener process under the risk neutral measure.8 This is the same process as is generally assumed within equity markets for the evolution of stock prices and has the property that the asset value of the firm can never go negative and that the random changes in the asset value increase proportionally with the asset value itself. As it is the same assumption as used by Black-Scholes for pricing equity options, it is possible to use the option pricing equa- tions developed by BSM to price risky corporate liabilities.
The company can default only at the maturity time of the debt when the payment of the debt (face value) is made. At maturity, if the asset value lies above the face value, there is no default, else the company is in bankruptcy and the recovery value of the debt is the asset value of the firm. While we shall discuss more complex cases later, for this simple one-period case, the probability of default at maturity is
(8.4)
where φ(⋅) represents the log normal density function, N(⋅) represents the cumulative normal probability, and
Equation (8.4) implies that the risk neutral probability of in the moneyN(d2) is also the survival probability. To find the current value of
8The discussions of the risk neutral measure and the change of measure using the Girsonav theorem can be found in standard finance texts. See, for example, Darrell Duffie,Dynamic Asset Pricing(Princeton, NJ: Princeton University Press, 2000), and John Hull, Options, Futures, and Other Derivatives (New York: Prentice Hall, 2002).
dA t( ) A t( )
--- = rdt+σdW t( )
p φ[A T( )]dA T( )
∞ –
K
∫
1–N d( )2= =
d2 lnA t( )–lnK+(r–σ2⁄2)(T t– ) σ T t–
---
=
the debt, D(t,T) (maturing at time T), we need to first use the BSM result to find the current value of the equity. As shown above, this is equal to the value of a call option:
(8.5) where . The current value of the debt is a covered call value:
(8.6)
Note that the second term in the last equation is the present value of probability-weighted face value of the debt. It means that if default does not occur (with probability N(d2)), the debt owner receives the face value K. Since the probability is risk neutral, the probability-weighted value is discounted by the risk-free rate. The first term represents the recovery value. The two values together make up the value of debt.
The yield of the debt is calculated by solving D(t,T) = Ke–y(T–t)for y to give
(8.7)
Consider the case of a company which currently has net assets worth $140 million and has issued $100 million in debt in the form of a zero-coupon bond which matures in one year. By looking at the equity markets, we estimate that the volatility of the asset value is 30%. The risk-free interest rate is at 5%. We therefore have
Applying equation (8.5), the equity value based upon the above example is
A(t) = $140 million K = $100 million
σ = 30%
T – t = 1 year
r = 5%
E t( ) = A t( )N d( )1 –e–r T t( – )KN d( )2 d1 = d2+σ T t–
D t T( , ) = A t( )–E t( )
A t( )–[A t( )N d( )1 –e–r T t( – )KN d( )2 ]
=
A t( )[1–N d( )1 ]+e–r T t( – )KN d( )2
=
y lnK–lnD t T( , ) T t– ---
=
and market debt value, by equation (8.6) is
Hence, the yield of the debt is, by equation (8.7):
which is higher than the 5% risk-free rate by 170 bps. This “credit spread”
reflects the 1-year default probability from equation (8.4):
and the recovery value of
if default occurs.
From above, we can see that, as the asset value increases, the firm is more likely to remain solvent, the default probability drops. When default is extremely unlikely, the risky debt will be surely paid off at par, the risky debt will become risk free, and yield the risk-free return (5% in our exam- ple). In contrast, when default is extremely likely (default probability approaching 1), the debt holder is almost surely to take over the company, the debt value should be the same as the asset value which approaches 0.
Implications of BSM Model
As we can see from this example, the BSM model captures some important properties of risky debt; namely, the risky yield increases with the debt-to- asset leverage of the firm and its asset value volatility. Using the above
d2 ln140 ln100– +(0.05 0.3– 2)×1 0.3 1
--- 1.4382
= =
d1 = 1.4382 0.30– = 1.1382
E t( ) = 140×N(1.1382)–e–0.05×100×N(1.4382)
$46.48 million
=
D t T( , ) = A t( )–E t( ) = 140 46.48– = $93.52 million
y ln100 ln93.52–
---1 6.70%
= =
p = 1–N(1.4382) = 12.75%
A t( )(1–N d( )1 ) = $17.85
equations, one can also plot the maturity dependency of the credit spread, defined as the difference between the risky yield and the risk-free rate.
What is appealing about this model is that the shapes of the credit spread term structures resemble those observed in the market. The highly leveraged firm has a credit spread which starts high, indicating that if the debt were to mature in the short term, it would almost certainly default with almost no recovery. However as the maturity increases, the likeli- hood of the firm asset value increasing to the point that default does not occur increases and the credit spread falls accordingly. For the medium leveraged firm, the credit spread is small at the short end—there are just sufficient assets to cover the debt repayment. As the maturity increases, there is a rapid increase in credit spread as the likelihood of the assets falling below the debt value rises. For the low-leveraged company, the initial spread is close to zero and so can only increase as the maturity increases and more time is allowed for the asset value to drop. The gen- eral downward trend of these spread curves at the long end due to the fact that on average the asset value grows at the riskless rate and so given enough time, will always grow to cover the fixed debt.
Empirical evidence in favor of these term structure shapes has been reported by Fons who observed similar relationships between spread term structure shapes and credit quality.9Contrary evidence was reported by Helwege and Turner who observed that the term structure of some low- quality firms is upward sloping rather than downward sloping.10