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Jump-Dynamics of Price

Dans le document INDIFFERENCE PRICING OF DEFAULTABLE CLAIMS (Page 31-37)

1.4 Quadratic Hedging

1.4.3 Jump-Dynamics of Price

We assume here that the price process follows

dSt=St−(νdt+σdWt+ϕdMt), S0>0

where the constant ϕ satisfyϕ > −1 so that the price St is strictly positive.

Hence, the primary market, where the savings account and the asset S are traded is arbitrage free, but incomplete (in general). It follows that the wealth process follows

dVtv(π) =πt(νdt+σdWt+ϕdMt), V0v(π) =v.

As in the previous subsection, our aim is, for a given initial endowmentv, solve the minimization problem:

minπ EP((VTv(π)−X)2).

In order to characterize the value function we proceed analogously as before.

That is, we are looking for processes X,Θ and Ψ such that the process (for simplicity we write Vt in place ofVtv(π))

J(t, Vt) = (Vt−Xt)2Θt+ Ψt

is a submartingale for any π and a martingale for some π, and such that ΨT = 0, XT = X,ΘT = 1. (Note that Mania and Tevzadze (2003a) did a similar approach for continuous processes, with a value function of the form

Jt = Φ0(t) + Φ1(t)Vt+ Φ2(t)Vt2.) Let us assume that the dynamics of these processes are of the form

dXt = ftdt+bxtdWt+extdMt, (1.28) t = Θttdt+ϑbtdWt+ϑetdMt) (1.29) t = ψtdt+ψbtdWt+ψetdMt (1.30) where the driftsft,ϑtandψthave to be determined.

From Itˆo’s formula we obtain

d(Vt−Xt)2= 2(Vt−Xt)(πtσ−bxt)dWt

+ £

(Vt+πtϕ−Xt−xet)2(Vt−Xt)2¤ dMt

+ ¡

2(Vt−Xt)(πtµ−ft) + (πtσ−bxt)2 + ξt

£(Vt+πtϕ−Xt−xet)2(Vt−Xt)22(Vt−Xt)(πtϕ−xet)¤¢

dt.

Process Θt(Vt−Xt)2+ Ψtis a (local) martingale iffk(πt, ft, ϑt, ψt) = 0 for all t, where

k(π, ϑ, f, ψ) =ψ+ Θt

£ϑt(Vt−Xt)2 + 2(Vt−Xt)

³

(πµ−f) +ϑbt(πσbxt)−ξt(πϕ−xet)

´

+ (πσ−xbt)2 + ξt(ϑet+ 1)¡

(Vt+πϕ−Xt−xet)2(Vt−Xt)2¢i .

In the first step, we findπ] such that the maximum ofk(π) is obtained. Then, one defines (f, ϑ, ψ) such that k(π], f, ϑ, ψ) = 0. This implies that, for anyπ,k(π, f, ϑ, ψ)0, and thatk(π], f, ϑ, ψ) = 0.

The optimalπ] is the solution of

(Vt−Xt)(µ−ξtϕ+ϑbtσ) +σ(πσ−xbt) + ξt(ϑet+ 1)ϕ(Vt+πϕ−Xtext) = 0. hence

π]t = 1

σ2+ϕ2ξt(ϑet+ 1)

³

(σbxt+ξtϕ(ϑet+ 1)ext)(µ+ϑbtσ+ξtϕϑet)(Vt−Xt)

´

= At−Bt(Vt−Xt) with

At =

³

σbxt+ξtϕ(ϑet+ 1)ext

´

−1t Bt =

³

µ+ϑbtσ+ξtϕϑet

´

−1t

t = σ2+ϕ2ξt(ϑet+ 1).

1.4. QUADRATIC HEDGING 33

After some computations the drift term of Θt(Vt−Xt) + Ψt is found to be Θt(Vt−Xt)2t−Bt2t) + 2Θt(Vt−Xt)

³

AtBtt−ϑbtbxt−ξtϑetext−ft

´

+ Θtξt(ϑet+ 1)(Atϕ−xet)2+ Θt(Atσ−bxt)2+ψt. Then, we choose

ϑt = Bt2t

ft = AtBtt−ϕbtbxt−ξtϑetext

ψt = −Θtξt(ϑet+ 1)(Atϕ−ext)2Θt(Atσ−bxt)2.

Let us suppose that with this choice of drifts equations (1.29)–(1.30) admit solutions (we shall discuss this issue below). Next, let us denote these solutions as (Θbe), (X,xb,xe) and (Ψbe); the corresponding processes A, B and ∆ will be denoted as A, B and ∆. Consequently, the drift term of Θt(Vt(π)−Xt) + Ψt is non-positive for any admissibleπand it is equal to 0 forπ=At−Bt(Vtv,∗)−Xt).

The three dimensional process (Θbe) is supposed to satisfy the BSDE t = Θt

Ã(µ+ϑbtσ+ξtϕϑet)2

σ2+ϕ2ξt(ϑet+ 1) dt+ϑbtdWt+ϑetdMt

!

(1.31) ΘT = 1.

We shall discuss this equation later.

The three dimensional process (X,xb,ex) is a solution of thelinear BSDE dXt = 1

t1,txbt+κ2,text)dt+xbtdWt+xetdMt

XT = X where

κ1,t=σµ+σϕξtϑet−ϕ2ϑbtξt(1 +ϑet), κ2,t=ϕξt(1 +ϑet)(µ+σϑbt)−σ2ξtϑet. Thus,

Xt=EQκ(X|Gt), where dQκ|Gt =L(κ)t dP|Gt and

dL(κ)t =−L(κ)t−(κ1,t

t

dWt+κ2,t

ξ∆t

dMt). The three dimensional process (Ψbe) is solution of

t = −Θt

³

ξt(ϑet+ 1)(Atϕ−ext)2+ (Atσ−bxt)2´

dt+ψbtdWt+ψetdMt

ΨT = 0.

Thus, noting that Ψt +

Z t

0

Θs

³

ξs(ϑes+ 1)(Asϕ−exs)2+ (Asσ−xbs)2

´ ds is aG-martingale, we obtain that

Ψt =E ÃZ T

t

Θs

³

ξs(ϑes+ 1)(Asϕ−xes)2+ (Asσ−bxs)2´ ds|Gt

!

. (1.32)

Discussion of equation (1.31): Duality approach

Our aim is here to prove that the BSDE (1.31) has a solution. We take the opportunuity to correct a mistake in Bielecki et al (2004b) where we claim that, in the particular case where the intensityγtis constant, we get a solution of the formθetconstant. The solution that appear in Bielecki et al. is valid only in the caseP(τ < T) = 1. We proceed using duality approach.

The set of equivalent martingale measure is determined by the set of densi-ties. From Kusuoka (1999) representation theorem, it follows that any strictly positive martingale in the filtrationGcan be written as

dLt=Lt−(`tdWt+χtdMt) (1.33) for aG-predictable processχ satisfyingχt>−1. In order thatL corresponds to the Radon-Nikodym density of an emm, a relation between`andχhas to be satisfied in order to imply that process LtSt is aP(local) martingale. (Recall thatr= 0.) Straightforward application of integration by parts formula proves that the drift term ofLS vanishes iff

ϕχtξt+σ`t+ν= 0

Recall that by definition the variance optimal measure for L is a probability measureQsuch that it minimizesEQ(L2T). At this moment we are unable to verify existence/uniqueness of such a measure in the context of our model. We thus assume that the measure exists,

Hypothesis: We assume that the variance optimal measure exists.

In what follows we shall use the same argument as in Bobrovnytska and Schweizer (2004). Towards this end we denote by L the Radon-Nikodym density of the variance optimal martingale measure. Let Z be the martingale Zt =EQ(LT|Gt) andU =L/Z. It is proved in Delbaen and Shachermayer (Lemma 2.2) that, if the variance optimal martingale measure exists, then there exists a predictable processzbsuch that

dZt/Zt−=bztdSt=zt(σdWt+ϕdMt+νdt)

wherezt=bztSt−(in the proof of lemma 2.2, the hypothesis of continuity of the asset is not required). The processL is a (P,G) martingale, hence there exist

`andχ such that

dLt =Lt−(`tdWt+χtdMt)

1.4. QUADRATIC HEDGING 35

From Itˆo’s calculus, settingU =L/Z, we obtain dUt=Ut− so that processU is a solution of

dUt=Ut−

µ(ν+σbut+ϕξtuet)2

σ2+ϕ2ξt(1 +eut)dt+butdWt+eutdMt

, UT = 1, which establishes that the BSDE (1.31) has a solution as long as the variance optimal martingale measure exists in our set-up.

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Dans le document INDIFFERENCE PRICING OF DEFAULTABLE CLAIMS (Page 31-37)

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