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Pricing of Defaultable Coupon Bonds When Firm Values and Interest Rates are Stochastic German Finance Association Sixth Annual Meeting, Aachen, Germany Mark C. W. Wong Financial Options Research Centre Warwick Business School, University of Warwick, Coventry, CV4 7AL, UK Off: 44 (024) 7657 2572 ext 24691 Res: 44 (024) 7657 2572 ext 30365 Fax: 44 (024) 7652 4650 Email: [email protected] Stewart D. Hodges Financial Options Research Centre Warwick Business School, University of Warwick, Coventry, CV4 7AL, UK Off: 44 (024) 7652 4118 Fax: 44 (024) 7652 4167 Email: [email protected] June, 17 Title: Name: Address: Phone: Fax: Email: Pricing of defaultable bonds when interest rates and firm values are stochastic Mark Wong, Stewart D. Hodges Warwick Business School, University of Warwick, CV4 7AL, UK (Off) 44 (0)1203 572572 ext 24465, (Res) 44 (0)1203 572572 ext 30365 44 (0)1203 524650 [email protected] ABSTRACT This paper develops a flexible binomial approach to valuing risky debt when interest rates and firm values are stochastic. By using a hypothetical asset as a numeraire, it can be shown that not only does the use of this numeraire significantly simplify the analytic valuation of risky discount bonds, but also gives an implication that the computations of two-factor model can be implemented easily. By extending the method employed by Ho, Stapleton and Subrahmanyam (1995), we suggest an efficient computation algorithm for the pricing of general risky coupon bonds. We investigate the properties of defaultable bonds in an economy of stochastic interest rates. Interactions of market and credit risk are also discussed in this paper. This approach has a number of interesting implications, including: Firstly, the recovery rate of a defaulted bond is generated endogenously depending on the remaining values of the firm. Secondly, our model integrates market and credit risk together to allow for a more complete picture of the underlying risks, as Longstaff and Schwartz (1995) provides strong evidence that both default risk and interest rate risk are necessary components for a valuation model for corporate debt. We show that when the firm value is low, its credit spread is more sensitive to the changes in interest rates. This confirms our intuition that firms with low credit quality should have more market risk than firms with high credit quality. Thirdly, while providing conceptual insights into default behaviour, the flexibility of our method allows for efficient pricing of bond options, credit risk put options on a general defaultable coupon bond and floating rate notes. Fourthly, this structural approach also paves the way for a further analysis of more complicated debt structures. Keywords: Defaultable Coupon Bonds, Stochastic Interest Rates, Stochastic Firm Values, Numeraire, Binomial Method, Market and Credit Risk. 1. INTRODUCTION In a seminal paper Black and Scholes (1973) show how to price simple European call and put options on a non-dividend paying stock. They suggest that almost all corporate liablities can be viewed as combinations of options, and hence their analysis can be used to value corporate liabilities such as shares, corporate loans and warrants. The literature on the valuation of defaultable term-structure started with Merton (1974). In this paper Merton adopt the Black and Scholes option pricing model to the pricing of risky discount bond. Under the assumption of a constant interest rates economy, Merton’s model yields important insight into the determinants of the risk structure, and shows how the default risk premium is affected by changes in the firm’s business risk, debt maturity and the prevailing interest rate. As a generalization to Merton’s model of defaultable discount bond, Geske (1977) applies the technique for valuing compound options to the problem of risky coupon bonds. He derives an analytic formula, which consists of multivariate normal integrals with dimensions up to the total number of contractual payments. It is shown that with a special auto-correlation structure, an application of an integral reduction may simplify the numerical computations. Subsequently, in a further paper Geske (1979) develop a general theory for pricing compound options in terms of multivariate normal integrals. In practice, a wide variety of important problems have turned out to be very closely related to the valuation of compound options. In 1983, Selby in his Ph.D. thesis generalizes Geske’s (1977) work on pricing risky discrete coupon bonds in three ways. First, a continuous dividend, as a known proportion of the firm value, is paid to the equity holders. Second, he derives a general valuation formula for valuing individual risky discrete coupon bonds, or tranches of such bonds with equal seniority. Third, he derives general formulae for valuing senior and subordinate bonds with discrete coupons when two alternative default clauses, liquidation and reorganization in the event of a default, are considered. By using the preference-independent approach first suggested by Cox and Ross (1976), the derivations become much simpler than in the alternative backwards recursion techniques employed by Geske (1977). Shimko, Tjima and Van Deventer (1993) generalizes Merton’s risky debt pricing model to allow for stochastic interest rates as in Vasicek (1977). The method of analysis is based on Merton’s (1973) earlier work on valuation of options with stochastic interest rates and timevarying volatility. They obtain a risky discount bond pricing formula, which yields comparative static results in consistence with Merton’s (1974). They also show that the combined effect of term structure of interest rates and credit variables are significantly important for risky bond pricing. Each of the valuation formulae for risky coupon bonds developed has a number of common features. From the viewpoint of economic and computational implementation, one common point of paramount importance is that each valuation formula gives rise to a sum of multinormal distribution functions. Moreover, the multinormal distributions are nested, in the sense that the integration region at any stage is dependent of those at the stage of higher dimension. Selby and Hodges (1987) prove a general identity relating sums of nested multinormal distributions. By reducing the number of integrals to be evaluated, the application of the identity significantly improves the computational aspects of both the Geske and Johnson’s (1984) analytical American put and the Roll’s (1977) formula for American call option. However, all these results are more of theoretical interest than practical use. Although mathematicians have come up with a lot of more efficient algorithms, the computation of high dimensional normal integrals has yet been a very challenging problem. On the numerical solution of Geske’s (1977) formula, the computation is not as onerous as would first appear. In the one-factor case where a firm’s value is modeled as a geometric Brownian motion and interest rate is taken as a constant, the prices of a defaultable coupon bond can be computed efficiently and accurately by building a binomial tree for the firm value. Instead of modeling a recovery rate as endogenously given, Longstaff and Schwartz (1995) adopt a new approach to valuing risky debt by extending Merton’s (1973) and Black and Cox’s (1976) models in two ways. Firstly, their model incorporates both default risk and interest rate risk. Secondly, they derive the model in such a way that allows for deviations from strict absolute priority. Given the recovery rate as a constant, analytic formulae for fixed rate debt and floating rate debt are derived. By construction the ratio of the firm value to the face amount of the debt is a sufficient statistic for default risk in this model, in order to value a coupon bond it is not necessary to condition on the pattern of cash payments to be made before the maturity. As a consequence, coupon bonds can be valued as simple portfolios of discount bonds. Though this is a result that provides much of the tractability of the model, it is highly questionable whether or not it is reasonable to retain the linear properties in cash flows for the modeling of defaultable coupon bonds. Furthermore, the implication that a firm has a constant value upon default in the typical diffusion approach is problematic. On the one hand, this approach emphasizes the central role of firm value in the determination of default. On the other hand, the approach can not allow the variation in the recovery rate of a risky bond to depend on the firm’s remaining value at default. In addition, the empirical results suggest that the implications of this valuation model are consistent with the properties of credit spreads implicit in Moody’s corporate bond yield averages, which in turn provide strong evidence that both default risk and interest rate risk are necessary components for a valuation model for corporate debt. This paper is structured as follows. Section 2 presents a set of assumptions under which risky discount bonds are priced. By using a hypothetical asset as a numeraire, we show that not only does the use of this numeraire significantly simplify the analytic valuation of risky discount bonds, but also gives an implication that the computations of two-factor model can be implemented easily. Section 3 derives some basic properties of two underlying processes. In section 4, we propose an algorithm for constructing the binomial processes in section 3 by extending a method suggested by Ho, Stapleton and Subrahmanyam (1995). The method is to approximate a bivariate lognormal distribution by a bivariate binomial process. Based on Geske’s idea, we can price a defaultable coupon bond efficiently by using a three dimensional lattice. A brief discussion on Geske’s method of evaluating defaultable coupon bonds is given in section 5. Section 6 presents an efficient computation algorithm for the pricing of general risky coupon bonds. We generalize Geske’s (1977) and Selby’s (1983) valuation models of risky coupon bonds to allow for stochastic interest rates as in Vasicek (1977). Section 7 illustrates the efficiency of our computation algorithm. We investigate the properties of defaultable bonds in an economy of stochastic interest rates. Discussions on interaction of market and credit risk are given in this section. Section 8 is devoted to further applications of the framework. Section 9 concludes. 2. VALUATION OF RISKY DISCOUNT BONDS 2.1 Assumptions and Notation In this section we shall consider a firm which has one discount bond and has no other form of loan. We make the standard assumptions as follows: (A1): Frictionless Markets Trading in assets takes place continuously in time, There are no taxes, bankruptcy, agency or transaction costs, nor are there problems of indivisibility of assets, Every individual acts as through the market price is independent of the amount bought or sold, Borrowing and lending are at the same cost, Short sales are permitted, as is full use of the proceeds. (A2): The Term Structure The term structure of interest rates, r , is assumed to follow Vasicek’s interest rate model. (A3): Firm Value Process The dynamics for the value of the firm, V , follows a Geometric Brownian Motion with instantaneous standard derivation, V , where is nonstochastic and is known. The firm value is assumed to be a traded asset, and the correlation coefficient between the interest rate process and the firm value process is a constant denoted by . (A4): Maturity Payment Maturity payment, , is financed by rights issues only taken up by existing shareholders. (A5): Dividends Shareholders are entitled to receive a continuous dividend which is a constant proportion of the value of the assets of the firm. 2.2 Models Merton (1974) first derives the value of a pure discount corporate bond by employing assumption (A3). Following Merton’s (1973) derivation of Black and Scholes model, he considers forming a three-securities portfolio containing the firm, the corporate bond and a riskless debt such that the total investment in the portfolio is zero. By usual no-arbitrage arguments, a parabolic partial differential equation for the corporate bond is derived. Having observed that the differential equation is identical to Black and Scholes equation (1973, p.643, (7)) for a European call option on a non-dividend paying stock, where firm value in his equation corresponds to stock price and maturity payment corresponds to exercise price, the isomorphic price relationship between equity of the firm and a call option immediately allows him to write down the solution to his equation directly. An alternative to Merton’s (1974) method of analysis is using a probabilistic approach to pricing corporate bonds, we discuss this approach as follows. By assumption (A2) and (A3), we can express the stochastic processes of interest rates and the firm values in the following forms drt ( rt )dt dBtr (1) dVt ( rt a )Vt dt Vt dBt (2) where Btr , Bt 1are two standard Brownian motions under an equivalent martingale measure Q 2, and satisfy cor ( Btr , Bt ) . is a constant depending on the market price of risk of interest rate risk 3. t Let t exp( rudu) be the money market account at time t by starting 1 unit of cash at 0 time 0. If D(Vt , rt , t ) 4represents the price of the risky discount bond at time t, then the relative price of D(Vt , rt , t ) to t is a Q -martingale. We write D(Vs , rs , s) s EQ[ D(Vt , rt , t ) t s ] where the expectation is under Q conditional on the information set s , for any t s 0 . 1 Mathematically Bt r Bt 1 2 Bt' , where Bt , Bt' are two independent standard Brownian motions. 2 A measure that is equivalent to the original objective measure. It is also known as a risk neutral measure. The existence of this measure is guaranteed by no-arbitrage arguments. See Harrison and Kreps (1979), and Harrison and Pliska (1981). 3 If rt is the drift term under the original objective measure, then where 0 is the market price of risk of interest rates. In general two market prices of risk are needed in our setting, however, by the assumption that the firm value is a traded asset, only the market price of interest rate risk appears. 4 D(Vt , rt , t ) must be a C 2 function with respect to the first and second coordinates. If the risky discount bond matures at time T , then the price of the bond at time 0 is given by D(V0 , r0 ,0) E Q [ Min (VT , K ) T 0 ] . This means that the probabilistic approach to pricing a corporate bond is essentially a problem of computing an expectation under Q , which is in turn a problem of computing multivariate normal integrals. The practicality of this approach lies in whether we can express VT and T 5 in form of simple expressions. In the case of Merton (1974) where interest rate is assumed to be constant, the expectation is a univariate normal integral, the computation of which is trivial. In our case of two-factor model, the problem becomes handling of bivariate normal integrals. Though it is known that when the integration regions are of some particular forms high dimensional integrals can be reduced to ones with lower dimensions, introduction of stochastic interest rates necessarily complicates the computation issues. 2.3 Change of Numeraire Most of the bond pricing models we mentioned before are based on the approach of using money market account as a numeraire, associated with which we have the risk-neutral measure Q . While this convenient choice of numeraire leads to an intuitive idea of risk-neutral pricing for asset pricing problems, it is by no means necessary. For example, Geman, Karoui and Rochet (1995) shows that many other probability measures can be defined in a similar way to solve option pricing problems. In this section, we shall discuss the use of an alternative numeraire which provides computational convenience for bond pricing problems. By assumption (A5) we assume that continuous dividends are paid at a constant rate of a per unit of the firm value. The solution of equation (2) is expressed as follows, 1 Vt V0 t exp( at 2t Bt ) . 2 Because of the presence of a which is in general non-zero, it is not difficult to see that the relative price of Vt to the money market account is not a Q -martingale. However, by assuming the existence of a hypothetical asset Vt Vt exp(at ) 6, we see that H Note that VT and T are log-normally distributed under Q . It can be regarded as the value of an identical firm that pays no dividends to its shareholders. This assumption is in fact not important, for we merely want to find out a stochastic process, which has nice 5 6 Vt H 1 exp( 2t Bt ) t 2 is a Q -martingale. Now let us quote without proof a well-known result in probability theory which is central to the analysis of changes of measure7. Let Given two equivalent measures P1 and P2 . LEMMA 1 dP2 be the RadondP1 Nikody`m derivative of P2 relative to P1 . Then for any random variable X integrable with respect to (, P2 ) , the following abstraction of Bayes theorem holds E P1 [ X ] E P1 [ ]E P2 [ X ] . This lemma suggests that under some regularity conditions if there exists an equivalent measure QV such that the following holds EQ[ D(Vt , rt , t ) t V s ] E Q [ t for any t s 0 , then the relative price H t s ] E Q [ V D(Vt , rt , t ) s ] , H Vt D(Vt , rt , t ) is a QV -martingale. The pricing problem H Vt is now turned into finding the new measure QV . The following theorem guarantees the existence of such a measure, which can be obtained by a simple transformation of standard Brownian motions. THEOREM 2 Under the above assumptions on the processes of the interest rates and the firm value, there exists a measure transformation BtV t Bt , Proof t Vt H QV equivalent to Q such that under the t V H is a Q -martingale. Vt 1 1 exp( 2t Bt ) . By Ito lemma we have V0 2 mathematical properties such that the relative price of the bond to this process is a martingale under another measure. 7 See Musiela and Rutkowski (1997), p458. d t Vt H t Vt H ( dt dBt ) . Then since is a constant, then by virtue of Girsanov’s theorem there exists an equivalent martingale d t Vt H QV measure t Vt H such BtV t Bt that V dBt is driftless, this implies that is a QV -martingale. Since t H is an exponential martingale. Hence Vt result follows. 2.4 Valuation of Risky Discount Bonds With the results in the last section, the valuation of the risky discount bond becomes fairly straightforward. By applying LEMMA 1, we have V EQ [ V D(Vt , rt , t ) D(Vt , rt , t ) s ] E Q [ Ht s ]E Q [ s ] . H t Vt Vt 1 2 T Define a normal random variable ZT rudu 2T BTV . Since by THEOREM 2 0 a QV -martingale and the relative price of D(Vt , rt , t ) to t is a Q -martingale, t H is Vt D(Vt , rt , t ) is a H Vt QV -martingale, which implies that D(V0 , r0 ,0) V0 E Q [ V V Min (VT , K ) K 0 ] V0 E Q [ Min (exp( aT ), exp( ZT )) 0 ] H V0 VT (3) Suppose that ZT has mean T and variance T2 , then the price of the defaultable discount bond at time T is given by8 D(V0 , r0 ,0) V0e aT N [ Log (V0 / K ) aT T T ] KP(T ) N [ Log (V0 / K ) aT T T T ] (4) where N [.] is the cumulative normal distribution function. If P(T ) is the price of a default-free zero coupon bond with the face amount of unity, then P(T ) exp[ R ( ) 8 1 e T ( R() r0 ) TR() 2 (1 e T )2 ] , 3 4 1 2 , 2 2 The computations are essentially the same as those required for a single factor model. aT bT2 9 1 (1 e T )[ r0 T e T 1 (e 1)] ( T) 2 2 2 2 2 T T ( 1 e ) (1 e T ) 2 3 3 2 1 2 1 e T T aT T ( T) 2 10 T2 bT2 2T 2 1 e T ( T) We can consider T2 as the integrated instantaneous variance of ZT over the life of the risky debt. It is interesting to note that T2 is a quadratic function of . For typical values of the parameters and T , this function has a positive coefficient in the leading term. This implies that T2 is an increasing function of when is positive, and T2 attains its minimum value at some positive value of when is negative. For small values of 11, an important implication on the bond prices is that a decreasing trend is expected when the firm value has a positive correlation with the interest rates. On the contrary when the firm value has a negative correlation with the interest rates, an increasing trend in bond prices is expected. Let B(T ) KP(T ) be the price of a risk-free discount bond, which pays K at maturity time T . Then we can rewrite (3) as follows: D(V0 , r0 ,0) V0e aT N [ h1T ] B(T ) N [h2T ] Log ( where h1T V0 1 ) aT T2 B(T ) 2 T (5) , h2T h1T T . Note that (5) resembles the solution derived by Merton (1974). The firm will be able to make its promised payment, K , if and only if its value at the bond maturity is at least K . The first term in (5) represents the expected present value of the firm if default happens at maturity, and the V The mean and variance are evaluated under the equivalent martingale measure Q . 10 See APPENDIX 1 for a proof. 11 To avoid a high probability that the interest rates go negative, we have to choose small values of use. 9 to second term is the expected present value of the promised payment if default does not occur at maturity. 3. BASIC PROPERTIES OF rt and Z t . In the last section, we derived the pricing formula for a defaultable discount bond. The method is appealing in its simplicity by assuming Z t as the only source of variability. Before going further into the pricing of general coupon bonds, it is tempting to conjecture that such nice properties of Z t is preserved in multi-period case so that analytic simplicity can be followed in the same line as the single period case. However, it turns out that this conjecture is wrong. The reason is as follows: t 1 2 Given the definition Z t ru du 2t BtV , we can easily see that Z t satisfies the 0 following stochastic differential equation 1 dZ t ( rt 2 )dt dBtV . 2 This implies that Z t is a Markovian process of two state variable rt and Z t . THEOREM 3 gives some more properties of Z t . THEOREM 3 For any time 0 s t , under the martingale measure12 QV we have the following results: E[rt | s ] as,t e ( t s ) rs , ( t s ) E[Zt | s ] bs,t Z s 1e (6) rs , where as,t (1 e ( t s ) ) bs,t 12 2 (t s) (t s 1e ( t s ) ) are deterministic functions. Furthermore, if we express the above two equations in the following forms: rt as,t e ( t s ) rs s,t ,r , ( t s ) Zt bs,t Zs 1e 12 rs s,t ,Z , (7) (8) From now on, we implicitly assume that all expectations and variances are computed under the martingale measure QV . where s ,t ,r and s ,t ,Z are two normal random variables independent of s satisfying E[ s ,t ,r | s ] 0 and E[ s ,t ,Z | s ] 0 , then Var[rt ] e2 ( t s )Var[rs ] s2,t ,r ( t s ) Var[Zt ] Var[Z s ] ( 1e ( t s ) )2Var[rs ] 2( 1e cov[ rt , Zt ] e ( t s ) cov[ rs , Zs ] e ( t s ) ( 1e ( t s ) ) cov( rs , Z s ) s2,t ,Z )Var[rs ] s,t ,r ,Z where we assume the conditional variances Var[ s ,t ,r | s ] s2,t ,r , Var[ s,t ,Z | s ] s2,t ,Z . Proof: See APPENDIX 2. This theorem shows that the value of Z t at time t can be predicted by using the knowledge of rs and Z s at time s . Although Z t is not a Markovian process on its own, its simple decomposition into rs , Z s suggests that it is possible to solve the problem of pricing defaultable coupon bonds by using binomial trees. In the following section, we shall extend the method employed by Ho, Stapleton and Subrahmanyam (1995) to build binomial trees for rs and Zs . 4. A METHOD FOR CONSTRUCTING BINOMIAL PROCESSES 4.1 One-Period Case Ho, Stapleton and Subrahmanyam (1995) shows how to construct a multivariatebinomial approximation to a joint lognormal distribution of variables with a recombining binomial lattice. Each variable is lognormally distributed and Markovian on its own. In the present case, we need to modify the procedure, allowing the value of Z t to depend on the values of rs and Z s . As interest rates are assumed to follow normal distributions, the construction of the interest rate tree is the same as they propose. For simplicity, we first consider a one-period case. For detailed construction of two-period case, see APPENDIX 3. Let X s exp( rs ) . Our method involves the construction of one binomial distribution X s . Exhibit 1 X s ,0 X s ,1 X s,2 A discrete process for X s ,3 X0 Xs . X s,4 X s ,5 X s ,6 X s ,7 0 s We approximate X s by a vector of ns 1 numbers: X s,i X 0usi d sns i , for i 0,..., ns , where ds 2( E[ X s ] / X 0 )1/ ns , 1 exp( 2 0,s ,r / ns ) us 2( E[ X s ] / X 0 )1 / ns d s . On the time-interval [0, s ] , we choose a transition probability qs of an up-movement at the initial node such that property (6) holds, that is r0 ns (qs Log[us ] (1 qs ) Log[d s ]) a0,s e s r0 , hence, qs a0,s e s r0 r0 ns Log[d s ] . ns ( Log[us ] Log[d s ]) Given the above interest rate tree, we are now ready to construct a second tree for Ys exp( Z s ) conditional on rs . At time s , a tree of similar structure Ys is approximated by a vector of (ms 1) numbers Ys,i usi d sms i , for i 0,..., ms , where ds 2( E[Ys ])1/ ms , 1 exp( 2 0,s ,Z |rs / ms ) us 2( E[Ys ])1 / ms d s , where 0,s ,Z |rs is the volatility of Z s conditional on rs . In order to incorporate the correlation structure into the trees, we relate the noise terms in property (7) and (8) through their correlation 0 s as follows 0,s ,Z 0,s 0,s ,r 1 02,s 0,s , 0,s , Z 0,s ,r where 0,s is a normal standard random variable which is independent of 0 , s,r . By property (8) we have E[ Z s | rˆs ] b0,s 1e r0 0,s s 0,s ,Z ˆ . 0, s ,r 0, s ,r We choose a transition probability p s of an up-movement at initial node such that the above equation holds, that is b0,s ps 1 e s r0 0,s 0, s , Z ˆ ms Log[d s ] 0, s ,r 0, s ,r ms ( Log[us ] Log[d s ]) . 4.2 Multiple-period case In general, both J -period trees can be constructed similarly. Suppose that there are J future dates t1 , t2 ,..., tJ on which we are interested in the asset prices. We are interested in the joint distributions of X t and Yt on these dates. Over each of the time interval [t j 1 , t j ] , we assume that there are exogenously given number of binomial steps nt j and mt j for X t and Yt respectively. The construction of both trees are summarized as follows: For j 1,2,..., J , we approximate X t j by a vector of N j 1 numbers N j i X t j ,i X 0u ij d j , for i 0,..., N j , where 1/ N j dj 2( E [ X t j ] / X 0 ) 1 exp( 2 t j 1 ,t j ,r / nt j ) u j 2( E [ X t j ] / X 0 ) 1/ N j , dj, j N j ntl . l 1 On the time interval [t j 1 , t j ] , we choose a transition probability q j of an up-movement at node i at time t j 1 such that property (6) holds, qj at j 1 ,t j e ( t j t j 1 ) rˆt j 1 r0 iLog [u j ] ( N j 1 i ) Log[d j ] nt j Log[d j ] nt j ( Log[u j ] Log[d j ]) . At time t j , Yt j is approximated by a vector of M j 1 numbers M j i Yt j ,i u ji d j , for i 0,..., M j , where dj 2( E[Yt j ]) 1/ M j 1 exp( 2 t j 1 ,t j ,Z |rt / mt j ) , j u j 2( E[Yt j ]) 1/ M j dj , j M j mtl l 1 where t j 1 ,t j ,Z |rt is the volatility of Z t j conditional on t j 1 and rt j . We choose a transition j probability p j of an up-movement at node i at time t j 1 as follows bt j 1 ,t j 1 e Zˆ t j 1 pj ( t j t j 1 ) rˆt j 1 t j 1 ,t j t j 1 ,t j , Z t j 1 ,t j ,r ˆt j 1 ,t j ,r iLog [u j ] ( M j 1 i ) Log[d j ] mt j Log[d j ] mt j ( Log[u j ] Log[d j ]) The following theorem shows the basic properties of the constructed r̂t and Ẑ t processes that the estimated means, variances and covariances converge to their true values in the limit. THEOREM 4 Suppose that the X t and Yt trees are constructed as above. j 1,2,..., J , we have Then for E[rˆt j ] E[rt j ] (9) E[ Zˆ t j ] E[ Z t j ] (10) ˆ t2 (11) j 1 ,t j ,r t2j 1 ,t j ,r as nt j , ˆ t2 j 1 ,t j , Z |rt j ˆ t2 j 1 ,t j , Z ˆ t (1 t2j 1 ,t j ) t2j 1 ,t j ,Z as mt j , t2j 1 ,t j ,Z as mt j , nt j ,13 j 1 ,t j ,r , Z (12) (13) t j 1 ,t j ,r ,Z as mt j , nt j , (14) Furthermore, E[ Xˆ t j ] E[ X t j ] as nt j ,..., nt1 , (15) E [Yˆt j ] E [Yt j ] as mt j , nt j ,..., mt1 , nt1 . (16) Proof: We can prove (9) by induction. Note that by the choice of the probabilities q j , we have E[rˆt j | t j 1 ] at j 1 ,t j e ( t j t j 1 ) rˆt j 1 . This implies that E[rˆt j ] at j 1 ,t j e ( t j t j 1 ) E[rˆt j 1 ] . It is easy to check that E[rˆt1 ] E[ rt1 ] . By induction, result (9) follows. Similarly we can prove (10). For (11), we only consider the case when j 1 . The proof for a general j is similar. At time t1 , a realization of rt1 can be estimated by rˆt1 r0 iLog [u1 ] (nt1 i ) Log[d1 ] , where i is a binomial random variable with parameters ( nt1 , q1 ) . Hence, 13 Here the order of operations is important. The limits are taken over mt j , nt j respectively. ˆ 02,t ,r nt q1 (1 q1 )( Log[u1 ] Log[d1 ]) 2 1 Since q1 1 1 as nt1 , we have ˆ 02,t1 ,r 02,t1 ,r as nt j . Result (11) obtains. 2 Similarly we can prove that ˆ t2 j 1 ,t j , Z |rt j t2j 1 ,t j ,Z |rt as mt j . j By property (8), we have Zt j bt j 1 ,t j Zt j 1 1e where t j 1 ,t j ,Z t j 1 ,t j t t j 1 ,t j , Z t j 1 ,t j , r ( t j t j 1 ) 1 t2 j 1 , t j t rs t j 1 ,t j ,Z , j 1 ,t j , Z t j 1 ,t j , for a standard normal random j 1 ,t j , r variable t j 1 ,t j that is independent of t j 1 ,t j ,r . This implies that t2j 1 ,t j ,Z |rt (1 t2j 1 ,t j ) t2j 1 ,t j ,Z . j Hence (13) holds. By construction, the choice of p j satisfies E[ Zˆ t j | t j 1 , rˆt j ] bt j 1 ,t j Zˆ t j 1 1e ( t j t j 1 ) rˆt j 1 t j ,t j t j 1 ,t j , Z t j 1 ,t j , r ˆt j 1 ,t j , r , which implies that Var[ E[ Zˆt j | t j 1 , rˆt j ] | t j 1 ] t2j 1 ,t j t2 j 1 ,t j , Z 2 t j 1 ,t j ,r ˆ t2 j 1 ,t j ,r t2j 1 ,t j t2j 1 ,t j ,Z as nt j by (11). As ˆ t2j 1 ,t j ,Z Var[ E[ Zˆ t j | t j 1 , rˆt j ] | t j 1 ] E[Var[ Zˆ t j | t j 1 , rˆt j ] | t j 1 ] , result (13) follows. Proof of (14) is similar. To justify (15), it suffices to prove by induction on j that for any real number E[exp( rˆt j )] E[exp( rt j )] as nt j ,..., nt1 . For j 1 , E[exp( rˆt j )] X 0 [q1ut1 (1 q1 )dt1 ] t1 n 1 nt1 X 0 exp[ (a0,t1 e t1 r0 r0 ) 2 02,t1 ,r ] 2 1 exp[ ( a0,t1 e t1 r0 ) 2 02,t1 ] 2 E[exp( rt1 )] Suppose that the proposition is true for some j . For any real number , conditional on the node i at time t j we consider E[exp( rˆt j 1 ) | t j ] X 0 (utj 1 )i (d tj 1 ) t j 1 X 0 exp[ (at j ,t j 1 e n N j i ( t j 1 t j ) [q j 1utj 1 (1 q j 1 )d tj 1 ] nt j 1 1 rt j r0 ) 2 t2j ,t j 1 ,r ] 2 1 ( t t ) exp[ ' rt j ] exp[at j ,t j 1 2 t2j ,t j 1 ,r ] , where ' e j 1 j 2 By assumption, we have nt j 1 ,..., nt1 1 2 1 E[exp( rˆt j 1 )] exp[ ' E[rt j ] ' Var[rt j ]] exp[at j ,t j 1 2 t2j ,t j 1 ,r ] 2 2 By THEOREM 3, right hand side of the above expression is of the form 1 exp[E[rt j 1 ] 2Var[ rt j 1 ]] . 2 Hence the proposition holds for j 1 . To prove (16), it suffices to prove an alternative proposition by induction on j for any real number , E[exp( Zt j rˆt j )] E[exp( Zt j rt j )] as mt j , nt j ,..., mt1 , nt1 . The method is similar to the proof of (15). COROLLARY 5 Suppose that the X t and Yt trees are constructed as above. Then for j 1,2,..., J , convergence in distribution is guaranteed rˆt j | t j 1 rt j | t j 1 as nt j . Zˆ t j | ( t j 1 , rˆt j ) Z t j | (t j 1 , rˆt j ) as mt j . Proof. The proof is followed by the constructions of q j and p j , and the properties of conditional variances. 5. GESKE’S METHOD OF EVALUATING DEFAULTABLE COUPON BONDS For ease of exposition, we first give a review on the methods of analysis given by Geske (1977) and Selby (1983). Under the assumption of constant interest rate, Geske demonstrated that risky securities could be valued as compound options. Selby generalized Geske’s pricing model by incorporating continuous dividends as a constant proportion of the total value of the firm assets. He used the “preference independent” approach first suggested by Cox and Ross (1976) to simplify the mathematical derivation of the model. An important assumption that underlay the analysis by them is that the firm finances each coupon payment with equity only taken up by shareholders, and that bankruptcy occurs when the firm fails to make a coupon payment because it is unable to raise enough money to fund the payment. Black and Cox (1976) argue that this will happen whenever the value of the equity, after payment is made, is less than the value of the payment. Black and Cox’s (1976) argument is intuitive, in that the firm will find no takers for its stock if they know the stock will become less valuable than the total value that they need to contribute to the promised payment. Suppose that the firm has an obligation to meet a coupon payment C t at time t . Let S t be the value of the stock immediately after time t . Then by Black and Cox’s (1976) argument, the firm will be able to finance the coupon payment by rights issues if St Ct (17) By the Modigliani and Miller theorem, the value of the firm is independent of its capital structure. Therefore the above inequality can be rewritten as Vt Ct Dt , (18) where Vt Vt Vt 14 is the firm value at time t . This implies that the value of the firm at time t should be greater than the total value of the coupon to be honored and the debt immediately after the coupon date. Looking from another viewpoint, the right hand side of inequality (18) is the value of the debt at time t . Hence by the Modigliani and Miller theorem again, if the firm is able to honor its obligation on interest payment by issuing new equity, then the stock has a 14 The money raised by rights issues is used to finance the coupon payment. Firm value remains unchanged before and after the payment. positive value. Inequality (18) is of particular importance when we are working on numerical computations of defaultable bonds. In the case of Geske’s (1977) model, it is fairly straightforward to imply from inequality (17) the existence of a critical value of the firm V ( below which a default happens ). The reason is that when the interest rate is assumed to be constant, there is only one stochastic variable Vt in the function of the stock price. Moreover, the monotonic increasing property of the stock prices on the firm value always guarantees the existence of such a point V . Mathematically we write St (V ) Ct . (19) Assuming the existence of such critical values of the firm allows us to price risky coupon bonds analytically. With the “risk-neutral technique” of Cox and Ross (1976), the pricing problems simply become finding expected present value of a stream of promised payments on a series of events on coupon dates that determine whether or not a default has happened 15. The derivation becomes much simpler than the alternative backward recursion techniques employed by Geske (1977). A formula containing multi-variate normal integrals can be derived. However, the result is more of theoretical interest than practical use. Even with the aid of fast computers in existence today, the computation of high dimensional normal integrals is still a very challenging problem16. Alternatively the Geske’s (1977) formula can be computed numerically. In the one-factor case where a firm’s value is modeled as a geometric Brownian motion and interest rate is taken as a constant, the prices of a defaultable coupon bond can be computed efficiently and accurately by building a binomial tree for the firm value. In a multi-factor framework, the method of solution for the pricing of defaultable coupon bonds is in general more complicated. One complication is the introduction of the stochastic interest rates that makes the determination of an analytic solution for coupon bond prices difficult, if not impossible. In our two-factor case, the stock price is a function of two variables, namely the firm value and the interest rate, and so the solution of equation (18) is a function of future interest rates on a coupon date, which are unknown at the time of issue. In other words, V is a moving default boundary. In the following section, instead of looking for an analytic 15 16 See Cox and Ross (1976), and Selby (1983). See Papageorgiou and Traub (1997), Paskov and Traub (1997). solution, we shall propose an efficient numerical technique to price risky coupon bonds by building two binomial trees. 6. APPLICATIONS TO PRICING OF DEFAULTABLE BONDS In section 2.4, we derived an analytic formula for a defaultable discount bond. Using the hypothetical asset Vt H as a numeraire, we showed that by combining two sources of variability together to form a single variable Z t the analytic valuation is significantly simplified. In section 3, we discussed the properties of Z t , which implies that the analytic simplicity in valuation of defaultable bonds can not be retained in multi-period case. We have to resort to compute coupon bond prices numerically. In principle computation is independent of the choice of a numeraire, however, appropriate choice of it does provide much of the computational simplicity. It is important and interesting to investigate the differences in computational efficiency between using the money market account t and the hypothetical asset Vt H as numeraires in bond pricing problems. In the case of a defaultable discount bond with maturity T , we consider the following cases. Suppose we use VTH as a numeraire to price the bond, the initial price is estimated by taking average17 of the following expression V0 Min (VT , K ) VTH Min (V0 exp( aT ), K exp( ZT )) With the results in section 4, the bond price can be approximated by building two binomial trees rT and ZT . On the other hand, if the money market account T is used as a numeraire, the bond price is estimated by taking average18 of Min (VT , K ) T T 1 Min (V0 exp( aT 2T BT ), K exp( ru du )) 0 2 T Note that exp( ru du) is a Markovian process on its own and the interest rate. By using the 0 T similar techniques as in section 4, exp( ru du) can be mimicked by two binomial trees. This 0 17 Under QV . result implies that with this choice of numeraire the construction of three binomial trees is necessary to price the bond. Therefore the use of the numeraire VTH is supported by the computational convenience by saving one binomial tree. The following theorem establishes the theoretical justification for numerical computation of discount bonds. THEOREM 6 With the construction of two one-period binomial trees rT and ZT with binomial steps n and m respectively, the price of a defaultable discount bond of maturity T can be estimated by the following two steps: (i) E[ Min (V0 exp( aT ), K exp( ZˆT )) | rˆT ] E[ Min (V0 exp( aT ), K exp( ZT )) | rˆT ] as m . (ii) E[ E[ Min (V0 exp( aT ), K exp( ZT )) | rˆT ]] E[ Min (V0 exp( aT ), K exp( ZT ))] as n , where the limit is the price of the defaultable discount bond. Proof (i) By COROLLARY 5, ZˆT | rˆT converges to ZT | rˆT in distribution. Since Min (V0 exp( aT ), K exp( ZT )) | rˆT is continuous at ZT , Billingsley(1995, p.334, corollary 1) implies that Min (V0 exp( aT ), K exp( ZˆT )) | rˆT Min (V0 exp( aT ), K exp( ZT )) | rˆT , as m . Note that Min (V0 exp( aT ), K exp( ZT )) | rˆT is uniformly integrable as it is bounded above by V0 . By Billingsley(1995, p.338, Theorem 25.12), (i) follows. (ii) Since r̂T converges to rT in distribution and E[ Min (V0 exp( aT ), K exp( ZT )) | rT ] is continuous at rT , by Billingsley(1995, p.334, corollary 1) again E[ Min (V0 exp( aT ), K exp( ZT )) | rˆT ] E[ Min (V0 exp( aT ), K exp( ZT )) | rT ] , as n . E[ Min (V0 exp( aT ), K exp( ZT )) | rˆT ] is uniformly integrable, and so by Billingsley(1995, p.338, Theorem 25.12) again, E[ E[ Min (V0 exp( aT ), K exp( ZˆT )) | rˆT ]] E[ E[ Min (V0 exp( aT ), K exp( ZT ))] | rT ] as n . Result follows. 18 Under Q . This theorem shows that computations of a discount bond price consist of three main steps. Firstly, we compute Min (V0 exp( aT ), K exp( ZˆT )) at each node Ẑ T of the ZT tree. Secondly, given each r̂T , we find the mean of the above expression. Thirdly, the bond price is estimated by taking the mean of the results in step 2 over all r̂T . Similarly we can estimate the price of a defaultable coupon bond by binomial methods. With the same notation used as before we first consider a two-period case where t1 and t2 are the coupon and maturity dates respectively. As similar to the proof of THEOREM 6, we can prove that (i) E[ Min (V0 exp( at2 ), K (1 c) exp( Zˆt2 )) | t1 , rˆt2 ] E[ Min (V0 exp( at2 ), K (1 c) exp( Z t2 )) | t1 , rˆt2 ] , as m2 , and (ii) E[ E[ Min (V0 exp( at2 ), K (1 c) exp( Z t2 )) | t1 , rˆt2 ] | t1 ] E[ Min (V0 exp( at2 ), K (1 c) exp( Z t2 )) | t1 ] , as n2 . Let B( rt1 , Z t1 ) E[ Min (V0 exp( at2 ), K (1 c) exp( Z t2 )) | t1 ] . Then Vt1H B(rt1 , Zt1 ) / V0 is the bond price at time t1 . In other words, the bond price at time t1 is approximated by (i) and (ii) THEOREM 6. By (18), the firm will be able to finance the coupon payment by a rights issue if Vt1 Vt1H B(rt1 , Zt1 ) / V0 Kc , and so the bond price immediately before the coupon date is Min (Vt1 ,Vt1H B(rt1 , Zt1 ) / V0 Kc) . By the martingale property, its initial price is given by V0 E[ Min (Vt1 ,Vt1H B(rt1 , Zt1 ) / V0 Kc)] E[ Min (V0 exp( at1 ), B( rt1 , Z t1 ) Kc exp( Z t1 ))] . As similar to THEOREM 6, we can estimate this bond price by the following two steps: (iii) E[ Min (V0 exp( at1 ), B( rˆt1 , Zˆt1 ) Kc exp( Zˆt1 )) | rˆt1 ] E[ Min (V0 exp( at1 ), B( rˆt1 , Z t1 ) Kc exp( Z t1 )) | rˆt1 ] as m1 , and (iv) E[ E[ Min (V0 exp( at1 ), B( rˆt1 , Z t1 ) Kc exp( Z t1 )) | rˆt1 ]] E[ Min (V0 exp( at1 ), B( rt1 , Z t1 ) Kc exp( Z t1 ))] as n1 . With one- and two-period cases, we have depicted the essence of numerical techniques for computations of general defaultable coupon bonds. For j 1,2,..., J , on each interval [t j 1 , t j ] , the relative price of the bond to the hypothetical asset Vt H agrees with martingale properties, and computations consist of three similar steps as described above. Firstly, we work out the default barrier and the payoff immediately before time t j in accordance with Geske’s idea. Secondly, we find the relative value of the payoff to Vt Hj and take the mean of it given each r̂t j . Thirdly, we take the average of the results in the second step over all r̂t j . 7. NUMERICAL COMPUTATIONS 7.1 Prices of Defaultable Bonds For a J -period model, our method of constructing binomial trees rt and Z t requires the choice of J -binomial steps nt1 , nt2 ,..., ntJ and mt1 , mt2 ,..., mtJ respectively. As discussed before, the order of operations is important. To guarantee the convergence of the estimated bond prices to the true value, the limits should be taken in the order of mtJ , ntJ ,..., mt1 , nt1 by letting them tend to infinity. In general, the estimation of the true value by a choice of realizations of mtJ , ntJ ,..., mt1 , nt1 is a difficult task, and depends on the essence of the problem. Improper choice of binomial steps may lead to a possibility that the sequence of approximated bond prices does not converge to the true limit. For J 1 , we compute the prices of one-year discount bonds with par value 70, 100 and 130 analytically. Exhibit 2 Example of bond prices with par value 70, 100 and 130. The prices are computed analytically using parameter values r =0.04, =0.06, =1, =0.031, =-0.25, =0.2, V0 =100, a =0.12. K =70 one-year discount 66.2571 K =100 84.314 K =130 88.3116 Exhibit 3 shows bond price convergence graphs for a one-year discount bond with par value K =70, 100 and 130. It shows that the prices of the discount bonds converge to the true values in Exhibit 2. The rate of convergence is fairly high, and suggests that it is not necessary to compute with many binomial steps. Binomial steps with n =5 and m =5 appear to be sufficiently large to give good estimates of bond prices. Exhibit 4 shows convergence graphs for the same instruments with one difference in the use of binomial steps that n =2 and m =2,3,…,20. However, it exhibits the same pattern as indicated in Exhibit 3. Exhibit 5 gives further details on the estimated prices of a discount bond with par value K =70. It shows that the bond prices are insensitive to the changes of interest rate depicted by the first tree rt , and the binomial steps m plays a more influential role in the precision of estimates. This can be explained by the fact that the variable Z t already captures most of the variability in rt , and so the increases in binomial steps n have merely negligible influence on the generated bond prices. Higher values of m tend to give a more accurate approximation of the bond prices. For this reason we shall keep using small values of n in the subsequent computations. Exhibit 3 Bond price convergence graphs against binomialsteps for a one-year discount bond. The bond prices are computed numerically assuming stochastic interest rate using parameter values r =0.04, =0.06, =1, =0.031, =-0.25, =0.2, V0 =100, a =0.12, n m =2,3,…,20 (i) K =70(___), (ii) K =100(- - -), (iii) K =70(---). Insert Exhibit 3 here Exhibit 4 Bond price convergence graphs against binomial steps for a one-year discount bond. The bond prices are computed numerically assuming stochastic r =0.04, interest rate using parameter values =0.06, =1, =0.031, =-0.25, =0.2, V0 =100, a =0.12, n =2, m =2,3,…,20. (i) K =70(___), (ii) K =100(- - -), (iii) K =70(---). Insert Exhibit 4 here Exhibit 5 Example of discount bond prices with par value 70 for the cases (i) n =2, m =2,3,…,20, (ii) The prices are computed n m =2,3,…,20. numerically using parameter values r =0.04, =0.06, =1, =0.031, =-0.25, =0.2, V0 =100, a =0.12. Case (i) Case (ii) m=2 66.7393 66.7393 m=3 66.3665 66.3665 m=4 66.2381 66.2381 m=5 66.3429 66.3429 m=10 66.257 66.2568 m=11 66.3263 66.3261 m=12 66.2616 66.2614 m=13 66.2727 66.2726 m=18 66.2678 66.2676 m=19 66.2886 66.2884 m=20 66.256 66.2558 It is worthwhile to note that the estimated discount bond prices converge to the true limit at a fairly fast fashion. Because of the similarity of the binomial construction of coupon bonds to discount ones, high rate of convergence is also expected in the computations of coupon bond prices over each coupon period as long as the binomial steps m is chosen to be sufficiently large. Therefore to resolve the problem we discussed above, in the computations of coupon bonds we choose mt1 mt2 ... mt J . In the following table, we illustrate the efficiency of our method by showing the time taken to compute a one-year discount bond, a one-year 8% coupon bond and a two-year 8% coupon bond. It shows that the computation time increases approximately linearly with the binomial step m . Exhibit 6 Example of bond prices and their computation times with par value 70 for n =2, m =2,4,…,10. The prices are computed numerically using parameter values r =0.04, =0.06, =1, =0.031, =0.25, =0.2, V0 =100, a =0.12. one-year one-year 8% two-year 8% discount m=2 66.7393 (0.16s) 71.395 (1.75s) 70.765 (13.79s) m=4 66.2381 (0.21s) 71.4713 (2.75s) 70.576 (24.02s) m=6 66.358 (0.27s) 71.3807 (3.9s) 70.6929 (35.51s) m=8 66.3159 (0.34s) 71.3782 (5.05s) 70.6299 (47.56s) m=10 66.257 (0.38s) 71.4299 (6.04s) 70.6068 (57.77s) Exhibit 7 Two-year 8% risky coupon bond prices with stochastic interest rates: r0 =0.04, =0.06, =1, =0.031, =-0.25, =0.2, V0 =100, K =70, a =0.12, c =0.04, unless otherwise stated. In Exhibit 7.1:- _____: K =70, - - -: K =100, ----: K =130. In Exhibit 7.3:- _____: =-0.25, - - -: =0.25. Insert Exhibit 7 here 7.2 Credit Spreads of Defaultable Bonds Having worked out the prices of a coupon bond, we can solve for the credit spread. Credit spread is defined as a spread level over the yield of a default-free bond with the same promised payments and maturity. In other words, a defaultable discount bond can be expressed as a default-free discount bond with an upward adjustment of a credit spread in its yield. In the case of a coupon bond, a credit spread can be solved numerically by assuming that a defaultable coupon bond takes a simple form of an associated default-free coupon bond where each term is of the same upward adjustment of a credit spread. The plots of credit spreads of a 2-year 8% coupon bond against different parameters are shown in Exhibit 8. Exhibit 8 Credit spreads of a two-year 8% risky coupon bond with stochastic interest rates: r0 =0.04, =0.06, =1, =0.031, =-0.25, =0.2, V0 =100, K =70, a =0.12, c =0.04 and n =2, m =2,4,…,10, unless otherwise stated. In Exhibit 8.1:- _____: K =70, - - -: K =100, ----: K =130. In Exhibit 8.3:- _____: =-0.25, - - -: =0.25. Insert Exhibit 8 here As expected, in most of the plots credit spreads move in a direction opposite to the bond prices. However, in Exhibit 7.4 and 8.4 bond prices and credit spreads are both decreasing functions of the interest rates. The reason for this phenomenon is attributable to the fact that in the risk neutral world the firm value tends to increase in the response to the increasing interest rates. Bankruptcy is less likely to happen as a result of higher stock prices, therefore credit spreads decrease with interest rates. This result agrees with the empirical findings in Longstaff and Schwartz (1995). 7.3 Interaction of Market and Credit Risk Market risk refers to changes in bond prices as a result of changes in interest rates. Credit risk refers to the risk that the issuer of a bond may default. As market events have shown, there is an important interplay between both concepts. Our model integrates market and credit risk together to allow for a more complete picture of the underlying risk. Exhibit 9, 10 show bond price sensitivity and credit spread sensitivity to parameters. Exhibit 9 Two-year r0 =0.04, 8% coupon bond price sensitivity: =0.06, =1, =0.031, =-0.25, =0.2, V0 =100, K =70, a =0.12, c =0.04, unless otherwise stated. Insert Exhibit 9 here Exhibit 10 Credit spread sensitivity of a two-year 8% coupon bond: r0 =0.04, =0.06, =1, =0.031, =0.25, =0.2, V0 =100, K =70, a =0.12, c =0.04, unless otherwise stated. Insert Exhibit 10 here Exhibit 10.1 shows that when the firm value is low, credit spread is more sensitive to the changes in interest rates. This confirms our intuition that firms with low credit quality should have more market risk than firms with high credit quality. On the contrary, firms with high credit quality are those which we expect have only a base level of interest rate exposure. Furthermore, it is clear from 10.2 and 10.3 that when dividend and coupon rates are higher, credit spread is more sensitive to firm value volatility. This implies that under the assumption that coupons are financed by right issues, the bond is of higher default risk as coupon rate increases. A similar trend holds for the case where shareholders are entitled to receive higher dividend rate as a proportion of firm value. Exhibit 11 Credit spread sensitivity of a one-year 8% coupon bond: r0 =0.04, =0.06, =1, =0.2, V0 =100, K =70, a =0.12, c =0.04, unless otherwise stated. Insert Exhibit 11 here Exhibit 11 plots the relation of credit spreads with respect to interest rate volatility and correlation . As shown, the effect of correlation can be very significant. When correlation is high, credit spread appears to be sensitive to the changes in interest rate volatility. When correlation is small, credit spread decreases slightly over a wide range of interest rate volatility. As in Longstaff and Schwartz (1995), these results are consistent with empirical evidence that credit spreads for bonds of equal rating vary across sections. 8. POSSIBLE EXTENSIONS In addition to providing conceptual insights into default behaviour, the flexibility of our method allows for efficient pricing of some other financial instruments. These include bond options, credit risk put options on a general defaultable coupon bond and floating rate notes. Many papers in financial literature have addressed the important topic of bond option valuation. Of these papers, Jamshidian (1989) and Longstaff (1993) provide an analytic formula for the value of an option on a coupon bond with stochastic interest rates 19. However, they derive the formulae under the assumption that the underlying coupon bonds are non-defaultable. In this section, we shall show that the pricing algorithm for risky coupon bonds can be modified to price options on a defaultable bond. Consider a -year European option on a T -year defaultable coupon bond D(Vt , rt , t ) with a fixed exercise price, where 0 t T . Here we are focusing on options on bonds with final maturity date T . Since the bond option is a derivative of the defaultable coupon bond D(Vt , rt , t ) , which is a function of Vt , rt and t , we can express its value at time s as 20 C (Vs , rs , s) , where 0 s . Note that that C (Vs , rs , s ) s D (Vs , rs , s ) s is a Q -martingale, Ito’s Lemma implies is also a Q -martingale. By LEMMA 1 and THEOREM 2 in section 2, it is easy 19 They assume that interest rates follow a Vasicek process and Cox, Ingeroll and Ross (1985) process respectively. 20 2 It has to be a C function with respect to the first and second coordinates. to prove that under the transformation BsV s Bs , C (Vs , rs , s ) is a QV -martingale. Now H Vs we state this result in the following lemma. LEMMA 7 The relative price of C (Vs , rs , s) to VsH is a QV -martingale for 0 s . The initial price of the bond option is given by C (V0 , r0 ,0) V0 E Q [ V C (V , r , ) 0 ] , H V where C (V , r , ) is the payoff of the option at its maturity. With this lemma, the numerical valuation of European bond options becomes straightforward. This is due to the fact that the relative price of bond option C (Vs , rs , s) to the same numeraire VsH (as what we employed for the pricing of defaultable coupon bonds) evolves in the same way as that of the underlying asset, and so immediately lends the numerical algorithm developed to the efficient pricing of bond options. The numerical valuation algorithm for defaultable coupon bonds can be easily modified to price European and American bond calls and bond puts. Exhibit 12 and 13 show bond option prices for different levels of initial interest rates. In Exhibit 12, the first six columns represent call and put option prices on a risky coupon bond, and the last three columns are the prices of a call option on a default-free coupon bond with the same payment schedule as the risky one. The corresponding put option prices are not shown in Exhibit 12 because of their extremely small values. This can be explained by the fact that when the underlying bond is non-defaultable, its price at maturity is likely to be greater than the exercise level, and so the put option becomes deep out-of-the money. As shown, the prices of the call option on a non-defaultable bond are uniformly decreasing with the interest rate levels, and increasing with the exercise prices. This result agrees with Longstaff’s (1993) findings on the valuation of options on default-free coupon bonds. Not surprisingly, when the underlying bond is defaultable, the corresponding calls become less valuable because of lower values of the underlying. More interestingly, the risky call prices move in a direction opposite to the riskless call prices. The reason for this is similar to the explanation we gave at the end of section 7.2. The gradually increasing feature of the bond call prices can be explained by the fact that, in the risk-neutral world, the value of the firm increases in the response to the increasing interest rates. As a result, the bond calls become more valuable as interest rates increase. Exhibit 12 Examples of prices for 2-year call and put options on a 5-year 8% coupon bond with face value 70 for different initial interest rate levels r, and exercise price E. The option values are computed using parameters =0.06, =1, =0.031, =-0.25, =0.2, V0 =100, K =70, a =0.12, c =0.04, and binomial steps n =2, m =5. Call Prices (Risky) Put Prices (Risky) Call Prices (Riskless) r E=60 E=65 E=70 E=60 E=65 E=70 E=60 E=65 E=70 0.01 3.411 3.472 3.354 3.586 3.623 3.659 3.697 3.735 3.774 3.813 3.852 3.894 3.935 3.981 4.027 1.340 1.368 1.400 1.424 1.441 1.458 1.475 1.491 1.514 1.537 1.560 1.583 1.606 1.629 1.657 0.317 0.325 0.332 0.337 0.339 0.340 0.344 0.347 0.352 0.357 0.363 0.368 0.374 0.380 0.385 2.716 2.590 2.468 2.359 2.256 2.154 2.055 1.961 1.867 1.775 1.686 1.600 1.515 1.436 1.358 5.281 5.081 4.887 4.714 4.551 4.392 4.234 4.079 3.933 3.788 3.645 3.504 3.364 3.227 3.095 8.893 8.633 8.376 8.142 7.926 7.713 7.504 7.298 7.095 6.900 6.699 6.503 6.310 6.119 5.929 15.01 14.79 14.57 14.36 14.15 13.94 13.74 13.54 13.34 13.13 12.94 12.75 12.56 12.38 12.19 10.37 10.20 10.02 9.846 9.674 9.505 9.338 9.174 9.011 8.851 8.694 8.538 8.384 8.233 8.084 5.739 5.601 5.464 5.329 5.197 5.066 4.938 4.812 4.687 4.565 4.444 4.326 4.209 4.093 3.980 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 Exhibit 13 Examples of prices for -year call and put options on a 5-year 8% risky coupon bond with face value 70 for different initial interest rate levels r. The option values are computed using parameters =0.06, =1, =0.031, =-0.25, =0.2, V0 =100, K =70, a =0.12, c =0.04, E =70 and binomial steps n =2, m =5. Call Prices r 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 =1 0.295 0.282 0.269 0.257 0.245 0.233 0.222 0.211 0.200 0.189 0.179 0.166 0.150 0.138 Put Prices =2 0.317 0.325 0.332 0.337 0.339 0.340 0.344 0.347 0.352 0.357 0.363 0.368 0.374 0.380 =5 0.280 0.289 0.299 0.308 0.315 0.322 0.330 0.338 0.347 0.356 0.365 0.374 0.384 0.394 =1 6.067 5.994 5.922 5.850 5.778 5.707 5.635 5.563 5.492 5.420 5.348 5.297 5.268 5.242 =2 8.893 8.633 8.376 8.142 7.926 7.713 7.504 7.298 7.095 6.900 6.699 6.503 6.310 6.119 =5 13.96 13.61 13.25 12.91 12.59 12.28 11.96 11.66 11.36 11.06 10.77 10.48 10.19 9.913 0.15 0.127 0.385 0.405 5.216 5.929 9.637 This shows that default risk has a significant effect on the pricing of bond options on a defaultable asset. Another interesting feature of defaultable coupon bond option prices is their relationship with the option maturity . Here we are focusing on the underlying bonds with maturity +5 years. In Exhibit 13, it is easy to see that both call and put prices are decreasing functions of initial interest rate when =1. This is because when is small, the option values are close to their intrinsic value, and so move in the same direction as the underlying bond does. However, when becomes large, option features play a more important role in call and put prices. Call prices are uniformly increasing, and put prices are uniformly decreasing with interest rates. This result agrees with those in Exhibit 12. Another possibility of generalization is through the introduction of bankruptcy costs. Jones, Mason, and Rosenfeld’s (1984) attempt to implement the Merton model on U.S. corporate bonds proves disappointing. The models do not fit very well and tend to systematically underestimate observed yields when plausible values of asset volatility were employed. To improve the levels of credit spreads, we suggest that bankruptcy results in costly liquidation, and so the creditor will receive the assets of the firm net of any liquidation costs. We assume that liquidation costs take the following form C (1 )Vt , where C represents a fixed initial liquidation cost, is a constant between 0 and 1, and t is the time at which liquidation happens. Thus at the liquidation barrier the value of the bond is given by Max (Vt C,0) . The flexibility of our computational method allows for easy incorporation of liquidation costs in our bond pricing model. With a slight modification of the default payoffs, recovery rates now become a consequence of the combined effects of the endogenously generated mechanism and the exogenously given structure as above. In Exhibit 14, we show that the deployment of the simple functional form of liquidation costs improves the levels of credit spreads generated by our framework, and hence provides a more pragmatic approach to the solution of defaultable bonds. Exhibit 14 Term structure of credit spreads for a 8% coupon bond: r0 =0.04, =0.06, =1, =0.2, V0 =100, a =0.12, c =0.04, unless otherwise stated. In Exhibit 14.1: K =50, _____: without liquidation costs ; - - -: with liquidation costs =0.05 and C =2 In Exhibit 14.2: K =70, _____: without liquidation costs ; - - -: with liquidation costs =0.05 and C =2 In Exhibit 14.3: K =90, _____: without liquidation costs ; - - -: with liquidation costs =0.05 and C =2 Insert Exhibit 14 here 9. CONCLUSION In this paper we have generalized, in computational aspects, Geske’s (1977) and Selby’s (1983) valuation models of risky coupon bonds to allow for stochastic interest rates proposed by Vasicek (1977). By using the hypothetical asset Vt H as a numeraire, it has been shown that not only does the use of this numeraire significantly simplify the analytic valuation of risky discount bonds, but also gives an implication that the two-factor model can be implemented easily. We have discussed computational efficiency when the hypothetical asset Vt H is used as a numeraire, and showed that this is an appropriate choice of numeraire. In addition, we have suggested an efficient computation algorithm for the pricing of general risky coupon bonds by generalizing the models proposed by Ho, Stapleton and Subrahmanyam (1995). Much of the simplicity of this method lies in the fact that two sources of variability, namely interest rate risk and asset value risk, are combined together to form a single stochastic process Z t . This approach to pricing general risky debt is particularly appealing when the underlying assets of a derivative instrument can be managed to be driven by Z t . This analysis can be extended in several ways. The Vasicek model suffers from its implicit assumption that interest rates can become negative with a positive probability at any given time. While it has been shown that this probability can usually be made small by properly adjusting the process parameters, the weakness of negative interest rates is perhaps offset by Hull and White’s (1990) observation that the extended Vasicek model can be used to fit any observable term structure. Moreover, the fact that the Vasicek process can be embedded in a framework of HJM’s (1992) model leaves us with an implication that the methods developed in this paper would be readily generalizable to incorporate a more general term structure consistent interest rate process. Despite this, it still retains much of the computational tractability. For example in the extended Vasicek model, apart from being consistent with initial term structure, any volatility term that is a deterministic function of time can readily be fitted into our framework. Our method of construction of binomial trees can be easily generalized to cope with more general Markovian processes dependent on several stochastic variables. This should be useful in pricing of some more complicated instruments, for example currency swaps. Another drawback is with the assumption that asset values are log-normally distributed. The traditional Merton’s (1973) approach to pricing risky debt has been criticized as being incapable of generating credit spreads consistent with those observed in corporate debt markets. One of the reasons is that the model lacks the fat-tailness properties that we normally observe in asset returns. The introduction of jump processes to the bond pricing model would be able to resolve part of the issues, though the mathematics for dealing with point processes is more involved. Another reason is the absence of a mechanism that allows for costly liquidation in the event of bankruptcy. We have shown that the pragmatic deployment of a simple functional form of liquidation costs improves the levels of credit spreads generated by our framework. Finally, we observe that while this traditional approach to modeling risky debt does not provide practical tools for valuing realistic types of corporate securities, it has been an indispensable tool for discussing the distribution of the firm’s value between shareholders and bondholders. In addition to providing conceptual insights into default behaviour, the flexibility of our method allows for efficient pricing of bond options, credit risk put options on a general defaultable coupon bond and floating rate notes. This structural approach also paves the way for a further analysis of more complicated debt structures. Incorporation of bankruptcy costs in the model is an important avenue that can be explored in our framework. Efficient numerical valuation of general risky debts when interest rates and firm’s values are stochastic should be a crucial step forward in understanding the full complexity of credit analysis. APPENDIX 1 Proof Since rt follows a Vasicek process, it is not difficult to prove that under the equivalent measure Q , T rsds aT 0 where aT 1 (1 e T )[ r0 T T s r r e e dBs BT 0 T e T 1 (e 1)] ( T ) and Bsr Bs 1 2 Bs' . By THEOREM 2, under the equivalent martingale measure QV and the transformation BtV t Bt we have T rsds aT 0 T T s 1 e T e e d [ BsV 1 2 Bs' ] [ BTV 1 2 BT' ] [ T] 0 which is a normal random variable. T 1 2 By definition ZT rudu 2T BTV , 0 T2 Var[ ZT ] bT2 2T 2 1 e T ( T) 2 2 2 2 2 T where b 2 T (1 e ) 3 (1 e T ) . 3 2 2 T APPENDIX 2 Under that assumption that rt follows a Vasicek process as described by (1), we have rt r0e t t (1 e t ) e t e s dBsr , 0 where Btr Bt 1 2 Bt' . By THEOREM 2, under the equivalent martingale measure QV and the transformation BtV t Bt , the above equation can be expressed as rt r0e t t (1 e t ) e t e s d [ BsV 1 2 Bs' ] . 0 (*) We can prove that E[rt | s ] r0e t s (1 e t ) e t e u d [ BuV 1 2 Bu' ] .(**) 0 Result follows after eliminating the integrals in (*), (**). t 1 2 By definition Z t ru du 2t BtV , we have 0 t 1 E[ Z t | s ] Z s 2 (t s ) E[ ru du | s ] . s 2 After integrating (*) and substituting the result and (*) into the above equation, result follows. The proofs for the rest of results are trivial. APPENDIX 3 In the text, we have constructed the one-period trees for X s , Ys . Now we consider the construction of the trees at a second time period t , with t s , by following the similar fashion to approximate X t by a vector of ns nt 1 numbers: X t ,i X 0uti dtns nt i , for i 0,..., ns nt , where dt 2( E[ X t ] / X 0 )1/(ns nt ) , 1 exp( 2 s ,t ,r / nt ) ut 2( E[ X t ] / X 0 )1 /(ns nt ) d t . On the time-interval [ s, t ] , we choose a transition probability qt of an up-movement at node i at time s such that property (6) holds, that is r0 iLog[ut ] (ns i ) Log[dt ] nt (qt Log[ut ] (1 qt ) Log[dt ]) as,t e ( t s ) rˆs , hence, qt as ,t e ( t s ) rˆs r0 iLog [ut ] ( ns i ) Log[d t ] nt Log[d t ] . nt ( Log[ut ] Log[d t ]) Given the above interest rate tree, we are now ready to construct a second tree for Yt conditional on rt . At time t , we create a vector of (ms mt 1) numbers Yt ,i uti dtms mt i , for i 0,1,..., (ms mt ) , where 2( E[Yt ])1/(ms mt ) , dt 1 exp( 2 s ,t ,Z |rt / mt ) ut 2( E[Yt ])1 /(ms mt ) d t , where s ,t ,Z |rt is the volatility of Z t conditional on s and rt . A transition probability pt of an up-movement at node i at time s is chosen such that the following property holds (ts ) E[ Z t | s , rt ] bs ,t Z s 1e rs s ,t s ,t ,Z s ,t ,r s ,t ,r that is iLog [ut ] (ms i ) Log[d t ] mt [ pt Log[ut ] (1 pt ) Log[d t ]] bs ,t Zˆ s 1 e bs ,t Zˆ s pt ( t s ) rˆs s ,t 1 e (t s ) rˆs s ,t s ,t ,Z ˆs ,t ,r s ,t ,r s ,t ,Z ˆ iLog[ut ] (ms i ) Log[d t ] mt Log[d t ] s ,t ,r s ,t ,r mt ( Log[ut ] Log[d t ]) References 1. Billingsley, P.(1994) Probability and Measure. Third Edition. Wiley Series in Probability And Mathematical Statisitcs. 2. Black, F., Scholes, M. (1973) The pricing of options and corporate liabilities. Journal of Political Economy 81, 637-654. 3. Black, F., Cox, J. 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