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) dΘt = Θt(ϑtdt+ϑbtdWt+ϑetdMt) (1.29) dΨt = 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)2−2(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(πσ−bxt)−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−Xt−ext) = 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)2(ϑt−Bt2∆t) + 2Θt(Vt−Xt)
³
AtBt∆t−ϑbtbxt−ξtϑetext−ft
´
+ Θtξt(ϑet+ 1)(Atϕ−xet)2+ Θt(Atσ−bxt)2+ψt. Then, we choose
ϑ∗t = Bt2∆t
ft∗ = AtBt∆t−ϕbtbxt−ξtϑetext
ψ∗t = −Θtξt(ϑet+ 1)(Atϕ−ext)2−Θt(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 (Θ∗,ϑb∗,ϑe∗), (X∗,xb∗,xe∗) and (Ψ∗,ψb∗,ψe∗); the corresponding processes A, B and ∆ will be denoted as A∗, B∗ and ∆∗. Consequently, the drift term of Θ∗t(Vt∗(π)−Xt∗) + Ψ∗t is non-positive for any admissibleπand it is equal to 0 forπ∗=A∗t−B∗t(Vtv,∗(π∗)−Xt∗).
The three dimensional process (Θ∗,ϑb∗,ϑe∗) is supposed to satisfy the BSDE dΘt = 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
∆t(κ1,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 (Ψ∗,ψb∗,ψe∗) is solution of
dΨt = −Θt
³
ξt(ϑet+ 1)(Atϕ−ext)2+ (Atσ−bxt)2´
dt+ψbtdWt+ψetdMt
ΨT = 0.
Thus, noting that Ψ∗t +
Z t
0
Θs
³
ξs(ϑes+ 1)(Asϕ−exs)2+ (Asσ−xbs)2
´ ds is aG-martingale, we obtain that
Ψ∗t =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 measureQ∗such 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∗(L∗T|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
dL∗t =L∗t−(`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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