LIANG Xue

（School of Mathematics and Physics，SUST，Suzhou 215009，China）

Hedging of defaultable claims in a Markov regime-switching model

LIANG Xue

（School of Mathematics and Physics，SUST，Suzhou 215009，China）

This paper presented a martingale representation for Markov chain,and investigated the hedging of defaultable claims in a Markov regime switching model.It derives the pricing dynamics of a defaultable claim and the self-financing trading strategy based on the savings account and defaultable bonds.

representation theorem；Markov chain；replicating strategy；regime switching

1 Introduction

Based on martingale representation of Brownian motion,Bielecki,Jeanbanc and Rutkowski[1]obtain the replicating strategies of defaultable claims in CDS markets.This paper provides a martingale representation theorem for a Markov chain,and deals with the problem of hedging the defaultable claim in bond markets in a Markov regime-switching setting.The set of hedging instruments consists of defaultable bonds with different maturities and recovery rates,which are issued by the same firm.And then we try to find the replication strategies based on the savings account and defaultable bonds.

There has been a large literature on the topic of hedging defaultable claims,such as[2－3],which construct rigorously dynamic hedging strategies of defaultable contingent claims.Blanchet-Scalliet，et al.[4]consider the hedging defaultable claims using BSDE with uncertain time horizon.Bielecki，et al.[5]deal with the problem of hedging the credit default swaption in the CDS markets.Fujii，et al.[6]provide explicit expressions of approximate hedging portfolio under the Bayesian and KalmanBucy frameworks.Their works are all in the case that the reference filtration is generated by a Brownian Motion.Laurent，et al.[7]derive dynamic hedging strategies of CDO tranches in a Markovian contagion model,but the reference filtration in their work is trivial.In this paper,we will consider the hedging defaultable claims in the case that the reference filtration is generated by a Markov chain,which is used to describe the economic state of the world.

In recent years,regime-switching models have become popular in finance.Elliott,Chan and Siu[8]considered Esscher transform in a regime-switching model.Liang and Wang[9]considered the pricing of portfolio credit derivatives in jump-diffusion model with regime-switching.Siu[10]considered bond valuation problems with regime-switching.Capponi and Figueroa-LópezIn[11]considered a portfolio optimization problem in a Markov regime-switching setting.In this paper,we will focus on the issue of hedging the defaultable claim in a regimeswitching model.We shall provide a martingale representation theorem for a Markov chain,and then we obtain the closed-form expressions for replicating strategy of the defaultable claim.

The organization of this paper is as follows.The next section provides the basic setup of our framework.In section 3,we prove the martingale representation theorem for a Markov chain.In section 4,we derive the pricing dynamics of a defaultable claim and obtain the self-financing trading strategy based on the savings account and defaultable bonds.Section 5 concludes.

2 The model

Given a filtered probability space（Ω，g，G，Q） withG=（gt）t≥0,where Q is the risk-neutral probability measure.Let τ be a strictly positive random variable,defined on（Ω，g，G，Q）,representing the default time of the credit name.We work here within a single credit name framework，so that τ is the moment of default of the reference credit name.We define the default process Ht=1{τ≤t}and denote by Ht=σ{Hs：0≤s≤t}the filtration generated by the default process Ht.We assume thatG=H∨F,where

andF=（Ft）t≥0is some reference filtration.All the above filtrations are assumed to satisfy the so-called usual conditions（see,for example,Chapter 1 of[12]）.

We assume that the states of the economy are modeled by a continuous-timeG-Markov process{Ct}defined on（Ω，g，G，Q） with a finite state space K={1，2，…，K}（see Chapter 11，Section 2 of[13]）,and

Without loss of generality,we assume C0=1.Define Λt=[λij（t）]1≤i，j≤Kto be the infinitesimal generator matrix function associated with the Markov chain C.For any fixed i≠j,let Hijstand for the number of jumps of the process C form i to j in the interval（0，t].Formally,for any i≠j,we set

The following result is classic（see Proposition 11.2.2 and Lemma 11.2.3 in[13]）.

Lemma 1Let C is aG-Markov chain with the infinitesimal generator matrix function Λ（·）,then for?i，j∈K，i≠j the process

are bothG-Martingales（andF-Martingales）.

We assume that the short-term interest-rate rtfollows anF-progressively measurable stochastic process.The price process{Bt}of the savings account is given by

We assume that

where{ht}is the risk-neutral default intensity and it is non-negative,F-progressively measurable.By Proposition 5.1.3 of[13],we know that

is aG-martingale.

Consider a defaultable contingent claim with three basic building blocks.（see[14]）.One is a payment X∈FTat maturity date T which occurs if no default has occurred prior to or at T；one is a stream of payments prior to default at a rate specified by the Ft-adapted process Yt；another is a recovery payoff Zτat the time of default,where{Zt}is predictable with respect to the reference filtrationF.Then,the price process S=（St）t＞0of the defaultable contingent claim（X，Y，Z，τ） maturing at T is given by,for t∈[0，T],

the cumulative price of the defaultable contingent claim is given by,for t∈[0，T],

and the discounted cumulative price the defaultable contingent claim equals,for t∈[0，T],

We now consider n defaultable bonds with certain maturities Ti（Ti≥T） and the recovery payoff δifor i=1，2，…，n.We assume that these bonds are traded over the time interval[0，T],and that the recovery processare predictable with respect to the reference filtrationFfor i=1，2，…，n.All these bonds are supposed to refer to the same underlying name,and thus they have the common default time τ.Then,the price processof the ith defaultable bond maturing at Tiis given by,for t∈[0，Ti]，

the cumulative price of the ith defaultable bond is given by,for t∈[0，Ti],

and the discounted cumulative price the ith defaultable bond equals,for t∈[0，Ti],

3 Representation theorem

In this section,we will establish a representation theorem forF-martingales,which can be used to establish the existence of a hedging strategy.

Proposition 1Let F is an FT-measurable square integrable random variable,then there exist FT-predictable stochastic processes{fij}（i≠j，i，j∈K）,such that

where{Mij}（i≠j，i，j∈K） are given by（3）.Furthermore,if（Ft）t≥0is a square integrableF-martingale,then,for all t≥0,there exist Ft-predictable stochastic processes{gij}（i≠j，i，j∈K），such that

where{gij}（i≠j，i，j∈K） are Ft-predictable stochastic processes.

Proof.Define

and thus the representation（8） follows.

Define

We first show that M?N.Fix F∈M,define

So{Xt}is a martingale and

since F0is trivial.Without loss of generality,we assume that 0=t0≤t1≤…≤tn=T.

?j∈{1，2，…，n},and ?t∈（tj-1，tj],

Set

and by Chapman-Kolmogorov equation,we have

therefore

and

From Lemma 1 and Kolmogorov backward equation,we have

After some simple calculations,we obtain

Combining with（10）,（11）,we have

therefore

Form（2）-（3）,we have

and hence

where

Note（9） and therefore

So F∈N.We have proved that M?N,and it is easy to check that N is a monotone vector space and M is multiplicative.（see Chapter 1 of[11]）.By Monotone Class Theorem we know that N contains all bounded,FTmeasurable functions.Hence,we have

The representation（8） follows.

Since（Ft）t≥0is aF-martingale,F0is F0-measurable and F0is trivial,we have E（Ft）=E（F0）=F0，and hence

4 Hedging of the defaultable claim

In this section,we shall find the hedging strategy for the defaultable claim（X，Y，Z，τ） in bond markets.We will consider trading strategies φ=（φ0，φ1，…，φn）,where φ0is aF-adapted process and the processes φ1，φ2，…，φnareF-predictable.The following definition is similar to Definition 2.7 of[1].

Definition 1The wealth process V（φ） of a strategy φ=（φ0，φ1，…，φn） in the savings account B and defaultable bonds Sk，k=1，…，n equals,for any t∈[0，T],

A strategy θ is said to be self-financing if

where Htis given by（4）.

Lemma 2Let φ=（φ0，φ1，…，φn） be a self-financing trading strategy in the savings account B and defaultable bonds Sk，k=1，…，n.Then the discounted wealth process V*（φ）=B-1V（φ） satisfies,for t∈[0，T]，

Proof. Since V*（φ）=B-1V（φ）,we have

Combining with（6）,（7）,we have

Proposition 2The price process S=（St）t＞0of a defaultable claim（X，Y，Z，τ） maturing at T equals,for t∈[0，T]，

and thus,the discounted cumulative price Sc，*satisfies,for t∈[0，T]，

where Mtis given by（4） and mtis given by

Proof.Note（4）,?t∈[0，T],we have

By Proposition 3.1 in[11],we have

If we define

then we have

and therefore

Since

we have

and therefore,

Similarly,we have the following proposition.

Proposition 3The price processof the k-th defaultable bond maturing at Tkequals,for t∈[0，Tk],

and thus,the discounted cumulative price Sc，k，*satisfies,for all t∈[0，T]，

Definition 2We say that a self-financing strategy φ=（φ0，φ1，…，φn）replicates a default-able claim（X，Y，Z，τ） if its wealth process V（φ） satisfies

for every t∈[0，T].

The following result is similar to Theorem 2.1 in[2],where the defaultable claim is replicated by CDSs with the reference filtration generated by a Brownian Motion.

Proposition 4Assume that there existF-predictable processes φ1，φ2，…，φnsatisfying the following conditions,for any t∈[0，T]，

Let the process V（φ） be given by（12） with the initial condition,and let φ0be given by,for t∈[0，T]，

then the self-financing trading strategy φ=（φ0，φ1，…，φn） in the savings account B and defaultable bonds Sk，k=1，…，n,replicates the defaultable claim（X，Y，Z，τ）.

Proof.Applying theorem 3.1 in[2]to theF-Martingales{m}，{nk},which are given by（13） and（14） respectively,we have that there existF-predictable processes

such that

From lemma 2,we have

So if（15）and（16） are satisfied for any t∈[0，T]，then the equality

holds for any t∈[0，T]， and hence

5 Conclusions

This paper studies the martingale representation problem in a Markov chain setting.Based on martingale representation of Markov chain,we obtain the replicating strategies of defaultable claims in bonds markets in a Markov regime switching model.

[1]BIELECKI T R，JEANBLANC M，RUTKOWSKI M.Pricing and trading credit default swaps in a hazard process model[J].The Annals of Applied Probability 2008，18（6）：2495-2529.

[2]BIELECKI T R，JEANBLANC M，RUTKOWSKI M.Paris-Princeton Lectures on Mathematical Finance 2003[M].Berlin：Springer，2004：1-132.

[3]BLANCHET-SCALLIET C，JEANBLANC M.Hazard rate for credit risk and hedging defaultable contingent claims[J].Finance and Stochastics，2004，8：145-159.

[4]BLANCHET-SCALLIET C，EYRAUD-LOISEL A，ROYER-CARENZI M.Hedging of defaultable contingent claims using BSDE with uncertain time horizon[J].Bulletin Francais d'Actuariat，2010，10（20）：102-139.

[5]BIELECKI T R，JEANBLANC M，RUTKOWSKI M.Hedging of a credit default swaption in the CIR default intensity model[J].Finance and Stochastics，2011，15（3）：541-572.

[6]FUJII M，TAKAHASHI A.Making mean-variance hedging implementable in a partially observable market[J].Quantitative Finance，2014，14（10）：1709-1724.

[7]LAURENT J P，COUSIN A，F(xiàn)ERMANIAN J D.Hedging default risks of CDOs in Markovian contagion models[J].Quantitative Finance，2011，11（12）：1773-1791.

[8]ELLIOTT R J，CHAN L，SIU T K.Option pricing and esscher transform under regime switching[J].Annals of Finance，2005，1：423-432.

[9]LIANG X，WANG G J，DONG Y H.A Markov regime switching jump-diffusion model for the pricing of portfolio credit derivatives[J].Statistics and Probability Letters，2013，83：373-381.

[10]SIU T K.Bond pricing under a Markovian regime-switching jump-augmented Vasicek model via stochastic ows[J].Applied Mathematics and Computation，2010，216：3184-3190.

[11]CAPPONI A，F(xiàn)IGUEROA-LóPEZ J E.Dynamic portfolio optimization with a defaultable security and regime-switching[J].Mathematical Finance，2014，24（2）：207-249.

[12]PROTTER P E.Stochastic Integration and Differential Equations[M].Berlin：Springer-Verlag，2004.

[13]BIELECKI T R，RUTKOWSKI M.Credit Risk：Modeling，Valuation and Hedging[M].Berlin：Springer-Verlag，2002.

[14]LANDO D.On Cox processes and credit risky securities[J].Review of Derivatives Research，1998，2：99-120.

責(zé)任編輯：謝金春

2016－10－24

國(guó)家自然科學(xué)基金資助項(xiàng)目（11401419）；江蘇省自然科學(xué)基金資助項(xiàng)目（BK20140279）

梁 雪（1978-），女，湖北麻城人，副教授，博士，研究方向：金融數(shù)學(xué)。

機(jī)制轉(zhuǎn)換模型中可違約權(quán)益的對(duì)沖

梁 雪
（蘇州科技大學(xué) 數(shù)理學(xué)院，江蘇 蘇州 215009）

給出了馬氏鏈的表示定理，并且在馬氏機(jī)制轉(zhuǎn)換模型下考慮了可違約權(quán)益的對(duì)沖問(wèn)題，利用馬氏鏈的表示定理得到了可違約權(quán)益的動(dòng)態(tài)對(duì)沖策略。

表示定理；馬氏鏈；復(fù)制策略；機(jī)制轉(zhuǎn)換

O211.6MR（2010） Subject Classification60J27；91B28

A

2096－3289（2017）04－0001－08