c Springer-Verlag 2004
Hazard rate for credit risk
and hedging defaultable contingent claims
Christophette Blanchet-Scalliet1, Monique Jeanblanc2
1 C.B. Laboratoire J.A.Dieudonne, Universite de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 2, France (e-mail: [email protected])
2 M.J. Equipe d’analyse et probabilit´es, Universit´e d’Evry Val d’Essonne, Rue du P`ere Jarlan, 91025 Evry Cedex, France (e-mail: [email protected])
Abstract. We provide a concise exposition of theoretical results that appear in modeling default time as a random time, we study in details the invariance martin- gale property and we establish a representation theorem which leads, in a complete market setting, to the hedging portfolio of a vulnerable claim. Our main result is that, to hedge a defaultable claim one has to invest the value of this contingent claim in the defaultable zero-coupon.
Key words: Default risk, representation theorem, hedging JEL Classification: G10
Mathematics Subject Classification (1991): 91B24, 91B29, 60G46
1 Introduction
We study an arbitrage free financial market, where a risk-free assetS0and a risky assetSare traded. The filtration generated by the discounted pricesS/S0is denoted byFS. The agents have some information on the asset prices modeled by a filtration F. In thestructural approach, the default timeτis a stopping time in the filtration FS and it is assumed that the agents have all the information contained on the prices, i.e.,F=FS, whereas in thereduced-form approach, the default arrives “by surprise", as in Cox modeling (see Lando [16]). An intermediary case is whenτis aFS-stopping time andF⊂FS, for example when the filtrationFis the trivial
The authors would like to thank D. Becherer and J.N. Hugonnier for interesting discussions and the anonymous referee whose pertinent questions on the first version of this paper help them to clarify the proofs. All remaining errors are ours.
Manuscript received: May 2001; final version received: April 2003
one or whenFis the filtration generated by the information of prices observed at discrete times, as in Duffie and Lando [10]. The reader can refer to Bielecki and Rutkowski [4] and Cossin and Pirotte [6] for a full treatment. Crucial problems of hedging are studied for example in B´elanger et al. [3]
In a first section, we present the model and we establish a general representation theorem. In a second section, we discuss the role of the hypothesis of invariance of martingales and establish that this hypothesis holds as soon as the default-free market is complete and arbitrage free and the defaultable market is arbitrage free.
We show that under some regularity condition, the default time is the first time when a stochastic barrier is reached, as in Cox process modeling. We give the hedging of vulnerable contingent claims using defaultable zero-coupon (DZC in short) and default-free assets, when the default-free market is complete. Our main result shows that to hedge a defaultable claim one has to invest the value of this contingent claim in the defaultable zero-coupon. This result is not surprising and is linked with the hazard process approach: the default arrives by surprise so that the jump in the hedging portfolio value is equal to the jump in the DZC.
2 The model
All the processes and random variables are constructed on a given probability space (Ω,G, P). In what follows, we limit our study, mainly for simplicity of notation, to the case where there is only one risky financial asset, whose price at timetis denoted bySt. As in Musiela and Rutkowski [17], the interest rate is supposed to be a non-negative processr, we denote byRt= exp
− t
0 rsds
the discounted factor and bySt0= exp
t
0 rsds
the savings account. The filtration generated by the discounted price processSt=St/St0is denoted byFS = (FtS =σ(Ss, s≤ t);t≥0). We refer to the market where the savings account and the risky assetSare traded up to timeTas thedefault-free market.We assume that there exists at least one probabilityQequivalent toPonFTSsuch that(St, t≤T)is aFS-martingale, so that the default-free market is arbitrage free.
A default occurs at a random time τ (i.e., a non-negative random variable).
In the defaultable world, the payment of a contingent claim depends whether or not the default has appeared before the maturity. In particular, a defaultable zero- coupon bond (DZC) with maturityT pays 1 unit at maturity if and only if the default has not appeared beforeT. More generally, we investigate the case where a promised payoffX ∈ FTSis paid at maturity if the default has not appeared and where a compensation is paid at hit (at the default time) if the default occurs before maturity.
2.1 Filtrations and equivalent martingale measures
We assume, as in [12] that the t-time information available to the agents in the default-free market is a σ-algebra Ft. We do not assume that the agents have
complete information on the prices: the filtrationF= (Ft, t≥0)can be smaller thanFS.
We denote byDthe filtrationD= (Dt;t≥0)withDt=σ(Ds, s≤t)where Dis the default process defined asDt= 11{τ≤t}. At timet, the agent’s information on the prices and on default time isGt = Ft∨ Dt: at any time the agent knows whether or not the default has appeared. Hence, the default timeτis aG-stopping time whereG= (Gt, t≥0). In fact,Gis the smallest filtration which containsF, satisfying the usual hypotheses, such thatτis aG-stopping time.
2.2 Hazard process
In this section, we work under a reference probabilityP. Later on, this probability will be either the historical probability, or a risk neutral one.
LetF be the right-continuous version of the submartingaleFt = P(τ ≤ t|Ft) andGthe conditional survival probabilityGt = 1−Ft. We assume thatFt<1 a.s. for anyt (In particular,τ is not aF-stopping time. See Andreasen [1] and B´elanger et al. [3] for generalization). We introduce theR+-valuedhazard process Γt=−ln(Gt). We assume for simplicity thatF0= 0so thatΓ0= 0. Obviously, the hazard process depends of the reference probability. For typographical reasons, we shall sometimes useF,GandΓ in the same formula.
We recall a key lemma stated as an example in Dellacherie [7] (p. 64) and its corollary.
Lemma 1 LetX ∈ FT be integrable. Then, E(X11T <τ|Gt) = 11{t<τ} 1
Gt
E(XGT|Ft). (1) Corollary 1 Lethbe aF-predictable bounded process. Then,
a)
E(hτ|Gt) = 11{τ≤t}hτ−11{t<τ} 1 GtE
∞ t
hudGu|Ft
. (2) b) In particular, ifFis increasing and continuous,
E(hτ|Gt) = 11{τ≤t}hτ+ 11{t<τ}E
∞ t
huexp Γt−Γu
dΓu|Ft
. (3) The proof of this lemma and its corollary is based on the important remark that anyGt-measurable random variable is equal, on the set{t < τ}, to anFt- measurable random variable.
It is also useful to note that for anyG-predictable process h, there exists an F- predictable processh∗such that both processes are equal on the set{t≤τ}, i.e., ht11{t≤τ} = h∗t11{t≤τ}. Moreover, under the hypothesis∀t, Ft <1, the process (h∗t, t≥0)is unique (see [8], p. 186).
As an application of Lemma 1, it is easy to check that the discontinuous process (Lt, t≥0)where
Lt= 11t<τeΓt = (1−Dt)eΓt is aG-martingale.
2.3 Decomposition of theF-martingales asG-semi-martingales
One major question is to describe the dynamics of the assets in the filtrationG.
As mentioned in Hull and White [13] “When we move from the vulnerable world to a default-free world, the stochastic processes followed by the underlying state variables may change." We study here the reverse case, i.e. we move from the default-free world to the vulnerable one.
The submartingaleFadmits a unique Doob-Meyer decomposition of the from Ft = Zt+At, where Z is an F-martingale andAan F-predictable increasing process. Moreover, the processMt = Dt−Λt∧τ wheredΛt = dAt/Gt− is a G-martingale.
We require now, as in [2] that(C)holds, where (C)One of the following conditions is satisfied (i) AnyF-martingale is continuous
(ii) For anyF-stopping timeθ,P(τ=θ) = 0.
Under this condition, anF-martingale has no common jump with the hazard pro- cess,τ is aG-totally inaccessible stopping time and for anyF-martingalem, the process
mt∧τ=mt∧τ+ t∧τ
0 eΓsd[m, Z]s
is a stoppedG-martingale, where[X, Y]is the quadratic covariation of two mar- tingalesXandY (see Az´ema et al. [2]; Dellacherie et al. [8], p. 188; Yor [18], p. 41 for proof of this result and comments on the condition (C)). We shall refer tomas theG-martingale part of theF-martingalem; in particularZis defined as
Zt∧τ=Zt∧τ+ t∧τ
0 eΓsd[Z, Z]s.
2.4 Representation theorem
Our aim is now to establish a representation theorem forG-martingales. Such a representation theorem is difficult to obtain for anyG-martingale. Az´ema et al.
[2] have studied the particular case whereFis the natural filtration of a Brownian motion W andτ is an honest time in F∞, that is the end of a predictable set (see Dellacherie et al. [8], p.188; Yor [18], p.41). They established that anyG- martingale can be written as a sum of a stochastic integral with respect to the G-martingaleW, a stochastic integral with respect toM and a third martingale of the formv11(τ≤t)wherev ∈ Fτ+such thatE(v|Fτ) = 0, whereFτ+is generated by the random variableshτ wherehisF-progressively measurable. For example, ifτ = sup{t≤1 : Wt= 0}, thenv =V sgne(W1)withV ∈L2(Fτ)(see Yor [18], p. 74).
In what follows, we restrict our attention to the class ofG-martingales of the formE(hτ| Gt),orE(X11T <τ| Gt).
Theorem 1 Suppose that(C)holds and letF =Z+Abe the Doob-Meyer de- composition ofF. Lethbe anF-predictable process such thathτ is integrable, andHt=E(hτ| Gt). Then, theG-martingaleH admits a decomposition in mar- tingales as follows
Ht=mh0+ t∧τ
0 eΓs−(dmhs−(hs−Js−h )dZs)+
]0,t∧τ]e∆Γs(hs−Js−h )dMs. (4) Heremhis theF-martingale
mht =E
∞
0 hudFu| Ft
=E
∞
0 hudAu| Ft .
The processes mh and Z are theG-martingale parts of the F-martingales mh andZ,Jth =eΓt(mht −
t
0 hudFu)andM is the discontinuousG-martingale Mt=Dt−Λt∧τwheredΛt= dAt
1−Ft−. Furthermore,
Jth11t<τ =Ht11t<τ.
Proof The rather technical proof is based on Itˆo’s calculus and property (C). We
give the proof in the Appendix.
Corollary 2 Suppose that (C)holds and thatF is continuous. LetY ∈ FT be integrable and
Yt=E(Y11T <τ|Gt) = 11t<τeΓtE(Y GT| Ft) = 11t<τeΓtmYt , wheremY is theF-martingale
mYt =E(Y GT| Ft). Then,
Yt=mY0 + t∧τ
0 eΓs
dmYs +Ys−dZs
−
]0,t∧τ]Ys−dMs. (5) Proof The proof follows from Theorem 1 withht=Y11T <t. Nevertheless, when Fis continuous, a direct proof can be established from the remark thatYt=LtmYt
and some Itˆo’s calculus which leads in particular to dLt=−dDt
Gt
+ (1−Dt−) dFt
G2t +d Ft
G3t
=− 1
GtdMt+(1−Dt−) G2t dZt.
It remains to apply integration by parts formula.
3 Invariance of martingale hypothesis
We assume that the knowledge of the default time does not induce arbitrages in the market,1hence there exists aG-equivalent martingale measure, i.e., a probability Q∗onGT, equivalent toP, such that(St, t ≤T)is a martingale. Let us remark that, ifF=FS, the restriction of anyGe.m.m. toFTS is anFS-e.m.m. Indeed, if
EQ∗(ST|Gt) =St
taking the expectation with respect toFtS of both members, we get EQ∗(ST|FtS) =St.
We introduce an invariance of martingale hypothesis denoted by (H), which implies that the dynamics of the asset price are the same in the default-free world and in the defaultable world. We discuss the meaning of that hypothesis, its stability under a change of probability measure, its links with absence of arbitrage opportu- nities in the defaultable world, and we study the hedging of defaultable contingent claims in that setting.
(H)AnyF-square integrable martingale is aG-square integrable martingale.
3.1 Arbitrage free markets
We discuss now the hypothesis on the modeling of default time that we require in order to avoid arbitrages in the defaultable market. We show that under completion of the default-free market, (H) hypothesis holds under anyG-e.m.m.
Proposition 1 Assume that there exists a unique probabilityQ, equivalent toP onFTS such that the discounted price process(St, t ≤ T)is anFS-martingale under the probabilityQ. Assume moreover that there exists at least one probability Q∗, equivalent to P onGT such that(St, t ≤ T)is aG-martingale under the probabilityQ∗. Then,(H)holds underQ∗.
Proof We give a “financial proof” of this obvious and important result. From the hypothesis, any square integrable r.v.X, such that XRT ∈ FTS (any contingent claim) can be written as a stochastic integral with respect to the discounted price, i.e., there existsxand a square integrableFS predictable processθsuch thatRTX = x+T
0 θsdSs. The t-time price of the contingent claim isEQ(XRT/Rt|FtS).
Obviously,Xis hedgeable by theG-adapted strategyθand, from the uniqueness of price for hedgeable claims, for anyG-e.m.m.Q∗,
EQ(XRT|FtS) =EQ∗(XRT|Gt).
Hence,EQ(Z|FtS) =EQ∗(Z|Gt)for any square integrableFT-measurable r.v.Z, therefore any square integrableFS-Q∗-martingale is aG-Q∗-martingale.
1 This is a restrictive assumption on the timeτ. See [12] for examples where this assumption is not satisfied.
Remark 1 It is easy to construct arbitrage free models where (H) does not hold.
For example, in a incomplete information case as in Duffie and Lando [10], a straightforward computation establishes that the hazard process is not increasing, hence (H) does not hold (see also Sect. 3.3).
3.2 Characterization of (H) hypothesis
It is well known [9] that(H)hypothesis holds underQ∗if and only if
∀t, Q∗(τ≤t|F∞) =Q∗(τ≤t|Ft). (6) In particular,F andΓ, evaluated underQ∗ are increasing processes. This is in particular the case for Cox processes (see e.g., Lando [16] ) whereτis defined via a given non-negativeF-adapted processγas
τ= inf
t≥0, t
0 γsds≥Θ
whereΘis a given random variable, independent ofF, generally chosen with an exponential law.
The following interesting lemma, which establishes that working under (H) hypothesis is equivalent to a Cox process modeling is proved in [11].
Lemma 2 If(H)hypothesis holds andF is continuous, then the random variable Γτis exponentially distributed and independent ofF∞. Hence,
τ= inf{t : Γt≥Θ}
whereΘis an exponential random variable, independent ofF∞. Proof We suppose that (H) holds, which implies that
Q∗(τ≤t|F∞) =e−Γt.
SettingΘdef=Γτ leads to
{t < Θ}={t < Γτ}={Ct< τ},
whereCis the right inverse ofΓ, so thatΓCt =t.Therefore Q∗(Θ > u|F∞) =e−ΓCu =e−u.
We have thus established thatΘis an exponential random variable, independent of theσ-fieldF∞.Furthermore,τ= inf{t:Γt> Γτ}= inf{t:Γt> Θ}.
3.3 Brownian filtration, stability of (H) hypothesis
We have seen that if the default-free market is complete, and both markets are arbitrage free (H) holds under any e.m.m. However, hypothesis onτare generally done under the historical probability measure. In general, (H) hypothesis is not stable under a change of probability (see Kusuoka [15] for a counterexample). We now check that if (H) holds under the historical probabilityP, and if theF-market is complete and arbitrage free, then the defaultable market is arbitrage free. Under our hypotheses, denoting byQthe e.m.m. for theF-market and by(ζt, t≥0)its Radon-Nikodym density, the process(ζtSt, t≥0)is aP-Fmartingale, hence aP- Gmartingale. Then, there exists at least aG-e.m.m. defined asdQ∗|Gt =ζtdP|Gt and theG-market is arbitrage free.
3.4 Dynamics of defaultable zero-coupon
We assume now that a defaultable zero-coupon bond of maturityTis traded on the market and we denote byρ(t, T)itst-time price.
We assume that the market where the asset, a default-free zero-coupon and the DZC are traded, is arbitrage free. Then, there exists at least oneG-e.m.m.Q∗ such that the discounted price of the DZC is aG-martingale, that is,ρ(t, T)Rt= EQ∗(RT11T <τ|Gt). We emphasize that the e.m.m. is chosen by the market which trades the defaultable zero-coupon at the market priceρ(t, T). We do not assume the uniqueness of the e.m.m.Q∗; the DZC being traded, for any e.m.m., the equality ρ(t, T)Rt=EQ∗(RT11T <τ|Gt) = 11t<τeΓtmt=LtmtRt (7) holds, wheremt = mtRt = EQ∗(RTGT|Ft)and where the hazard process is Γt=−ln(Gt) =−lnQ∗(τ > t|Ft).
Proposition 2 Suppose that(H)holds underQ∗and thatF is continuous. Then dρ(t, T) =Lt−dmt−ρ(t−, T)dMt.
Proof The result follows fromρ(t, T) =LtmtanddLt=−Lt−dMt.
3.5 Representation theorem
In the case whereFis continuous and(H)holds, the representation Theorem 1 and the Corollary 2 write
Proposition 3 Suppose that (H)holds under Q∗ and F is continuous, and let Mt = Dt−Γt∧τ whereΓt = −lnQ∗(τ > t|Ft). Lethbe an F-predictable process such thathτis integrable, andHt=E(hτ| Gt). Then,
Ht=mh0+ t∧τ
0 eΓsdmhs+
]0,t∧τ](hs−Hs−h )dMs. (8)
Heremhis theF-martingale mht =EQ
∞
0 hudFu| Ft
,
LetY ∈ FTbe integrable. Then, theG-martingaleYt=EQ∗(Y11T <τ| Gt)admits a decomposition as follows
Yt=mY0 + t∧τ
0 eΓudmYu −
]0,t]Yu−dMu
wheremY is theF-martingale
mYt =EQ
Y GT|Ft .
3.6 Hedging strategies
We suppose thatF=FS, and that the default-free market is complete and arbitrage free. We assume that a defaultable zero-coupon is available on the market and (H) holds under theG-e.m.m.Q∗. We also assume that the processF is continuous.
We now make precise the hedging of a vulnerable claim.
3.6.1 Terminal payoff
We study in a first step the hedging strategy forX11T <τbased on savings account, risky asset and defaultable zero-coupon.
We recall that a pair(α, β)of FS-adapted processes is an hedging strategy for the contingent claimY withY RT ∈ FTS if, denoting byVt =αtSt0+βtSt
the value of this strategy, the self-financing relationdVt=αtdSt0+βtdStholds andY =αTST0 +βTST. A self-financing strategy is characterized by its initial valuexand the parameterβviaRtVt =x+t
0βsdSs=EQ(RTY|cFtS).The number of shares of riskless asset for this strategy isαt = Rt(Vt−βtSt). We shall call “hedging portfolio" ofY the numberβof shares of the asset held in the self-financing portfolio.
In the same way, a triple(at, bt, ct;t ≥0)ofG-adapted processes is an hedging strategy forY withY RT ∈ GT if the value processVt = atSt0+btSt+ctρt
satisfiesVT =Y and the self-financing relationdVt=atdSt0+atdSt+ctdρt. The default-free market being complete andXGTRT ∈ FTS, there exists a predictable process(µXt , t ≥0)(the hedging portfolio) and a constantmX0 (the price) such that
XGTRT =mX0 + T
0 µXs dSs, (9)
LetQbe theF-e.m.m. andmXt =EQ(XGTRT|Ft)the discountedt-time price ofXGT. The strategy(Rt(mXt −µXt St), µXt )is a self-financing strategy hedging the contingent claimXGT.
We denote, as in (7)
mtRt=m1t =EQ(GTRT|Ft) =m0+ t
0 µsdSs, (10) the discounted price ofGT.
Theorem 2 Assume thatF=FS, theF-market is complete, theFandG-markets are arbitrage free, and thatFt=Q∗(τ ≤t|FtS)is continuous. LetXdbe the value of the defaultable contingent claimX11T <τ, i.e.
XtdRt=EQ∗(XRT11T <τ|Gt).
The self-financing hedging strategy(at, bt, ct;t≥0)for the defaultable contingent claimX11T <τ, based on the riskless bond, the asset and the defaultable zero-coupon satisfies
atSt0+btSt+ctρ(t, T) =Xtd and consists of, ont < τ
(i) ct= Xtd ρ(t, T),
(ii) a position on the savings account and the risky asset such that atSt0+btSt= 0.
More precisely, the self financing hedging strategy is made of (i) a long position ofmXt
mt
defaultable zero-coupon, (ii) a number of asset’ shares equal toeΓt
µXt −mXt mtµt
(iii) an amount in the riskless bond equal to
−eΓt
µXt −mXt mtµt
St,
where the different values(mt, µt)and(mXt , µXt )are defined in (10) and (9).
Proof One can prove this theorem as an application of the representation theorem.
However, we prefer to give a direct proof. The price of the defaultable claimX11T <τ
isXtddefined by
RtXtd=EQ∗(XRT11T <τ|Gt) = 11t<τeΓtEQ(XRTGT|Ft)
= 11t<τeΓtmXt =LtmXt and the price of the DZC satisfies
Rtρ(t, T) = 11t<τeΓtmt. Hence
Xtd= mXt
mtρ(t, T)
We now check that there exists a triple(a, b, c)determining a self-financing portfolio such thatct= mXt
mt
andatSt0+btSt= 0. From Proposition 2 the self financing condition reads
atSt0rtdt+btdSt+ct(Lt−dmt+mtdLt) =dXtd=Lt−dmXt +mXt dLt. (11) UsingatS0t+btSt= 0, equality (11) reduces to
−btStrtdt+btdSt+ctLt−dmt=Lt−dmXt . or, in terms of discounted processes
btdSt+ctLt−µtdSt=Lt−µXt dSt.
Hence the form ofbgiven in the theorem.
It is difficult to make explicit the hedging, since it requires the hedging of claims of the formexp T
0 γsds
whereγis a random process. This problem is similar to hedging under a stochastic interest rate. However, in the very particular case whereFandrare deterministic we get
ct=EQ(X|Ft), µt= 0. Let∆be the hedging strategy ofX:
XRT =x+ T
0 ∆sdSs. Then,
RTX11T <τ=h+ T∧τ
0 eΓs−ΓT∆sdSs+
]0,T∧τ]EQ(X|Fs)dρs. and the hedging strategy of the vulnerable claimX11T <τconsists of holding (a) EQ(X|Ft)DZC, so that the wealth invested in DZC isEQ(X|Ft)ρ(t, T) (b) eΓt−ΓT∆tshares of risky asset.
3.6.2 Rebate part
We compute the quantity EQ∗(hτ11τ≤TRτ|Gt), which corresponds to the dis- counted price of the rebate, when the compensation is payed at hit.
Proposition 4 LetCthbe the value of the rebate, i.e.
CthRt=EQ∗(hτRτ11τ <T|Gt) =hτRτ11τ≤t+11t<τeΓtEQ
T
t
RuhudFu|Ft
. (12) The hedging strategy before default time of the rebate part consists of
(i) ct= 1 ρt
(Cth−ht)defaultable zero-coupon
(ii) a position on savings account and risky asset such that atSt0+btSt=ht
and the strategy(at, bt, ct)is self financing.
More precisely the hedging strategy of the rebate part is made of (before the default time)
(i) 1
ρ(t, T)(Cth−ht)defaultable zero-coupon (ii) eΓt
µht − 1
mtµtCth
+ 1
mtµthtshares of the asset (iii) a cash amount ofht+St
eΓt 1
mtµtCth− 1
mtµtht+eΓtµht
, whereµhis defined by
E T
0 RuhudFu|Ft
=mh0+ t
0 µhsdSs. (13) Proof We denote byCththe price of the rebate defined by (12) and byµh, defined in (13), the hedging strategy for the discounted claimT
0 RuhudFuin the default-free world. The representation theorem states that
EQ∗(hτRτ11τ≤T|Gt) =C0h+ t∧τ
0 eΓuµhudSu+
[0,t∧τ[Ru(hu−Cu−h )dMu. Hence, introducingmt, the discounted price ofGT
EQ∗(hτRτ11τ≤T|Gt) =C0h+ t∧τ
0 eΓuµhudSu
−
[0,t∧τ[(huRu−Cuh) 1 Lumu
[dρu−Ludmu]
which leads to
EQ∗(hτRτ11τ≤T|Gt) =mh0 + t∧τ
0
eΓu
µhu−Cuhµhu mu
+µuhu
mu
dSu
−
[0,t∧τ[(hu−Cu−h ) 1 Lumudρu
Corollary 3 Under(H), ifFis continuous, the defaultable-market is complete as soon as a defaultable zero-coupon is traded.
4 Conclusion
We have proved that, if the vulnerable market is complete, hedging strategies are trivial and give a great importance to the DZC. It remains to study the case with several correlated defaults. In the case where defaults are independents, the result has the same form. The general case is a work in progress.
Appendix Proof of theorem
As usual,Gcis the martingale continuous part of the semi-martingaleG. We recall thatd[G]t=d[Gc]t+ (∆Gt)2. Then, Itˆo’s formula leads to
d(G−1t ) =− 1
(Gt−)2dGt+ 1
(Gt−)3d[Gc]t+
eΓt−eΓt−+ 1 (Gt−)2∆Gt
= 1
(Gt−)2
−dGt+ 1
Gt−d[Gc]t
+ 1
Gt(Gt−)2(∆Gt)2
= 1
(Gt−)2
−dGt+ 1
Gt−d[Gc]t+ 1 Gt
(∆Gt)2
(14) The quadratic variation of the processes
Yt=mht − t
0 hudFu=mht + t
0 hudGu
andeΓt =G−1t is
d[eΓ, Y]t=d[eΓ, mh]t+htd[eΓ, G]t
= 1
(Gt−)2
−d[G, mh]t+ 1
Gt(∆Gt)2∆mht −htd[G]t+ ht
Gt(∆Gt)3
= 1
(Gt−)2
−d[G, mh]t+ 1 Gt
(∆Gt)2∆mht −htd[Gc]t
−ht(∆Gt)2+ ht
Gt
(∆Gt)3
From integration by part formula, the dynamics ofJth=YteΓt are dJth=eΓt−dYt+Yt−deΓt+d[eΓ, Y]t
=−eΓt−(Jt−h −ht)dGt+Jt−h −ht
(Gt−)2 d[Gc]t+Jt−h −ht
GtGt− (∆Gt)2 +eΓt−
dmht + 1
GtGt−(∆Gt)2∆mht
− 1
(Gt−)2d[G, mh]t
=eΓt−(Jt−h −ht)
−dGt+eΓt−d[Gc]t+ 1 Gt
(∆Gt)2
+eΓt−
dmht + 1
GtGt−(∆Gt)2∆mht − 1
Gt−d[G, mh]t
The decomposition ofF in the filtrationGis Ft∧τ =Zt∧τ+At∧τ=Zt∧τ−
t∧τ
0
1
Gsd[Z]s+At∧τ
Then, on the set{τ > t}
dJth=eΓt−
(Jt−h −ht)dZt+dmht + (Jt−h −ht)dCt+dKt
where
dCt= 1
Gt(∆Gt)2+eΓt−d[Gc] +dAt− 1 Gtd[Z]t
dKt= 1
GtGt−(∆Gt)2∆mht − 1
Gt−d[G, mh]t+ 1 Gt
d[G, mh]t
IfGhas no jump at timet,dKt= 0, ifGhas a jump dKt=−(∆Gt) (∆mht) 1
GtGt−
−∆Gt+Gt−Gt−
= 0.
The first term is equal to dCt= 1
Gt(∆Gt)2+eΓtd[Gc]t+dAt− 1 Gtd[Z]t
= 1
Gtd[G]t+dAt− 1 Gtd[Z]t
= 1 Gt
d[A]t+dAt= 1 Gt
(∆At)2+dAt
=e∆ΓtdAt
where we have used that, if theFmartingales are continuous∆A=∆F and that Ais continuous in the caseP(τ = θ) = 0(see Jeulin [14], p. 65). It remains to compensate the jump at timeτin order to obtain the result.
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