Analysis on the Pricing model for a Discrete Coupon Bond with Early redemption provision by the Structural Approach
aa r X i v : . [ q -f i n . P R ] J u l Analysis on the Pricing model for a Discrete Coupon Bond with Earlyredemption provision by the Structural Approach
Hyong-Chol O , Tae-Song Kim Faculty of Mathematics,
Kim Il Sung
University, Pyongyang, D.P.R.Korea
Abstract
In this paper, using the structural approach is derived a mathematical model of the discrete coupon bond with theprovision that allows the holder to demand early redemption at any coupon dates prior to the maturity and based on thismodel is provided some analysis including min-max and gradient estimates of the bond price. Using these estimatesthe existence and uniqueness of the default boundaries and some relationships between the design parameters of thediscrete coupon bond with early redemption provision are described. Then under some assumptions the existence anduniqueness of the early redemption boundaries is proved and the analytic formula of the bond price is provided usinghigher binary options. Finally for our bond is provided the analysis on the duration and credit spread, which are usedwidely in financial reality. Our works provide a design guide of the discrete coupon bond with the early redemptionprovision.
Keywords: corporate bond, structural approach, coupon, early redemption
1. Introduction
Issueing bond is a kind of financing methods of firms and among firm bonds there is a bond with the provisionunder which the bond holder can demand early redemption prior to the maturity. This is a kind of defaultable cor-porate bonds and the pricing problem of defaultable corporate bonds is one of the most promising areas in financialmathematics [1].It is well known that there are two main approaches among methods to price defaultable corporate bonds: one isthe structural approach and another one is the reduced form approach. In the structural approach, it is thought that thedefault events occur when the firm value is not su ffi cient to repay debt, that is, the firm value attains a certain lowerthreshold( default barrier or default boundary ) from the above [16, 12]. In the reduced form approach, they think thatit is possible for the default event to occur at any time and the default event is an unpredictable event without anyrelation to the firm value. In the reduced form approach, if the default probability in time interval [ t , t + ∆ t ] is λ ∆ t ,then λ is called default intensity or hazard rate [6, 10]. The third approach is to unify the structural and reduced formapproaches [2, 5, 13, 14]. As for the history of the above approaches and their advantages and shortcomings, readerscan refer to the introductions of [2, 5].The related information such as default barrier and default intensity is related to the internal business informationof companies and the structural and reduced form approaches can be used to design the corporate bonds.In reality it is very hard for investors out of the company to get the information of the company in the wholelife time interval of the bond. They might probably know only the discretely (for example, every year or everythree months etc.) declared informations. Hence the modelling of corporate bonds using only the discrete defaultinformation was proposed with the purpose of making the study of credit risk close to the financial reality. In thisdirection, [16, 18] gives some results of zero coupon bonds using higher binary options([17]).There have been many studies of theoretical modelling of the price of zero coupon bonds which are originatedin [15], whereas studies of realistic payout structure providing fixed discrete coupons are relatively less[1]. Geske[7] Email address: [email protected] (Hyong-Chol O , Tae-Song Kim ) Preprint submitted to Elsevier July 6, 2020 tudied this problem at first, where the author models discrete interest payouts prior to maturity as determinants ofdefault risk. Agliardi[1] generalized the formula of [7] for defaultable coupon bonds and studied a stochastic risk freeterm structure and the e ff ects of bankruptcy cost and government taxes on bond interest and calculated the duration ofdefaultable bonds. Agliardi’s approach in [1] is a kind of structural approach.[19, 21] studied the problem of generalizing the structural model of [1] into the comprehensive unified model ofstructural and reduced form approaches. [19] obtained the pricing formula of the corporate bond with discounteddiscrete coupon in unified two-factor model of structural and reduced form approaches. [21] obtained the pricingformula of the corporate bond with fixed discrete coupon in unified one-factor model of structural and reduced formapproaches.In [1, 19, 21], they studied the discrete coupon bonds without the early redemption provision. However, manyfirms issue and use discrete coupon bonds with the provision that allow the holder to demand early redemption priorto the maturity.Generally speaking, discrete coupon bonds with the provision that allow the holder to demand early redemptionprior to the maturity are included in the class of puttable bonds (or bond options)[3, 8] and widely used in manycompanies but it seems di ffi cult to find studies on the their concrete pricing models and price estimates, there are onlysome works on general pricing bond option on zero coupon bonds[9, 23, 24].In this paper we derive a PDE model for the price of a discrete coupon bond with the provision that allows theholder to demand early redemption at any coupon dates prior to the maturity, and based on the financial analysis onthe relationships between the design parameters of the bond we prove the existence and uniqueness of the default andearly redemption boundaries. Then we give the analytic pricing formula of the bond using higher binary options, andsome applications including the analysis on duration and credit spread. Our works provide some design guide of thediscrete coupon bond with the early redemption provision.The remainder of the article is organized as follows. In Section 2 we consider the pricing model of a discretecoupon bond with the early redemption provision. In Section 3 we prove the existence and uniqueness of the defaultboundaries and describe some relationships between the design parameters of the discrete coupon bond with earlyredemption provision. In Section 4 we prove the existence and uniqueness of the early redemption boundaries andSection 5 gives the analytic pricing formula of the bond. Section 6 provides some applications including the analysison duration and credit spread.
2. The mathematical model of the bond priceAssumptions
1) The short rate r is a constant. Then the price of default free zero coupon bond with maturity T and face value 1is Z ( t ; T ) = e − r ( T − t ) .
2) The firm value process V ( t ) follows a geometric Brownian motion dV ( t ) = ( r − b ) V ( t ) dt + s V V ( t ) dW ( t )under the risk neutral martingale measure. Here b ≥ = T < T < · · · < T N − < T N = T and let T be the maturity of our corporate bond with face value F (unitof currency). At time T i ( i = , · · · , N − C i (unit of currency) fromthe firm and at time T N = T , the bond holder receives the face value F and the last coupon C N (unit of currency).4) The bond holder has the right to demand early redemption at any coupon dates T i , ( i = , · · · , N −
1) priorto the maturity. If the bond holder demand early redemption, the firm does not pay the coupon of the day and bondholder receives the face value deducted the coupons he had already received. That is, if the bond holder demand earlyredemption at T i , the bond holder receives F − P i − j = C j (unit of currency)5) The default occurs only at time T i when the firm value is not su ffi cient to pay the debt and the coupon or theearly redemptive money. If the default occurs, the bond holder receives δ · V as default recovery. Here 0 ≤ δ < default recovery .6) In the subinterval ( T i , T i + ], the prices of our corporate bond are given by su ffi ciently smooth functions B i ( V , t ) , ( i = , · · · , N − athematical model for bond price We will use the following notations for simplicity:¯ c N = F + C N ; ¯ c i = C i , i = , · · · , N − . First we analyse the default event. If the default event occurs or the bond holder demands early redemption before T i or at T i ( i = , · · · , N − T i < t ≤ T i + . Hence B i ( V , t ) , T i < t ≤ T i + is the bond(debt) price on the interval ( T i , T i + ] under the condition that the bond holder doesn’t demand earlyredemption and the default event doesn’t occur at T i or before T i . Therefore, the fact that the default event doesn’toccur at T i ( i = , · · · , N −
1) means that the firm value is not smaller than B i ( V , T i ) after paying the coupon ¯ c i , that is, V ≥ B i ( V , T i ) + ¯ c i . And the fact that the default event doesn’t occur at T i ( i = , · · · , N −
1) means that the firm valueis enough to pay the early redemptive money to the bond holder at T i . From the assumption 4), if the bond holderdemands early redemtion at T i , the firm gives the bond holder the face value deducted the coupons ¯ c j ( j = , · · · , i − F − P i − j = ¯ c j (unit of currency). Thus the fact that the default event doesn’t occurmeans that V ≥ F − P i − j = ¯ c j . On the whole, the fact that the default event doesn’t occur at T i means that V ≥ max F − i − X j = ¯ c j , B i ( V , T i ) + ¯ c i . (1)On the other hand, the fact that the default event occurs at T i means that V < max F − i − X j = ¯ c j , B i ( V , T i ) + ¯ c i . (2)Next, we analyse whether it is advantageous for the bond holder to keep the bond contract or to demand earlyredemption at T i . If the bond holder demands early redemption at T i , then the holder receives F − P i − j = ¯ c j (unit ofcurrency), whereas if the bond holder keeps the contract, then the holder receives coupon ¯ c i (unit of currency) and alsopossesses the bond with the value of B i ( V , t ) after T i . Thus, at time T i , the bond holder compares B i ( V , T i ) + ¯ c i with F − P i − j = ¯ c j and if F − P i − j = ¯ c j is larger, then the holder will demand early redemption immediately but if B i ( V , T i ) + ¯ c i is larger, then the holder will keep the contract. As a result, it is reasonable to think that the bond holder compares theproposal of keeping the contract with the proposal of demanding early redemption and then choose the better proposal.Therefore under the assumption that the default event didn’t occur and the holder didn’t demand early redemption atthe coupon dates prior to T i , if the default event doesn’t occur at T i , then the bond price at T i ismax F − i − X j = ¯ c j , B i ( V , T i ) + ¯ c i , and if the default event occurs at T i , then the bond holder receives δ V as default recovery by the assumption 5). Hencethe bond price at T i ( i = , · · · , N −
1) is as follows:if V ≥ max F − i − X j = ¯ c j , B i ( V , T i ) + ¯ c i , then max F − i − X j = ¯ c j , B i ( V , T i ) + ¯ c i , if V < max F − i − X j = ¯ c j , B i ( V , T i ) + ¯ c i , then δ V . (3)In particular the bond price at the maturity T N = T is B N − ( V , T N ) = ¯ c N · { V ≥ ¯ c N } + δ V · { V < ¯ c N } . (4)3rom the assumptions 1), 2), 6), it follows that the bond price B i on the subinterval ( T i , T i + ) ( i = , · · · , N − ∂ B i ∂ t + s V V ∂ B i ∂ V + ( r − b ) V ∂ B i ∂ V − rB i = , T i < t < T i + , V > . (5)From (1)-(4), we get terminal conditions of the bond price. B N − ( V , T N ) = ¯ c N · { V ≥ ¯ c N } + δ V · { V < ¯ c N } , V >
0; (6) B i ( V , T i + ) = max B i + ( V , T i + ) + ¯ c i + , F − i X j = ¯ c j · V ≥ max B i + ( V , T i + ) + ¯ c i + , F − i X j = ¯ c j + δ V · V < max B i + ( V , T i + ) + ¯ c i + , F − i X j = ¯ c j , V > , i = , · · · , N − . (7)So our model of the bond price is (5), (6), (7), that is, we must find B i satisfying (5) i = N − and (6) on the interval T N − < t ≤ T N and (5) and (7) on the interval T i < t ≤ T i + ( i = , · · · , N − T N − < t ≤ T N is just the same one as in [21]. But on the interval T i < t ≤ T i + ( i = , · · · , N − T i + and makes a decision, thus we must first find the early redemption boundary and the problembecomes American option-like pricing problem, or more precisely, Bermudan option-like pricing problem (at pages193 of [11] and 253 ∼
255 of [17]).
3. The existence and uniqueness of the default boundaries
The following notations are used: f N − ( V ) = ¯ c N · { V ≥ ¯ c N } + δ V · { V < ¯ c N } , (8) f i ( V ) = max B i + ( V , T i + ) + ¯ c i + , F − i X j = ¯ c j · V ≥ max B i + ( V , T i + ) + ¯ c i + , F − i X j = ¯ c j + δ V · V < max B i + ( V , T i + ) + ¯ c i + , F − i X j = ¯ c j , V > , i = , , · · · , N − . (9)The supremum and infimum of the function f defined on interval [0 , + ∞ ) are denoted by M ( f ) , m ( f ), repectively.First consider the case when i = N −
1. The bond price B N − ( V , t ) on the interval T N − < t ≤ T N is the solution ofthe following problem: ∂ B N − ∂ t + s V V ∂ B N − ∂ V + ( r − b ) V ∂ B N − ∂ V − rB N − = , T N − < t < T N , V > , B N − ( V , T N ) = ¯ c N · { V ≥ ¯ c N } + δ V · { V < ¯ c N } , V > . This is a terminal value problem for Black-Scholes equation with interest rate r , dividend rate b and volatility s V . Bythe terminal condition, D N = ¯ c N is the default boundary at T N .By the pricing formula of the first order binary option [4, 17] we have B N − ( V , t ) = ¯ c N B + ¯ c N ( V , t ; T N ; r , b , s V ) + δ A − ¯ c N ( V , t ; T N ; r , b , s V ) . (10)Here B + K ( x , t ; T ; r , b , s V ) , A − K ( x , t ; T ; r , b , s V ) is the price at t of the bond and asset binary options with maturity T ,exercise price K , interest rate r , dividend rate b and volatility s V , respectively [17].4 heorem 1 (The gradient estimates and the existence and uniqueness of the default boundaries) . Assume that thevolatility s V is enough large, that is, there exists a sequence δ = d N < d N − < · · · < d < such thats V ≥ (1 − δ ) √ π · ( T i + − T i )( d i − d i + ) if b = s V ≥ (1 − δ ) e − b ( T i + − T i ) √ π · ( T i + − T i )( d i − d i + e − b ( T i + − T i ) ) if b > , i = , · · · , N − . Then for the solution B i ( V , t ) , i = , · · · , N − of (5) , (6) , (7) , we have < ∂ V B i ( V , T i ) < d i < and the equation V = max F − i − X j = ¯ c j , B i ( V , T i ) + ¯ c i has unique root D i and we have V ≥ max B i ( V , T i ) + ¯ c i , F − i − X j = ¯ c j ⇔ V ≥ D i . (12) Proof.
We use induction. First when i = N −
1, we consider properties of B N − ( V , t ). From (8), the terminal payo ff f N − ( V ) is an discontinuous function with jump ∆ f N − (¯ c N ) = (1 − δ )¯ c N at V = ¯ c N . Now estimate ∂ V B N − . FromTheorem 4 of [20], we have m ( f ′ N − ) e − b ( T N − T N − ) + [ ∆ f N − (¯ c N )] − ¯ c N · e − b ( T N − T N − ) q π · s V ( T N − T N − ) < ∂ V B N − ( V , T N − ) << M ( f ′ N − ) e − b ( T N − T N − ) + [ ∆ f N − (¯ c N )] + ¯ c N · e − b ( T N − T N − ) q π · s V ( T N − T N − ) . Here [ x ] + = max { x , } , [ x ] − = min { x , } . From (8) M ( f ′ N − ) = δ = d N , m ( f ′ N − ) = < ∂ V B N − ( V , T N − ) < d N e − b ( T N − T N − ) + (1 − δ ) e − b ( T N − T N − ) q π · s V ( T N − T N − ) . From the assumption of our theorem we have (11) for i = N − V = max B N − ( V , T N − ) + ¯ c N − , F − N − X j = ¯ c j . From (11) for i = N −
1, the function max B N − ( V , T N − ) + ¯ c N − , F − N − X j = ¯ c j is monotone increasing on V and its derivative is strictly less than 1 at all the potins except for the only indi ff erentiablepoint (the intersecting point of graphs of B N − ( V , T N − ) + ¯ c N − and F − P N − j = ¯ c j ). Thus the equation V = max B N − ( V , T N − ) + ¯ c N − , F − N − X j = ¯ c j D N − . And from (11) for i = N − i = k + V = max B k + ( V , T k + ) + ¯ c k + , F − k X j = ¯ c j has unique root D k + and we have (12) for i = k +
1. Then consider properties of B k ( V , t ). From (9), the terminalpayo ff f k ( V ) is an discontinuous function with jump ∆ f k ( D k + ) = (1 − δ ) D k + at V = D k + . Now estimate ∂ V B k . FromTheorem 4 of [20], we have m ( f ′ k ) e − b ( T k + − T k ) + [ ∆ f k ( D k + )] − D k + · e − b ( T k + − T k ) q π · s V ( T k + − T k ) < ∂ V B k ( V , T k ) << M ( f ′ k ) e − b ( T k + − T k ) + [ ∆ f k ( D k + )] + D k + · e − b ( T k + − T k ) q π · s V ( T k + − T k ) . From (9) and (11) for i = k +
1, we have M ( f ′ k ) = d k + , m ( f ′ k ) = < ∂ V B k ( V , T k ) < d k + e − b ( T k + − T k ) + (1 − δ ) e − b ( T k + − T k ) q π · s V ( T k + − T k ) . From the assumption of our theorem we have (11) for i = k . Now consider roots of the non-linear equation V = max B k ( V , T k ) + ¯ c k , F − k − X j = ¯ c j . From (11) for i = k , the function max B k ( V , T k ) + ¯ c k , F − k − X j = ¯ c j is monotone increasing on V and its derivative is strictly less than 1 at all the potins except for the only indi ff erentiablepoint (the intersecting point of graphs of B k ( V , T k ) + ¯ c k and F − P k − j = ¯ c j ). Thus the equation V = max B k ( V , T k ) + ¯ c k , F − k − X j = ¯ c j has unique root D k . And from (11) for i = k we have (12). Remark 1.
From (12) , D i is called the default boundary at T i . Lemma 1 (The minimum estimate) . Under the assumption of Theorem 1, for the solution B i ( V , t ) , i = , · · · , N − of (5) , (6) , (7) , we have the estimate: min V B i ( V , T i ) = B i (0 , T i ) = . The proof of Lemma 1 is provided in appendix. In what follows, the proofs are provided in appendix if theirmathematical proofs are not directly related to the expansion of the paper.The following lemma shows that for our bond, if at one intermediate coupon date early redemption is alwaysadvantegeous regardless of firm value , then early redemption is always advantegeous at all coupon dates prior to that.6 emma 2.
Under the assumption of Theorem 1, if for some i = , · · · , N − V [ B i ( V , T i ) + ¯ c i ] ≤ F − i − X j = ¯ c j , (13) then we have sup V [ B i − ( V , T i − ) + ¯ c i − ] ≤ F − i − X j = ¯ c j . Corollary 1.
Under the assumption of Theorem 1, if sup V [ B ( V , T ) + ¯ c ] > F , (14) then we have sup V [ B i ( V , T i ) + ¯ c i ] > F − i − X j = ¯ c j , i = , , · · · , N − . From Lemma 2, if (13) holds for some i ∈ { , · · · , N − } , then (13) is also true for i =
1, that is, B ( V , T ) + C < F for all V ∈ [0 , + ∞ ). The financial meaning of this expression is that it is always advantageous for the bond holderto demand early redemption regardless of the firm value at the first coupon date ( T ). In the viewpoint of the bondissuing company, the significance of issuing bond is reduced in full width. Indeed, in this case the bond exists only onthe interval [0 , T ] and does not exist after the time T . And the bond price on the interval [0 , T ] satisfies ∂ B ∂ t + s V V ∂ B ∂ V + ( r − b ) V ∂ B ∂ V − rB = , ≤ t < T , V > , B ( V , T ) = F · { V ≥ F } + δ V · { V < F } . This problem is just the same as the problem (2.8) and (2.9) of [21] and thus the bond price at t ∈ [0 , T ] is providedas B ( V , t ) = FB + F ( V , t ; T ; r , b , s V ) + δ A − F ( V , t ; T ; r , b , s V ) , ≤ t ≤ T . Therefore in this case the bond is a zero coupon bond with the maturity T and the face value F .It is reasonable for the coupon bond with early redemption to assume that (14) is satisfied. Lemma 3 (The maximum estimate) . Under the assumption of Theorem 1 and the assumption (14) , the solutionB i ( V , t ) , i = , · · · , N − of (5) , (6) and (7) satisfies sup V B i ( V , T i ) = B i ( + ∞ , T i ) = N X j = i + h ¯ c j e − r ( T j − T i ) i . Now using Lemma 3 we analyse what relations between the parameters of the bond the assumption (14) requires.From Lemma 3, (14) becomes ¯ c + N X j = h ¯ c j e − r ( T j − T ) i > F . Multiplying e r ( T N − T ) to the both sides of this inequality, we have N X j = h C j e r ( T N − T j ) i > F · e r ( T N − T ) − F . (15)7his relation gives us the lower bound for the bond coupons. In particular, if the intervals between the adjoiningcoupon dates and the coupons are always the same, i.e, ∆ T = T i + − T i , i = , · · · , N − C i = C j = C , ≤ i , j ≤ N ,then (15) becomes C N X j = h e r ( T N − T j ) i > F · [ e r ( T N − T ) − . This yields CF > ( e r ∆ T − · e ( N − r ∆ T − e Nr ∆ T − ≈ r ∆ T · ( N − r ∆ TNr ∆ T = N − N · ( r ∆ T ) . (16)This relation gives us the lower bound to the ratio of the coupon to the face value in the bond with early redemptionprovision. Remark 2.
From the process of deriving (15) , under the assumption of Theorem 1, the assumption (15) becomes anecessary condition for the assumption (14) to hold.
Now in order to show that under the assumption of Theorem 1, the assumption (15) is a su ffi cient condition forthe assumption (14) to hold, we prove the following lemmas. Lemma 4.
If for some ≤ m ≤ N − , N X j = m h ¯ c j e r ( T N − T j ) i > F − m − X j = ¯ c j · e r ( T N − T m ) is satisfied then we have N X j = m + h ¯ c j e r ( T N − T j ) i > F − m X j = ¯ c j · e r ( T N − T m + ) . The financial meaning of Lemma 4 is that if a risk free discrete coupon bond has early redemption provision andkeeping the risk free bond at some coupon date T m is more advantageous than early redemption, then keeping the riskfree bond is advantageous at the date T m + , too. Corollary 2. If (15) holds, then for ≤ m ≤ N − we have N X j = m h ¯ c j e r ( T N − T j ) i > F − m − X j = ¯ c j · e r ( T N − T m ) , or equivalently, N X j = m h ¯ c j e − r ( T j − T m − ) i > F − m − X j = ¯ c j · e − r ( T m − T m − ) . (17) Lemma 5.
Under the assumption of Theorem 1 and assumption (15) , we have sup V B i ( V , T i ) = B i ( + ∞ , T i ) = N X j = i + h ¯ c j e − r ( T j − T i ) i . Corollary 3.
Under the assumption of Theorem 1, (15) is a necessary and su ffi cient condition for the assumption (14) to hold. This shows that it is valid to set up the parameters of the discrete coupon bond with early redemption provisionsuch that the assumption (15) is satisfied.
Remark 3.
In the financial reality, there are such bonds that (15) is not satisfied. For example, see zero couponbonds. In the case of such bonds that the assumption (15) is not satisfied (that is, the bonds with too small coupons),in the viewpoint of the bond issuing firm, the early redemption provision should be canceled. If the early redemptionprovision is canceled, then the bond becomes discrete coupon bond without early redemption provision (studied in[21], already).
From now, we consider the bonds with early redemption provision satisfying (15).8 . The early redemption boundaries
So far, it is not clear whether the problem (5) and (7) can be solved by using higher order binary options or not.To make it clear, we study the structure of the terminal functions f i ( V ) for B i ( V , t ) , i = , · · · , N − y = B i + ( V , T i + ) + ¯ c i + and the line of y = F − P ij = ¯ c j . See Figure1. There are 3 cases: the case that the inequalitysup V { B i + ( V , T i + ) + ¯ c i + } = ¯ c i + + N X j = i + h ¯ c j e − r ( T j − T i + ) i ≤ F − i X j = ¯ c j (18)holds(see line 1 in Figure 1), the case that the inequalitymin V { B i + ( V , T i + ) + ¯ c i + } = ¯ c i + > F − i X j = ¯ c j (19)holds(see line 2 in Figure 1) and the case that the graph of y = B i + ( V , T i + ) + ¯ c i + and the line of y = F − P ij = ¯ c j intersect at only one point(see line 3 in Figure 1). line 1line 2line 3 y=sup V [B i+1 (V,T i+1 )+C i+1 ] y=B i+1 (V,T i+1 )+C i+1 (black) C i+1 Figure 1. The positions of the graph of y = B i + ( V , T i + ) + ¯ c i + and the line of y = F − P ij = ¯ c j . Here the lines 1, 2, 3are the graphs of y = F − P ij = ¯ c j in di ff erent cases.Consider the first case (with (18)). Since we assumed (15), Corollary 3 implies that (14) holds and Corollary 1implies that sup V [ B i ( V , T i ) + ¯ c i ] > F − i − X j = ¯ c j , i = , , · · · , N − . i + X j = ¯ c j > F (20)and if for some i = m (19) holds, then for all m < i ≤ N − c i ≥
0. That is, for all m < i ≤ N − V ∈ [0 , + ∞ ), we have B i + ( V , T i + ) + ¯ c i + > F − i X j = ¯ c j . Now let M = min ≤ k ≤ N − k + X j = ¯ c j > F . (21)Then for 0 ≤ i < M , neither (18) nor (19) holds and we have only the third case (see line 3 in Figure 1). Remark 4.
The bond holder should keep always the contract at T M + and the later coupon dates. Thus at T M + andthe later coupon dates, our discrete coupon bond becomes the discrete coupon bond without early redemption [21].Therefore, on the interval T M + ≤ t ≤ T N , our bond price B i ( V , t ) ( T i ≤ t < T i + , M ≤ i ≤ N − is given by theformula (2.10) in the case of λ i = in [21]. The result is as follows:B i ( V , t ) = N − X k = i h ¯ c k + B + ··· + + D i + ··· D k D k + ( V , t ; T i + , · · · , T k , T k + ; r , b , s V ) . + δ A + ··· + − D i + ··· D k D k + ( V , t ; T i + , · · · , T k , T k + ; r , b , s V ) i , ( T i ≤ t < T i + , M ≤ i ≤ N − . (22)Thus, we have evaluated the bond price on the interval T M < t ≤ T N .Now we only need to evaluate the bond price on the interval T ≤ t ≤ T M . Theorem 2 (Existence and uniqueness of early redemption boundaries) . Suppose that (15) is satisfied. Then fori = , · · · , M the nonlinear equation B i ( V , T i ) + ¯ c i = F − i − X j = ¯ c j has unique root E i and we have B i ( V , T i ) + ¯ c i ≥ F − i − X j = ¯ c j ⇔ V ≥ E i . Proof.
From (21), for all i = , · · · , M we have ¯ c i < F − i − X j = ¯ c j . On the other hand, since (15) is satisfied, Corollay 3 and Lemma 2 implies F − i − X j = ¯ c j < sup V { B i ( V , T i ) + ¯ c i } . Now note that ¯ c i = min V { B i ( V , T i ) + ¯ c i } we havemin V { B i ( V , T i ) + ¯ c i } < F − i − X j = ¯ c j < sup V { B i ( V , T i ) + ¯ c i } (see line 3 in Figure 1) . B i ( V , T i ) + ¯ c i is continuous and strictly increasing (see (11)), the nonlinear equation B i ( V , T i ) + ¯ c i = F − i − X j = ¯ c j , i.e., B i ( V , T i ) = F − i X j = ¯ c j has unique root E i and from (11) we have B i ( V , T i ) + ¯ c i ≥ F − i − X j = ¯ c j ⇔ V ≥ E i . Remark 5.
For all i = , · · · , M we haveB i ( V , T i ) + ¯ c i ≥ F − i − X j = ¯ c j ⇔ V ≥ E i . Thus E i is called the early redemption boundary at T i . Remark 6.
If between the face value and coupons there is a relation N − X j = C j < F , then M = N − by (21) . Thus the early redemption boundary E i uniquely exists for any i = , · · · , N − and ourbond becomes the bond with early redemption in the whole interval ≤ t ≤ T N . On the other hand, if N − X j = C j ≥ Fthen from (21) we have M < N − and E i uniquely exists only for i = , · · · , M. And our bond becomes the bondwith early redemption on the interval ≤ t ≤ T M but on the interval T M < t ≤ T N , it becomes the bond without earlyredemption and we calculate the bond price by (22) .
5. The pricing formula of our bond
Now in order to get the formula of the bond price in the interval 0 ≤ t ≤ T M , we calculate the bond price B i ( V , T i + )at the coupon dates T i ( i = , · · · , M − y = max { B i + ( V , T i + ) + ¯ c i + , F − P ij = ¯ c j } ; the blue curveis the graph of y = B i + ( V , T i + ) + C i + ; the black line is the graph of y = F − P ij = ¯ c j ; the pink line is the graph of y = V . As you can see in Figure 2 and Figure 3, there are two cases of the positions of D i + and E i + : In first case theintersection point of the graph of y = V and the graph of y = max (cid:8) [ B i + ( V , T i + ) + ¯ c i + ] , F − P ij = ¯ c j (cid:9) is on the branchof y = B i + ( V , T i + ) + ¯ c i + so D i + < E i + . In second case the intersection point of the graph of y = V and the graphof y = max (cid:8) [ B i + ( V , T i + ) + ¯ c i + ] , F − P ij = ¯ c j (cid:9) is on the branch of y = F − P ij = ¯ c j so D i + > E i + .First we consider the case when the intersection point of the graph of y = V and the graph of y = max (cid:8) [ B i + ( V , T i + ) + ¯ c i + ] , F − P ij = ¯ c j (cid:9) is on the branch of y = B i + ( V , T i + ) + ¯ c i + . (Figure 2). Then D i + is the solution of the equation V = B i + ( V , T i + ) + ¯ c i + and E i + ≤ D i + . As you can see in Figure 2, in this case, the default boundary D i + corre-sponds to the default event that occurs because of that the firm value is less than debt when the bond holder keeps thecontract and we can rewrite the terminal payo ff function at T i + as follows: B i ( V , T i + ) = [ B i + ( V , T i + ) + ¯ c i + ] · { V ≥ D i + } + δ V · { V < D i + } . (23)11 i+1 D i+1 y=F- j=1i C j (black) y=max{B i+1 (V,T i+1 )+C i+1 , F- j=1i C j } (red) y=V (pink) y=B i+1 (V,T i+1 )+C i+1 (blue) C i+1 Figure 2. The intersection point is on the branch of y = B i + ( V , T i + ) + ¯ c i + . E i+1 D i+1 y=max{B i+1 (V,T i+1 )+C i+1 , F- j=1i C j } (red) y=V (pink) C i+1 y=F- j=1i C j (black) y=B i+1 (V,T i+1 )+C i+1 (blue) Figure 3. The intersection point is on the branch of y = F − P ij = ¯ c j .Next we consider the case when the intersection point of the graph of y = V and the graph of y = max (cid:8) [ B i + ( V , T i + ) + c i + ] , F − P ij = ¯ c j (cid:9) is on the branch of y = F − P ij = ¯ c j (Figure 3). Then D i + = F − P ij = ¯ c j and D i + < E i + . As youcan see in Figure 3, in this case, the default boundary D i + corresponds to the default event that occurs because of thatthe firm value is less than the early redemption money when the bond holder demands early redemption and we canrewrite the terminal payo ff function at T i + as follows: B i ( V , T i + ) = [ B i + ( V , T i + ) + ¯ c i + ] · { V ≥ E i + } + ( F − Σ ij = ¯ c j ) · { D i + ≤ V < E i + } + δ V · { V < D i + } = [ B i + ( V , T i + ) + ¯ c i + ] · { V ≥ E i + } + ( F − Σ ij = ¯ c j ) · (1 { V ≥ D i + } − { V ≥ E i + } ) + δ V · { V < D i + } . (24)Putting (23) and (24) together, then we have B i ( V , T i + ) = [ B i + ( V , T i + ) + ¯ c i + ] · { V ≥ max { E i + , D i + }} + ( F − Σ ij = ¯ c j ) · { D i + < E i + }· [1 { V ≥ min { E i + , D i + }} − { V ≥ max { E i + , D i + }} ] + δ V · { V < D i + } . Now we use the following notations: U i + = max { D i + , E i + } , L i + = min { D i + , E i + } , i = , · · · , M − . Then we have B i ( V , T i + ) = [ B i + ( V , T i + ) + ¯ c i + ] · { V ≥ U i + } + ( F − Σ ij = ¯ c j ) · { D i + < E i + } · { V ≥ L i + }− ( F − Σ ij = ¯ c j ) · { D i + < E i + } · { V ≥ U i + } + δ V · { V < D i + } . (25)Thus for i = M − B M − ( V , T M ) = [ B M ( V , T M ) + ¯ c M ] · { V ≥ U M } + ( F − Σ M − j = ¯ c j ) · { D M < E M } · { V ≥ L M }− ( F − Σ M − j = ¯ c j ) · { D M < E M } · { V ≥ U M } + δ V · { V < D M } . Substituting B M ( V , T M )((22) for i = M −
1) to the above, we get B M − ( V , T M ) = N − X k = M (cid:2) ¯ c k + B + ··· + + D M + ··· D k D k + ( V , t ; T M , · · · , T k + ; r , b , s V ) + δ A + ··· + − D M + ··· D k D k + ( V , t ; T M , · · · , T k , T k + ; r , b , s V ) (cid:3) · { V ≥ U M } ++ ¯ c M · { V ≥ U M } + ( F − Σ M − j = ¯ c j ) · { D M < E M } · { V ≥ L M }−− ( F − Σ M − j = ¯ c j ) · { D M < E M } · { V ≥ U M } + δ V · { V < D M } By using the pricing formula of higher order binary options[17], we can get B M − ( V , t ) = N − X k = M (cid:2) ¯ c k + B + + ··· + + U M D M + ··· D k D k + ( V , t ; T M , · · · , T k + ; r , b , s V ) + δ A + + ··· + − U M D M + ··· D k D k + ( V , t ; T M , T M + , · · · , T k + ; r , b , s V ) (cid:3) + ( F − Σ M − j = ¯ c j ) · { D M < E M } · B + L M ( V , t ; T M ; r , b , s V ) − (26) − ( F − Σ M − j = ¯ c j ) · { D M < E M } · B + U M · ( V , t ; T M ; r , b , s V ) ++ ¯ c M B + U M ( V , t ; T M ; r , b , s V ) + δ A − D M · ( V , t ; T M ; r , b , s V ) . We can rewrite this as follows: B M − ( V , t ) = N − X k = M − (cid:2) ¯ c k + B + + ··· + + U M D M + ··· D k D k + ( V , t ; T M , · · · , T k + ; r , b , s V ) + δ A + + ··· + − U M D M + ··· D k D k + ( V , t ; T M , T M + , · · · , T k + ; r , b , s V ) (cid:3) + ( F − Σ M − j = ¯ c j ) · { D M < E M } · B + L M ( V , t ; T M ; r , b , s V ) −− ( F − Σ M − j = ¯ c j ) · { D M < E M } · B + U M · ( V , t ; T M ; r , b , s V ) .
13y induction, we assert that the following theorem holds.For convenience, we use the following notations: D N = E N = ¯ c N ; U i = D i , M < i ≤ N . (27) Theorem 3 (The pricing formula) . Under the assumptions of Theorem 1 and Theorem 2, the solution of (5) and (6) and (7) is given as follows: For i = , · · · , N − ,B i ( V , t ) = N − X k = i (cid:2) ¯ c k + B + ··· + + U i + ··· U k U k + ( V , t ; T i + , · · · , T k + ; r , b , s V ) + δ A + ··· + − U i + ··· U k D k + ( V , t ; T i + , · · · , T k , T k + ; r , b , s V ) (cid:3) ++ M X k = i + ( F − Σ k − j = ¯ c j ) · { D k < E k } · (cid:2) B + ··· + + U i + ··· U k − L k ( V , t ; T i + , · · · , T k ; r , b , s V ) (28) − B + ··· + + U i + ··· U k − U k ( V , t ; T i + , · · · , T k ; r , b , s V ) (cid:3) , T i < t < T i + . Here B + ··· + K ··· K m ( x , t ; T , · · · , T m ; r , q , σ ) and A + ··· + K ··· K m ( x , t ; T , · · · , T m ; r , q , σ ) is the price of m-order bond andasset binary options [17] with the free risk rate r, the dividend rate q and the volatility σ , respectively. Corollary 4.
Under the assumptions of Theorem 1 and Theorem 2, the solution of (5) , (6) , (7) is given as follows: Fori = , · · · , N − ,B i ( V , t ) = N − X k = i (cid:2) e − r ( T k + − t ) ¯ c k + N k − i + ( d − i + ( t ) , · · · , d − k + ( t ); A k − i + ) + e − b ( T k + − t ) δ VN k − i + ( d + i + ( t ) , · · · , d + k ( t ) , − ˜ d + k + ( t ); A − k − i + ) (cid:3) + (29) + M X k = i + ( F − Σ k − j = ¯ c j ) · { D k < E k } · e − r ( T k − t ) (cid:2) N k − i ( d − i + ( t ) , · · · , d − k − ( t ) , ˜˜ d − k ( t ); A k − i ) − N k − i ( d − i + ( t ) , · · · , d − k − ( t ) , d − k ( t ); A k − i ) (cid:3) , T i < t < T i + . Here d ± j ( t ) = s V p T j − t " ln VU j + r − b ± s V ! ( T j − t ) , ˜ d ± j ( t ) = s V p T j − t " ln VD j + r − b ± s V ! ( T j − t ) , ˜˜ d ± j ( t ) = s V p T j − t " ln VL j + r − b ± s V ! ( T j − t ) . And the matrix A n , A − n are the inverse A n = ( R n ) − , A − n = (cid:0) R − n (cid:1) − of the n × n dimensional matrix R n , R − n , which aregiven as follows: if we use the notation ˜ T j = T j + i + , ≤ j < n then the elements r lm ( t ) of R n are given asr ll ( t ) = , r lm ( t ) = r ml ( t ) = s ˜ T l − t ˜ T m − t , l < m , ( l , m = , · · · , n − . And R − n is the matrix, in which,r − m , n − ( t ) = − r m , n − ( t ) , r − n − , m ( t ) = − r n − , m ( t ) , ( m = , · · · , n − and the other elements are as in R n . And N n ( d , · · · , d n ; A n ) is n-dimensional normal distribution function withcorrelation matrix R n = ( A n ) − , i.e.,N n ( d , · · · , d n ; A n ) = Z d −∞ · · · Z d n −∞ √ π ) n p det A n exp − y ⊥ A n y ! dy . orollary 5 (The initial bond price) . Under the assumptions of Theorem 1 and Theorem 2, the initial bond price isdenoted as follows:B = B ( V , = N − X k = (cid:2) e − rT k + ¯ c k + N k + ( d − (0) , · · · , d − k + (0); A k + ) + e − bT k + δ V N k + ( d + (0) , · · · , d + k (0) , − ˜ d + k + (0); A − k + ) (cid:3) + (30) + M X k = ( F − Σ k − j = ¯ c j ) · { D k < E k } · e − rT k (cid:2) N k ( d − (0) , · · · , d − k − (0) , ˜˜ d − k (0); A k ) −− N k ( d − (0) , · · · , d − k − (0) , d − k (0); A k ) (cid:3) Now we denote the initial leverage ratio F / V by L and the kth coupon ratio C k / F by c k . Then the initial price of thebond is as follows:B ( V , = B ( L , F , c , · · · , c N ; δ ; r , b ) = F n e − rT N N N ( d − (0) , · · · , d − N (0); A N ) ++ N − X k = h e − rT k + c k + N k + ( d − (0) , · · · , d − k + (0); A k + ) + e − bT k + δ L N k + ( d + (0) , · · · , d + k (0) , − ˜ d + k + (0); A − k + ) i + (31) + M X k = (1 − Σ k − j = c j ) · { D k < E k } · e − rT k (cid:2) N k ( d − (0) , · · · , d − k − (0) , ˜˜ d − k (0); A k ) − N k ( d − (0) , · · · , d − k − (0) , d − k (0); A k ) (cid:3)o In what follows, we give numerical examples for the bond price calculated using the pricing formula (29) andMatlab. We use the function mvncdf of Matlab in order to calculate multi-dimensional normal distribution functionin the pricing formula (29). The basic data are given as follows: N = , T = , T = , T = annum ) , r = . , b = , s V = . , δ = . , F = , C = C = C = . Figure 4 shows the default boundary D and the early redemption boundary E at T when the firm value V variesfrom 0 to 16000. Here the red line represents the graph-( V , max { B ( V , T ) + C , F } ), the blue curve represents the( V , B ( V , T ) + C )-graph and the black line represents ( V , F )- graph and the pink line represents ( V , V )-graph,respectively.Figure 5 shows the default boundary D and the early redemption boundary E at T when the firm value V variesfrom 0 to 16000. Here the red line represents the graph-( V , max { B ( V , T ) + C , F − C } ), the blue curve representsthe ( V , B ( V , T ) + C )-graph and the black line represents the ( V , F − C )-graph and the pink line represents the( V , V )-graph, respectively.As you can see in Figure 4 and Figure 5, for i = , ≤ V < D i , then the firm value is in the default region. That is, the default event occurs at T i ;If D i ≤ V < E i , then the firm value is in the early redemption region. That is, the bond holder should demandearly redemption at T i ;If E i ≤ V , then the firm value is in the continuous region. That is, the bond holder should keep the contract at T i .15 Firm Value B ond P r i c e E1=11945D1=1000 Continuous Region Early redemption RegionDefaultRegion
Figure 4. The default boundary D and the early redemption boundary E at T . Firm Value B ond P r i c e E2=5099D2=960DefaultRegion EarlyRedemption Region Continuous Region
Figure 5. The default boundary D and the early redemption boundary E at T .16 Firm Value B ond P r i c e E1D1 t = 0t = 0.25t = 0.5t = 0.75t = 1.0
Figure 6. ( V , B ( V , t ))-graphs Firm Value B ond P r i c e E2D2 t = 1.0t = 1.25t = 1.5t = 1.75t = 2.0
Figure 7. ( V , B ( V , t ))-graphs17 V = Firm Value B = B ond P r i c e D3 t = 2.0t = 2.25t = 2.5t = 2.75t = 3.0 Figure 8. ( V , B ( V , t ))-graphs Time B ond P r i c e T1 T2
V = 5000V = 10000V = 15000
Figure 9. ( t , B ( V , t ))-graphsFigure 6 shows the ( V , B ( V , t ))-graphs at the times t = , . , . , . , .
0, respectively when the firm value18aries from 0 to 16000.Figure 7 shows the ( V , B ( V , t ))-graphs at the times t = . , . , . , . , .
0, respectively when the firmvalue varies from 0 to 16000.Figure 8 shows the ( V , B ( V , t ))-graphs at the times t = . , . , . , . , .
0, respectively when the firmvalue varies from 0 to 16000.Figure 9 shows the ( t , B ( V , t ))-graphs for firm values V = , , , T ] = [0 , E = , E = , D = , D = V = D < V < E , the default event doesn’t occur andthe bond holder demands early redemption at T . Thus, as you can see in the red graph on [0 , T ] in Figure 9, we get B ( V , T ) = F , and thus the bond does not exist on the interval ( T , T ]. So the real bond price is B ( V , t ) = , T < t ≤ T . On the other hand, the graphs of the interval ( T , T ] in Figure 9 represent the bond price under the condition thatthe default event didn’t occur and the bond holder didn’t demand early redemption at T . Under this assumption, since D < V < E , the default event doesn’t occur and the bond holder demands early redemption at T . Thus, as you cansee in the red graph on ( T , T ] in Figure 9, we get B ( V , T ) = F − C and thus the bond does not exist on the interval( T , T ].And the graphs of the interval ( T , T ] represent the bond price under the condition that the default event didn’toccur and the bond holder didn’t demand early redemption at T , T . Since V > D = F + C , the default event doesn’toccur at T and as you can see in the red graph on ( T , T ] in Figure 9, B ( V , T ) = F + C .2) We consider the case when V = D < V < E the default event doesn’t occurand the bond holder demands early redemption at T . Thus, as you can see in the blue graph on [0 , T ] in Figure 9, B ( V , T ) = F and the bond does not exist on the interval ( T , T ] like the first case.On the other hand, the blue graphs of the interval ( T , T ] in Figure 9 represent the bond price under the condition that the default event didn’t occur and the bond holder didn’t demand early redemption T . Under this assumption,since V > E > D , the default event doesn’t occur at T and the bond holder keeps the contract. Thus, as you can seein the blue graph on ( T , T ] in Figure 9, B ( V , T ) = B ( V , T ) + C . And the bond price on the interval ( T , T ] isgiven by the blue curve on the last interval of Figure 9 and B ( V , T ) = F + C .3) Next we consider the case when V = V > E > D the default event doesn’t occur at T and the bond holder keeps the contract. Thus, as you can see in the black graph on the first interval in Figure 9, B ( V , T ) = B ( V , T ) + C . The bond price on the interval ( T , T ] is given by the black graph of the second intervalof Figure 9 and B ( V , T ) = B ( V , T ) + C .And the bond price on ( T , T ] is given by the black graph of the lastinterval in Figure 9 and B ( V , T ) = F + C . 19 Firm Value C1=C2=C3=20C1=C2=C3=30C1=C2=C3=40
Figure 10. The early redemption boundary at T . Firm Value C1=C2=C3=20C1=C2=C3=30C1=C2=C3=40
Figure 11. The early redemption boundary at T .Figure 10 and Figure 11 show the relation between coupons and early redemption boundaries.20s you can see in Figure 10, in the case when C = C = C =
20, the assumption (15) (or (16)) is not satisfiedand thus the bond holder should demand early redemption at T . In other cases, the assumption (15) is satisfied andthus the early redemption boundary exists. On the other hand, Figure 11 shows the early redemption boundary at T under the condition that the default event doesn’t occur or the holder doesn’t demand early redemption at T . Theresults show that increasing coupons makes the early redemption boundary smaller. This is compatible with theirfinancial meaning. Time B ond P r i c e T1 T2
C = 80C = 90C = 100
Figure 12. The ( t , B ( V , t ))-graphs for V = ff ect of coupons on the bond price. As you can see in Figure 12, in the case of C = C = C =
80, the initial bond price is less than the face value, in the case of C = C = C =
90, the initial bond price isslightly larger than the face value, and in the case of C = C = C =
6. Some applications of the pricing formula
A duration is a measure of average life of a bond[8] and defined as follows: D ( V , t ) = − B ( V , t ) ∂ r B ( V , t ; r ) . Now we will use the following notations: f k ( r ) = N k ( d − , · · · , d − k ; A k ) , g k ( r ) = N k ( d + , · · · , d + k − , − ˜ d + k ; A − k ) , h k ( r ) = N k ( d − , · · · , d − k − , ˜˜ d − k ; A k ) . B ( V , = N − X k = h ¯ c k + e − rT k + f k + ( r ) i + e − bT k + δ V N − X k = g k + ( r ) + M X k = (cid:18) F − X k − j = ¯ c j (cid:19) · { D k < E k } · e − rT k [ h k ( r ) − f k ( r )] . Thus we have − ∂ r B = N − X k = (cid:2) ¯ c k + e − rT k + ( T k + f k + ( r ) − ∂ r f k + ) (cid:3) − e − bT k + δ V N − X k = ∂ r g k + + M X k = ( F − Σ k − j = ¯ c j ) · { D k < E k } · e − rT k (cid:2) T k ( h k − f k ) − ( ∂ r h k − ∂ r f k ) (cid:3) . (32)The lemma on the derivative of multi-dimensional normal distribution function is as follows: Lemma 6. [21] ∂ x N m ( a ( x ) , · · · , a m ( x ); A ) = m X i = ¯ N m , i ( a ( x ) , · · · , a m ( x ); A ) a ′ i ( x ) is satisfied. Here ¯ N m , i ( a ( x ) , · · · , a m ( x ); A ) = Z a ( x ) −∞ · · · Z a i − ( x ) −∞ Z a i + ( x ) −∞ · · · Z a m ( x ) −∞ √ det A ( √ π ) m exp − ⌢ y i ( x ) ⊥ A ⌢ y i ( x ) ! d ¯ y i , ⌢ y i ( x ) ⊥ = ( y , · · · , y i − , a i ( x ) , y i + , · · · , y m ) , d ¯ y i = dy · · · dy i − dy i + · · · dy m ; i = , · · · , m . Using Lemma 6 and ∂∂ r d ± i (0) = ∂∂ r ˜ d ± i (0) = ∂∂ r ˜˜ d ± i (0) = T i s V √ T i , we can get ∂ r f k + ( r ) = ∂ r N k + ( d − , · · · , d − k + ; A k + ) = k + X i = ¯ N k + , i ( d − ( r ) , · · · , d − k + ( r ); A k + ) T i s V √ T i ,∂ r g k + ( r ) = ∂ r N k + ( d + , · · · , d + k , − ˜ d + k + ; A − k + ) == k X i = ¯ N k + , i ( d − ( r ) , · · · , d + k ( r ) , − ˜ d + k + ( r ); A k + ) T i s V √ T i − ¯ N k + , k + ( d − ( r ) , · · · , d + k ( r ) , − ˜ d + k + ( r ); A k + ) T k + s V √ T k + ,∂ r h k ( r ) = ∂ r N k ( d − , · · · , d − k − , ˜˜ d − k ; A k ) = k X i = ¯ N k , i ( d − ( r ) , · · · , d − k − , ˜˜ d − k ( r ); A k ) T i s V √ T i . (33)Substituting these expressions into (32), then we have the following theorem: Theorem 4 (Duration) . ˜ D = − ∂ r B B = B N − X k = (cid:2) ¯ c k + e − rT k + ( T k + f k + ( r ) − F k + ) − e − bT k + δ V G k + (cid:3) ++ B M X k = ( F − Σ k − j = ¯ c j ) · { D k < E k } · e − rT k [ T k ( h k − f k ) − ( H k − F k )] . ere F k = k X i = ¯ N k , i ( d − ( r ) , · · · , d − k ( r ); A k ) T i s V √ T i , G k = k − X i = ¯ N k , i ( d − ( r ) , · · · , d − k − ( r ) , − ˜ d + k ( r ); A k ) T i s V √ T i − ¯ N k , k ( d − ( r ) , · · · , d − k − ( r ) , − ˜ d + k ( r ); A ) T k s V √ T k , H k = k X i = ¯ N k , i ( d − ( r ) , · · · , d − k − ( r ) , ˜˜ d − k ( r ); A k ) T i s V √ T i . The credit spread is defined in every subintervals as follows: CS i = − ln( B i ( V , t )) − ln Z i ( t ; T ) T − t , i = , · · · , N − . Here Z i ( t ; T ) = N X j = i + ¯ c j · e − r ( T j − t ) . In what follows, we give numerical examples of credit spread calculated by using Matlab. The basic data are asfollows: N = , T = , T = , T = annum ) , r = . , b = , s V = . , δ = . , F = , C = C = C = . C r ed i t S p r ead V = 5000V = 10000V = 15000
Figure 13. ( t , CS )-graphs for di ff erent firm values C r ed i t S p r ead sv = 0.5sv = 1.0sv = 1.2 Figure 14. ( t , CS )-graphs for di ff erent volatilities23 C r ed i t S p r ead delta = 0.3delta = 0.5delta = 0.8 Figure 15. ( t , CS )-graphs for di ff erent recoveries C r ed i t S p r ead C1=C2=C3=30C1=C2=C3=40C1=C2=C3=50
Figure 16. ( t , CS )-graphs for di ff erent couponsFigure 13 shows the e ff ect of firm value on credit spread. ( t , CS )-graphs for firm values V = , , , T ] = [0 , ff ect of volatility on credit spread. ( t , CS )-graphs for the volatilities s V = . , . , . , T ] = [0 , V = ff ect of recovery rate on credit spread. ( t , CS )-graphs for recovery rates δ = . , . , . , T ] = [0 , V = ff ect of coupon on credit spread. ( t , CS )-graphs for coupons C = , ,
50 are providedon time interval. Here other parameters except for coupon are the same as the above and the firm value is fixed as V =
7. Conclusion
In this paper is derived the structural model of a discrete coupon bond with early redemption provision, usingsome analysis including min-max and gradient estimates of the bond price we studied the existence and uniquenessof default boundary and relationships between the design parameters of the bond, under some assumptions whichare valid in finance we proved the existence and uniqueness of early redemption boundary and provided the analyticformula of the bond price by using higher binary options, and we gave the analysis on the duration and credit spread.Our works provide some design guide for the discrete coupon bond with early redemption provision.24irst, for the coupon bond with early redemption provision, coupons must be set up appropriately large suchthat (15) is satisfied, and if the firm does not want to pay so large coupon and thus (15) is not satisfied, then earlyredemption provision must be removed. In particular, if coupons are all the same, the face value and coupon must beset up such that (16) is satisfied.Second, if the coupon is set up too large, early redemption is disadvantageous for the bond holder and the fair initialprice of bond might be higher than the face value. In this case, if the firm sells the bond for the face value, then thefirm may have a loss. Thus if the firm wants to sell the bond for the face value, the coupon must be set up not toolarge. If the firm set up the coupon too large, then the firm must set up the initial selling price of the bond higher thanthe face value.
8. AppendixProof of Lemma 1.
We use induction. First we consider the case when i = N −
1. By the gradient estimate ofTheorem 1 and Corollary 2 of [20], we havemin V B N − ( V , T N − ) = B N − (0 , T N − ) = f N − (0) · e − r ( T N − T N − ) . On the other hand, from (8), f N − (0) = V B N − ( V , T N − ) = B N − (0 , T N − ) = . Next in the case when i = k we assume thatmin V B k ( V , T k ) = B k (0 , T k ) = i = k −
1. By the gradient estimate of Theorem 1 and Corollary 2 of [20], we havemin V B k − ( V , T k − ) = B k − (0 , T k − ) = f k − (0) · e − r ( T k − T k − ) . Using induction assumption, we havemax n [ B k ( V , T k ) + ¯ c k ] , F − Σ k − j = ¯ c j o ≥ [ B k ( V , T k ) + ¯ c k ] ≥ ¯ c k > . Then from (9) f k − (0) = δ · = V B k − ( V , T k − ) = B k − (0 , T k − ) = . ✷ Proof of Lemma 2.
By the gradient estimate of Theorem 1 and Corollary 2 of [20], we havesup V [ B i − ( V , T i − )] = B i − ( + ∞ , T i − ) = f i − ( + ∞ ) · e − r ( T i − T i − ) . From (8) and (9), we have f i − ( + ∞ ) = max n B i ( + ∞ , T i ) + ¯ c i , F − Σ i − j = ¯ c j o and from (13), we have sup V [ B i − ( V , T i − )] = ( F − Σ i − j = ¯ c j ) · e − r ( T i − T i − ) . On the other hand,sup V [ B i − ( V , T i − ) + ¯ c i − ] − ( F − Σ i − j = ¯ c j ) = [( F − Σ i − j = ¯ c j ) · e − r ( T i − T i − ) + ¯ c i − ] − ( F − Σ i − j = ¯ c j ) = ( F − Σ i − j = ¯ c j )[ e − r ( T i − T i − ) − . F − P i − j = ¯ c j ≥ V [ B i − ( V , T i − ) + ¯ c i − ] − ( F − Σ i − j = ¯ c j ) ≤ . Therefore, the lemma is proved. ✷ Proof of Lemma 3.
We use induction. First we consider the case when i = N −
1. From the gradient estimate ofTheorem 1 and Corollary 2 of [20], we havesup V [ B N − ( V , T N − )] = B N − ( + ∞ , T k − ) = f N − ( + ∞ ) · e − r ( T N − T N − ) . From (8) we have f N − ( + ∞ ) = ¯ c N and thus we have B N − ( + ∞ , T N − ) = ¯ c N · e − r ( T N − T N − ) . Thus the assertion of our lemma holds in this case.Next in the case when i = k we assume thatsup V B k ( V , T k ) = B k ( + ∞ , T k ) = N X j = k + h ¯ c j e − r ( T j − T k ) i and we will prove in the case when i = k −
1. From the gradient estimate of Theorem 1 and Corollary 2 of [20], wehave sup V [ B k − ( V , T k − )] = B k − ( + ∞ , T k − ) = f k − ( + ∞ ) · e − r ( T k − T k − ) . And from (9) we have f k − ( + ∞ ) = max n B k ( + ∞ , T k ) + ¯ c k , F − Σ k − j = ¯ c j o . Thus from Corollary 1, we havesup V [ B k − ( V , T k − )] = [ B k ( + ∞ , T k ) + ¯ c k ] · e − r ( T k − T k − ) . Substituting the induction assumption of the case when i = k into the above expression, we havesup V [ B k − ( V , T k − )] = ¯ c k + N X j = k + h ¯ c j e − r ( T j − T k ) i · e − r ( T k − T k − ) = N X j = k h ¯ c j e − r ( T j − T k − ) i . Lemma is proved. ✷ Proof of Lemma 4.
From the assumption N X j = m + h ¯ c j e r ( T N − T j ) i > ( F − Σ m − j = ¯ c j ) · e r ( T N − T m ) − ¯ c m e r ( T N − T m ) = ( F − Σ mj = ¯ c j ) · e r ( T N − T m ) > ( F − Σ mj = ¯ c j ) · e r ( T N − T m − ) . Lemma is proved. ✷ Proof of Lemma 5.
The proof in the case when i = N − i = k , we assume thatsup V B k ( V , T k ) = B k ( + ∞ , T k ) = N X j = k + h ¯ c j e − r ( T j − T k ) i i = k −
1. From the gradient estimate of Theorem 1 and the Corollary 2 of [20],we have sup V [ B k − ( V , T k − )] = B k − ( + ∞ , T k − ) = f k − ( + ∞ ) · e − r ( T k − T k − ) . (A.1)On the other hand, from (9) we have f k − ( + ∞ ) = max n B k ( + ∞ , T k ) + ¯ c k , F − Σ k − j = ¯ c j o . Substituting the above expression into (A.1) and using the induction assumption of the case of i = k , we havesup V [ B k − ( V , T k − )] = max N X j = k + h ¯ c j e − r ( T j − T k ) i + ¯ c k , F − k − X j = ¯ c j · e − r ( T k − T k − ) = max N X j = k h ¯ c j e − r ( T j − T k − ) i , F − k − X j = ¯ c j · e − r ( T k − T k − ) . From Corollary 2 we have sup V [ B k − ( V , T k − )] = B k − ( + ∞ , T k − ) = N X j = k h ¯ c j e − r ( T j − T k − ) i . ✷ Proof of Corollary 3.
The necessity has already been mentioned in Remark 2. We prove the su ffi ciency. FromLemma 5, (15) implies sup V B ( V , T ) = N X j = h ¯ c j e − r ( T j − T ) i . (A.2)Multiplying e − r ( T N − T ) to the both sides of (15) and then rewriting it, (15) is equivalent to P Nj = [¯ c j e − r ( T j − T ) ] > F , andconsidering (A.2), we have sup V B ( V , T ) + ¯ c > F . ✷ Proof of Theorem 3.
First we consider the case when M ≤ i ≤ N −
1. In this case B i ( V , t ) is given by (22), i.e., B i ( V , t ) = N − X k = i h ¯ c k + B + ··· + + D i + ··· D k D k + ( V , t ; T i + , · · · , T m + ; r , b , s V ) + δ A + ··· + − D i + ··· D k D k + ( V , t ; T i + , · · · , T m , T m + ; r , b , s V ) i . Now considering (27), then the above expression can be written into B i ( V , t ) = N − X k = i h ¯ c k + B + ··· + + U i + ··· U k U k + ( V , t ; T i + , · · · , T m + ; r , b , s V ) + δ A + ··· + − U i + ··· U k D k + ( V , t ; T i + , · · · , T m , T m + ; r , b , s V ) i . B i ( V , t ) = N − X k = i (cid:2) ¯ c k + B + ··· + + U i + ··· U k U k + ( V , t ; T i + , · · · , T k + ; r , b , s V ) ++ δ A + ··· + − U i + ··· U k D k + ( V , t ; T i + , · · · , T k , T k + ; r , b , s V ) (cid:3) ++ M X k = i + ( F − Σ k − j = ¯ c j ) · { D k < E k } · (cid:2) B + ··· + + U i + ··· U k − L k ( V , t ; T i + , · · · , T k ; r , b , s V ) −− B + ··· + + U i + ··· U k − U k ( V , t ; T i + , · · · , T k ; r , b , s V ) (cid:3) , T i < t < T i + . Indeed, i + > M and thus in the above expesssion, the term of second sum does not exist. Thus we have (28).Next we consider the case when 0 ≤ i ≤ M −
1. We use induction. First we prove in the case when i = M − B M − ( V , t ) = N − X k = M − (cid:2) ¯ c k + B + + ··· + + U M D M + ··· D k D k + ( V , t ; T M , · · · , T k + ; r , b , s V ) + δ A + + ··· + − U M D M + ··· D k D k + ( V , t ; T M , T M + , · · · , T k + ; r , b , s V ) (cid:3) ++ ( F − Σ M − j = ¯ c j ) · { D M < E M } · B + L M ( V , t ; T M ; r , b , s V ) −− ( F − Σ M − j = ¯ c j ) · { D M < E M } · B + U M · ( V , t ; T M ; r , b , s V ) . Recalling (27), this can be rewritten as follows: B M − ( V , t ) = N − X k = M − (cid:2) ¯ c k + B + + ··· + + U M U M + ··· U k U k + ( V , t ; T M , · · · , T k + ; r , b , s V ) + δ A + + ··· + − U M U M + ··· U k D k + ( V , t ; T M , T M + , · · · , T k + ; r , b , s V ) (cid:3) + ( F − Σ M − j = ¯ c j ) · { D M < E M } · B + L M ( V , t ; T M ; r , b , s V ) −− ( F − Σ M − j = ¯ c j ) · { D M < E M } · B + U M · ( V , t ; T M ; r , b , s V ) . Thus we obtained (28).Next in the case when i = m , we assume that B m ( V , t ) = N − X k = m (cid:2) ¯ c k + B + ··· + + U m + ··· U k U k + ( V , t ; T m + , · · · , T k + ; r , b , s V ) ++ δ A + ··· + − U m + ··· U k D k + ( V , t ; T m + , · · · , T k , T k + ; r , b , s V ) (cid:3) + (A.3) + M X k = m + ( F − Σ k − j = ¯ c j ) · { D k < E k } · (cid:2) B + ··· + + U m + ··· U k − L k ( V , t ; T m + , · · · , T k ; r , b , s V ) −− B + ··· + + U m + ··· U k − U k ( V , t ; T m + , · · · , T k ; r , b , s V ) (cid:3) , T m < t < T m + . and then we prove in the case when i = m −
1. From (25), B m − ( V , T m ) = [ B m ( V , T m ) + ¯ c m ] · { V ≥ U m } + ( F − Σ m − j = ¯ c j ) · { D m < E m } · { V ≥ L m }−− ( F − Σ m − j = ¯ c j ) · { D m < E m } · { V ≥ U m } + δ V · { V < D m } . B m − ( V , T m ) = (cid:26) N − X k = m (cid:2) ¯ c k + B + ··· + + U m + ··· U k U k + ( V , T m ; T m + , · · · , T k + ; r , b , s V ) ++ δ A + ··· + − U m + ··· U k D k + ( V , T m ; T m + , · · · , T k , T k + ; r , b , s V ) (cid:3) ++ M X k = m + ( F − Σ k − j = ¯ c j ) · { D k < E k } · (cid:2) B + ··· + + U m + ··· U k − L k ( V , T m ; T m + , · · · , T k ; r , b , s V ) −− B + ··· + + U m + ··· U k − U k ( V , T m ; T m + , · · · , T k ; r , b , s V ) (cid:3)(cid:27) · { V ≥ U m } ++ ( F − Σ m − j = ¯ c j ) · { D m < E m } · { V ≥ L m } − ( F − Σ m − j = ¯ c j ) · { D m < E m } · { V ≥ U m } ++ ¯ c m · { V ≥ U m } + δ V · { V < D m } . Using the pricing formula of higher order binary option [17], we have B m − ( V , t ) = N − X k = m (cid:2) ¯ c k + B + + ··· + + U m U m + ··· U k U k + ( V , t ; T m , T m + , · · · , T k + ; r , b , s V ) ++ δ A + + ··· + − U m U m + ··· U k D k + ( V , t ; T m , T m + , · · · , T k , T k + ; r , b , s V ) (cid:3) ++ M X k = m + ( F − Σ k − j = ¯ c j ) · { D k < E k } · (cid:2) B + + ··· + − U m U m + ··· U k − L k ( V , t ; T m , T m + , · · · , T k ; r , b , s V ) −− B + + ··· + − U m U m + ··· U k − U k ( V , t ; T m , T m + , · · · , T k ; r , b , s V ) (cid:3) ++ ( F − Σ m − j = ¯ c j ) · { D m < E m } · B + L m ( V , t ; T m ; r , b , s V ) −− ( F − Σ m − j = ¯ c j ) · { D m < E m } · B + U m · ( V , t ; T m ; r , b , s V ) ++ ¯ c m B + U m ( V , t ; T m ; r , b , s V ) + δ A − D m · ( V , t ; T m ; r , b , s V ) . 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