Valuation of Credit Contingent Claims an Arbitrage Free Credit Model

Valuation of Credit Contingent Claims: An Arbitrage-Free Credit Model

Thomas S. Y. Ho1

President Thomas Ho Company, Ltd

Sang Bin Lee2

Professor of Finance Hanyang University

June 2008

1 Correspondence Information : Thomas S. Y. Ho, President Thomas Ho Company, Ltd , 55 Liberty Street, 4 B New York NY 10005-1003, tel: , Mail to : [email protected]

2 Correspondence Information : Sang Bin Lee, Professor of Finance Hanyang University, 17 Haengdang-dong, Seongdong-ku, Seoul, 133-791, Korea, tel : 82- 2-2290-1057, Mail to : [email protected]

1 Abstract

This paper extends the generalized Ho-Lee model to the CDS curve movements that ensures the hazard rate movement is arbitrage-free for any given CDS curve. This paper shows that the generalized Ho-Lee model is not limited to pricing the interest contingent claims. The Ho-Lee model can be equally applicable to pricing the credit contingent claims. This model can value a broad range of credit contingent claims. These credit contingent claims include the American and Bermudan CDS options, make-whole and callable bonds.

The model features the separation of the specification of volatilities of the hazard rate from the fitting of the model to the CDS curve. This separation enables the model to have several advantages over other models.

We have also shown that many interest rate analytics can be extended to credit analysis. For example, we use the model to depict the credit performance profile of a bond, by plotting the credit contingent claim values over a range of hazard curves. The performance profiles can identify the impact of the credit risks on the contingent claims. In this way, we can apply the tools to analyze the interest risk to the credit risk, because the discount function and the survival function is essentially the same concept.

2 Valuation of Credit Contingent Claims: An Arbitrage-Free Credit Model

A. Introduction

Credit contingent claims are financial instruments whose stochastic movements are driven by the credit risk of the underlying securities. Credit contingent claims are prevalent in the financial market. The credit provisions of corporate bonds, such as the American, European, Bermudan and other types of call and put make- whole options are some examples. There are many over-the-counter credit derivatives such as the call and put credit options, which can be options on the credit derivative swaps. Asset-backed securities and collateralized debt obligations consist of tranches backed by a portfolio of financial instruments, the collaterals, with credit risk. The payments to the tranches are based on pre- specified priority rules. And therefore, each tranche has an embedded credit contingent claim that relate the value and risk of the tranche to the underlying credit risk of the collaterals. Indeed, more generally, all the liabilities and the equity of a firm are credit contingent claims on the credit risk of the firm.

Recently, the credit derivative markets have grown significantly. In particular the credit derivative swap (CDS) market is now reasonably liquid. In 2008, the CDS market has a notional value of $40 trillions. Today, the CDS market provides a term structure of the default risk of a reference name as the tenors of the credit derivative swaps have lengthened beyond five years to 10 years for many reference names. As a result, the CDS curve, which is defined by the premiums of credit derivative swaps over a range of tenors, can be used for dynamic hedging of credit contingent claims.

For this reason, the CDS curve can be used for valuing credit contingent claims in an arbitrage-free framework. Since a credit contingent claim can be dynamically replicated using CDSs, the cost of the dynamic hedging strategy can be viewed as the value of the contingent claim. Therefore, the valuation methodology is analogous to that of using the arbitrage-free movements of the term structure of interest rates to value interest rate contingent claims. In essence, the current active CDS market enables us to use an arbitrage-free credit model that “takes the CDS curve as given” to value a broad range of credit contingent claims. This paper provides such a model.

3 This paper’s arbitrage-free credit model is a reduced form model. The CDS valuation model specifies the movements of the CDS curve over time which can be used to value credit contingent claims satisfying the arbitrage-free condition: when the proposed valuation model is used to value a CDS as a special case, the model value of the CDS must be consistent with the observed CDS curve. This approach offers a relative valuation model of credit contingent claims, based on the market observed CDS curves. As a result, our model offers a valuation methodology consistent with the observed market prices, and therefore the model provides a valuation consistent with the cost of hedging and avoids a valuation model not consistent with the arbitrage-free condition.

There are papers proposing valuation methodologies of credit contingent claims such as Duffie (2005), Das (2006) and Longstaff and Rajan (2006). These models assume that the hazard rate follows a mean-reversion process similar to that of the Cox, Ingersoll and Ross model. And the model is then calibrated to the observed CDS curve. This approach must necessarily calibrate the hazard rate movement model to the CDS curve and the curve volatilities, as specified by the mean reversion process, jointly. These models cannot separate the calibration of the CDS curve and its volatilities. By way of contrast, this paper extends the generalized Ho-Lee model to the CDS curve movements that ensures the hazard rate movement is arbitrage-free for any given CDS curve. This feature separates the specification of volatilities of the hazard rate from the fitting of the model to the CDS curve. This separation enables the model to have several advantages over other models.

The use of the generalized Ho-Lee model has several advantages. The model does not require calibrating to the CDS curve. The model ensures that the valuation is arbitrage-free relative any arbitrary shapes of the CDS curve. The model can be used to determine the credit duration or DV01 of a credit contingent claim in a straightforward manner. Therefore, the model can be used to provide analytical measures for hedging and other portfolio management purposes. Second the generalized Ho-Lee model for credit risks can be implemented easily to value American, Bermudan options where the roll-back approach provides an efficient valuation algorithm. By way of contrast, the mean reverse process provides a

4 closed form solution for European options but not options with the American feature.

The paper proceeds as follows. In Section B we begin with the valuation of a coupon bond and its CDS. This model does not require a stochastic model of the hazard rate. It relates the CDS premium to the pricing of a coupon bond. Then in Section C we provide a valuation model based on a stochastic movement of the hazard rate and we show that the model is arbitrage-free.

B. Credit Derivative Swap Curve and Credit Volatility Function

Specification of volatilities is central to valuation of options, or contingent claims more generally. In valuing interest rate contingent claims, a class of arbitrage-free models takes the observed yield curve as given. In addition, the models also assume a term structure of volatilities. They are for example, Ho-Lee, Black- Derman-Toy, BGM models. This specification of the term structure of volatilities is independent of the shape of the yield curve. For example, it can be calibrated from the observed at-the-money swaption prices, and the estimated term structure of volatilities is called the “implied volatility function.” This approach can relative value the interest rate contingent claims to the market conditions in terms of both the yield curve and the market volatilities.

By way of contrast, Cox-Ingersoll-Ross (CIR) type of models assume the interest rate for follow a mean reversion process dr= a( b - r ) dt + rs dz where r is the short term interest rate; a is the positive constant adjustment rate; b is the constant long term rate, s s is the constant instant interest rate volatility and dz is the Wiener process. Therefore, the term structure of volatilities is constrained by the specification of the mean reversion process and it cannot take on a broad range of shapes.

The valuation of credit option is analogous to that of interest rate options. The term structure of hazard rates or the conditional default rates can be represented by the credit default swap (CDS) curve. The CDS curve can take on a broad range

5 of shape depending on the market perception of the credit risk of the reference name over the investment horizon.

To illustrate the movements of the CDS curves, we consider the CDS curves of a selected reference names over a historical period. The daily data is taken from August 1, 2007 to February 28, 2008. This period depicts the beginning of the subprime crisis. During this period, the CDS curves typically rise significantly with an increase of volatilities after a period of tight credit spreads and relative calm. Also, during this period, there was an increase in trading activities which result in better reported prices as the markets were more liquid across the reference names. Therefore, this period can demonstrate the possible CDS stochastic movements with reasonable observed CDS prices. The data was obtained from a large Wall Street firm.

Table 1 below shows that the average CDS curves for each reference name over the sample period. The results show that the CDS curves can take on a broad range of shapes. They are typically upward sloping, with the credit premium increases with the term. However, as the subprime crisis deepened, the short term premium of some reference names rose faster and some curve became inverted or humped with the peak at the five year term.

The credit risk of a firm is affected by the “crisis of maturity” of the debt structure, the company’s forecasted earnings and liquidity. The anticipation of these combined factors result in a broad range of shapes of the CDS curves and the credit volatilities.

Table 1 Average CDS Curves (basis points) CDS Terms Average Symbols Reference Name 3 5 7 10 Premium CAEP American Electric Power Co 30.81 43.99 52.50 62.60 47.47 CACE ACE Ltd 34.59 46.76 52.07 60.45 48.47 CCCL Carnival Corp 32.78 51.08 67.49 79.08 57.61 CARW Arrow Electrics Inc 37.08 57.53 69.89 88.20 63.17 CAIG AIG 76.06 77.78 75.53 77.56 76.73 CCOF Capital One Bank 158.92 180.37 173.01 168.98 170.32 CCTX Centex Corp 393.00 369.71 343.84 320.87 356.86

6 Table 2 presents the historical volatilities of the CDS curves in lognormal and normal measures. The lognormal measure is the standard deviation of the daily percentage change of the CDS premiums. The normal measure is the standard deviation of the daily shift of the CDS premiums.

Table 2 Historical Volatilities: Lognormal versus Normal

Symb Reference Historical Volatility Function (basis ol Name Historical Volatility Function (%) point) 3 5 7 10 3 5 7 10 CAEP AEP 6.61 4.35 4.52 4.12 2.04 1.91 2.37 2.58 CACE ACE Ltd 8.73 7.06 7.21 6.62 3.02 3.30 3.75 4.00 11.3 13.1 CCCL Carnival Corp 2 4 4.90 5.77 3.71 6.71 3.30 4.57 CARW Arrow Electrics Inc 9.89 8.40 8.37 8.22 3.67 4.84 5.85 7.25 10.7 CAIG AIG 2 7.50 8.45 8.65 8.15 5.84 6.38 6.71 12.3 18.2 10.2 10.2 19.6 17.6 CCOF Capital One Bank 4 3 1 1 1 32.89 6 17.25 20.5 14.8 CCTX Centex Corp 5.23 4.80 4.32 4.59 6 17.73 6 14.72

The results show that the term structure of volatilities tends to be downward sloping, with the volatilities higher for the short term. However, the volatility curves can peak as in the case of CCOF or trough as in the case of CAIG at the five year term.

These historical experiences suggest that an arbitrage-free credit model should be able to accept the CDS curve and the term structure of credit volatilities independently.

C. Valuation of a Coupon Bond and its Corresponding CDS

The model is a discrete time one factor binomial recombining lattice model where the number of time step is given by T = 0, 1, 2,…, N. The model follows the standard perfect capital market assumptions. For clarity of the exposition, we assume that there is no interest rate uncertainty, and the market term structure of

7 interest rate is represented by the spot yield curve r*(T), and the discount function P(T) is related to the spot yield curve given by

P( T )= exp( - r* ( T ) T ) (1)

Based on the discrete time bond arithmetic, we often find that the yield curve r*(T) can depict the interest rate behavior, but, it is less convenient to use in valuation. Instead we further define, the one period discount factor at time T to be p(T), which is often used in valuation models, where

P( T ) p( T ) = P( T - 1) (2)

A credit derivative swap (CDS) is defined as a swap between the buyer and the seller of protection. Let the swap has tenor T* and a constant recovery ratio of R. The protection seller receives a constant premium ξ(T*) from the protection buyer at the end of each binomial step till T*. At any time, if the reference name defaults at any binomial period, the protection seller pays $ (1 - R) the protection buyer and the swap is terminated. If the reference name survives beyond time T*, then, the swap terminates at time T* without any payments to the protection buyer.

The survival function S(T) is defined as the probability of the reference name to survive till time T. Therefore we note that the function S(T) is analogous to the discount function P(T), in that both functions have value 1 initially (when T=0) and both functions decline monotonically, as the interest rate and the hazard rate are always positive.

We now define the hazard curve h*(T) to be:

S( T )= exp( - h* ( T ) T ) (3)

8 Again the hazard curve is analogous to the yield curve and it is useful in depicting the term structure of credit risk of a bond. Given the survival function S(T), we can define the one period survival factor at time T, s(T), as

S( T ) s( T ) = S( T - 1) (4)

The one period survival factor at time T is the marginal probability of survival, conditional on the bond surviving to period T-1, and therefore

We define the hazard rate h(T) to be the marginal probability of an event of default during the period T,

h( T) =1 - s( T ) (5)

It is also called the conditional default rate (CDR).

Proposition 1 Valuation of a defaultable coupon bond given the survival function

Let the discount function of time value be given by P(n), n = 0, 1, 2…T. Suppose the face value of a coupon bond is 1 with a coupon rate c per binomial period and the survival function of the bond is S(n), n = 0, 1, …. T, where T is the maturity of the bond.

The value of the bond is given as a recursive equation, where the bond value B(n) is given by:

B( n-1) = p( n) 臌轾 s( n)( B( n) + c) +( 1 - s( n)) (1 + c ) R for n = 1, …, T (6) with the terminal condition B(T) = 1

9 The one period discount factor at time n, p(n) and the survival factor at the period n, s(n) are given by equations (2) and (4) respectively.

Proof:

If the bond survives till the end of period T, the bond value is the face value together with the interests. At the beginning of period n, the risk neutral expected value of the bond at the end of the period is the sum of the values of the bond conditional of surviving Bs ( n ) and defaulting Bd ( n ) that period.

Bs ( n )= B ( n ) + c and

Bd ( n )= (1 + c ) R

Then, the present value at the beginning of period n is the expected value discounted by the time value. QED

Proposition 2 Equilibrium Spread of a CDS

The equilibrium CDS spread of ξ(T) is the solution to the set of recursive equations

V( n- 1) = p ( n )[ s ( n ) V ( n ) - (1 - s ( n ))(1 + c )(1 - R ) +x ( T )] (7) for n = 1, … T with the terminal conditions V(T) = V(0) = 0.

Proof:

10 The argument is similar to that of proposition 1. We simply need to note that if the bond survives till time T, the protection seller receives z (T ). For the spread to be “equilibrium”, the initial agreement of the swap has no value. QED

We note that the equilibrium CDS spread is often quoted as

x =(1 -s )(1 - R ) (8)

This equation is correct only under stringent conditions, which are V(n) = 0 , n= 1… T, and c = 0

Corollary:

Given a bond with maturity T and the equilibrium spreads quoted for each tenor n = 1,… T for the bond, then the survival function S(n), n = 1… T is uniquely defined. That is, the CDS curve can be used to value a defaultable bond.

Proof:

Consider the set of T equations (7), there are exactly T unknowns s( n ) , n = 1,…, T, and (T-1) unknowns V(n), n= 1,… T-1. But, there are CDS with tenors n = 1, … T. For each tenor t, we have (t-1) V(n) unknowns. Therefore, including the s( n ) unknowns, there are altogether (T+1)T/2 unknowns.

Now consider equation (7) again. Proposition 2 shows that there are exactly t equations for each CDS with tenor t. Therefore, with CDS tenors n = 1.. T, there are (T+1)T/2 equations and the CDS equilibrium spreads uniquely determine the survival function of a defaultable bond. According to Proposition 1, the survival function also determines the valuation of the defaultable bond. Appendix A provides the detail solutions. QED

Suppose a bond has maturity T and its CDS with tenor T. Then the bond value can be determined relative to the CDS premium and the time value of money. The precise relationship is provided below.

11 Proposition 3 A bond with a CDS protection is equivalent to a bond with scheduled payments.

Proof:

Consider equation (6) and (7), let V* ( n ) be the value of a coupon bond with a CDS protection at time n, then, V* ( n ) is a bond value together with the CDSs,

V* ( n )= B ( n ) - V ( n )

* = p ( n )臌轾 s ( n ) V ( n+ 1) +( 1 - s ( n )) + c -x ( T ) (9) where V* ( T )= 1. The cash flows of the bond is the same as a bond with a prepayment of principal with a conditional prepayment rate given as (1- s ( n )) . That is, let F( n ) be the unpaid balance of the bond at the beginning of the period n, then at the end of the period, the prepayment of principal is F( n )(1- s ( n )) with the interest amount F( n )( c-x ( T )) . The cash-flow of the bond with a CDS at time n before maturity T is s( n ) cF ( n )+( 1 - s ( n )) (1 + c ) RF ( n ) - F ( n )x ( T ) = F ( n )( 1 - s ( n )) + F ( n )( c - x ( T )) where R is one because of the CDS protection.

Proposition 3 shows that a bond with a CDS protection is equivalent to lowering the coupon rate by the CDS spread and imposing a scheduled prepayment of the principal.

C. Arbitrage-free Stochastic Survival Function Models

12 We now extend Section B to incorporate a stochastic survival function, with non- stochastic interest rates. We assume a recombining binomial lattice of the movements of the survival function in that the initial survival function S(n) may move up or down over each binomial step. We say that the survival function movement is arbitrage-free if and only if at any node point, the expected return of a coupon bond equals to the risk free rate, and there is no arbitrage opportunity at any node point on the binomial lattice.

For clarity of the exposition, we first apply the Ho-Lee (1984) arbitrage-free interest rate model, which is a simple arbitrage-free model to describe, to the hazard rate arbitrage-free movements. The Ho-Lee model is a one factor short rate model. Given the interest rate discount function P( T ) and volatility 0.5lnd , the one period discount factor at time n and state i is given by:

P( n + 1) d i p( n , i ) = P( n ) 1+d n (10)

Theorem A shows that Equation (10) can be extended to formulate an analogous arbitrage-free survival function movement model.

Theorem A Let s( n , i ) be the one period survival factor at the node in time n and state i, with i = 0,…, n. Let s( n , i ) be defined by

i S( n ) ds s( n , i ) = n S( n - 1) 1 + ds (11)

where ds is a positive real number less than 1. Then s( n , i ) defines an arbitrage- free survival function movement. That is, when we apply the standard recursive process in valuing a contingent claim to value a zero coupon bond, the value is consistent with valuation without using the stochastic arbitrage-free movements of the one period survival factor, such that the value

13 B(T) = P(T)S(T-1). where B(T) is a zero coupon defaultable bond; P(t) is the value of the zero coupon risk-free bond with maturity t; S(t) is the survival function.

The recursive valuation process is given as follows. We set the terminal condition of the bond value at the maturity T to be X(T,i) = 1. Then, the bond value is determined by the recursive equation below,

P( n + 1) X( n , i )= 0.5[ X ( n + 1, i + 1) + X ( n + 1, i )] s ( n , i ) P( n )

And the initial bond value B(T) = X(0,0).

Proof:

First, we show that the value of a coupon bond derived from the lattice is identical to that derived by the initial survival function. Then we show that at any point on the node, the return of a bond subject to the survival function binomial movement equals to the risk free rate.

We assume that the arbitrage-free movement of the default bond price is given by:

轾(1δ+)(1nδ-1) +(1δ n) - 2⋯ + 1 n S(n-1+T)臌 s s s iT P(n+T) Xi (T)= 2δ T+ n -1 T s (12) S(n-1) (1δ+)s (1δ)⋯ + s P(n) where S(T) is a survival function and P(T) is a discount function.

0 First, we show that X0 (T)=S ( T - 1) P ( T ) , which means that the T-period default bond is at the node in time 0 and state 0 is consistent with the initial survival function and the initial discount function.

14 轾(1δ+) 0- 1 0S(0-1+T)臌 s 0 T P(0+T) X0 (T)= 2δ( 1) ( ) T +0 - 1 s =S T - P T , because S(0-1) (1δ+)s P(0) 0- 1T + 0 - 1 S(0-1)=1, δs =0, δ s =1, and P(0)=1by definition

Second, we have

轾(1δ+)(1nδ) + (1 nδ-1) ⋯ + 1 n+1S(n+T-1)臌 s s s i ( T - 1) P(n+1+T-1) Xi (T-1)= 2δ T+ n -1 T - 1 s (13) S(n) (1δ+)s (1δ)⋯ + s P(n+1)

轾(1δ+)(1nδ) + (1 nδ-1) ⋯ + 1 n+1S(n+T-1)臌 s s s (i+ 1)( T - 1) P(n+1+T-1) Xi+1 (T-1)= 2δ T+ n -1 T s (14) S(n) (1δ+)s (1δ)⋯ + s P(n+1)

nS(n-1+1) 1 i P(n+1) Xi (1)= 2δ n s S(n-1) (1δ+)s P(n) (15)

What we have to show that the arbitrage condition holds in (12) is prove that the following equation holds.

1 Xn (T)= X n (1) [ X n+1 (T-1)+X n+1 (T-1)] (16) i i2 i i+1

Substituting Equation (13), (14) and (15) into the right side of (16) and simplifying, we have

15 n Xi (T) 禳轾(1δ+)(1nδ) + (1 n-1δ) ⋯ + 1 S(n) 1i 1镲 S(n+T-1)臌 s s s i(T-1) (i+1)(T-1) P(n+T) =2δ2δδ ns睚 T+n-1 T-1 ( s + s ) S(n-1) (1δ+) 2 S(n) (1δ) + (1δ)⋯ + P(n) s铪镲 s s 禳轾(1δ+ )(1n δ + n-1)(1 δ + n-2)⋯ (1 δ + ) 1 S(n-1+T) 1镲臌 s s s s iT (T-1) P(n+T) =2δ (1 δ ) 睚 + S(n-1) (1δ+ )n (1 δ+ T+n-1 )⋯ (1 δ+ )(1 T δ+ T-1 )s s P(n) s铪镲 s s s 禳轾(1δ+n-1 )(1 δ+ n-2)⋯ (1 δ + ) 1 S(n-1+T)镲臌 s s s iT P(n+T) = 2δ 睚 s , which is Equation (12) S(n-1) (1δ+T+n-1 )⋯ (1 δ+ ) T P(n) 铪镲 s s QED

It is well known that the Ho-Lee model has certain limitations. The model generates negative interest rates, and hence, the hazard rates can be negative based on the Ho-Lee model in this case. And also, the model has a constant volatility number and does not accept a more general implied volatility function. But these and other limitations have been dealt with by the Generalized Ho – Lee model (2007). Theorem A provides the insights into the extension of the interest rate models to the credit risk model. Theorem B shows that the Generalized Ho- Lee model can also be extended to describe the hazard rate movements.

Theorem B. Let s(n,i) be the one period survival factor at the node in time n and state i. Let s(n,i) be defined analogously to the Generalized Ho-Lee model. Then the survival factor generates an arbitrage-free movement.

Theorem B follows directly by applying the Generalized Ho-Lee model as opposed to the Ho-Lee model can ensure that the hazard rate is always positive and that the model can take any term structure of volatilities. The Generalized Ho- Lee model is provided in Appendix B and the proof that the General Ho-Lee model can be extended to hazard rate movements is provided in Appendix C.

Section D will show that this hazard rate movement model can be used to value credit derivatives, including credit derivative options. Based on the credit

16 derivative option prices, we can use the standard calibration procedure to determine the implied credit volatilities of the reference bond.

D. Valuation of a CDS Option and a Callable Bond.

The valuation of a CDS and a coupon bond does not require the use of stochastic hazard rate. But in general, credit contingent claim requires the use of a stochastic hazard rate model. This section presents such a model. For this section, we assume that the interest rate is certain with the observed discount function P(T). The initial survival function is S(T) and the stochastic movement of the survival function is specified by Theorem 1 or 2, where the one period survival factor is denoted by s(n,i). The reference name is a coupon bond with maturity T and coupon rate c.

A CDS call option gives the holder the right but not the obligation to buy a CDS protection as a strike spread. The CDS option will knock out if the bond defaults before the expiration of the option.

Proposition 4 Valuation of a Credit Derivative Swap Option

We assume that the underlying CDS terminates at time T and the reference name is a coupon bond with maturity T * , where T* > T. The CDS option expires at time t with a strike spread ξ. The valuation model of a CDS option is given by the terminal conditions:

T P( TT ) S ( TT ) B( t , i )= Max轾xt ( T ) - x ,0 c臌 i TT= t P( t ) S ( t ) (17)

t where xi (T ) is the CDS premium at the option expiration date t and state i.

The recursive condition to derive the CDS option value is given below.

17 P( n + 1) Xc( n , i )= 0.5[ B c ( n + 1, i + 1) + B c ( n + 1, i )] s ( n , i ) P( n ) (18) where n=0, ,t-1 and i=0,  n:

where Xc ( n , i ) is defined by rolling back the option value from the maturity date of the option .

The valuation methodology is analogous to the valuation of interest rate contingent claims. The premium leg can be written as the sum of the premium paid by the protection buyer, discounted by the risk-free and the survival probability.

T PVpremiumleg = x (T)鬃 P ( m ) S ( m ) m=1 where P( m ) is the discount function and S( m ) is the survival probability up to time m.

PV premiumleg can be expressed in a binomial lattice denoted by PL( n , i ) . At n= T , PL ( T , i ) = x ( T )� s ( T , i ) where i 0, , T

1P ( n + 1) At n构 T or n 0, PL ( n , i )={ PL ( n+ 1, i + 1) + PL ( n + 1, i )} s ( n , i ) + x ( T ) s ( n , i ) 2P ( n ) where i= 0, , n 1P ( n + 1) At n= 0, PL (0,0)={ PL ( n + 1, i + 1) + PL ( n + 1, i )} s (0,0) where i = 0 2P ( n )

The default leg will be the discounted sum paid by the protection seller, in case of default. It is therefore weighted by the marginal default probability. We assume that the default can occur only on a series of discrete dates. T PVdefaultleg =(1 - R )� { S ( m - 1) S ( m )} P ( m ) m=1

18 PV defaultleg can be expressed in a binomial lattice denoted by DL( n , i ) .

T At n= T , DL ( T , i ) = (1 - R )(1 - si (1)) where i = 0, , T

1P ( n + 1) At n构 T or n 0, DL ( n , i )={ DL ( n+ 1, i + 1) + DL ( n + 1, i )} s ( n , i ) + (1 - R )(1 - s ( n , i )) 2P ( n ) where i= 0, , n 1P ( n + 1) At n= 0, DL (0,0)={ DL ( n + 1, i + 1) + DL ( n + 1, i )} s (0,0) where i = 0 2P ( n )

By imposing the equality between the two legs, we can solve for x (T) at year 0.

T (1-R )� { S ( m - 1) S ( m )} P ( m ) m=1 x (T) = T P( m ) S ( m ) m=1

i x (T) also can be expressed in a premium binomial lattice denoted by xn (T ) at each node.1 DL( n , i ) x i (T ) = n PL( n , i )

Once we generate the premium lattice and the default lattice, we can determine the CDS premium at each node by dividing the default leg by the premium leg.

To illustrate the valuation of a CDS European call option, we use the following assumptions in this numerical example. (1) The yield curve and the hazard curve are flat with continuously compounding rates of 5% and 1% respectively. The s hazard rate volatility ( h* ) is assumed to be 10%. (2) The bond face value is 1and recovery rate is 40%. (3) The option expires in year 3. (4) The CDS

1 0 We can see that x0 (T ) is equal tox (T) .

19 terminates in year 5 with a strike price of 50bp; (5) the CDS curve moves according to the Ho-Lee model (1984).

Based on these assumptions, Proposition 2 shows that the CDS curve is constant at 60.301bp.

Figure 1 The CDS premium Binomial Lattice

90.75 84.65 78.55 78.58 72.46 72.49 66.33 66.41 66.44 60.301 60.33 60.36 54.26 54.28 54.31 48.22 48.25 42.19 42.22 36.16 30.15 Today (yr) 1 2 3 4 5

Figure2 The CDS Call Option Binomial Lattice

158.63 117.39 80.69 91.16 53.04 54.14 30.82 23.82 11.24 0.00

Today (yr) 1 2 3

20 The terminal conditions (Equation (17)) are applied to the CDS call option at the end of year three, and then Equation (18) is used to rolling back to determine the CDS call option price at time 0.

We compare the CDS premium with the exercise premium at each node at year 3 and roll back until year 0 to get the CDS call option premium. The first lattice and the second lattice show the CDS premium from year 3 to year 5 and the CDS call option premium from year 0 to year 3, respectively.

This same methodology can be extended to value defaultable callable bonds. We consider two kinds of callable bond here: the refinancing callable bond and the make whole callable bond. A refinancing callable bond has an embedded interest rate option as well as a credit risk option as it gives the issuer the right but not the obligation to buy back the bonds according to a fixed call price schedule over time. The make whole callable bond gives the right to the issuer to replace the bond by a Treasury bonds with the same maturity and a pre-specified coupon rate.

The bond issuer would call the refinancing callable bond when the interest rate falls or the credit risk has improved (the hazard rate has fallen) resulting in a higher price. By way of contrast, since a make whole bond requires the issuer to replace the bond with an equivalent treasury bond, the issuer has no incentive to call a make-whole bond for the refinancing purpose and therefore the interest rate option has negligible value. However, when the firm improves in credit, then the issuer may exercise the make whole option. Therefore, the make whole provision provides a credit risk option to the issuer but not an interest rate option.

A callable bond gives the issuer the right but not the obligation to buy back the bond at a pre-specified schedule of prices, the call schedule, kn , n =1, … T. Then the bond can be valued by the recursive equations

21 P( n + 1) Xc( n , i )= 0.5[ B c ( n + 1, i + 1) + B c ( n + 1, i )] s ( n , i ) , P( n ) n=1, , T and i = 0,  , n with the boundary condition,

Bc( n , i )= Min[ X c ( n , i ), k n ] + c

A make-whole callable bond gives the issuer the right but not the obligation to buy back the bond by replacing the bond that has the same promised payment with a fixed make-whole premium (MWP). Then the bond can be valued by the recursive equations

P( n + 1) Xmwc( n , i )= 0.5[ B mwc ( n + 1, i + 1) + B mwc ( n + 1, i )] s ( n , i ) , P( n ) n=1, , T and i = 0,  , n with the boundary condition,

Bmwc( n , i )= Min[ X mwc ( n , i ), kk n ] + c T- n c F where kkn =i + T- n and MWP = make-whole i=1 (1+r + MWP ) (1 + r + MWP ) premium and r is the appropriate risk free rate.

The make-whole redemption price is equal to the sum of the present value of the remaining coupon and principal payments discounted at an Adjusted Treasury Rate (ATR)2 plus the make-whole premium. We assume that ATR is the risk-free rate at each node.

E. Performance Profiles of Credit Contingent Claims

2 The Adjusted Treasury Rate is calculated by selecting a US Treasury security having a maturity comparable to the remaining maturity of the bond to be redeemed. An average price over multiple primary dealers is used to calculate a bond equivalent yield which becomes the Adjusted Treasury Rate.

22 Since the valuation of credit contingent claims presented in this paper is based on the market observed term structures of interest rates and the hazard rates, we can extend some of the standardized analytical tools used by the arbitrage-free interest rate model framework to analyze the credit options or credit embedded options. For example, we can analyze the CDS call options using the performance profile which depicts the impact of the shifts of the hazard curves on the credit contingent claim value. Performance profile can depict the behavior of the behavior of the contingent claims.

In our simulations, we use the same market assumptions in Section D. In addition, we assume that the face value of the bond is $100 with a 6% coupon rate and maturity of 10 years.

For the credit default swap, we assume that the CDS has a tenor of five years. For the option on the credit derivative swap, we assume that the time to expiration is three years. The call exercise price is 0.0005. The put exercise price is 0.0015.

We simulate the CDS call option to generate the performance profiles. We change the flat hazard curve from 0 to 3% with an increment of 0.2%. We use the Ho-Lee s (1984) model with a constant hazard rate volatility ( h* ) of 10% over the d=Exp -2 s . binomial period, and therefore ( h* )

Figure 3 The Performance Profile of the CDS Call Option,

23 Call Option on CDS

700 600 500

e 400 c Call OPtion on i r

p 300 CDS 200 100 0 0 0.01 0.02 0.03 hazard rate

The performance profile shows that the CDS call option falls with the hazard rate but the value does not become negative, as an option would behave.

We now use the credit contingent claim model to study the performance profiles of bonds. The callable bond and the make-whole callable bond have a five year call protection period and are callable at every coupon payment date at a semi-annual interval. The call price of the callable bond is constant at 100. The make-whole premium is assumed to be 45 basis points.

The call prices of the make-whole can be calculated using the risk-free rate and the make-whole premium. Since we have assumed that there is no interest rate risk, a make-whole callable provision can be viewed as a bond with a call schedule. The call schedule is (102.425, 102.211, 101.992, 101.534, 101.295, 101.05, 100.798, 100.539, 100.273, 100.000) at six months apart from the first call date to the maturity date.

Using the valuation model, we can determine the value of the bonds. The price of the coupon bond, the make-whole callable bond and the callable bond at the base case are 103.034, 103.011 and 101.926, respectively. The performance profiles are presented below.

24 Figure 4 The Performance Profile of the coupon Bond, Callable, Make- Whole Callable Bond

Performance Profile

s Straight Bond

e 104 c i

r Make-Whole p 102 Callable 100

98 0.000 0.004 0.008 0.012 0.016 0.020

hazard rate

The results show that the coupon bond value rises linearly with the fall of the hazard rates. However, the make whole bond and the callable bond exhibit a “negative convexity” when the hazard rates are low, as the bonds would be called or made-whole. The value of the call options in both cases would depend on the call provisions. In this paper, we have assumed that there is no interest rate risk and therefore a make-whole bond is equivalent to a callable bond, in that they both have a preset call schedule. However, when there is interest rate risk, then the make-whole callable bond would isolate the interest rate risk from the credit risk. The results further show that the specification of the call schedules of a callable bond and the make-whole premium of a make whole bond can ensure that the two types of bonds can be made equivalent from the credit risk perspective.

F. Conclusions

This paper proposes an arbitrage-free valuation model for credit contingent claims based on a binomial lattice model. The model can value a broad range of credit

25 contingent claims by ensuring the stochastic movements of the hazard curve are arbitrage-free relative to the given yield curve and the hazard curve.

This model can value a broad range of credit contingent claims. These credit contingent claims include the American and Bermudan CDS options, make-whole and callable bonds.

We have also shown that many interest rate analytics can be extended to credit analysis. For example, we use the model to depict the credit performance profile of a bond, by plotting the credit contingent claim values over a range of hazard curves. The performance profiles can identify the impact of the credit risks on the contingent claims.

The proposed model can be extended to provide many applications. The discussions of these extensions are beyond the scope of this paper, but are explained in other papers. For example, by taking the hazard curve as given in the valuation of credit contingent claims, Ho and Lee (2008) can show that the model can be used to extend the specification of key rate durations, widely used in managing interest rate risk, to credit key rate durations. As a result, credit derivatives swaps can provide effective dynamic hedging strategies to credit options. Ho and Lee (2008) shows that the model can also be extended to a portfolio of credit securities and the model can be used to value CDX and other structured products, providing an alternative approach to the copula model in valuation. Finally, the model can be extended to incorporate arbitrage-free interest rate movement, valuing securities that are subject to both interest rate and credit risks. See Ho and Lee (2008). This unified model suggests new approaches to manage the combined interest rate risks and credit risks, and not managing the risks separately.

26 Appendix A Determining the Survival Function from the CDS Spreads

To determine the survival factors from the portfolio of CDS, we use a bootstrap procedure taking the CSD spreads and a recovery rate as inputs. Starting from year 1, we can solve the following equation. T =1

V1(0)= p (1)臌轾 s (1) V 1 (1) -( 1 - s (1)) (1 + c )(1 - R ) +x (1) (A1)

From Equation (A1), we can uniquely determine s(1) , because V1(0) and V1 (1) are zero.

Then, we use the survival factor during year 1 to obtain the survival factor during for year 2.

V2(0)= p (1)臌轾 s (1) V 2 (1) -( 1 - s (1)) (1 + c )(1 - R ) +x (2) (A2)

V2(1)= p (2)臌轾 s (2) V 2 (2) -( 1 - s (2)) (1 + c )(1 - R ) +x (2) (A3)

From Equation (A2), we can uniquely determineV2 (1) , because V2 (0) is zero and s(1) is previously determined.

From Equation (A3), we can uniquely determine s(2) , because V2 (2) is zero and

V2 (1) is previously determined.

Then, we use the survival factor during year 1 and year 2 to obtain the survival factor during year 3.

27 V3(0)= p (1)臌轾 s (1) V 3 (1) -( 1 - s (1)) (1 + c )(1 - R ) +x (3) (A4)

V3(1)= p (2)臌轾 s (2) V 3 (2) -( 1 - s (2)) (1 + c )(1 - R ) +x (3) (A5)

V3(2)= p (3)臌轾 s (3) V 3 (3) -( 1 - s (3)) (1 + c )(1 - R ) +x (3) (A6)

From Equation (A4), we can uniquely determineV3 (1) , because V3 (0) is zero and s(1) is previously determined. Similarly, from Equation (A5), we can uniquely determineV3 (2) , because V3 (1) and s(2) are previously determined. From Equation

(A6), we can uniquely determine s(3) , because V3 (3) is zero and V3 (2) is previously determined.3

We can continue until we get s( T ) , which is the survival factor during year T. Once we determine the survival factor, s( n ), n= 1,2, T , we can get the survival function S( n ), n= 1,2, T using Equation (4)

3 At year 3, we have 6 unknowns ( s(1), s (2), s (3), V2 (1), V 3 (1), V 3 (2) ) and 6 equations, which uniquely determine 6 unknowns.

28 Appendix B: Two Factor Credit Arbitrage-Free Model

n Let si, j ( T ) be the T year survival probability at time n, at state (i, j). Then the survival probability is specified by combining two one-factor models. Specifically, we have

S( n- 1 + T ) n骣1+dk-1 (n - k ) n 骣 1 + d k - 1 ( n - k ) i-1 j-1 sn( T )= 0,1 0,2 d n-1 ( T ) d n - 1 ( T ) i, j照琪k-1 琪 k - 1 照 k ,1 k ,2 S( n- 1)k=1桫 1 +d0,1 ( n - k + T ) k = 1 桫 1 + d 0,2 ( n - k + T ) k = 0 k = 0 (B.1) where 骣1+d n+1 (T - 1) dn(T )= d n d n+1 ( T - 1)琪 i+1,1 i,1 i ,1 i ,1 琪 n+1 桫1+di,1 (T - 1)

骣1+d n+1 (T - 1) dn(T )= d n d n+1 ( T - 1)琪 i+1,2 i,2 i ,2 i ,2 琪 n+1 桫1+di,2 (T - 1) (B.2) and the one period forward volatilities are given by definition,

m m m 3/ 2 di,1(1)= d i ,1 = exp( - 2譊 s 1 (m ) min( h i ,1 , H) t )

m m m 3/ 2 di,2(1)= d i ,2 = exp( - 2譊 s 2 (m ) min( h i ,2 , H) t ) (B.3)

where the functions s j (n )= ( a + bn )exp( - cn) + d is specified by the parameters a, b, c, and d, which can be obtained from the calibration to the market price of an option on CDS. This specification of the implied volatility function allows for a broad range of shapes including downward sloping or dumped shape.

Using the direct extension, we can specify the one period hazard rates for the two- m m factor model for any future period m and state i, and hi,1 and hi,2 are defined by

29 骣 S( m ) m骣1+d k-1 (m - k ) i-1 hmD t = -log - log0,1 - logd m-1 (1) i,1琪邋琪 k-1 ( k ,1 ) 桫S( m- 1)k=1桫 1 +d0,1 ( m - k + 1) k = 0 . (B.4) 骣 S( m ) m骣1+d k-1 (m - k ) i-1 hmD t = -log - log0,2 - logd m-1 (1) i,2琪邋琪 k-1 ( k ,2 ) 桫S( m- 1)k=1桫 1 +d0,2 ( m - k + 1) k = 0

For a security with no coupons or other interim payments, we can apply the recursive equation below to determine the initial value X(0,0,0), given the terminal conditions X(T, i, j) for 0 < i, j < T

X( n , i , j )= 0.25[ X ( n + 1, i + 1, j + 1) + X ( n + 1, i , j + 1) + X ( n + 1, i + 1, j ) + X ( n + 1, i , j )] P( n + 1) s( n , i , j ) P( n )

30 Appendix C The Arbitrage-free Condition of the Credit Model.

To show that the credit model is arbitrage-free, we need to show that 0 0 X0 ( T )= S ( T - 1) P ( T ) , where DB0 ( T ) is a zero-coupon default bond price with the time to maturity T at time n and state i.

The zero coupon default bond price with the time to maturity T at time n and state i

nk -1 i-1 nS( n- 1 + T )(1+d0 (n - k )) n-1 P ( n + T ) Xi( T )= 照 k -1 d j ( T ) (C.1) S( n- 1)k=1 (1 +d0 ( n - k + T )) j = 0 P ( n )

The zero-coupon default bond with the time to maturity 1 at time n and state i

nk -1 i-1 nS( n )(1+d0 (n - k )) n-1 P ( n + 1) X i(1)= 照 k -1 d j (1) S( n- 1)k=1 (1 +d0 ( n - k + 1)) j = 0 P ( n )

n Xi ( T ) Divide Equation (C.1) by Equation (C.2) to get n X i (1)

i-1 d n-1(T ) nn k -1 j Xi ( T )S( n- 1 + T ) (1+d0 ( n - k + 1)) j=0 P ( n + T ) n= k -1 i-1 (C.3) Xi (1) S ( n )k =1 (1+d0 ( n - k + T ))n-1 P ( n + 1) d j (1) j=0

The zero-coupon default bond with the time to maturity T-1 at time n+1and state i

31 n+1k -1 i - 1 n+1 S( n+ T - 1)(1+d0 (n + 1 - k )) n P ( n + T ) Xi( T- 1) =照 k -1 d j ( T - 1) S( n )k=1 (1+d0 ( n + 1 - k + T - 1)) j = 0 P ( n + 1)

nk-1 n i-1 S( n+ T - 1)(1+d0 (n + 1 - k )) (1 + d 0 ( n + 1 - n - 1)) n P ( n + T ) = 照 k-1 � n d j (T 1) S( n )k=1 (1+d0 ( n + 1 - k + T - 1)) (1 + d 0 ( n + 1 - n - 1 + T - 1)) j = 0 P ( n + 1)

nk-1 n i-1 S( n+ T - 1)(1+d0 (n + 1 - k )) (1 + d 0 (0)) n P ( n + T ) = 照 k-1 � n d j (T 1) S( n )k=1 (1+d0 ( n - k + T )) (1 + d 0 ( T - 1)) j = 0 P ( n + 1)

nk-1 i-1 n-1 n S( n+ T - 1)(1+d0 (n + 1 - k )) 2dj(T ) (1+ d j ( T - 1)) P ( n + T ) = 照 k-1 n n - 1 n S( n )k=1 (1+d0 ( n - k + T )) (1 + d 0 ( T - 1)) j = 0 dj (1 + d j+ 1 ( T )) P ( n + 1)

(1+d n (T - 1)) n-1 n - 1 n j+1 4 Qdj(T )= d j d j ( T - 1) n from Equation (13) (1+d j (T - 1))

nk-1 i-1 n-1 S( n+ T - 1)(1+d0 (n + 1 - k )) d j (T ) 2 P ( n + T ) = 照 k-1 n - 1 n (C.5) S( n )k=1 (1+d0 ( n - k + T )) j = 0 dj (1 + d i ( T - 1)) P ( n + 1)

4 Thomas S. Y. Ho and Sang Bin Lee, "Generalized Ho-Lee Model: A Multi- Factor State-Time Dependent Implied Volatility Function Approach", Journal of Fixed Income, Vol. 17, No.3, pp.18-37, 2007, U.S.A.

32 i-1 d n-1(T ) nn k -1 j Xi ( T )S( n- 1 + T ) (1+d0 ( n - k + 1)) j=0 P ( n + T ) n= k -1 i-1 Xi (1) S ( n )k =1 (1+d0 ( n - k + T ))n-1 P ( n + 1) d j (1) j=0

Comparing Equation (C.5) with Equation (C.3), we see that

n n+1 Xi ( T ) 2 Xi ( T - 1) = n n (C.6) Xi(1) (1+d i ( T - 1))

Similarly, we can show that

n n n+1 Xi( T ) 2 d i ( T - 1) Xi+1 ( T - 1) = n n (C.7) Xi(1) (1+d i ( T - 1))

We need to show that the arbitrage-free condition holds, that is,

n1 n n+1 n + 1 Xi( T )= X i (1){ X i ( T - 1) + X i+1 ( T - 1)} (C.8) 2

Plugging Equation (C.6) and Equation (C.7) into Equation (C.8), we can see that

Equation (C.8) holds.

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