Introduction to Malliavin calculus andApplications to Finance

Part II

Giulia Di Nunno

Finance and Insurance,Stochastic Analysis and Practical Methods

Spring SchoolMarie Curie ITN - Jena 2009

PART II

1. Short on Levy processes

2. Malliavin calculus for Poisson random measures• Iterated Ito integrals• Wiener-Ito chaos expansion• Skorohod integral• Malliavin derivative• Clark-Ocone formulae

3. Incomplete markets: Minimal variance hedging

4. Minimal variance hedging under partial information

References

1. Short on Levy processesDefinition A Levy process is a stochastically continuous stochastic processη(t), t ∈ [0,T ], (η(0) = 0) with stationary and independent increments.

I Levy-Khintchine formula. For every t, the random variable η(t) isdistributed according to the law given by:

E[e iuη(t)

]= e iΨ(u),

with

Ψ(u) := iαu−1

2σ2u2+

∫|z|<1

(e iuz−1−iuz

)ν(dz)+

∫|z|≥1

(e iuz−1

)ν(dz).

Here α ∈ R and σ2 ≥ 0 and ν = ν(dz), z ∈ R0 := R \ 0 , is aσ-finite measure on B(R0) satisfying∫

R0

min(1, z2)ν(dz) <∞.

I The law of a Levy process is characterized by the characteristictriplet (α, σ2, ν).

I This process admits a modification with right-continuous andleft-limited (cadlag) trajectories.

We assume to work with such a modification.

I The Levy-Ito decomposition theorem. A Levy process η(t),t ∈ [0,T ], admits the integral representation in the form

η(t) = at + σW (t) +

∫ t

0

∫|z|<1

zN(ds, dz) +

∫ t

0

∫|z|≥1

zN(ds, dz)

for some constants a ∈ R, σ ∈ R.Here W (t), t ∈ [0,T ] (W (0) = 0), is a standard Wiener processand N(dt, dz), (t, z) ∈ [0,T ]× R0, is the Poisson random measuredescribing the jumps of the process andN(dt, dz) := N(dt, dz)− ν(dz)dt where E

[N(dt, dz)

]= ν(dz)dt.

I For the applications we have in mind we will assume that∫R0

z2ν(dz) <∞.

This actually implies that the Levy process takes values in L2(P).

I We will consider Levy process with zero-expectation.

I Then the Levy-Ito decomposition can be written as:

(1) η(t) = σW (t) +

∫ t

0

∫R0

zN(ds, dz).

Ito-Levy processes. Inspired by the representation (1), we will considerprocesses admitting integral representation of form

η(t) =

∫ t

0

σ(s)dW (s) +

∫ t

0

∫R0

γ(s, z)N(ds, dz), t ∈ [0,T ],

where σ and γ are F-adapted processes (predictable for the cadlagmodification) such that:

E[ ∫ t

0

σ2(s)ds +

∫ t

0

∫R0

γ2(s, z)ν(dz)ds]<∞.

A combination of Gaussian and pure jump Levy noises. Denote theprobability space on which W (t), t ∈ [0,T ], is a Wiener process by(ΩW ,FW ,PW ) and denote the one on which

N(dt, dz) = N(dt, dz)− ν(dz)dt is a compensated Poisson random

measure by (ΩeN ,F eN ,P eN ).We set

Ω = ΩW × ΩeN , F = FW ⊗F eN , P = PW × PeN .

Correspondingly, let FW := FWt , t ∈ [0,T ] and

FeN := F eNt , t ∈ [0,T ] be the P-augmented filtrations generated by the

values of W and N.

Set FW = FWT and F eN = F eN

T .

I Following this approach we study separately the Malliavin calculusfor the Brownian motion and the compensated Poisson randommeasure and then we merge the results on the space (Ω,F ,P) justdefined.

I One could follow another approach and consider the calculus withrespect to the whole process η directly. See Di Nunno (2007).

In the coming sections we concentrate on the calculus with respect to the

Poisson kind of noise only.

2. Malliavin calculus for Poisson random measuresIterated Ito integrals. Define

Gn :=

(t1, z1, ..., tn, zn) : 0 ≤ t1 ≤ ... ≤ tn ≤ T , zi ∈ R0, i = 1, ..., n

For any g ∈ L2(Gn), i.e.

‖g‖2L2(Gn) :=

∫Gn

g 2(t1, z1, ..., tn, zn)dt1ν(dz1) · · · dtnν(dzn) <∞,

the n-fold iterated integral Jn(f ) is the random variable in L2(P) definedas

Jn(g) :=

∫ T

0

∫R0

· · ·∫ t2

0

∫R0

g(t1, z1, ..., tn, zn)N(dt1, dz1) · · · N(dtn, dzn).

We set J0(g) := g for g ∈ R.

Directly from the properties of Ito integrals we have:

I Jn(g) ∈ L2(P), by the Ito isometry ‖Jn(g)‖2L2(P) = ‖g‖2

L2(Gn).

I If g ∈ L2(Gm) and f ∈ L2(Gn) (m < n), then E[Jm(g)Jn(f )

]= 0.

The symmetrization f of f is defined by

f (t1, z1, ..., tn, zn) =1

n!

∑σ

f (tσ1 , zσ1 , ..., tσn , zσn ),

the sum being taken over all permutations σ = (σ1, ..., σn) of 1, ..., n.A function f is symmetric if f = f . We denote the space of squareintegrable symmetric functions by L2((λ× ν)n).

Note that the symmetrization is over the n pairs (ti , zi ) and not over the2n variables.

For any symmetric f ∈ L2((λ× ν)n) we define its n-fold iterated integralby

(2) In(f ) :=

∫([0,T ]×R0)n

f (t1, z1, ..., tn, zn)N⊗n(dt, dz) = n!Jn(f ).

Wiener-Ito chaos expansion for Poisson random measuresAny F ∈ L2(P) FT -measurable random variable admits the representation

F =∞∑

n=0

In(fn)

via a unique sequence fn ∈ L2((λ× ν)n), n = 1, 2, .... Here we setI0(f0) := f0 for f0 ∈ R.Moreover, we have that

‖F‖2L2(P) =

∞∑n=0

n!‖fn‖2L2((λ×ν)n).

Skorohod integral

Let X (t, z), 0 ≤ t ≤ T , z ∈ R0, be a stochastic process withFT -measurable values such that

E[ ∫ T

0

∫R0

X 2(t, z)ν(dz)dt]<∞.

Then for each (t, z), the random variable X (t, z) has an expansion of theform

X (t, z) =∞∑

n=0

In(fn(·, t, z)) where fn(·, t, z) ∈ L2((λ× ν)n).

Let fn(t1, z1, ..., tn, zn, tn+1, zn+1) be the symmetrization of

fn(t1, z1, ..., tn, zn, t, z) as a function of the n + 1 pairs.

Suppose that∞∑

n=0

(n + 1)!‖fn‖2L2((λ×ν)n+1) <∞.

Then the Skorohod integral δ(X ) of X with respect to N is defined as

(3) δ(X ) :=

∫ T

0

∫R0

X (t, z)N(δt, dz) :=∞∑

n=0

In+1(fn).

Moreover,

‖δ(X )‖2L2(P) =

∞∑n=0

(n + 1)!‖fn‖2L2((λ×ν)n+1) <∞.

Next we consider Skorohod integrals with respect to

η(t) =

∫ t

0

∫R0

zN(dt, dz), t ∈ [0,T ].

Let Y (t), t ∈ [0,T ], be a measurable stochastic process such that

X (t, z) := Y (t)z , (t, z) ∈ [0,T ]× R0,

is Skorohod integrable with respect to N.Then we define the Skorohod integral of Y with respect to η by∫ T

0

Y (t)δη(t) :=

∫ T

0

∫R0

Y (t)zN(δt, dz).

Some basic properties of the Skorohod integral

I The Skorohod integral is a linear operator

I E[δ(u)

]= 0

Theorem: Skorohod integral as extension of the Ito integral

(a) Let X (t, z), t ∈ [0,T ], z ∈ R0, be a predictable process such that

E[ ∫ T

0

∫R0

X 2(t, z)ν(dz)dt]<∞. Then X is both Ito and Skorohod

integrable with respect to N and∫ T

0

∫R0

X (t, z)N(δt, dz) =

∫ T

0

∫R0

X (t, z)N(dt, dz).

(b) Let Y (t), t ∈ [0,T ], be a predictable process such that

E[ ∫ T

0Y 2(t)dt

]<∞. Then Y is both Ito and Skorohod integrable

with respect to η and∫ T

0

Y (t)δη(t) =

∫ T

0

Y (t)dη(t).

ExampleWhat is

∫ T

0η(T )δη(t)? Since

η(T ) =

∫ T

0

∫R0

zN(dt, dz) = I1(f1(t1, z1)),

where f1(t1, z1) = z1, we have∫ T

0

η(T )δη(t) =

∫ T

0

∫R0

I1(z1z)N(δt, dz) = I2(z1z2)

= 2

∫ T

0

∫R0

∫ t−2

0

∫R0

z1z2N(dt1, dz1)N(dt2, dz2)

= 2

∫ T

0

∫R0

z2η(t−2 )N(dt2, dz2) = 2

∫ T

0

η(t−2 )dη(t2)

= η2(T )−∫ T

0

∫R0

z2N(dt, dz).

Malliavin derivativeIn the case of the compensated Poisson random measure the set D1,2

consists of all FT -measurable random variables F ∈ L2(P) with chaosexpansion

F =∞∑

n=0

In(fn), fn ∈ L2((λ× ν)n),

such that

‖F‖2D1,2

:=∞∑

n=1

nn!‖fn‖2L2((λ×ν)n) <∞.

The Malliavin derivative Dt,z F of F ∈ D1,2 at (t, z) is defined as

(4) Dt,z F =∞∑

n=1

nIn−1(fn(·, t, z)).

Here In−1(fn(·, t, z)) means that the (n − 1)-fold iterated integral of fn is

regarded as a function of its (n − 1) first pairs of variables

(t1, z1), ..., (tn−1, zn−1), while the final pair (t, z) is kept as a parameter.

Theorem: Closability of the Malliavin derivative.Suppose F ∈ L2(P) and Fk , k = 1, 2, ..., are in D1,2 and that

(i) Fk −→ F , k →∞ in L2(P)

(ii) Dt,z Fk , k = 1, 2, ..., converges in L2(P × λ× ν).

Then F ∈ D1,2 and

Dt,z Fk −→ Dt,z F , k →∞, in L2(P × λ× ν).

Theorem: Duality formulaLet X (t, z), t ∈ [0,T ], z ∈ R0, be Skorohod integrable and F ∈ D1,2.Then

E

[F

∫ T

0

∫R0

X (t, z)N(δt, dz)

]= E

[∫ T

0

∫R0

X (t, z)Dt,z Fν(dz)dt

].

Thus the Skorohod integral is the adjoint to the Malliavin derivative.This also yields the closability of the Skorohod integral as operator.

I About this definition of the Skorohod integral, see e.g. Kabanov(1975 - Poisson type noise) inspired by Hitsuda (1972, 1979 -Brownian noise)

I About this presentation of the Malliavin derivative, see e.g. Løkka(2004)

I More details in the reference list

About the calculus.The Skorohod integral and the Malliavin derivative for the Poissonrandom measure share many of the properties that we have alreadymentioned in the case of the Brownian motion. However, there areimportant aspects of difference as it will appear clear with the chain rulehere below.

In the following we will mention only some of the most important results.Theorem: Chain rule.Let F ∈ D1,2 and let ϕ be a real continuous function on R. Ifϕ(F ), ϕ(F + Dt,z F ) ∈ L2(P), then ϕ(F ) ∈ D1,2 and

Dt,zϕ(F ) = ϕ(F + Dt,z F )− ϕ(F ).

Example.

Dt,zη2(T ) =

(η(T ) + Dt,zη(T )

)2 − η2(T )

=(η(T ) + z

)2 − η2(T ) = 2η(T )z + z2,

The following result relies on the duality formula.Theorem: Integration by partsLet X (t, z), t ∈ [0,T ], z ∈ R0, be a Skorohod integrable stochasticprocess and F ∈ D1,2 such that the product X (t, z)(F + Dt,z F ),t ∈ [0,T ], z ∈ R0, is Skorohod integrable.Then

F

∫ T

0

∫R0

X (t, z)N(δt, dz)

=

∫ T

0

∫R0

X (t, z)(F + Dt,z F

)N(δt, dz) +

∫ T

0

∫R0

X (t, z)Dt,z Fν(dz)dt.

Fundamental theorem of calculusLet X (s, ζ), (s, ζ) ∈ [0,T ]× R0, be a process such that

E[ ∫ T

0

∫R0

X 2(s, ζ)ν(dζ)ds]<∞. Assume that X (s, ζ) ∈ D1,2 for all

(s, ζ) ∈ [0,T ]× R0, and that Dt,z X (·, ·) is Skorohod integrable with

E[ ∫ T

0

∫R0

(∫ T

0

∫R0

Dt,z X (s, ζ)N(δs, dζ))2

ν(dz)dt]<∞.

Then ∫ T

0

∫R0

X (s, ζ)N(δs, dζ) ∈ D1,2

and

Dt,z

∫ T

0

∫R0

X (s, ζ)N(δs, dζ) =

∫ T

0

∫R0

Dt,z X (s, ζ)N(δs, dζ) + X (t, z).

In particular, if X (s, ζ) = Y (s)ζ, then

Dt,z

∫ T

0

Y (s)δη(s) =

∫ T

0

Dt,z Y (s)δη(s) + zY (t).

I Clark-Ocone formula. Let F ∈ D1,2. Then

(5) F = E [F ] +

∫ T

0

∫R0

E [Dt,z F |Ft ]N(dt, dz).

Observation. Suppose F ∈ D1,2 has the form F = ϕ(η(T )) for somecontinuous real function ϕ(x), x ∈ R. Then by the Clark-Oconetheorem combined with the Markov property of the process η, weget

ϕ(η(T )) = E[ϕ(η(T ))

]+

∫ T

0

∫R0

E [ϕ(y + η(T − t) + z)− ϕ(y + η(T − t))]|y=η(t)N(dt, dz).

Clark-Ocone theorem for Gaussian and Poisson type noisesRecall that we can combine the Brownian motion and the Poisson settingon:

Ω = ΩW × ΩeN , FT = FWT ⊗F

eNT , P = PW × P

eN .I Then we can also consider

Hα(ω) := Iα(W ) (fW ,α(W ) )(ωW ) · Iα(eN) (feN,α(eN) )(ωeN )

for any α = (α(W ), α(eN)) with α(k) = 0, 1, ..., for k = W , N. HereIα(k) (fk,α(k) ) is the α(k)-fold Ito integral with respect to the Brownianmotion, if k = W , or to the compensated Poisson random measure,if k = N.

I Any F ∈ L2(P) be a real FT -measurable random variable can bewritten as

F =∑α

Hα

for an appropriate choice of deterministic symmetric integrands.

I We say that F ∈ D1,2 := DW1,2 ∩ DeN

1,2 if

‖F‖2D1,2

:=∞∑

α(W )=0

α(W )α(W )!‖fW ,α(W )‖2

L2([0,T ]α(W )

)

+∞∑

α(eN)=0

α(eN)α(eN)!‖feN,α(eN)‖2

L2(([0,T ]×R0)α(eN)

)<∞.

I If F ∈ D1,2 we define the Malliavin derivative DF of F as thegradient

DF =(DtF ,Dt,z F

)where DtF is the derivative with respect to the W and Dt,z F the

one with respect to N.

Clark-Ocone theorem for combined Gaussian-pure jump Levy noise.Let F ∈ D1,2. Then

(6) F = E [F ] +

∫ T

0

E[DtF |Ft

]dW (t) +

∫ T

0

∫R0

E[Dt,z F |Ft ]N(dt, dz).

Comments:

I Leon et al. (2002) suggests a different approach to prove aClark-Ocone formula and a Malliavin derivative for pure jump Levyprocesses. The Clark-Ocone formula in this case is basicallyequivalent to the one presented here.

I The domain of the Malliavin derivatives, D1,2, may be a constraintin the applications. An extension of the Malliavin derivativesoperators to the whole L2(P) is achieved using techniques of whitenoise analysis. Ref. e.g. Aase et al. (2000), Di Nunno et al. (2004),Øksendal et al. (2004).

I A different approach to obtain an integral representation theoremfor all random variables in L2(P) and for general martingales asintegrators was introduced by Di Nunno (2002, 2007). Thisapproach introduces a non-anticipating stochastic derivative anddoes not use Malliavin calculus techniques.

3. Incomplete markets: Minimal variance hedgingLet us consider a market model of the form:

risk free asset S0(t) = 1, t ∈ [0,T ],

risky assets dS1(t) = σ(t)dW (t) +

∫R0

γ(t, z)N(dt, dz), t ∈ (0,T ],

where σ(t), and γ(t, z), t ∈ [0,T ], z ∈ R0, are predictable processeswhich might depend on S1.We denote by F = Ft , t ∈ [0,T ] the filtration generated by theBrownian motion and the Poisson random measure.

The market is assumed frictionless and with no trading constrains.In the framework of Ito stochastic calculus we remember that the set ofintegrands A is given by F-adapted stochastic processes such that

E[ ∫ T

0

ϕ2(t)(σ2(t) +

∫R0

γ2(t, z)ν(dz))dt]<∞.

The set of these processes constitutes the set A of admissible portfolios.

We can show that this is an incomplete market (e.g. F = η2(T ) is notreplicable).Then there will be an infinite choice of possible risk-neutral probabilitymeasures all equivalent from the non-arbitrage point of view.

Let us suppose to have chosen one and to work in a risk neutral

environment. Namely, P is a risk neutral probability measure and with

S0(t) ≡ 1, for all t our prices are already discounted.

Minimal variance hedging

Question: given an FT -measurable claim F ∈ L2(P), how close can we

get to F at time T by hedging?

Minimal variance hedging

Question: given an FT -measurable claim F ∈ L2(P), how close can we

get to F at time T by hedging?

Of course, if the claim F is replicable, then we can perfectly replicate.But in an incomplete market, the claim F might be not replicable andthen?

Minimal variance hedging

Question: given an FT -measurable claim F ∈ L2(P), how close can we

get to F at time T by hedging?

Of course, if the claim F is replicable, then we can perfectly replicate.But in an incomplete market, the claim F might be not replicable andthen?

If we consider closeness in term of variance, the question is then:Given F ∈ L2(FT ) find ϕ∗ ∈ A such that(7)

infϕ∈A

E[(

F−E [F ]−∫ T

0

ϕ(t)dS(t))2]

= E[(

F−E [F ]−∫ T

0

ϕ∗(t)dS(t))2]

.

Here A is the set of admissible portfolios, i.e. set of integrands.

The portfolio ϕ∗ is called a minimal variance hedging portfolio

Minimal variance hedging

Question: given an FT -measurable claim F ∈ L2(P), how close can we

get to F at time T by hedging?

Of course, if the claim F is replicable, then we can perfectly replicate.But in an incomplete market, the claim F might be not replicable andthen?

If we consider closeness in term of variance, the question is then:Given F ∈ L2(FT ) find ϕ∗ ∈ A such that(7)

infϕ∈A

E[(

F−E [F ]−∫ T

0

ϕ(t)dS(t))2]

= E[(

F−E [F ]−∫ T

0

ϕ∗(t)dS(t))2]

.

Here A is the set of admissible portfolios.

The portfolio ϕ∗ is called a minimal variance hedging portfolio

We characterize the minimal variance hedging portfolio by means of Malli-avin calculus.

Case γ ≡ 0: Brownian motion case

First of all we consider the complete market case.This is the original example/application of the Clark-Ocone theorem infinance. Ref. e.g. Ocone et al. (1991).Theorem Any claim F ∈ D1,2 = DB

1,2 admits representation:

(8) F = EF +1

E[σ2(t)

] ∫ T

0

E[DtF |Ft

]σ(t)dW (t).

[This is just a variation of the original Clark-Ocone theorem for aBrownian motion with volatility (equiv. variance) σ.]Thus we have that the minimal variance hedging strategy is given by:

(9) ϕ∗ =σ(t)

E[σ2(t)

]E[DtF |Ft

], t ∈ [0,T ].

General case

TheoremThe minimal variance hedging strategy ϕ∗ of a contingent claim F ∈ D1,2

is given by

(10) ϕ∗ =1

λ(t)·σ(t)E

[DtF |Ft

]+

∫R0

γ(t, z)E[Dt,z F |Ft

]ν(dz)

,

where

λ(t) := E[σ2(t) +

∫R0

γ2(t, z)ν(dz)].

Examples

I Suppose the dynamics of the risky security is given by:dS1(t) = dW (t) +

∫R0

zN(dt, dz). Let us consider

F :=

∫ T

0

∫R0

zN(dt, dz).

Then, since DtF ≡ 0 and Dt,z F = z = E[Dt,z F |Ft

], we get that

the minimal variance hedging strategy is:

ϕ∗(t) =1

λ

∫R0

z2ν(dz), t ∈ [0,T ],

with λ = 1 +∫

R0z2ν(dz). (Note that this process is actually

constant).

I Suppose the dynamics of the risky security is given by:dS1(t) =

∫R0

zN(dt, dz). Let us consider

F = S21 (T ) = E [F ] +

∫ T

0

2S1(t)dS1(t) +

∫ T

0

∫R0

z2N(dt, dz).

Then Dt,z F =(S1(T ) + Dt,z S1(T )

)2 − S21 (T ) = 2S1(T )z + z2 and

since S1 is an F-martingale, the minimal variance hedging strategy is

ϕ∗(t) =1∫

R0z2ν(dz)

∫R0

(2z2E

[S1(T )|Ft

]+ z3

)ν(dz)

= 2S1(t) +

∫R0

z3ν(dz)∫R0

z2ν(dz).

We can also compute the claim F which gives the minimal variancehedge of F . This is

F = E [F ] +

∫ T

0

2S1(t)dS1(t) +

∫ T

0

∫R0

z

∫R0ζ3ν(dζ)∫

R0ζ2ν(dζ)

N(dt, dz).

(Note that F 6= F , in fact F is not perfectly replicable.)

Sketch of the proofLet ϕ∗ be the minimal variance hedging strategy and let F be theminimal variance hedge:

F := E [F ] +

∫ T

0

ϕ∗(t)dS1(t)

= E [F ] +

∫ T

0

ϕ∗(t)σ(t)dW (t) +

∫ T

0

∫R0

ϕ∗(t)γ(t, z)N(dt, dz).

To prove the statements it is enough to show that

(11) E[(F − F ) · G

]= 0

for all FT -measurable G ∈ L2(P), of the form

G := E [G ] +

∫ T

0

ψ(t)dS1(t)

= E [G ] +

∫ T

0

ψ(t)σ(t)dW (t) +

∫ T

0

∫R0

ψ(t)γ(t, z)N(t, z) (ψ ∈ A).

By the Clark-Ocone formula we have

F = E [F ] +

∫ T

0

E[DtF |Ft

]dW (t) +

∫ T

0

∫R0

E[Dt,z F |Ft

]N(t, z).

Substituting this into (11) and applying some calculus, we obtain that

E[(F − F ) · G

]= E

[ ∫ T

0

ψ(t)L(t)dt]

= 0,

with

L(t) = σ(t)E[DtF |Ft

]− ϕ∗(t)σ2(t)

+

∫R0

[γ(t, z)E

[Dt,z F |Ft

]− ϕ∗(t)γ2(t, z)

]ν(dz).

This holds for all ψ ∈ A if and only if L(t) ≡ 0.Note that we can write

L(t) =σ(t)E

[DtF |Ft

]+

∫R0

γ(t, z)E[Dt,z F |Ft

]ν(dz)

− λ · ϕ∗(t),∀t.

Hence we conclude that:

ϕ∗ =1

λ

σ(t)E

[DtF |Ft

]+

∫R0

γ(t, z)E[Dt,z F |Ft

]ν(dz)

.

Clark-Ocone formula under change of measureGirsanov theorem for Levy processes. Let θ(s, x) ≤ 1, s ∈ [0,T ], x ∈ R0

and u(s), s ∈ [0,T ], be F−predictable processes such that∫ T

0

∫R0

| log(1 + θ(s, x)) | +θ2(s, x)ν(dx)dt <∞ P − a.e.,

∫ T

0

u2(s)ds <∞ P − a.e.

Let

Z (t) = exp

−∫ t

0

u(s)dW (s)−∫ t

0

u2(s)ds

+

∫ t

0

∫R0

log(1− θ(s, x)) + θ(s, x)ν(dx)ds

+

∫ t

0

∫R0

log(1− θ(s, x))N(ds, dx)

, t ∈ [0,T ].

Assume Z (t), t ∈ [0,T ], is a martingale and define dQ = Z (T )dP onFT . Set

NQ(dt, dx) = θ(t, x)ν(dx)dt + N(dt, dx)

anddWQ(t) = u(t)dt + dW (t).

Then NQ(·, ·) and WQ(·) are compensated Poisson random measure of

N(·, ·) and Brownian motion under Q, respectively.

Clark-Ocone formula under change of measure. Let F ∈ D1,2 andF ∈ L2(Q). Under appropriate integrability assumptions, the result canbe formulated and the integral representation of F with respect to WQ

and NQ is as follows:

F = EQ [F ] +

∫ T

0

EQ

[DtF − F

∫ T

t

Dtu(s)dWQ(s)|Ft

]dWQ(t)

+

∫ T

0

∫R0

EQ

[F (H − 1) + HDt,x F |Ft

]NQ(dt, dx),

where

H = exp

∫ t

0

∫R0

[Dt,xθ(s, x) + log(1− Dt,xθ(s, x)

1− θ(s, x))(1− θ(s, x))

]ν(dx) ds

+ log(1− Dt,xθ(s, x)

1− θ(s, x))NQ(ds, dx)

.

4. Minimal variance hedging under partial information

In many situations, one cannot rely on the complete knowledge of theinformation contained in

F :=Ft : t ∈ [0,T ]

.

It is often more realistic to consider the case of a dealer who has only apartial information at disposal for his financial decisions. We denote by

E = Et : t ∈ [0,T ], Et ⊆ Ft ∀t,

the flow of information available to the trader.Examples of such a situation include:

I Delayed information: Et = Ft−δ, t ≥ δ, and Et = E0, 0 ≤ t < δ

I Trivial information (no information): Et = trivial , t ∈ [0,T ]

I Price information: Et = σS1(s) : 0 ≤ s ≤ t, t ∈ [0,T ].

In this situation it is natural to think of a modified minimal variancehedging problem and study the minimal variance hedging under partialinformation:

Given F ∈ L2(FT ) find ϕ∗ ∈ AE such that

infϕ∈AE

E[(

F−E [F ]−∫ T

0

ϕ(t)dS(t))2]

= E[(

F−E [F ]−∫ T

0

ϕ∗(t)dS(t))2]

.

Here AE is defined as the set of the E-adapted processes ϕ such that

E[ ∫ T

0

ϕ2(t)(σ2(t) +

∫R0

γ2(t, z)ν(dz))dt]<∞.

AE is the set of admissible portfolios.

TheoremThe minimal variance hedging strategy ϕ∗E under partial information of acontingent claim F ∈ D1,2 is given by(12)

ϕ∗E =1

λE·

E[σ(t)E

[DtF |Ft

]+

∫R0

γ(t, z)E[Dt,z F |Ft

]ν(dz)

∣∣∣Et

],

where

λE := E[σ2(t) +

∫R0

γ(t, z)ν(dz)|Et

].

ExampleSame as before: suppose the dynamics of the risky security is given by:dS1(t) =

∫R0

zN(dt, dz). Let us consider

F = S21 (T ) = E [F ] +

∫ T

0

2S1(t)dS1(t) +

∫ T

0

∫R0

z2N(dt, dz).

The minimal variance hedging strategy under partial information is

ϕ∗E(t) =1∫

R0z2ν(dz)

∫R0

(2z2E

[S1(T )|Et

]+ z3

)ν(dz)

= 2E[S1(t)|Et

]+

∫R0

z3ν(dz)∫R0

z2ν(dz).

In this case the minimal variance hedge under partial information is

FE = E [F ] +

∫ T

0

2E[S1(t)|Et

]dS1(t) +

∫ T

0

∫R0

z

∫R0ζ3ν(dζ)∫

R0ζ2ν(dζ)

N(dt, dz).

References

This presentation follows:

G. Di Nunno, B. Øksendal and F. Proske, Malliavin Calculus for Levy Processes with Applications to Finance, inProgress, Springer 2008.

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F. E. Benth, G. Di Nunno, A. Løkka, B. Øksendal and F. Proske, Explicit representation of the minimal varianceportfolio in market driven by Levy processes, Math. Finance, 13 (2003), 54-72.

G. Di Nunno, Stochastic integral representations, stochastic derivatives and minimal variance hedging, StochasticsStochastics Rep., 73 (2002), 181-198.

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G. Di Nunno, B. Øksendal and F. Proske, White noise analysis for Levy processes, Journal of Functional Analysis,206 (2004), 109-148.

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