The pricing of credit risk derivatives

Journal

ELSEVIER

of Economic Dynamics 21 (1997) 1579-161

and Control

I

The pricing of credit risk derivatives Yiannos Department

A. Pierides

of Public and Business Administration, Universih, oJ’C\prus. P.O. Box 537. Kullipoleos 75 Cyprus

Received

20 February

1995; final version

received

6 December

Nicosia.

1678

1996

Abstract This paper considers the pricing of derivatives that protect holders of corporate bonds from a reduction in their value because of a deterioration in their credit quality. These derivatives are structured as either puts on the bond price or calls on the bond spread (above the risk free rate) in the context and Cox (1976). The pricing properties and numerical methods.

of models developed by Merton (1974) and Black of these options are derived using both analytical

Keywords: Credit risk derivatives; Barrier options; Numerical methods JEL c/ass(jiccition: C63; G 13

1. Introduction The credit risk derivatives market has grown dramatically in the last two years. Credit risk derivatives are instruments whose payoff is tied to the credit characteristics of a particular asset. A portfolio manager holding some corporate bonds that trade at a spread of 100 basis points above Treasuries could buy a credit risk derivative that would compensate him if the securities start trading at a spread greater than 100 basis points. This paper examines the structuring and pricing of credit risk derivatives that protect the holders of corporate bonds from a deterioration in their value (or equivalently from a rise in the spread above Treasuries at which they trade). A number of articles have already

This paper is based on Chapter 1 of the author’s doctoral dissertation at the MIT Sloan School of Management. The author is indebted to his advisors, Professors John C. Cox and Stewart C. Myers, for many helpful comments on earlier versions of this paper. 0165-1889/97/$17.00 C 1997 Elsevier Science B.V. All rights reserved PII SO1 65- I889(97)00049-3

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YA. Pierides /Journal

of Economic Dynamics

appeared on this topic; accordingly, order to explain how this paper derivatives. 1. I. Literature

and Control 21 (1997) 1579-1611

the next subsection differs from other

describes this work in papers on credit risk

review

The paper by Flesaker et al. (1994) is the first published paper on credit risk derivatives. It presents an approach to pricing credit risk derivatives which is based on the assumption that the actual time of default on a corporate bond always comes as a surprise to investors and corporate bonds of all maturities will be subject to non-trivial default risk. As Flesaker et al. (1994) point out this feature of the models they study contrasts markedly with another school of thought on the default process of corporate bonds represented by Merton (1974) and Black and Cox (1976). Merton (1974) and Black and Cox (1976) model corporate debt by regarding equity as a call option on the assets of the firm. Default is modelled as occurring on the first scheduled payment date on the corporate bond when the firm’s assets are no longer high enough to make it rational for the shareholders to provide the necessary cash to make the payment. Default can occur only at maturity in the case of a zero coupon bond and on a coupon date (or at maturity) in the case of a coupon bond. Since the asset value is presumed to be an observable process, the default can be predicted with increasing precision as the time for payment draws near. As a result, in this approach default does not come as a surprise (unlike Flesaker et al., 1994). Das (1995) was the first to apply this approach to the pricing of credit risk derivatives. His paper deals only with the pricing of options on the bond price, even though credit risk derivatives can also be structured as options on the bond spread. A paper that deals with the pricing of options on the bond spread is Longstaff and Schwartz (1995b). Another difference between the papers of Das (1995) and Longstaff and Schwartz (1995b) is that Das (1995) considers the pricing of options on the price of a bond that was issued by a particular company whereas Longstaff and Schwartz (1995b) consider the pricing of options on the bond spread (above the risk-free rate) of a class of bonds (for example, all bonds rated Baa). Another paper that follows the approach of Longstaff and Schwartz (1995b) is Jarrow and Turnbull (1995). 1.2. The contribution

of this paper

This paper is similar to Das (1995) in that it considers the pricing of derivatives on the price of a bond that was issued by a particular company. It differs from Das (1995) in three ways: First, it considers derivatives on both the bond price and bond spread; in both cases, the underlying bond is presumed to have been issued by a particular company and as a result the paper adopts a different

Y.A. Pierides 1 Journal of Economic Dynamics

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approach from Longstaff and Schwartz (1995b). Second, Das (1995) investigates only pricing issues, whereas in this paper we not only investigate pricing issues but also establish a number of no early exercise results for credit risk derivatives of the American type. Third, the coupon corporate bonds that Das (1995) considers (he defines derivatives on their price) are priced by assuming that the firm that issued the bonds declares bankruptcy when the value of its assets drops below a certain exogenous boundary. This assumption is also used in a recent paper by Longstaff and Schwartz (1995a) on valuing risky debt. In contrast, the default boundary for the coupon bonds in this paper is endogenously determined. Finally. a major difference between this paper and Das (1995) or Longstaff and Schwartz (1995b) is that these two papers assume that the interest rate is stochastic whereas in this paper we assume that the interest rate is constant. The extent to which interest rates affect the prices of corporate bonds is an issue on which the evidence is not clear. For example, Longstaff and Schwartz (1995a) argue that observed credit spread volatilities are not mimicable with constant interest rate models. In contrast, Cornell and Green (1991) show empirically that low grade corporate bonds have much lower sensitivity to interest rates than high grade corporate bonds. They also show that low grade bonds are much more sensitive to the issuing firm’s asset value changes than high grade bonds. In light of this, the model in this paper is more applicable to the pricing of credit risk derivatives on lower grade debt. This paper is organized as follows: Section 2 presents the Merton (1974) model of zero coupon bond pricing. Section 3 shows how to define and price credit risk derivatives in the context of this model. Section 4 presents the Black and Cox (1976) model of coupon bond pricing and Section 5 structures and prices derivatives in the context of this model. Section 6 concludes.

2. Merton’s model of zero-coupon corporate bond pricing The following assumptions were made: 1. r = riskless interest rate is constant; 2. Firm value (V) dynamics: dV = al/ dt + sV dz; a and s are constants and dz is the increment of a standard Wiener process; 3. Firm has outstanding a single issue of debt promising B at Tg. If payment is not made, the bondholders take over the company; f(V, t) = debt value at time r, is the solution to the following p.d.e. and boundary conditions; o.5s2 V’j”” + rvf” - 7-f+ft = 0 .f(O, t) = 0, .f(V,t)l

Vforanyt,

f( V, TB) = min( I/, B)

(1)

Y.A. Pierides 1 Journal of Economic Dynamics

1582

The closed-form f(v,

solution

can be written

t) = B exp( - r(T, + d-’

@(hl(d,

where Q(x) = standard

normal

and Control 21 (1997) 1579-1611

as

- t)) (@ (h2 (4 s2 (Ts - t))) s2(G

(2)

- t)))),

distribution

function

d=V-‘Bexp(-r(TB-t)), hl (d, s2(TB - t)) = - s-l (T, - t)P0.5 (0.5~~ (TB - t) - log d), h2(d, s’(TB - t)) = - s-l (TB - t)-‘.’

(0.5~~ (T, - t) + logd).

The above is the solution for t < TB, withf(V, TB) = min (V (TB), B). Defining e (I/, t) as the spread above the risk-free rate at which the debt trades at t, we can rewrite the solution for t < TB as .fV,Q=Bexp(-(r+e(v,Q)(T,-0) wheree(V,t)= -(T,t)-‘log(@(h2(d,s2(TBNote that e(V, t) is not defined for t = TB.

(3) t)))+dp1@(hl(d,s2(TB-

3. Structuring and pricing credit risk derivatives for zero-coupon

t)))).

bonds

We now consider the structuring and pricing of derivatives which protect holders of zero-coupon bonds from a reduction in their value. One such derivative is a put option on the bond’s price; another is a call option on the spread above the risk-free rate at which the bond trades. We begin with the put option. 3.1. An American put option on the zero-coupon

bond price

Consider the following American put option: At any t -e T, or at t = T, where T = option maturity < TB its owner can sell the bond at a time varying exercise price K(t) = Bexp( - (r + g)(TB - t)), i.e., the exercise price is the price of the bond for a fixed spread g( > 0) above the risk-free rate. To see why this option qualifies as a credit risk derivative, suppose that we know that a AA credit rating implies that the bond is priced at a spread g; furthermore, assume that a portfolio manager bought the bond when it was rated AA. If this portfolio manager also purchases this American put, he is protected from a downgrading: For example, if the bond is downgraded to A, its price will decrease and hence the put will be in the money; by exercising it, the portfolio manager is guaranteed a value equal to the price at which the bond should trade if it had not been downgraded. We first present a no early exercise result for this put.

Y.A. Pierides

1 Journal

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Dynamics

and Control

21 (1997)

1579-16/l

1583

Proposition I. Early exercise ofthe American put on the zero coupon bond price can never be optimal. Hence, its price will be the same as the price of an otherwise identical European put. A formal proof is in Appendix A. Here, we provide some intuition as to the reasoning that leads to Proposition 1. The two factors against early exercise are: (1) the exercise price increases with the passage of time and therefore, the longer we wait to exercise, the higher the exercise price we get; (2) if we wait, we retain the option to change our mind. The one factor in favor of early exercise is that the earlier we exercise the earlier we start earning interest on the exercise price. It can be shown that the first factor against early exercise more than offsetts the factor in favor of early exercise and, as a result, early exercise is never optimal. We now derive the p.d.e. and boundary conditions that this option’s price must satisfy. Let F [f(V, t), t] be the option price at time t. It is assumed that the option price depends on debt value and time. Since debt value depends on the value of the firm, I/, we can also think of the option price as a function of V and t, i.e..

F [.f‘(V, t), tl = MCI/, tl.

(4)

Using the usual no arbitrage argument of Black and Scholes easily show that M(V, t) must obey the following p.d.e.: 0.5M,.,s2 V2 + M, - rM + rM,V = 0. The boundary

conditions

(1973)

we can

(5)

are:

M[IV, Tl = max CO,K(T) -f(V, M[V, t] 2 max CO,K(t) -f(v,

T)], t)],

(6) (7)

(8) aM !,?I, af IS finite.

(9)

Eq. (6) says that at option maturity the option value will be equal to its exercise value if the latter is positive and zero otherwise. Eq. (7) says that the price of an American put cannot fall below its exercise value. Eq. (8) says that when firm value is very low and hence bond value is very low, a $1.00 increase in bond value will reduce option value by $1.00. The reason for this is that under such circumstances, exercise of the option is virtually guaranteed. Eq. (9) says that when firm and bond value are very high and hence the option is way out of the money, a marginal increase in bond value will only have a finite impact on option value.

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Y.A. Pierides /Journal

of Economic Dynamics and Control 21 (1997) 1579-1611

We employ standard numerical techniques (finite difference methods) as in Ames (1992) to obtain a solution to this valuation problem. The details are in Appendix B. 3.2. Pricing results for American put on zero-coupon

bond

The parameter values chosen were: T = 5, TB = 10, I = 0.1 per year, B = 50, g = 0.03, s = 0.2 or 0.4

Fig. 1 gives bond and option prices for different firm values. As expected, the bond price increases with firm value. Also, the bond price for given firm value is higher the lower the volatility (s). This confirms the analytical result of Merton (1974) that higher volatility reduces bond price. The put option price is a decreasing function of firm value. Moreover, for given firm value, the option price is higher the higher the volatility (s). This is the result of two separate effects. The first effect relates to the previously mentioned relation between bond price and volatility. For given firm value, the higher the volatility, the lower the bond price. Since this is a put option, lower bond price increases the option price. The second effect is a direct effect of higher volatility on put prices analogous to the effect of higher volatility on the price of a put option on a common stock. Another feature of Fig. 1 is that the option becomes virtually worthless when firm value is high enough (approximately 60 for the s = 0.2 case) that the bond

20 -f18 -lh--

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12 -*

n

.

.

n

.

A

. *

.

.

.

*

A

A

A

.A

.

>

l

8 --

Bood(s=O.Z) Option(s=O.Z)

. [email protected]) 6 -5

- 0pti0n(!F0.4) cl

-_ 4 --

-

_

2 -,

-

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=a ‘r....

0, 20

30

_ 40

_ 50

60

-

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_,

_ 70

I_ 80

I 90

100

110

YIRM VALUE

Fig. 1. Put on price of zero coupon

bond (T = 5, T, = 10, B = 50, r = 0.1, y = 0.03).

120

Y.A. Pierides I Journal of Economic Dvnamics

becomes

effectively

Bexp(

risk free. The risk-free

and Control 21 (1997) 1579. 1611

bond

1585

price for the s = 0.2 case is

- (r)(TR)) = 18.39.

This is a reasonable result because once the bond becomes risk free it remains risk free until its maturity and, therefore, there is no possibility that the option will be exercised. The computer program that implements the numerical valuation scheme was written in a way that permits us to check for early exercise. It confirmed our earlier no early exercise result (Proposition 1). Various other sensitivity results were established by varying the parameter values used as input. For example, it was confirmed that the option price is lower for a lower maturity and a lower striking price (i.e. higher 9). The influence of interest rates on put option prices for given firm value is ambiguous. The reason for this is that a higher interest rate has two opposing effects on option price: First, a higher interest rate reduces bond prices (Merton, 1974) and hence increases put prices for given firm value; second, a higher interest rate reduces both the exercise price and the present value of the exercise price, thereby reducing put prices. It is not clear which effect will dominate. Numerical analysis confirmed that depending on the parameter values chosen, a higher interest rate may result in either higher or lower option prices for given firm value. 3.3. Another credit risk derivative: bond ‘.sspread e (V, t)

nn

American

cull option on the zero-coupon

In the context of Merton’s model described above, consider an American call option on c(V, t) (the spread above r at time t) with maturity T < TH and exercise price K = LJ.If the option is exercised, the payoff is max(O, e (V, t) ~ q)). To avoid co&sing percentages with monetary units we shall ussume that e( V. t) und g ~lre expressed in numerical form and, hence, the pa\,off’is in monetary units. _ ,. This option qualifies as a credit risk derivative because its value increases with increases in the bond’s spread, i.e., with decreases in the bond’s price. We cannot obtain a no early exercise result for this call option because the option is written on e(V, t) which is not traded. In the earlier case of the put option, we managed to get a no early exercise result because the option was written on the corporate bond - a traded asset; of course, e(V, t) is a function of the bond pricefand we would be able to get a no early exercise result if e( I/. t) were a linear function of ,f’- since it is non-linear. we cannot. This is stated formally below. The proof is in Appendix A. Proposition 2. The price of the American cull on the spreud of’ the zero coupon bond con he equal to its exercise value at unq’ time, in purticulur, even h
1586

Y.A. Pierides /Journal

of Economic Dynamics and Control 21 (1997) I579-1611

Let F (e (V, t), t) be the option price at time t. It is assumed that the option price depends on the value of the spread e(V, t) and time. Since the spread depends on firm value, we can also think of option price as a function of I/ and t, i.e., F(e(V, t), t) = Z(V,

t).

(10)

Using the usual no arbitrage argument, we can show that Z [V, t] must obey the following p.d.e. and boundary conditions: OSZ,” s2 V 2 + z, - rz + rVZ, = 0,

(11)

Z(V, T) = max[e(V,

(12)

T) -g,O],

Z(V, t)kmax[e(V,t)-g,Ol,

(13) (14)

(15) Eq. (12) says that at option maturity, the option value will be equal to its exercise value if the latter is positive and zero otherwise. Eq. (13) says that the price of an American call can never fall below its exercise value. Eq. (14) says that when firm value is very low and hence the spread is very high, a $1.00 increase in the spread will increase option value by $1.00. The reason for this is that under such circumstances, exercise of the option is virtually guaranteed. Eq. (15) says that when firm value is very high and the spread is very low, a marginal increase in the spread will only have a finite impact on option value because the option is way out of the money. 3.4. Pricing results for American call on spread of zero-coupon

bond

Numerical results were obtained for the same parameter values as in Fig. 1, i.e., T = 5, TB = 10, r = 0.1 per year, B = 50, g = 0.03, s = 0.2 or 0.4.

Fig. 2 gives the bond spread and the option price for different firm values. As expected the bond spread decreases with increases in firm value (since the bond price increases with increases in firm value). Also the bond spread for given firm value is higher the higher the volatility (s). This confirms the analytical result of Merton (1974) that higher volatility reduces the bond price and therefore increases the bond spread. As expected, the call option price is a decreasing function of firm value (and an increasing function of bond spread). Moreover, for given firm value the option price is higher the higher the volatility (s). This is the result of two separate

Y.A. Pierides 1 Journal of Economic Dynamics

and Control 21 (1997) 1579-1611

1587

T

0.07

0.06

.

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0.05 t--

s

0.04

a s

0.03

A -

i

I * !

. -

0.02 -- .

* -

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0.01 --

0.00 I 20

*

.

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n

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= l

30

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n

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50

g

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I 70

II 80

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I 90

100

FIRM VALUE

Fig. 2. Call on spread

of zero coupon

bond (T = 5, Te = 10, B = 50, r = 0.1, g = 0.03).

effects. The first effect relates to the previously mentioned relation between bond spread and volatility. For given firm value, the higher the volatility, the higher the bond spread. Since this is a call option, higher bond spread increases the option price. The second effect is a direct effect of higher volatility on call prices analogous to the effect of higher volatility on the price of a call option on a stock. The computer program that implements the numerical scheme was written in a way that permits us to check for early exercise. For the parameter values above, there is no early exercise. However, it was shown that for a different set of parameter values early exercise may be optimal. This confirmed Proposition 2. A distinguishing feature of the results in Fig. 2 is the fact that option price sometimes exceeds the current value of the spread. For example, when s = 0.4, this happens for firm values lower than 40. This is due to the non-linear payoff function of this option. For an American call on a stock with constant exercise price we know from Cox and Rubinstein (1985), that the option price can never exceed the contemporaneous price of the stock. Nevertheless, the arbitrage proof of this result cannot be extended to situations in which the payoff is a non-linear function of the underlying state variable as shown below. This leads to Proposition 3, which is formally proven in the Appendix A. The price of the American call on the spread of the zero coupon Proposition 3. bond may exceed the current value of the spread.

1588

Y.A. Pierides /Journal

of Economic Dynamics

and Control 21 (1997) 1579-161 I

Various other sensitivity results were obtained by varying the parameter values used as input. For example, it was confirmed that the option price is higher the higher the option maturity and the lower the exercise price. To analyze the effect of the interest rate on option prices consider first its effect on the value of e(V, t) for given firm value. The equation linking the two is derived from Merton’s model Eq. (3) as e(V, t) =

&

{log B - logf(l/,

t) - I (Ta - t)}.

An increase in Y has two effects on e: A direct negative’ effect through the last term in the above equation and an indirect positive effect through the negative effect of higher r on the bond price f: These two effects oppose each other and as a result, an increase in Y could either increase or reduce e(V, t) for given firm value I/. As a result, an increase in r may either increase or decrease option prices. 3.5. How can an investor use the call option on the spread It is obvious that this call option qualifies as a credit risk derivative because its value increases with increases in the bond’s spread, i.e., with decreases in the bond”s price. However, it is less obvious, how a portfolio manager holding corporate bonds would use this option. To explore this, recall from the earlier discussion that a portfolio manager purchasing a bond rated AA (which is assumed to trade at a spread above Y of 1%) and a put on the bond with an exercise price K (T) = bond price for a spread above r of g = 1%, is guaranteed at option maturity T a portfolio value equal to MAX (f( I’, T ), K (T )). In other words, the least portfolio value at T will be the bond value for a 1% spread above r but if the actual bond price ends up being higher than this value, the portfolio value will be higher. So, the portfolio manager insures his portfolio from a credit deterioration but benefits from a credit improvement. If instead the portfolio manager purchasing the AA bond also buys y calls on the spread with exercise price g = lo?, the portfolio value at T will be f(v,

T) + Y MAX(eW,

T) - 9,

0).

A portfolio manager could choose y so that the portfolio value at T is also equal to MAX (f( I/, T ), K (T )) since this payoff has a straightforward economic interpretation, The optimal y is therefore given by the equation MAX (f(v,

T), K (T)) =f(v,

T) + y MAX (e (V, T) - g, 0).

Note that irrespective of choice of y, iff( I/, T ) > K (T ) this equation is always satisfied (since in this case (e (I’, T) < g). Hence, we focus on the choice of y if f(k’, T) < K(T). In this case, K(T)

=f(v,

T) + y(e(V,

T) - 9).

Y.A. Pierides /Journal of Economic Dynamics and Control 21 (1997) 3579-1611

This equation

can be rewritten

1589

as

Bexp(-(r+g)(T,-T))=Bexp(-(r+e(V,T))(T,-T)) +Y(cV,T)-g). exp(z) = 1 + z for small z we can show that the

Using the approximation optimal _r is given by J’ = B(TH - T). This discussion

is summarized

in Proposition

4.

Proposition 4. A portfolio manager holding u bond trading at a spread g ooer r and wanting to protect it against a rise in its spread aboor g should either: (1) huq one put on the bond with exercise price equal to the bond price for a spreud g over r or (2) buy y culls on the spread with exercise price g, Mshere y = B ( TH _ T ).

4. Coupon bonds: the Black-Cox

model

The above analysis considered the structuring and pricing of options on zero-coupon bonds. We now consider coupon bonds. We first present a pricing model for coupon bonds. Merton’s model assumed that the firm had outstanding a single issue of debt promising B at T,. Black and Cox (1976), extended Merton’s model to allow for the existence of coupon paying debt. They make the same assumptions as Merton except that they postulate the existence of coupon paying debt promising B at Tn and discrete coupons in the amount c (fixed) at discrete points in time, say, at 0.2T,, 0.4Ts, 0.6T, and 0.8TB. Furthermore, assume that these coupon payments must be financed by issuing new subordinated securities, i.e., no asset sales are allowed. If a coupon payment is not made, the firm is in default and the promised payment of B becomes due immediately. Black and Cox show that the value of the coupon paying debt at time t, denoted,f(V, t) will satisfy the following p.d.e. 0.5s’ v’,k

+

r Vf; -

rf+.ft

+ i

(16)

c6 (t - (0.2)iTB) = 0,

i=l

where 6( ) is the Dirac delta function. f(v,

TB) = min(B,

lim ,f‘= V, 1-O lim

df‘=(),

I”ll

av

v(T&,

The boundary

conditions

are: (17) (18) (19)

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and Control 21 (1997) 1579-1611

Boundary condition Eq. (17) says that at bond maturity, the bond value is equal to the promised amount B if the latter does not exceed firm value and to firm value otherwise. Boundary condition Eq. (18) says that when firm value is very low, bond value equals firm value. Boundary condition Eq. (19) says that when firm value is very high and the bond is effectively risk-free, a marginal increase in firm value will have no impact on bond value. The above-described stopping condition completes the specification of the program. We use standard numerical procedures as in Ames (1992) to obtain a solution to this valuation problem. The details are in Appendix C.

5. Structuring and pricing credit risk derivatives for coupon bonds

It stands to reason that the ex-coupon bond price (i.e. the price of the bond after payment of the coupon) is not defined on a coupon date if the firm goes bankrupt on that coupon date. For this reason, any option contract must be based on the cum-coupon bond price. Let G(V, t) be the cum-coupon bond price at t. We now consider the structuring and pricing of credit risk derivatives which protect holders of coupon bonds from a reduction in their value. Unlike zerocoupon bonds, coupon bonds can default prior to their maturity and as a result any credit risk derivative on a coupon bond will necessarily involve a barrier option

which ceases to exist in the event of bankruptcy. We consider two types of barrier options: A put on the coupon bond’s price and a call on the coupon bond’s spread. We begin with the put. 5.1. An American put option on the cum-coupon bond price (G)

Consider the following American put: At any t < T, where T = option maturity < TB, assuming the firm is not bankrupt, the option’s owner can sell the bond at a time-varying exercise price K(t) equal to the cum-coupon price of the bond for a fixed spread g above the risk-free rate. For example, consider the coupon bond defined earlier: It promised B at TB and coupons in the amount c at 0.2T,, 0.4Ta, 0.6Ta, and 0.8T,. Then: K(0) = c exp( - (r + g)(0.2T,))

+ c exp( - (r + g)(0.4T,))

+ c exp ( - (r + 9) (0.6T,)) + c exp( - (r + g)(O.gT,))

+ Bexp(

- (r + g) (Ts)),

(20)

K(0.2TB) = c + cexp( - (r + g)(0.4TB - 0.2TB))

+cexp(

- (r + g)(0.6TB - 0.2T,))

+cexp( -(r + g)(0.8Ts -0.2TB)) + B exp ( - (r + g) ( Ts - 0.2 TB)).

(21)

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Note that K(t) is defined in a way consistent with the bond price on which the option is defined ~ they are both cum-coupon. K(0.4TB), K(0.6TB), and K(0.8Ta) are similarly defined as the the cum-coupon price of the bond for a fixed spread y above the risk-free rate at times 0.4TB, 0.6T, and 0.8T,. The rationale for this put is the same as the rationale for the put on the zero coupon bond price which was discussed in Section 3.1. Furthermore, the option contract provides that if the firm goes bankrupt at any time prior to option maturity, the option will be immediately exercised. This is a necessary condition to impose since the bond will cease to exist if the firm goes bankrupt and hence there will be no way of calculating the payoffs to the option. This provision makes this option contract a barrier option. Since the firm can only go bankrupt on a coupon date prior to bond maturity, it is only on coupon dates that the option can be terminated. We now present a no-early exercise result. Proposition

5.

The American

[fit

exercised

early

is a forced

Americmn

and European

barrier

put on the coupon

exercise,

put calues

i.e., if the option

bond price

will only he

is terminated.

Hencr,

will he the sumr.

A proof is in Appendix A. The intuition is similar to the one provided earlier for Proposition 1. We now consider the pricing of this put. Let F [G, t] be the option price at time t. It is assumed that the option price depends on the cum-coupon bond price G and time t. Since G depends on firm value I/, we can also think of option price as a function of V and t, i.e. FIG,

t] = N[V,

Using the usual satisfy:

t].

arbitrage

argument,

we can show that the option

0..5N,.,s2V2+N,-rN+rN,V=O,

(22)

N[V.

T]

N[V,

t] 2 max [0, K(t) - G(V, t)]

lim v-o

= max[O,

!?!- _ aG -

1

’

lim !I!! is finite V-r ac

K(T)

price will

- G(V,

T)],

(23) (24) (25)

(26)

Eq. (23) says that option value at maturity will be the larger of exercise value and zero. Eq. (24) says that the price of an American put can never fall below its exercise value. Eq. (25) says that when firm and bond value are very low, a $1 .OO

1592

Y.A. Pierides / Journal of Economic Dynamics and Control 21 (1997) l-579-161 I

increase in bond value will reduce option value by $1.00 since the option is way in the money. Eq. (26) says that when firm and bond value are very high, a marginal increase in the bond price will only have a finite impact on option price because the option is way out of the money. Also, the option price is influenced by the termination condition imposed earlier. The values of V for which the option is terminated are given by the coupon bond numerical valuation model of Appendix C. For these values of I/, the option payoff is set equal to the exercise value. We solve this valuation problem using the techniques in Appendix B.. Note that a fundamental difference between the put on the coupon bond and the put on the zero-coupon bond is that in the case of the latter there exists a closed form solution for the zero-coupon bond price whereas no such closed form solution exists for the coupon bond price. For this reason, to price the put on the coupon bond we need two numerical schemes: The first scheme, as described above, gives us the coupon bond price G (see Appendix C); the second scheme uses the G calculated form the first scheme to calculate the option price using the techniques of Appendix B. 5.2. Pricing results for American The output

is presented

barrier put on coupon bond price

in Fig. 3. The parameter

values chosen

were:

T = 5, TB = 10, r = 0.1 per year, B = 50, g = 0.03, s = 0.2 or 0.4, c = 3 at t = 2,4, 6, 8. Fig. 3 gives bond and option prices for different firm values. As expected the bond price increases with increases in firm value. Furthermore, note that for the low volatility case (s = 0.2) bond price is only slightly less than firm value when firm value is low (V = 20); the reason is that in such circumstances, the probability that the firm will avoid bankruptcy either on the next coupon date or at maturity is very low. As a result, the equity of the firm is not worth much. The (cum-coupon) bond price is lower the higher the volatility. This is analogous to Merton’s analytical result that higher volatility reduces zero coupon bond prices; however, in the case of the coupon bond, no such result can be established analytically because there is no closed form formula for the coupon bond price. As expected, the put option price is a decreasing function of firm value. Moreover, for given firm value, the option price is higher the higher the volatility(s). This is the result of two separate effects. The first effect relates to the relation between bond price and volatility. For given firm value, the higher the volatility, the lower the bond price. Since this is a put, a lower bond price increases its price. The second effect is a direct effect of higher volatility

Y.A. Pierides /Journal

. 20

-

l*

?*

0

of Economic Dynamics

30

.I.

_

40

.

50

_

,_

60

and Control 21 tlYY7)

157%1611

1293

-

_,

_

70

(_ 80

I 90

100

110

120

FIRM VALOE

Fig. 3. Put on price of coupon

bond (T = 5. TB = 10, B = 50. r = 0.1. 6, = 0.03, c = 3 at

r = 2,4,6,8.

on put prices analogous to the effect of higher volatility on the price of a put on a stock. Another feature of Fig. 3 is that the option becomes virtually worthless when firm value is high enough (approximately I/ = 60 for the s = 0.2 case) that the bond becomes effectively risk-free. The risk free-bond price is

This is a reasonable result because once the bond becomes risk-free it remains risk-free until its maturity and therefore, there is no possibility that the option will be exercised. As a result it is certain that the option will expire out of the money and its price is close to zero. The computer program that implements the numerical scheme was written in a way that permits us to check for early exercise and termination of the option. It confirmed our earlier analytical result (Proposition 5) that the put will only be exercised early if it is a forced exercise i.e. if the option is terminated. For the parameter values used this barrier option is terminated early as follows: (i) when s = 0.2, it is terminated at t = 2 (the first coupon date) whenever I/ < 27.3 and at r = 4 (the second coupon date) whenever V < 30.2; (ii) when s = 0.4, it is terminated at t = 2 whenever V < 18.3 and at t = 4 whenever I/ < 20.2. These

1594

Y.A. Pierides

/ Journal

of Economic

Dynamics

and Control

21 (1997)

1579-161

I

termination values are endogenously determined from the coupon bond numerical valuation model in Appendix C. Various other sensitivity results were established by varying the parameter values used as input. For example, it was confirmed that the option price is consistently lower for a lower maturity. Finally, as expected, a lower strike price (i.e. higher g) lowers the option price. For reasons similar to those given in the case of the put on the zero coupon bond price the influence of higher interest rates on put prices for given firm value is ambiguous. 5.3. Another credit risk derivative for coupon bonds: A call option on the bond’s spread

Consider the following American call option: at any t 5 T, where T = option maturity < TB, assuming the firm is not bankrupt, the option’s owner can exercise the option and receive an amount equal to max (0, E( I/, t) - g) where E(V, t) is the spread above the risk-free rate at which the coupon bond trades at time t (on a cum-coupon basis) and g is a fixed exercise price. For example, consider the coupon bond defined earlier. It promised B at TB and coupons in the amount c at 0.2TB, 0.4TB, 0.6T,, and 0.8TB. The spread on a non-coupon date, for example when t = 0, defined as .$I/, 0) solves the equation: G(V, 0) = cexp( - (Y+ &(I/, 0)) (0.2TB)) + cexp( -(r

+ E(V, 0)) (0.4TB))

+ cexp( - (r + &(I/, 0)) (0.6Ta)) + c exp( - (r + E(V, 0)) (0.8TB)) + Bexp(

- (r + G”, 0)) (T,))

(27)

where G( I’, 0) equals cum-coupon bond price at time 0. Hence, E( I/, 0) equates the right-hand side of Eq. (27) to the numerically calculated cum-coupon bond price G( I/, 0). The spread on a coupon date, for example when t = 0.2 ( Ts), E( I/, 0.2 ( TB)) solves: G(V, 0.2(TB)) = c + cexp( -(r

+ &(I/, 0.2(TB)))(0.2T,))

+ c exp( - (r + E(V, 0.2(TB))) (0.4T,)) + c exp( - (r + &(I/, 0.2(T,))) (0.6T,)) + B exp( - (r +

E( I/,

0.2 ( TB))) (0.8 TB))

(28)

where G( I/, 0.2(TB)) equals cum-coupon bond price at time 0.2TB. Note that a fundamental difference between the call option on the zerocoupon bond spread and the call option on the coupon bond spread is that in the case of the former, the spread can be calculated from the closed-form solution, whereas in the case of the latter, the spread must be calculated from

Y.A. Pierides J Journal of Economic Dynamics and Control 21 (1997) 1579- 161 I

I595

Eqs. (27) and (28) using the coupon bond price provided by the numerical scheme. Eqs. (27) and (28) can be solved for s(V, t), using the Newton-Raphson root finding scheme. Furthermore, the option contract provides that if the firm goes bankrupt at any time prior to option maturity, the option will be immediately terminated and the payoff will be the exercise value. This is a necessary condition to impose since the bond will cease to exist if the firm goes bankrupt and hence there will be no way of calculating the payoffs to the option. This provision makes this option contract a barrier option. Since the firm can only go bankrupt on a coupon date prior to bond maturity, it is only on coupon dates that the option can be terminated. Another complication to consider is that on a coupon date, Eq. (28) may have no root; this will happen whenever G < c (as is obvious from inspection of Eq. (28)). In such circumstances, the value of E(V, t) is set equal to an arbitrarily defined upper bound smax and option payoffs are calculated using smax. Eq. (27) which gives the spread on a non-coupon date will always have a root. Nevertheless, for consistency reasons, whenever the calculated value of F.exceeds I:max, E is automatically set to smax and we calculate option payoffs using Pax. We now consider the pricing of this put. Let F [E( I/, t), t] be its price at t. It is assumed that option price depends on the spread s(C’, t) and time t. Since I: depends on I/, we can think of option price as a function of I/ and t, i.e., F[E(!‘,

t), r] = Q[V, t].

Using the usual arbitrage argument we can show that satisfy the following p.d.e. and boundary conditions:

(29) the option

price will

OSQ,.,. s2 I’ ’ + Qt - rQ + rQC V = 0,

(30)

QCV, Tl = max[O, s(V, T) - g],

(31)

Q CV, tl 2 max[O, s(V, t) - g],

(32)

lim v-0

aQ= 1 ac

’

lim aQ is finite. V-x as

(33)

(34)

The interpretation is similar to the one provided earlier for the boundary conditions of the call on the zero-coupon bond spread. No closed-form solution exists to this valuation problem. For this reason, we employ a numerical method of solution as in Appendix B.

YA. Pierides / Journal of Economic Dynamics and Control 21 (1997) 1579-I 61 I

1596

5.4. Pricing results for call on coupon bond’s spread

The parameter values chosen were: T = 0.8, TB = 1.2, Y= 0.06 per year, B = 20,

g = 4.0, s = 0.3 or 1.0, CAP = 5.0, c = 3 at t = 0.3, 0.6, 0.9. Fig. 4 gives the bond spread and the option price for different firm values. As expected the bond spread decreases with increases in firm value. Also, the bond spread for given firm value is higher the higher the volatility because higher volatility reduces bond prices. The call option price is a decreasing function of firm value (and an increasing function of bond spread). The influence of higher volatilty on the call price is ambiguous. When the option is in the money or not way out of the money (that is for I/ < 4) lower volatility increases the option price for given firm value. Note that this result holds even though lower volatility increases bond price and reduces the spread for given firm value - the latter should tend to reduce option price for given firm value. This result is obtained because of the capped nature of the option we are

3.00 -

* l

2.50 --

+ l l

2.00 --

*

1

A 3 $

&

A

1.50 --

A

> 1.00 i:--

l.*

- _ . .

A

A

* A A

.4

A

.

.

.

. .

.

.

I 2

A

-

0.50 --

0.00

A

,

4

6

8

.

. 10

.,

. 12

. 14

., 16

FJJW VALUE Fig. 4. Call on spread of coupon c = 3 at t = 0.3,0.6,0.9).

bond

(T = 0.8, T, = 1.2, r = 0.06, q = 4.0, CAP = 5.0, E = 20,

YA. Pierides /Journal

of Economic Dynamics and Control 21 (1997) 1579-1611

1591

considering. Note that for purposes of calculating the option price, the spread is capped at 5. Since the exercise price is at 4, when the option is in the money, the upside potential is limited whereas the downside risk is unlimited. In such circumstances, higher volatility reduces option prices because it implies that it is more likely that the option will end up out of the money without providing significant upward potential to counterbalance the former effect. For similar reasons, higher volatility reduces option prices for given firm value when the option is out of the money provided the option is not way out of the money. However, when the option is way out of the money (that is for V > 4) higher volatility increases option price for given firm value. Various other sensitivity results were obtained by varying the parameter values used as input. For example, it was confirmed that the option price is higher the higher the option maturity and the lower the exercise price. Also a higher cap (max spread) results in higher option prices. Finally, for reasons similar to those given for the call on the zero coupon bond’s spread, an increase in r may either increase or decrease option prices.

6. Conclusion This paper examined the structuring and pricing of credit risk derivatives for both zero-coupon and coupon bonds. Both puts on bond prices and calls on bond spreads can serve as credit risk derivatives. For coupon bonds, these put and call options must be of the barrier type. Furthermore, the call on the coupon bond’s spread must be capped; otherwise there will be no way of calculating its price. The pricing of these options was investigated using a numerical method of solution for the partial differential equation and associated boundary conditions. For zero-coupon bonds, both the bond price and the bond spread can be calculated analytically; the calculated price and spread are then used in the numerical scheme that gives the option prices. For coupon bonds, there is no way of calculating the bond price and spread analytically; instead, they must be calculated numerically. The numerically calculated price and spread are then used in a second numerical scheme that gives the option prices. We therefore have two numerical schemes that are interdependent. The pricing properties of these options were established. A number of unexpected results were obtained. American puts on the zero-coupon bond price will never be exercised early. American puts on the coupon bond price will be exercised early only if it is a forced exercise. The price of the call on the spread can exceed the current value of the spread. There are circumstances in which higher volatility can reduce the price of the call on the spread of the coupon bond. The influence of the interest rate on the price of these options is ambiguous.

Y.A. Pierides /Journal

1598

of Economic Dynamics and Control 21 (1997) 1579-1611

Appendix A. Proof of propositions Proposition 1. Early exercise of the American put on the zero coupon bond price can never be optimal. Hence, its price will be the same as the price of an otherwise identical European Option. Proof I.

It is shown that the price of this put can never be lower than, or equal to, the exercise value. Hence the put should never be exercised early. To show that the put price can never be lower than, or equal to, the exercise value, we show that if it is, an arbitrage opportunity exists. Assume, therefore, that the put price is lower than, or equal to, the exercise value at some time t prior to option maturity, i.e., assume that: F(t) I K(t) -f(V,

t),

where F(t) = put price, K(t) = exercise price = Bexp( - (r + g)(TB - t)), f(V,

t) = bond price.

Adopt the following arbitrage strategy: Buy the put, buy the bond and borrow an amount K(t) at the risk-free rate. The time t payoff is clearly non-negative. We wait until the maturity date of the put and then close our position. Assume that the option is in the money at maturity. We exercise it by selling the bond which gives us a payoff of K (T), the exercise price. To repay our time t borrowing of K(t) we need K(t) exp (r (T - t)). Therefore, Net payoff at T = K(T) - K(t)exp(r (T -t)) =Bexp(-r+g)(T,-T))-Bexp(-(r+g) x (TB - t) + r(T - t)) = Bexp( -(r

+ g)(Ts - T))[l

- exp(g(t - T))].

Note that since g > 0 and t < T, exp(g(t - T)) < 1. Hence net payoff at T is positive. Therefore, an arbitrage opportunity exists. Now suppose that the option is not in the money at T. It follows that the bond is worth more than K(T). We sell the bond for an amount greater than K(T) and repay K (t)exp (r (T - t)) for the amount we borrowed at time t. Hence, Net Payoff at T > K(T)

- K(t)exp(r(T

- t)).

Since we showed earlier that the right-hand side of the above inequality is positive, it follows that our net payoff at T is positive. Again, an arbitrage opportunity exists.

Y.A. Pierides

The implication need:

/Journal

of

Economic

D_vnantics and Control

21 (IY97)

of the above result is that to avoid arbitrage

1579--16/i

opportunities

1599

we

Hence, the price of the put can never be lower than or equal to the exercise value. The put should never be exercised early. I7 Proposition 2. The Price of the American call on the spread of the zero coupon bond can he equal to its exercise value at any time, in particular, even hgfore the expiration date. In such a case, the call should he immediately exercised. Proof 2. It is shown that when price equals exercise value, opportunity exists. Hence, this can be an equilibrium situation. Suppose that at t, < T,

no arbitrage

FCe(v, tl), t11 = e(V, tl) -g > 0, where F = call price at tl. To investigate whether arbitrage opportunities exist, at time t 1, buy the call short e( V, t,)/f (V, t 1) corporate bonds worth e (V, t ,) and lend g. The time tl payoff is 0. To see what type of payoff we can expect from this strategy in the future, suppose we exercise the call at t2 > t , . We pay the exercise price by using the time t, proceeds from lending g at time t r. Clearly, we gain the interest rate on g. By exercising the call we receive e(V, tJ which enables us to buy e( V, tJ/f(V, tJ corporate bonds at tl. This may be lower or higher than the number of bonds we need to buy to cover our short position. Hence, the time t2 payoff can be either positive or negative. It follows that no arbitrage opportunity exists at tl. Note that if r (I/, t) were a linear function off; i.e., if e ( V, t) = cf( V, t) where c = constant, we would always be assured a profit from this strategy and hence an arbitrage opportunity would exist. The reason is that at tr we short e( V, tI)/f(V, tI) = c bonds and at tz we buy e(V, t2)/f(V, t2) = c bonds, i.e., the number of bonds bought is equal to the number of bonds shorted. Hence the time t2 payoff upon exercising the call will be positive and equal to the interest earned on 9. q Proposition 3. The price of the call on the spread of the zero coupon bond may exceed the current value of the spread. Proqf‘3. It is shown that when call price exceeds the spread, no arbitrage opportunity exists. Hence, this can be an equilibrium situation. Suppose that at any tl < T, Z( I/, t,) > e (V, tI), i.e., option price exceeds the spread. To investigate whether an arbitrage opportunity exists, let us try to follow the arbitrage strategy described in Cox and Rubinstein (1985), to exploit situations

1600

Y.A. Pierides / Journal of Economic Dynamics

and Control 21 (I 997) 1579% I61 I

where the price of a call on a stock exceeds the current stock price. In our context, that strategy involves selling the call and buying e(V, ti)/f(V, ti) corporate bonds (notef(V, ti) = bond price) worth e(V, tJ. The time t, payoff is clearly positive. However, if the call we sold is exercised on or before the expiration date we may be in trouble: We have to pay an amount e(I’, t2) - g where t2 equals time of exercise of the option and it is not clear that the bonds we purchase at ti will be worth this much at t2. Hence our payoff may be negative and no arbitrage opportunity exists. 0 Proposition 5. The American barrier put on the coupon bond price will only be exercised early if it is a forced exercise, i.e., if the option is terminated. Hence, American and European put values will be the same. Proof 5.

For simplicity, assume current time is t, bond pays a coupon at t,, option matures at T and bond matures at TB with t
We shall prove that at either t or t, the option will never be exercised early (unless it is a forced exercise at t,). Prove that option will not be exercised early at t Suppose that at t, put price < K(t) - G(t), where K(t) = exercise price =L?exp(-(r+g)(TB-t))+cexp(-(r+g)(t,-t)), G(t) = cum-coupon

bond price.

Arbitrage strategy: Buy put, buy bond and borrow K(t). The time t payoff is positive. Wait until time t,. Two possibilities at time t,: 1. If forced to exercise at time t,, we deliver the bond and receive the option’s exercise price (K(Q) if the latter exceeds the bond price; if the option is not in the money, the bond must be worth at least K(t,). Therefore, our payoff from our bond and option position is at least K(t,). Also, we repay our borrowings which amount to K(t) exp(r(t, - t)) at time t,. Hence, payoff at t, 2 K (tJ - K(t) exp(r (tC - t)) = B exp( - (r + g)(T,

- t,)) + c - Bexp( - (r + g)(TB) + gt + rtJ

- c exp( - g (tC - t)) = c(1 - exp( - g(tC - t))) + Bexp( -(r

+ g)(TB - t,))

x (1 - exp( - g(t, - t))).

Clearly, 1 - exp( - g(tC - t)) > 0, hence payoff at t, 2 0. An arbitrage opportunity exists. We must have put price > K(t) - G(t).

Y.A. Pierides / Journal of’ Economic Dynamics and Control 21 (1997) 1579-. Ilil I

1601

2. If we are not forced to exercise at time t,, we wait until the expiration date T. Since we were not forced to exercise at t,, the firm did not go bankrupt and therefore we received a coupon of c at t, from our long bond position. By T, this coupon is worth c exp(r(T - t,)). At time T, our option and bond position must be worth at least K(T). The reason for this is that if the option is in the money, we exercise it by delivering the bond and receive K(T). If the option is not in the money, the bond must be worth at least K(T). Also, we repay our borrowings at time T which amount to K (t) exp (r(T - t)). Finally, as explained in the previous paragraph, we have an additional amount c exp(r(T - t,)) at T. Therefore, payoff at T 2 K(T)

+ cexp(r(T

- t,)) - K(t)exp(r(T

= Bexp( - (I” + g)(TB - T)) + cexp(r(T - Bexp( -(r

+ s)(TB - T) - s(T

- t)) - t,))

- 0)

- c’exp (r( T - tJ - g(& - t)) = Bexp(

-(r

+ g)(TB - TM1

- exp( - .4(T - 0))

+ c exp(r (T - t,)) (1 - exp ( - g( t, - t))). Clearly exp( - g(tc - t)) < 1.

expt - g(T - 4) < 1,

Hence, payoff at T 2 0. An arbitrage opportunity exists. We must have put price > K(t) - G(t). Prove that option will not be exercised early at t,. unless it is a,forced e.uercisr. Suppose at t, option is not terminated and put price

< K(t,)

- G(t,).

Adopt the following arbitrage strategy: Buy put, buy bond, borrow K(t,). The time t, payoff is clearly positive. We wait until the expiration date T. Note that we bought the bond cum-coupon at t, and hence receive a payoff of c at t,. At T, our long bond and long option positions will be worth at least K(T). As before, the reason is that if the option is in the money, we exercise it by delivering the bond and receive the exercise price K (T ); if the option is not in the money, the bond must be worth at least K(T). We repay our borrowings at T which amount to K (tJ exp(r(T - tJ). Also the coupon of c we received at t, is now worth c exp (r(T - t,)). Hence payoff at T 2 K(T)

+ cexp(r(T

- tJ) - K(t,)

exp(r(T

- t,))

==Bexp(-(r+g)(T,-T))+cexp(r(T-t,)) - B exp( - (r + g)(T, = Bexp(

-(r

- t,) + r(T - tc)) - cexp(r(T

+ g)(TR - T)) [l - exp( - g(T - t,))].

- t,))

1602

Y.A. Pierides /Journal

of Economic Dynamics and Control 21 (1997) 1579-16/l

Clearly, exp( - g(T - t,)) < 1. Hence payoff at T >_0.An arbitrage opportunity exists and we must have Put price > K(t,) - G(t,). Since we have shown that the put price must exceed the exercise value at both t and t, (unless the latter is a forced exercise situation), we have established that

the put will never be exercised early unless it is a forced exercise.

Appendix B. Calculating

0

the option price numerically

Note that the p.d.e. that gives the option price is the same for all four types of options considered. What differentiates one case from another are the boundary conditions. Here we present the numerical method of solution for the boundary conditions of the put option on the price of the zero coupon bond. The other three types of options are treated in a similar way. The valuation problem we want to solve is given by Eqs. (5)-(9) in the text: 0.5M,,s2~2++M,-rM+rM,I/=0.

(5)

The boundary conditions are: MV’, Tl = maxCO, K(T) -.W,

MIIV,cl 2 maxCO,K(t) -fW,

TN,

Ql,

(6) (7)

lim aM = - 1 v+o af

lim E is finite “-+co af .

(9)

The above p.d.e. has variable coefficients. This means that an explicit method of solution (which is computationally easier than an implicit method) may not converge (Ames, 1992). For this reason we use a transformation of the state variable which will give us a p.d.e. with constant coefficients. For such a p.d.e., we can give conditions such that an explicit method of solution converges. We define: Y = log V, M(V, r) = H (Y, r), M,=H,exp(-

Y),

MVV = (H,, - H,)exp(

M, = H,.

- 2Y),

Y.A. Pierides 1 Journal of Economic Dynamics and Control 21 (1997) 1579-161 I

Substituting

in (5), we obtain

the transformed

p.d.e. with constant

coefficients:

0.5 s2 H,, + (r - 0.5 s2)Hy + H, - rH = 0. The boundary H[Y, wheref’(Y,

conditions

(B.1)

in terms of Y are:

T] = max[O, K(T)

-f(Y,

T)]

(J3.2)

t) = B exp( - r(TB - t)(@(h2) + dK’@(h,));

d = Bexp(

- r(TB - t) - Y)

h, and h2 are as defined H[Y,

1603

r] 2

in the text.

max[O, K(t) -f’(Y,

t)l

03.3) (B.4)

lim E is finite. y++X af

(B.5)

These have a similar interpretation as the boundary conditions in terms of I/. To solve the p.d.e. numerically, we consider a finite number of discrete values of Y and t. Let Y,,, (Y,in) be the maximum (minimum) value of Y considered and, also, consider n - 1 other discrete values between Y,,, and Ymin such that dy represents the difference between two consecutive values, i.e., the step size. Then, any discrete value of Y can be represented as Y = Ymin + j(dy), where .j consists of the integers between 0 and n. Likewise, let T be the maximum value of t and 0 the minimum value. Consider a number of discrete values between 0 and T, such that dt is the step size. Then, any discrete value of t can be represented as f = k(dt) where k consists of the integers between 0 and T/(dt). The choice of dy and dt will be discussed later. The value of H for the discrete values of Y and t can be written as: H(t, Y) = H(k(dt), B.I.

Explicit

numerical

Ymin + j(dy)) scheme

In this scheme, the partial derivatives differences: H, = (H(k

are approximated

by the following

+ 1, j + 1) - H(k + 1, j - 1))/2 (dy),

H,, = (H(k + 1, j + 1) - 2H(k

H, = WV

= H(k, j).

+ 1,j) - HP, MW.

+ 1, j) + H (k + 1,j -

l))/(dy)‘,

finite

1604

Y.A. Pierides /Journal

Substituting these difference equation:

finite

of Economic Dynamics

differences

into

and

Control

21 (1997) 1579-1611

the p.d.e. we obtain

H(k j) = (PUH(k + Lj + 1) + J’,H(k + Lj) + P,H(k > I( + r(4)

the following

+ Lj - 1))

VW

where

Pu=(s2+h(+)))(&)> Cc=(s2-dy(r-(;)))(&) P, = 1 - sz -

dt

(dy)’ ’ The boundary H(T,j)

at maturity

(B.2) can be approximated

by

for early exercise (B.3) is approximated

by

= maxLO, K(T)

The boundary H&j)

condition

condition

-SV,,dl

2 max CO,K(k) -f(k

31,

where K(k) = B exp( - (r + g)(T, f(k,j)

= Bexp( - r(r,

4dt))) ,

- k W)

(@ (4

+ de’ @h)L

d = 13exp( - rVB - k (dt))) - (Ymin +j(dyh h, and h, are evaluated

at t = k (dt).

Since this is an explicit method of solution, the boundary conditions describing the behavior of N for very large and very small Y are not used in the numerical procedure. The above difference equation (B.6) and maturity boundary condition (B.2) give the time t option price only if their solution exceeds the exercise value. Otherwise, the option price is set equal to the exercise value. In particular, the solution is obtained as follows: Given H(T,j) for all j from the maturity boundary condition, we obtain H(T - dt, j) for all but two of the j’s we considered at time T from the finite difference equation. H(T - dt, j) can be thought of as the price of the option at T - dt for given j, assuming that we decide to keep the option alive, i.e., it is a temporary option price. We then compare the latter with the exercise value of the option at T - dt for givenj to take account of boundary condition Eq. (B.3). We set the option price at T - dt for givenj, denoted M(T - dt,j) equal to the maximum of the two. We then use

Y.A. Pierides 1 Journal of Economic Dynamics

and Control 21 (1997) 3579~1611

1605

M (T - dt,j) as the value for H(T - dt,j) in the difference equation to obtain the temporary option price at T - 2(dt), H(T - 2(dt), j). We compare the latter with the exercise value at T - 2 (dt) and set the option price at T - 2(dt), M (T - 2 (dt), j) equal to their maximum. We proceed in this way until we reach time 0. From Ames (1992) a sufficient condition for the convergence and stability of this numerical scheme is the positivity of P,, Pd, and P,. For this, dy and dt must be chosen so that

s2

dt

i-f_. dy< @p s2’

B.2. Crank-Nicholson For this scheme finite differences: H,

=

J

scheme (combination the partial

derivatives

qf an implicit and explicit method) are approximated

(H(k + l,j + 1) - H(k + Lj - 1)) +

by the following

(H(k,j + 1) - H(k, j - 1))

4 (dy)

N,, = (~(k~l~j+l)-2~(k+l,j)+~(k+l,j-l))+(H(k,j+1)-2N(k,j)+H(k,j-1))

2 (d1.)’ H

t

=

H (k + Lj) - H(k, j) dt

Substituting these finite differences into the p.d.e. we obtain the following of n - 1 equations in n + 1 unknowns H(k, j), j = 0,. . . , n:

set

u1 H(k + 1,j + 1) + b, H(k + 1, j) + !x2H(k + 1, j - 1) +rr,H(k,j+l)+b,H(k,j)+a,H(k,j-l)==O

forj=l,...,

n-l (t-3.7)

b, = -

g2 2(dy)Z-dt+.

1

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Y.A. Pierides / Journal

of Economic Dynamics and Control 21 (I 997) 1579- I61 I

The first two boundary conditions (B.2) and (B.3) can be approximated as in the explicit scheme. It can be shown that boundary condition (B.4) is equivalent to

This condition is approximated H(k, 1) = H(k, 0)

by

Vk.

(B.8)

Finally, it can be shown that boundary condition (B.5) is equivalent to lim y++‘x

E =0 ay

This condition is approximated H(k,n)=H(k,n-1)

by

Vk.

H(k 0) 1

Define, HK =

(B.9)

H(k, 1)

[ H (k n)

We can write the system of Eqs. (B.7) and the boundary conditions (B.8) and (B.9) as BHk+AHK+’

=O,

where A and B are the following tridiagonal matrices: r

A=

0

0

a2

81

a1

and

a1

a2 a2

Pl 0

a1 0

B=

1

-1

@2

B2

Ml El

x2 @2

B2 1

a1

-1

We can rewrite this system of equations as BHk=

-AHK+l,

This system of equations together with the maturity boundary condition (B.2) gives the time f option price only if their solution exceeds the exercise value given by boundary condition (B.3). Otherwise, the option price is set equal to the

Y.A. Pierides 1 Journal

ofEconomic

Dynamics

and Control 21 1I997j 1579~-1611

1607

exercise value. In particular, the solution is obtained as follows: Given H7‘from the maturity boundary condition, we solve the above system of equations to obtain Hrmdr. As before, HTmdtcan be thought of as the price of the option at T - dt (for variousj’s) assuming that we decide to keep the option alive, i.e., it is a temporary option price. We then compare the latter with the exercise value of the option at time T - dt (for given j) to take account of boundary condition (B.3). We set the option price at T - dt for given j, denoted MTddr equal to the maximum of the two. We then use MTmd’ as the value for HTmdrin the above system of equations to obtain the temporary option price at T - 2(dt), HT-2(d'). We compare the latter with the exercise value at T - 2(dt) and set M’- 2(d” equal to their maximum. We proceed in this way until we reach time 0. From Ames (1992), this scheme is unconditionally stable.

Appendix C. Calculating the coupon bond price numerically The valuation

problem

we want to solve is given by Eqs. (16))(19) in the text:

0.5~~ V2.f,, + rV,fv - rf+.ft + t

c6 (t - (0.2)i TB) = 0,

(16)

i=l

where 60 is the Dirac delta function.

.fl I/, Td = lim,f=

The boundary

conditions

are:

min(B, v (Td),

(17)

I/

(18)

1-O

lim ,.+7

af’= av

0

The stopping condition on a coupon payment date described completes the specification of the problem. We first transform the p.d.e. into one with constant coefficients. Y =logV,

.f’tv, f) =

w (Y, t),

,f;. = W, exp( - Y), &=(W,,.f; =

w,

(19)

.

W,)exp(-2Y),

in the text Define

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Y.A. Pierides /Journal

Making the appropriate

of Economic Dynamics and Control 21 (1997) 1579-1611

substitutions, we obtain the transformed equation

OS.? W,, + (r - 0.5s’) IV, + W, - rW + i

c6 (t - (0.2)iT,) = 0.

(C.1)

i=l

The boundary conditions in terms of Y are W (Y, TB) = min(exp(Y), B)

lim

W = exp(Y),

Y+-CE

lim W=O. Y-m

aY

(C.2) (C.3) (C.4)

The interpretation of these boundary conditions is identical to the interpretation provided earlier for the boundary conditions in terms of I/. To solve the p.d.e. numerically, we consider a finite number of discrete values of Y and t. Let Y max (Y min) be the maximum (minimum) value of Y considered and also consider n - 1 other discrete values between Y,,, and Yminsuch that dy represents the difference between two coqsecutive values, i.e., the step size. Then, any discrete value of Y can be represented as Y = Y,in + j (dy), where j consists of the integers between 0 and n. Likewise, let Ta be the maximum value oft and 0 the minimum value. Consider a number of discrete values between 0 and T, such that dt is the step size. Then, any discrete value oft can be represented at t = k (dt) where k consists of the integers between 0 and T,/dt. The choice of dy, dt will be discussed later. The value of W for the discrete values of Y and t can be written as

W(t, Y) = W(k(dt), Ymin+j(dy)) = W(kyj). C. 1. Explicit numerical scheme

For the explicit scheme, the partial derivatives are approximated by the finite differences described in Appendix B. We substitute these finite differences into the p.d.e., to obtain a difference equation. Between coupon dates, the Dirac delta function term does not afSect the evolution of W. Hence, between coupon dates, the evolution of W can be described by this p.d.e. without the Dirac delta term, i.e., by 0.5s’ W,, + (r - 0.5 s2) W, + W, - rW = 0

(C.5)

This p.d.e. describes the evolution of W between coupon dates, but not on coupon dates. The behavior of W on coupon dates will be described below. After

Y.A. Pierides /Journal

substituting

ofEconomicqvnamics

the finite differences

and Control 21 (1997) 1579-16/l

1609

we obtain

w(ki)=(P,W(k+l,j+l)+P,W(k+l,j)+P,W(k+l,j-l)) (1 + r(W) 1.

(C.6) where

K=(s2+&(r-(;)))(&)> f’d=(s2-dy(r-(;)))(&), dt P" = 1 - s2 (dy)2’ For stability, dt m

need P, > 0, Pd > 0, P, > 0. Hence,

1 <-?;dy< s

s2 r - (~‘12)’

The boundary conditions at low and high firm values are not used in the explicit scheme. The boundary condition at bond maturity can be approximated by W(TR, j) = min CB, exp(Y,i,

+ j(dJ)l.

(C.7)

To analyze the behavior of W on coupon dates, assume that the coupon payment dates coincide with discrete points on the time grid of the numerical scheme. To illustrate the calculations, assume that between time TH and time TB - 5(dt) only time TB - 3(dt) is a coupon payment date as shown below Time TR TR - dt TH - 2(dt) TU - 3(dt) TB - 4(dt) TR - 5(dt)

Coupon No No No Yes No No

The solution is obtained as follows: Given W ( TB, j) from the maturity boundary condition, we obtain W (TB - d&j) from the difference equation for all but two of the ,j’s considered at TB. At time TB - 2(dt), given W ( TB - dt, ,j), we obtain W (T,, - 2(dt), j) from the difference equation for all but two of the j’s considered at time TR - dt. At time T, - 3(dt), given W(TB - 2(dt)) we obtain W (TH - 3(dt)) from the difference equation for all but two of the j’s we considered at time TB - 2(dt). Furthermore, given TR - 3(dt) is a coupon date,

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Y.A. Pierides /Journal

of Economic Dynamics and Control 21 (1997) 1579-161 I

we calculate the total payoff to the bond at TB - 3(dt) defined as G((T, j) as follows:’

- 3(dt),

G ( TB - 3(dt), j) = W( TB - 3(dt), j) + c if exp[ Y,,,i, + j(dy)]

2 W ( TB - 3(dt), j) + c i.e., if firm can pay c.

G(TB - 3(d@ j) = exp CY,i, + if exp[Y,,,

+ j(dy)]

(C.8)

j(d.~)l

< W (TB - 3(dt), j) + c i.e., if firm bankrupt.

(C.9)

Note that W(TB - 3(dt), j) will be the ex-coupon bond price at time TB - 3(dt) only if the firm does not go bankrupt, i.e., if firm value > W + c. If the firm does go bankrupt (i.e., if firm value < W + c) at TB - 3(dt) because the firm cannot pay the coupon, the ex-coupon bond price cannot be defined; in such circumstances only the total payoff G(Ta - 3(dt), j) is defined. At time TB - 4(dt), we calculate W (Ts - 4(dt), j) from the difference equation using G(T, - 3(dt),j) instead of W (T, - 3(dt), j) in the difference equation. As before, we do this for all but two of thej’s considered at TB - 3(dt). Note that since TB - 4(dt) is not a coupon date the total payoff at T, - 4(dt), i.e., G(TB - 4(dt), j) = W (TB - 4(dt), j). At time TB - 5(dt), given W (TB - 4(dt),j) we calculate W (T, - 5(dt),j) from the difference equation for all but two of the j’s considered at TB - 4(dt). Again, G( TB - 5(dt), j) = W ( TB - 5(dt), j). The calculations proceed in this way and depend on whether the time grid point at which they are performed is a coupon date or not. We proceed in this way until we reach time 0. C.2. Explanation of total payoff calculation on a coupon date The bond contract requires that shareholders finance the coupon payment through a new equity issue. The shareholders will be able to pay the coupon only if, after a cash infusion from new equity in the amount c (equals coupon) and payment of the coupon, the combined value of the old and new stock (which will be equal to I/ - W) exceeds or is equal to c, i.e., only if v-w2c To see this, note that if V - W < c, new shareholders buy stock at any price and hence the firm goes bankrupt. if new shareholders are offered 100% of the company, the coupon is paid ( = V - W) will be worth less than not willing to pay c for such a claim. In such a case, the the total payoff to the bond is V.

’ An explanation

follows in the next section;

on non-coupon

will not be willing to The reason is that even their shares’ value after c and, clearly, they are firm goes bankrupt and

dates, G = W.

Y.A. Pierides /Journal

of Economic Dynamics and Control 21 (1997) 1579-161 I

161 I

Alternatively, consider a situation in which V - W exceeds c by a tiny amount. In this case the old shareholders will be able to finance the coupon provided they allow new shareholders to buy shares at a price such that the stake of old shareholders is diluted to a tiny amount. Of course, this is better than bankruptcy. Finally, if V - W exceeds c by a big amount, old shareholders will be able to finance the coupon without diluting their stake substantially. The above discussion suggests that the coupon will be determined as follows: coupon

= c if I/ - W 2 c

firm bankrupt

if I/ - W < c.

Hence. total payoff to bond = c + W, if I’ - W 2 c

(C.10)

= I/ if I/ - W’ < c. C.3. Crank-Nicholson

scheme (combination

of an implicit and explicit scheme)

The use of this scheme in calculating the coupon bond price is similar to its use in the calculation of the option price, which was described in Appendix B. For this reason, the details are not given here.

References Ames W., 1992, Numerical Methods for Partial Differential Equations, Academic Press, New York. Black F. and J. Cox, 1976, Valuing corporate securities: Some effects of bond indenture provisions, Journal of Finance 31, 351-367. Cornell B. and K. Green, 1991, The investment performance of low grade bond funds, Journal of Finance 46, 523-539. Cox J. and M. Rubinstein, 1985, Options Markets, Prentice-Hall, Englewood Cliffs, NJ. Das S., 1995. Credit risk derivatives, Journal of Derivatives 2, 7723. Flesaker B., L. Hughston, L. Schreiber and L. Sprung, 1994, Credit derivatives: Taking all the credit, Risk Magazine, September issue. Jarrow R. and S. Turnbull, 1995, Pricing derivatives on financial securities subject to credit risk. Journal of Finance 50, 53-85. Longstaff F. and E. Schwartz, 1995a, A simple approach to valuing risky fixed and floating rate debt, Journal of Finance 50, 7899819. Longstaff F. and E. Schwartz, 1995b, Valuing credit derivatives, Journal of Fixed Income 5, 25 - 32. Madan D. and H. Unal, 1994, Pricing the risks of default, Working paper, University of Maryland. Merton R., 1974, On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance 29. 4499470.