Credit derivatives with recovery of market value for multiple firms Keiichi Tanaka ∗ Graduate School of Social Sciences Tokyo Metropolitan University September 2006

Abstract It is well known that a defaultable bond subject to recovery of market value (RMV) is priced by discounting the payoff with an adjusted short rate by the loss rate and the default intensity rate of the issuer. We show that the formula can be generalized for a defaultable contract subject to RMV with heterogeneous multiple reference firms. The discounting short rate is adjusted by sum of the loss rate times the default intensity of each firm. Associated with the multiple defaultable firms with the RMV rule, a survival contingent forward measure is constructed. As applications tractable pricing formulae are derived for a vulnerable option on a defaultable bond and a vulnerable option contract to enter into a CDS.

Key words: survival contingent measure, default risk, recovery of market value, vulnerable option, credit default swap option JEL Classification: G12, G13, E43 Mathematics Subject Classification (2000): 91B28, 60G40

∗ Address: 1-1 Minamiohsawa Hachiouji Tokyo 192-0397 Japan, E-mail: [email protected]

1

1

Introduction

Credit derivatives are used for a hedging of a default risk of a third party. However, all of the over-the-counter transactions are subject to default risk of the counterparty by definition. So the hedger is exposed to the default risk of another party. In principle these default risk must be appropriately priced in each transaction. In the literature of mathematical finance the events of default are described by either structural models or reduced form models. Especially intensity based models have been intensely studied because of the intuition and tractability. Most of papers pay attention to the pricing of the default risk of one entity and only a few papers study the default risk of the counterparty of an OTC transaction. Not only the mechanism of default, but also the recovery rule after the default is one of the important assumptions for the pricing. Most of derivative transactions are documented based on ISDA (International Swap Dealer Association) which applies the recovery of market value (RMV) as the basic recovery scheme. In that sense the assumption of fractional RMV is very close to the reality in the markets of interest rates and foreign exchange. In this paper we consider an arbitrage-free price of a contingent claim subject to default risk of multiple firms with RMV in an intensity-based model. Upon the first default of the basket of firms, the contract is terminated by paying a recovery amount which is the pre-default market value times the fractional recovery ratio applied to the first defaulting firm. Our contributions are two folds. One is the derivation of the pricing formula of such derivatives, and another one is the construction of a survival contingent forward measure. The examples include a vulnerable option on a defaultable bond and a defaultable option to enter into a credit default swap (CDS). We show that a pricing formula applied to the case of RMV for one firm can be generalized to more general cases of RMV with heterogeneous multiple reference firms. The discounting short rate is adjusted by a summation of the loss rate times the default intensity of each firm. Associated with the multiple defaultable firms with the RMV rule, a survival contingent forward measure is constructed. Pricing of defaultable securities and contingent claims have been extensively studied in intensity-based models thanks to the intuition and the tractability. Jarrow and Turnbull (1995) analyze a defaultable zero-coupon bond subject to fractional recovery of treasury (RT). It seems that many market conventions applied to derivative transactions follow RMV. A basic pricing technique for fractional RMV is presented by Duffie and Singleton (1999). The result is intuitive; the payoff should be discounted by a short rate adjusted by the default risk. Duffie and Huang (1996) analyzed asymmetric recovery rule applied to an interest rate swap. In the relation to the feature of RMV, they use several results shown by Duffie, Schroder and Skiadas (1996) for backward equations to be satisfied by prices. We apply these basic techniques to our analysis. Jamshidian (2004) develops a general pricing framework of CDS and swaptions by making fully use of technique of num´eraire. But he treats one reference firm while we consider multiple reference entities. For a convenient calculation of derivatives prices, the technique of change of measure associated with a change of num´eraire is often applied. For each num´eraire, a martingale measure is defined as a probability measure under which the relative price of any asset with respect to the num´eraire is a martingale. It may be natural to think of a defaultable security as a num´eraire in a defaultable environment. The basic idea is that a defaultable

2

security can be a num´eraire until it defaults. The corresponding martingale measure is called a survival measure. Collin-Dufresne, Goldstein and Hugonnier (2004) studies a survival spot measure with a num´eraire of a defaultable money market account, and Sch¨onbucher (2000b) does a survival forward measure with a defaultable zero-coupon bond. A technical reason of introducing the survival measures is that neither papers take a subfiltration approach, and hence a default indicator remains inside the expectation operator. An advantage of using such non-equivalent measures is that we can neglect events of the default in the pricing of the relevant securities under the survival measure. However, these measures are not equivalent to the original risk-neutral measure since these num´eraire can be worthless with a positive probability. We face a critical issue whether we can justify to call the figure calculated under such a measure an arbitragefree price of a security. If a subfiltration approach is taken, we don’t need a survival measure and a survival contingent measure avoids such a problem. The idea of a survival contingent measure is first proposed by Sch¨onbucher (2000a). For the construction of a survival contingent measure, we focus on a pre-default price of a fictitious defaultable asset. Note that since the pre-default price does not represent an actually traded price, the relative price may not be a martingale. Our idea is that the pre-default price can be a num´eraire after modifying so that the relative price be a martingale under the original risk-neutral measure. We find that a slightly modified pre-default price of a defaultable bond can play the same role as the price of a num´eraire and define an equivalent martingale measure which we call a survival contingent measure. The contribution of this paper is to add another role in the change of measure. Namely, the change of measure to the survival contingent forward measure QT in this paper has two meanings. One meaning is that the change of measure is associated with the change of num´eraire to the pre-default price of a fictitious bond. Another one is that the change of measure reflects the possibility of a change of cash flow schedule before the maturity due to a default and the recovery rule. Bravely speaking, in the new num´eraire F (·, T )Λ, F (·, T ) works for the former purpose and Λ appears for the latter purpose. The choice of num´eraire relies on not only the non-defaultability but also the recovery rule. For practical calculation of complicated option prices it is convenient to add more assumptions about the state variables that drive the risk-free rate and the intensities of the defaults. Affine term structure models are characterized by Duffie and Kan (1996). Collin-Dufresne and Goldstein (2002) utilize the useful features of affine models to an approximation method of swaption prices. Tanaka, Yamada and Watanabe (2005) improve the accuracy and the computational efficiency by focusing on a Gram–Charlier expansion under a forward measure and bond moments. As applications we study a vulnerable option on a defaultable bond and a vulnerable option contract to enter into a CDS. Affine term structure models are well-fitted to the approximation method of option prices with the Gram–Charier expansion and bond moments. The rest of this paper is organized as follows. In Section 2, we develop a valuation of a defaultable contingent claim with RMV on multiple firms. In Section 3, we construct a survival contingent forward measure. A vulnerable option on a defaultable bond and a vulnerable option on CDS are discussed. Section 4 concludes the paper.

3

2

Valuation with RMV for multiple firms

We consider an economy where the uncertainty is represented by a complete probability space (Ω, F, Q) on a finite horizon [0, T ∗ ] with some T ∗ < ∞. The probability measure Q is assumed to be a spot martingale measure under a given default-free short rate process r which will be described shortly. E denotes an expectation with respect to Q. There are three firms A, B, C subject to default risk. The default time of firm i (i = A, B, C) is denoted by τ i and the default indicator is denoted by Hti = 1{τ i ≤t} . At time t = 0 all of these three firms are solvent almost surely. The followings are the standing assumptions throughout the paper although more additional assumptions will be made later to study more specific cases. Assumption 1. 1. A given filtration G = {Gt : t ∈ [0, T ∗ ]} satisfies the rightcontinuity and the completeness with respect to (Q, F). 2. The default-free short rate process r is a G-predictable process such that a process t exp(− 0 rs ds) is square-integrable. 3. The filtration F = {Ft : t ∈ [0, T ∗ ]} is generated by the default indicators and G, Ft = Gt ∨ σ(HsA , HsB , HsC : 0 ≤ s ≤ t). 4. Hti satisfies the Doob-Meyer decomposition  t i i 1{τ i >s} his ds Ht = Mt + 0

with a (Q, F)-martingale M i and a non-negative G-predictable process hi . 5. An adjusted short rate R is a G-predictable process defined by B B C C Rt = rt + δA hA t + δ ht + δ ht t such that a process exp(− 0 Rs ds) is square-integrable.

(1)

6. Simultaneous defaults by any two (hence more) firms do not occur almost surely; Q(τ i = τ j ) = 0 (i = j). 7. Any (Q, G)-martingale is also a (Q, F)-martingale. Under Assumption 1, both the filtration F and G satisfy the usual conditions of right continuity and completeness. It will be convenient to introduce an auxiliary notation, τ ABC = min{τ A , τ B , τ C }. We study a defaultable security which pays the cumulative dividend D until the maturity T and Y at T . In case of default of firm i (i = A, B, C) prior to the maturity T , the security holder receives 1 − δi times the pre-default price and the contract is terminated without further payments. In what follows, we claim that under certain conditions the price of the contract at time t is given by St = 1{τ ABC >t} Vt , where  Vt = E

t

T

  exp − t

s

   Ru du dDs + exp −

4

T t

  Ru du Y | Gt ,

(2)

which is a natural extension of the well known result by Duffie and Singleton (1999) by B B C C recalling that Rt = rt + δA hA t + δ ht + δ ht . The proof can be accomplished in a similar fashion as Duffie and Huang (1996). In order to prove it, we restate the following two auxiliary results with a right-continuous filtration H = {Ht : t ∈ [0, T ∗ ]} which is either G or F. Lemma 1. (Duffie and Huang) Let f : R × Ω × [0, T ∗ ] → R be a measurable function such that f is k-Lipschitz in its v argument for some k > 0: |f (x, ω, t) − f (y, ω, t)| ≤ k|x − y|,

∀ω, t, x, y,

and f (v, ·, ·) is H-predictable for all v. Let Y be an HT -measurable random variable, and let D be a finite variation process. Suppose that there is some p ∈ [1, ∞) such that T T p 0 |f (0, ω, t)|dt, Y and 0 |dDt | are all in L . Then there exists a unique solution V to the recursive stochastic integral equation  Vt = E

T

t

 f (Vs , ω, s)ds + dDs + Y | Ht ,

t ≤ T,

T in the space V p of all RCLL H-adapted processes that satisfy E[( 0 |Vt |dt)p ] < ∞. Proof. See Duffie and Huang (1996) Lemma 1. Lemma 2. (Duffie and Huang) T Let V be a semimartingale satisfying E( 0 |Vt |dt) < ∞, let D be a semimartingale T satisfying E( 0 |dDt |) < ∞, and let G be a progressively measurable process such that T E( 0 |Gt |dt) < ∞. There exists an H-martingale m such that dVt = −Gt dt − dDt + dmt ,

t ∈ [0, T ],

if and only if  Vt = E

t

 Gs ds + dDs + VT | Ht ,

T

t ∈ [0, T ].

Proof. See Duffie and Huang (1996) Lemma 3. Then we obtain the following result. Theorem 1. Under Assumption 1, consider a defaultable security which pays the cumulative dividend D until the maturity T and  T Y at T, twhere D is a G-predictable bounded process of finite variation satisfying E( 0 exp(− 0 rs ds)|dDt |) < ∞, and Y is a GT measurable, square-integrable random variable. In case of default of firm i (i = A, B, C) prior to the maturity T , the security holder receives 1 − δi times the pre-default price and the contract is terminated without further payments. Suppose that V defined by (2) does not jump at the time of default. Then the price St of the security at time t is given by St = 1{τ ABC >t} Vt .

5

Proof. Let Ht = 1{τ ABC ≤t} = HtA HtB HtC . By definition the security has the cumulative payoff U  t  t (1 − Hu )dDu + Su− (γuA + γuB + γuC )dHu , (3) Ut = 0

0

and γti = 1{τ i =t} (1 − δi ), (i = A, B, C). First, we will show that the price S exists. Consider a recursive integral equation on S for t ≤ T  T     B C St = E − h − h ds + 1 dD Y | F S − Rs − hA + 1 . (4) ABC ABC s s t {τ >s} {τ >T } s s s t

By Lemma 1 there exists a unique solution S to this equation. Let Zti = St− (1 − δi ) (i = A, B, C). Then by noting that St = St 1{τ ABC >t} , Lemma 2 shows that S satisfies  B C Rt − hA t − ht − ht St dt − 1{τ ABC >t} dDt + dmt    B B C C = rt St − 1{τ ABC >t} ZtA hA + Z h + Z h dt − 1{τ ABC >t} dDt + dmt , t t t t t

dSt =

where m is a (Q, F)-martingale. Since we observe that Ss− γsi dHs = 1{τ i =s} Ss− (1 − δi )dHs = 1{τ i =s} Zsi dHs = 1{τ ABC ≥s} Zsi dHsi , t by applying Ito’s formula to St exp(− 0 rs ds) we have with some (Q, F)-martingales m , m  t Ê   Ê

s − 0t rs ds St e = − e− 0 ru du 1{τ ABC >s} dDs + Zsi his ds + mt 0



i=A,B,C

T

= E

e−

Ês 0

t



T

= E t

 = E

T

 ru du 1{τ ABC >s} dDs + 1{τ ABC ≥s}

 Ês e− 0 ru du (1 − Hs )dDs + Ss− e−

Ês 0

ru du

t

dUs + mT | Ft



  Zsi dHsi | Ft + mt

i=A,B,C



  γsi dHs | Ft + mt

i=A,B,C



t where on the second equality we use the decomposition Hti = Mti + 0 1{τ i >s} his ds. Since T T mT = ST exp(− 0 rs ds) = exp(− 0 rs ds)1{τ ABC >T } Y , S satisfies the equation  St = E

t

T

  exp − t

s

   ru du dUs + exp −

t

T

  ru du 1{τ ABC >T } Y | Ft ,

This equation justifies that S is the arbitrage-free price of the defaultable security. Next we define V  by the unique solution of an equation  T     −Rs Vs ds + dDs + Y | Gt , t ≤ T, Vt = E t

6

(5)

whose existence and uniqueness is guaranteed by Lemma 1. Lemma 2 implies that dVt = RtVt dt − dDt + dnt for some (Q, G)-martingale n. By applying Ito’s formula to t Vt exp(− 0 Ru du), we obtain Vt



T

=E t

  exp −

s t

   Ru du dDs + exp −

t

T

  Ru du Y | Gt ,

which is identical to Vt defined by (2). Since V does not jump at the time of default ΔVτ ABC = 0 and no simultaneous defaults occur almost surely, it follows that using integration by parts, d(1{τ ABC >t} Vt )   = d (1 − HtA )(1 − HtB )(1 − HtC )Vt

 A B C B C )(1 − Ht− )(1 − Ht− )dVt − Vt− (1 − Ht− )(1 − Ht− )dHtA = (1 − Ht−  A C A B + (1 − Ht− )(1 − Ht− )dHtB + (1 − Ht− )(1 − Ht− )dHtC   = 1{τ ABC >t} dVt − Vt− dHtA + dHtB + dHtC    B C + h + h ˆt = 1{τ ABC >t} dVt − Vt hA t t t dt + dm  B C = Rt − hA ˆ t , t − ht − ht 1{τ ABC >t} Vt dt − 1{τ ABC >t} dDt + dm for some (Q, F)-martingales m, ˆ m ˆ  , where we use the assumption of no simultaneous default on the third equality. Again by Lemma 2, 1{τ ABC >t} Vt satisfies the same equation (4) for S. Hence we have St = 1{τ ABC >t} Vt due to the uniqueness.

7

3

Survival contingent forward measure for RMV

The pre-default price Vt of a vulnerable contract with RMV can be further simplified by using the survival contingent forward measure for RMV that is equivalent to the original measure. Moreover, when the intensities are affine, the approximated price can be easily obtained under the survival contingent forward measure. As the applications we will consider two defaultable option contracts.

3.1

Change of measure

We focus on a model which is described by a state vector X taking the values in a set D ⊂ Rn . Under the following Assumption 2, Assumptions 1.1, 1.4, and 1.7 are satisfied. Assumption 2. 1. An n-dimensional standard Brownian motion W is supported on the probability space (Ω, F, Q). The filtration G = {Gt : t ∈ [0, T ∗ ]} is the augmented filtration generated by the Brownian motion Gt = σ(Ws : 0 ≤ s ≤ t) ∨ N , where N is the Q-null sets in F. 2. A Markov-state vector X satisfies a stochastic differential equation (SDE) dXt = μ(Xt )dt + σ(Xt )dWt ,

(6)

where μ : D → Rn and σ : D → Rn × Rn are regular enough for (6) to have a unique strong solution. 3. The default-free rate rt is given by rt = r(Xt ) with a measurable function r : D → R of the state vector Xt . 4. The default intensity hi is given by hit = hi (Xt ) with a non-negative measurable function hi : D → R+ of the state vector Xt . The default time of firm i (i = A, B, C) τ i is defined by  t i hi (Xs )ds ≥ η i }, τ = inf{t ≥ 0 : 0

where η A , η B , and η C are independent random variables with a unit exponential law under Q. When D ≡ 0, Vt is rewritten as   t     R(Xu )du = E exp − Vt exp − 0

T 0

  R(Xu )du Y | Gt ,

which is G-martingale. Hence V is continuous by recalling that any G-martingale is continuous. Corollary 1. Under Assumptions 1 and 2, the time-t price St of the security stated in Theorem 1 with D ≡ 0 is given by    T   R(Xu )du Y | Gt . St = 1{τ ABC >t} E exp − t

8

When considering the applications, a typical form of Y is a payoff from an option  Y = max(G, 0) where G = i ai P (T, Ti ) with a pre-default price of some defaultable bonds P (T, Ti ). If the payoff is complicated, the valuation is difficult in general and a change of measure to a kind of a forward measure turns out useful. For the construction of an equivalent measure, it is natural to consider a fictitious defaultable bond which pays 1 at maturity T if all of firms A, B and C are solvent. If either firm i defaults prior to the maturity, the bond pays 1 − δi times the pre-default price as the recovery. By Theorem 1, the pre-default price F (t, T ) at time t is given by      T R(Xs )ds | Gt , (7) F (t, T ) = E exp − t

which satisfies a SDE dF (t, T ) = F (t, T )R(Xt )dt + F (t, T )σ F (t, T ) dWt , for some G-progressively measurable process σ F (·, T ) : [0, T ∗ ]×Ω → Rn . For an arbitrary fixed T we define a process LT by LTt Λt

F (t ∧ T , T )Λt∧T ,  t∧T exp 0 r(Xs )ds F (0, T )   t  = exp − δA hA (Xs ) + δB hB (Xs ) + δC hC (Xs ) ds . =

(9)

0

Then we observe that LTt

(8)

  1 E exp − = F (0, T )

 0

T

  R(Xs )ds | Gt ,

(t ≤ T ),

and LTt = LTT for t > T . Λ is constructed so that the relative price of a modified fictitious bond price F (t, T )Λt is a martingale under Q. The modification is made due to the RMV feature with non-zero recovery and the difference between the price and the pre-default price (or the jump on the default). Moreover, by using Ito’s lemma, we see that dLTt

 dF (t, T ) dΛt  + = 1{T >t} LTt −r(Xt )dt + F (t, T ) Λt = 1{T >t} LTt σ F (t, T )dWt = LTt dMtT ,

 t∧T F T  Thus we have LTt = where  T Mt = 0 σ (s, T ) dWs is a local (Q, G)-martingale. T E M t . By the above observation and the assumptions, L can be a density process to define a probability measure. Lemma 3. Under Assumptions 1 and 2, the process LT is a uniformly integrable (Q, G)martingale and hence (Q, F)-martingale, which is almost surely strictly positive. By Lemma 3, an equivalent measure QT with respect to Q can be defined by the Radon-Nykodim derivative dQT

= LTt . dQ Ft 9

(10)

We call QT the T -survival contingent forward measure for the RMV. It should be emphasized that the measure QT depends on the recovery rule upon default of firms A, B and C although we omit the dependence in the notation. By construction, QT is the corresponding martingale measure with a num´eraire of a modified pre-default bond price F (t, T )Λt . To obtain a better understanding of the change of the measure, the following lemma is useful. It states that the change of measure transforms the Brownian motion only and keeps the jump process unaffected. We denote by M, N  the Q-compensator of the quadratic covariation process [M, N ]. Lemma 4. Let N be an arbitrary (Q, F)-local martingale. Then, under Assumptions 1 and 2, the process defined by  t d N, LT s  = Nt − N, M T t Nt = Nt − T L 0 s is a local martingale with respect to (QT , F). In particular, (i) The process hi is the intensity of H i under QT . t (ii) WtT = Wt − 0 σ F (s, T )ds is a Brownian motion under QT . Proof. Although this is a straightforward application of the Girsanov theorem, we show · t the proof. By Lemma 3 we see that N, LT t = N, 0 LTs− dMsT t = 0 LTs− d N, M T s . By integration by parts we have  t  t 0 LT = s + [N  , LT ]t s− dLT + t LT − N LT dN N N t

0

t

0



t

= 0



t

= 0



t

= 0

 =

0

t

s− dLTt + N s− dLT + N t s− dLT + N t s− dLTt + N

s−

0



t

0



t

0



t

0



0

t



LTs− dNs

−

t 0

LTs−

d N, LT s  , LT ]t + [N LTs−

 , LT ]t − N, LT t LTs− dNs + [N  − N, LT ]t + a local (Q, F)-martingale LTs− dNs + [N LTs− dNs + a local (Q, F)-martingale.

 LT is a local (Q, F)-martingale since LT is a local (Q, F)-martingale. Hence Therefore, N  is a local (QT , F)-martingale by a Bayes’ rule we conclude that N τn | Ft ]LTt = E T [N τn | Ft ]E[LTτ | Ft ] = E[N τn LTτ | Ft ] = N t LTt E T [N n n  LT . for a localizing sequence (τn ) for the local (Q, F)-martingale N The pre-default price of a contingent claim with the RMV is expressed under the survival contingent forward measure QT as if F (t, T ) were a bond price. Theorem 2. Let Y be a GT -measurable and integrable random variable. Then under Assumptions 1 and 2, the price S of a contingent claim which pays Y at time T with

10

the recovery payoff based on fractional recovery of market value for firms A, B, and C is given by   St = 1{τ ABC >t} F (t, T )E T Y | Gt , t < T, where E T is the expectation with respect to QT Proof. By Theorem 1, for t < T we have  T   = 1{τ ABC >t} E exp − R(Xs )ds Y | Gt t     t R(Xs )ds F (0, T )E LTT Y | Gt = 1{τ ABC >t} exp 0   = 1{τ ABC >t} F (t, T )E T Y | Gt , 

St

where the third equality follows from a Bayes’ rule E[LTT Y | Gt ] = LTt E T [Y | Gt ]. The importance of change of num´eraire cannot be overstressed. By construction, for any asset price S, the relative price S/(F (·, T )Λ) is a (QT , F)-martingale. On the other hand, the pre-default price V of a contingent  claim paying Y with maturity T with the same RMV is written as Vt = F (t, T )E T Y | Gt by Theorem 2. Hence, the relative price V /F (·, T ) is a (QT , G)-martingale thus it is also a (QT , F)-martingale. The fictitious bond price F (·, T ) plays the same role as a num´eraire for pre-default prices of T -maturity contingent claims with RMV. However, it is not a num´eraire for other claims. The idea of a survival contingent measure is first proposed by Sch¨onbucher (2000a) for the case of zero-recovery. However, the contribution of this paper is to add another role in the change of measure. Namely, the change of measure to the survival contingent forward measure QT in this paper has two meanings. One meaning is that it is associated with the change of num´eraire to the pre-default price of a fictitious bond. Another one is that it reflects the possibility of a change of cash flow schedule before the maturity due to a default and the recovery rule. Bravely speaking, in the new num´eraire F (·, T )Λ, F (·, T ) works for the former purpose and Λ appears for the latter purpose. The choice of num´eraire relies on not only the non-defaultability but also the recovery rule. The survival contingent forward measure QT should not be confused with a survival T forward measure Q which is introduced by Sch¨onbucher (2000b). A survival contingent measure is equivalent while a survival measure is only absolutely continuous and is not T equivalent. The survival forward measure Q is defined by the following density process T L T

Lt

= 1{τ ABC >t∧T }

F (t, T )Λt F (t, T )Λt = ,  t exp 0 r(Xs )ds F (0, T ) exp 0 r(Xs )ds F (0, T )  t

where F (t, T ) = 1{τ ABC >t∧T } F (t, T ) is the price of the fictitious zero coupon bond with the RMV feature and  t   Λt = exp (1 − δA )hA (Xs ) + (1 − δB )hB (Xs ) + (1 − δC )hC (Xs ) ds 0

11

is a modifying term reflecting the RMV feature. Namely, the survival forward meaT sure Q is the corresponding martingale measure with a num´eraire of a fictitious bond T F (t, T )Λt . Since the num´eraire becomes worthless upon default, Q is not equivalent to the original measure Q. Sch¨onbucher (2004) discusses a justification of the name of a survival measure. In a similar way we can give a justification of the name of a survival contingent measure. It is clear that the survival contingent forward measure QT and the survival forward measure T Q are constructed in a different way. However, these two measures are identical if the domain is restricted to the sub σ-field Gt (t ≤ T ) because we see that for any event G ∈ Gt  T T Q (G) = E 1G Lt   F (t, T )Λt = E 1{τ ABC >t∧T } 1G  t exp 0 r(Xs )ds F (0, T )  t

   F (t, T )Λt hi (Xs )ds 1G = E exp −  t exp 0 r(Xs )ds F (0, T ) 0 i=A,B,C   F (t, T )Λt = E 1G  t exp 0 r(Xs )ds F (0, T )   = E 1G LTt = QT (G), where on the third equality we used a well-known formula  

   E 1{τ ABC >T } Y | Ft = 1{τ ABC >t} E exp − i=A,B,C

t

T

  his ds Y | Gt ,

(11)

which holds for any GT -measurable, integrable random variable Y and any t ≤ T , under Assumption 1 (see, e.g, Bielecki and Rutkowski, 2004). It implies that given no default, these two measures assign the same probability. But for other events involving a default they assign different probabilities. That is a justification of the name of a survival contingent measure. The relationship (11) is an instructive expression that leads us to a survival contingent forward measure rather than a survival measure. A survival measure is not necessary for the purpose of pushing out the default indicator within the expectation when a subfiltration approach is taken.

3.2

Vulnerable option on a defaultable coupon bond

In this subsection and the subsequent one we adopt the following affine structure assumption in addition to Assumptions 1 and 2. We will see that regardless of the complicated valuation form of vulnerable options, the pricing formulae become tractable in the computation owing to the affine structure of Assumption 3 and the change of measure to the survival contingent forward measure. Assumption 3.

1. X is an affine diffusion process, μ(x) = K0 + K1 x, (K0 , K1 ) ∈ Rn × Rn×n ,    σ(x)σ(x) ij = H0ij + H1ij x, (H0ij , H1ij ) ∈ R × Rn .

12

2. The short rate and intensities are given as rt = r(Xt ) and hit = hi (Xt ) where r : D → R and hi : D → R are affine functions, r(x) = ρ0 + ρ 1 x,

(ρ0 , ρ1 ) ∈ R × Rn ,

hi (x) = l0i + l1i x,

(l0i , l1i ) ∈ R × Rn .



In this subsection we consider a vulnerable option on a defaultable bond. • The firm A buys from the firm B a defaultable option (either call or put) with strike price K and the expiry T0 on a defaultable coupon bond issued by the firm C. • The bond pays aj at Tj (j = 1, · · · , N ) as long as C is solvent. The firm C also issues zero coupon bonds with maturity Tj (j = 1, · · · , N ). In case of the default of C the recovery payoff of these bonds is subject to RT, that is, the bond of unit face amount is assumed to be replaced by the corresponding default-free bonds with C upon the default of C. 1 face amount of 1 − δbond • If a firm i (i = A, B, C) defaults before the expiry T0 , the seller B pays 1 − δi times the market value prior to the default to the buyer A and the option contract is terminated. Time t-price of the zero coupon bond with the maturity date T issued by C is denoted by P C (t, T ). Since the bond is subject to RT, it is well known that P C (t, T ) is represented as P C (t, T ) = 1{τ C >t} P C (t, T ) where P C (t, T ) is the pre-default price of the bond that is a linear combination of default-free bond price and defaultable bond price with zero recovery  ÊT  ÊT   − t r(Xs )+hC (Xs ) ds C C − t r(Xs )ds C | Xt + δbond E e | Xt . P (t, T ) = (1 − δbond )E e This bond price is expressed as a linear combination of exponentially affine functions of the state variable Xt . A defaultable coupon bond has cash flows of aj at Tj (j = 1, · · · , N ) as long as C is solvent. Then we obtain the price St of the vulnerable call option on a defaultable coupon bond as St = 1{τ ABC >t} Vt with    Vt = E exp −

T0

t

= F (t, T0 )E T0



N  

  R(Xs )ds max aj P C (T0 , Tj ) − K, 0 | Xt j=1

N  

 aj P C (T0 , Tj ) − K, 0 | Xt . max

(12)

j=1

This formula has the same form as swaption pricing so that we can make use of the approximation technique shown by Tanaka, Yamada and Watanabe (2005). Since X is an affine diffusion, the price F (t, T ) can be written as   F (t, T ) = exp α(t, T ) + β(t, T ) Xt 1 Other recovery rules such as RMV can be applied to these bonds issued by C if appropriate modifications are made on the successive discussion.

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with some deterministic functions α, β. Thus, the diffusion term of the bond price is written as σ F (t, T ) = σ(Xt ) β(t, T ). Hence the state vector X is also an affine diffusion under QT satisfying  dXt = μ(Xt ) + σ(Xt )σ(Xt ) β(t, T ) dt + σ(Xt )dWtT . Then we can calculate the approximated price of the option (12) with the Gram–Charier expansion and bond moments as discussed in Tanaka, Yamada and Watanabe (2005). For the convenience of the readers, we will outline the basic procedure of the approximation by Tanaka, Yamada and Watanabe (2005). As the first step, we need to calculate the bond moment of the bonds involved in the valuation of the cash flow upon the exercise of the option in question. For a given set of dates T, T0 , U1 , . . . , Um (T ≤ T0 ≤ Ui for all i = 1, . . . , m), the bond moment is defined under the T -survival contingent forward measure as μT (t, T0 , {U1 , · · · , Um }) ≡ E T

m 

P C (T0 , Ui ) | Xt



i=1

and it can be obtained as a function of Xt either analytically or numerically. By regarding N the cash flow i=0 ai P C (T0 , Ti ) with a0 = −K as the value of a swap, the second step is to obtain the m-th swap moment with the bond moments and the cash flows as Mm (t) = E T0

N 

m  ai P C (T0 , Ti ) | Xt

i=0



=

ai1 · · · aim μT0 (t, T0 , {Ti1 , · · · , Tim }).

0≤i1 ,... ,im ≤N

Then we know the k-th cumulant ck (t) from the set of the moments {Mm (t)}m . Define the weighted cumulant Ck = ck (t)F (t, T0 )k (we omit the argument of time t to save the space) for k ≥ 1, and coefficients qk as q0 = 1, q1 = q2 = 0, and for k ≥ 3 [k/3]

qk =





m=1 k1 +···+km =k,ki ≥3

[k/3]

ck1 · · · ckm  1 k = √ m!k1 ! · · · km ! c2



m=1 k1 +···+km =k,ki ≥3

Ck1 · · · Ckm  1 k √ . m!k1 ! · · · km ! C2

The definition of qk looks complicated but the calculation is easy to do, for example, q3 =

C3

, 3/2

3!C2

q4 =

C4 , 4!C22

q5 =

C5

, 5/2

5!C2

q6 =

C6 + 10C32 , 6!C23

q7 =

C7 + 35C3 C4 7/2

7!C2

.

Lastly, since by the Gram–Charlier expansion the density function f of a random variable can be expanded as f (x) =

∞

 x − c1  x − c1 qn φ √ , √ Hn √ c2 c2 c2

n=0

the pre-default price can be approximated with a positive integer L as L  C1   C1   C1

 C1 (−1)k qk Hk−2 √ + C2 φ √ + C2 φ √ , Vt ≈ C1 N √ C2 C2 C2 k=3 C2

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where N and φ is the distribution function of a standard normal distribution N (0, 1) and the density function, respectively, and Hn is the n-th Hermite polynomial defined by dn Hn (x) = (−1)n φ(x)−1 dx n φ(x). Tanaka, Yamada and Watanabe (2005) suggests either L = 3 or L = 7 for a practical application.

3.3

Vulnerable option on CDS

Let’s consider a vulnerable option contract to enter into the following credit default swap (CDS): • A buys from B an option to enter into a CDS which starts at T0 . • The reference bond on the CDS is a zero coupon bond with maturity U issued by C (U ≥ TN ) subject to fractional recovery of Treasury. • As the premium the CDS buyer A pays κ to the CDS seller B at Tj (j = 1, · · · N ) if C survives at Tj ; τ C > Tj . • As the protection B pays a + bP (Tj , U ) to A at Tj if Tj−1 < τ C ≤ Tj . • If a firm i (i = A, B, C) defaults before the expiry T0 , the seller B pays 1 − δi times the market value prior to the default to the buyer A and the option contract is terminated. In order to make our discussion simple, the vulnerability is only up to the exercise time T0 of the option and we don’t assume the default of A and B after the CDS starts. Although the above option is a call-type option in the sense that the option buyer becomes the CDS buyer upon the exercise, a put-type option contract can be discussed in a similar fashion. The payoff at T0 upon the exercise of the call option is given by E

N 

−

e

Ê Tj T0

r(Xu )du

    −κ1{τ C >Tj } + 1{Tj−1 T0 } G where G =

N 

 −κG(T0 , Tj , Tj ) + a G(T0 , Tj , Tj−1 ) − G(T0 , Tj , Tj )

j=1

  + b G(T0 , U, Tj−1 ) − G(T0 , U, Tj ) .

Hence the price of the option on the CDS is given by    St = 1{τ ABC >t} F (t, T0 )E T0 max G, 0 | Xt .

(14)

Due to the form of G, this option price can also be approximated by the Gram–Charlier expansion and bond moments as have been explained briefly in the previous subsection.

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4

Conclusion

We demonstrate that the pricing formula with the default-risk adjusted short rate can be generalized for a defaultable contract subject to RMV with heterogeneous multiple reference firms. Associated with the multiple defaultable firms with the RMV rule, a survival contingent forward measure is constructed. As applications we study a vulnerable option on a defaultable bond and an option contract to enter into a CDS.

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