LOCAL RISK-MINIMIZATION UNDER A PARTIALLY OBSERVED MARKOV-MODULATED EXPONENTIAL LÉVY MODEL

Pure Mathematics No 5 ISSN 0806–2439 March 2011

LOCAL RISK-MINIMIZATION UNDER A PARTIALLY OBSERVED 1

MARKOV-MODULATED EXPONENTIAL L ´EVY MODEL 2

OLIVIER MENOUKEU-PAMEN AND ROMUALD MOMEYA 3

Abstract. In this paper, the option hedging problem for a Markov-modulated exponential L´evy model is examined. We employ the local risk-minimization approach to study optimal hedging strategies for Europeans derivatives under both full information and then partial information.

1. Introduction 4

Unpredictable structural changes in the trend of asset prices or stock indexes on financial 5

markets is a current reality nowadays. They are not usually caused by internal events of the 6

market itself but are more related to the global socioeconomic and political environment. To 7

account for these features, Markov-modulated (or regime-switching) models have since been 8

widely used in econometrics and financial mathematics. See for instance, Hamilton [24] for 9

exhibiting the non-stationarity of macroeconomic times series, Elliott and Van der Hoek [14]

10

for asset allocation, Pliska [29] and Elliott et al. [10] for short rate models, Naik [26], Guo 11

[23] and Buffington and Elliott [2] for option valuation.

12

The Markov-modulated exponential L´evy model is very attractive as alternative to the 13

classical Black-Scholes model because they couple the benefit of an exponential L´evy model 14

(notably the presence of jumps) with the possibility, thanks to the Markov chain, to having 15

long-term variability of some characteristics of the return distribution. However, in the con- 16

text of derivative pricing these models lead to incomplete markets. Therefore, the question 17

of hedging becomes a crucial one.

18

In this paper, we consider the problem of optimal quadratic hedging of an European de- 19

rivative contract in a market driven by a Markov-modulated L´evy model. Typically, in this 20

model the full information on the modulating factorX is not available in the market and the 21

agent has only access to the information contained in past asset prices. Consequently, we will 22

deal with an optimal quadratic hedging problem for a partially observed model (or partial 23

information scenario).

24

This kind of problem has been extensively studied in the literature. Di Masi, Platen 25

and Runggaldier [7] were the first to discuss the problem of risk-minimizing (mean-variance) 26

hedging under restricted information when the stock price is a martingale and the prices are 27

observed only at discrete time instants. In [33], Schweizer explicited for general filtrations 28

Date: First Version: July, 2010. This Version: April 6, 2011.

2010Mathematics Subject Classification. 60G51, 60H05, 91G10.

Key words and phrases. Local risk-minimization, partial information, L´evy process, regime-switching, hedging strategy.

Corresponding author: [email protected]; Phone: 0015145446440 . 1

G:= {G_t}_t∈[0,T] ⊆ {F_t}_t∈[0,T] := F a risk-miminizing strategy based on G-predictable pro- 29

jections. Pham [28] solved the problem of mean-variance hedging for partially observed drift 30

processes. Frey and Runggaldier [17] determined a locally risk-minimizing hedging strategy 31

when the asset price process follows a stochastic model and is observed only at discrete ran- 32

dom times. Frey [18] considered risk-minimization with incomplete information in a model 33

for high-frequency data. In the same framework but for more general model, Ceci [3] com- 34

puted the optimal hedge strategy under the criterion of risk-minimization. In all these papers, 35

the methodology consists first, to determine the optimal strategy under the full information 36

and second, determine the final solution by projecting on the filtration available in to the 37

investor. Then a natural question arises that given a Markov-modulated L´evy model, can we 38

applied the above methodology to study the problem of local risk-minimization under partial 39

information?

40

The aim of this paper is to give an answer to the previous question. In fact, we show that 41

under some restrictive conditions on our L´evy model, we can apply the same technics used 42

by the precedent authors to obtain an optimal hedging strategy for local risk-minimization 43

under partial information. In fact, we first derive a martingale representation for the wealth 44

process under full information. Then we proceed as in the classical setting by solving a local 45

risk minimization under full information. Let us mention that the optimal strategy obtained 46

under full information is quit explicit. Finally, using the fact that our processes do not jumps 47

simultaneously, we can deduce an orthogonal projection of the claim with respect to smaller 48

filtration and therefore the optimal strategy.

49

The paper is organized as follows. Section 2 describe in details our model setup and build 50

two different filtrations that characterized the situation where investor have full or partial 51

information. In Section 3, we recall some basic notions and results on risk-minimization.

52

Section 4 contains the main results, namely the martingale representation property for the 53

value process and the existence of optimal strategies in our market model under full and 54

partial information.

55

2. The model 56

2.1. Framework.

57 58

We consider a financial market with two primary securities, namely a money market account 59

B and a stock S which are traded continuously over the time horizon T := [0, T], where 60

T ∈(0,∞), is fixed and represents the maturity time for all economic activities. To formalize 61

this market, we fix a (complete) filtered probability space (Ω,F,F = (F_t)t∈T,P) satisfying 62

the usual conditions. We suppose also thatF_T =F and thatF₀ contains only the null sets of 63

F and their complements. All processes are defined on the stochastic basis above. Further, 64

we will add to this setup a filtration which specifies the flow of informations available for the 65

investors.

66

Let X := {X_t:t∈ T } an irreducible homogeneous continuous-time Markov chain with a 67

finite state space S = {e₁,e₂, . . . ,e_M} ⊂ R^M characterized by a rate (or intensity) matrix 68

A:={a_ij : 1≤i, j≤M}. Following Dufour and Elliott [8], we can identifyS with the basis 69

set of the linear space R^M. From now, we set e_i = (0,0, . . . , 1

|{z}

i−th

, . . . ,0). It follows from 70

Elliott [11] thatX admits the following semimartingale representation 71

X_t=X₀+ Z t

0

AX_s+ Γ_t, (2.1)

where Γ :={(Γⁱ_t)^M_i=1 :t ∈[0, T]} is a vector-martingale in R^M with respect to the filtration 72

generated byX.

73

Let r_t denote the instantaneous interest rate of the money market accountB at timet. If 74

we suppose that rt := r(t, Xt) = hr|X_ti, where h·|·i is the usual scalar product in R^M and 75

r = (r1, r2, . . . , rM)∈R+M, then the price dynamics ofB is given by:

76

dBt=rtBtdt, B(0) = 1 for t∈ T. (2.2) The appreciation rateµt and the volatilityσtof the stock S at time time tare defined by 77

µt :=µ(t, Xt) = hµ|X_ti,

σt :=σ(t, Xt) = hσ|X_ti, t∈ T (2.3) whereµ= (µ₁, µ₂, . . . , µ_M)∈R^M and σ = (σ₁, σ₂, . . . , σ_M)∈R+M.

78

The stock price process S is described by this following Markov modulated L´evy process:

79

dS_t=S_t⁻

µ_tdt+σ_tdW_t+ Z

R\{0}

(e^z−1)Ne^X(dt;dz)

, S(0) =S₀ >0 (2.4) Here W := (W_t)_t∈T is a one-dimensional standard Brownian motion or Wiener process on 80

(Ω,F,P), independent ofX andN^X, 81

Ne^X(dt, dz) :=

N^X(dt, dz)−ρ^X(dz)dt if|z|<1

N^X(dt, dz) if|z| ≥1, (2.5)

withN^X(dt, dz) is the differential form of a Markov-modulated random measure onT ×R\{0}.

82

We recall from Elliott and Osakwe [12] and Elliott and Royal [13] that a Markov-modulated 83

random measure onT ×R\{0} is a family{N^X(dt, dz;ω) :ω∈Ω} of non-negative measures 84

on the measurable space (T ×R\{0},B(T)⊗B(R\{0})), which satisfiesN^X({0},R\{0};ω) = 0 85

and has the following compensator, or dual predictable projection 86

ρ^X(dz)dt:=

M

X

i=1

hX_t−|e_iiρ_i(dz)dt, (2.6) where ρ_i(dz) is the density for the jump size when the Markov chain X is in state e_i and 87

satisfying 88

Z

|z|≥1

(e^z−1)²ρ_i(dz)<∞. (2.7) The general setting considered here can be seen as an extension of the exponential-L´evy model 89

described in Cont and Tankov [6] where a factor of modulation is introduced. Hence, we can 90

retrieve in a simple way most of some current models which exist in the literature as for 91

example the classical Black-Scholes model and the family of exponential-L´evy models.

92

The subsequent assumption will be fundamental for obtaining our results, particularly in 93

Section 4.1 to obtain a martingale representation for the value process.

94

Assumption 2.1. We assume that a transition of Markov chainX from stateej to stateek

95

and a jump ofS do not happen simultaneously almost surely.

96

Let ξ:={ξ_t}t∈T denoting the discounted stock price. Then, 97

ξt:= St

B_t =e⁻

Rt

0ruduSt.

IfR_t=e^R0trudu, for each t∈ T. Then, the discounted stock price process is given by : 98

dξt=Fµ(t, ξ_t⁻, Xt)dt+Fσ(t, ξ_t⁻, Xt)dWt+ Z

R\{0}

Fγ(t, ξ_t⁻, Xt)Ne^X(dt;dz),

ξ(0) =S₀>0P a.s, (2.8)

or the following integral decomposition 99

ξt=S0+ Z t

0

Fµ(s, ξ_s⁻, Xs)ds

| {z }

finite variation part

+ Z t

0

Fσ(s, ξ_s⁻, Xs)dWs+ Z t

0

Z

R\{0}

Fγ(s, ξ_s⁻, Xs)Ne^X(ds;dz)

| {z }

local-martingale part

,

(2.9) where

100







F_µ(t, ξ_t, X_t) :=

µ(t, R_tξ_t, X_t)−r(t, R_tξ_t, X_t) ξ_t F_σ(t, ξ_t, X_t) := σ(t, R_tξ_t, X_t)ξ_t

Fγ(t, ξt, Xt) := ξt(e^z−1),

(2.10) The theory of stochastic flows will also be used to identify the integrands in the stochastic 101

integrals involved in the martingale representation property in Section 4.1. Let now consider 102

a general form of stochastic differential equation (SDE) (2.8):

103

(

dξt =Fµ(t, ξ_t⁻, Xt)dt+Fσ(t, ξ_t⁻, Xt)dWt+R

R\{0}Fγ(t, ξ_t⁻, Xt)Ne^X(dt;dz),

ξ_s =x >0 P a.s. f or 0≤s < t≤T. (2.11)

We assume that the coefficients Fµ, Fσ, Fγ are smooth enough to guaranty the existence 104

and uniqueness of a strong adapted c`adl`ag solution ξ_{s, t}(x) (see Fujiwara and Kunita [21]).

105

Furthermore, this solution forms a stochastic flow of diffeomorphisms Φs, t : (0,+∞)×Ω→ 106

(0,+∞) given by 107

Φ_{s, t}(x, ω) =ξ_{s, t}(x)(ω), (2.12)

for each (s, t) such that 0 ≤ s < t ≤ T, x ∈ (0,+∞) and ω ∈ Ω. (Φs, t)s<t verifies the 108

following properties:

109

• Φs, t = Φ0, t◦Φ⁻¹_{0, s} for all s < t;

110

• Cocycle property : Φ_{s, u} = Φ_{t, u}◦Φ_{s, t} for alls < t < u;

111

• Conditional independent increments: for t0 ≤ t1 ≤ . . . ≤ tn, 112

Φ_{t0, t1},Φ_{t1, t2}, . . . ,Φ_{tn−1, tn} are conditionally independent givenF_T^X. 113

Letx=ξ_{0, t}(x₀), for eacht∈[0, T]. By the uniqueness of solutions of SDE and the semi-group 114

property, we get 115

ξ0, T(x0) =ξt, T(ξ0, t(x0)) =ξt, T(x). (2.13) Differentiating (2.13) with respect tox0, we obtain:

116

∂ξ_0,T(x₀)

∂x0

= ∂ξ_{t, T}(x)

∂x

∂ξ_{0, t}(x₀)

∂x0

. (2.14)

2.2. Market information.

117 118

In general, the Markov-modulated L´evy model as described by Equation (2.4) is based on 119

the mathematical framework of the Markov additive processes (MAP). This last object is 120

an old and widely studied subject in stochastic analysis (see, e.g, [4, 5, 16, 22] for a few.) 121

In particular, the couple (X, S) is a Markov additive process and yields to two important 122

filtrations as we will see below.

123

Let F^X := {F_t^X}t∈T and F^S :={F_t^S}t∈T denote the right-continuous, P−complete filtra- 124

tions generated byX etS respectively. We define fort∈ T, 125

G_t:=F_t^S (2.15)

and 126

G_t:=F_T^X ∨ F_t^S. (2.16)

The filtration G := {G_t}_t∈T represents all the information up to time t gained from the 127

observations of the price fluctuationsS. The strict larger filtrationG:={G_t}_t∈T denotes the 128

information about the stock price history up to time t and the information about the entire 129

path F_T^X of the modulation factor processX.

130

We will assume in the last section of is paper that the investors in the market only have 131

access to the first filtration which is thus the one used practically whereas the last serves 132

mainly theoretical purposes.

133

2.3. Esscher transform change of measure.

134 135

One of the main features of the Markov-modulated L´evy model is that it leads to an 136

incomplete market. We shall therefore employ the regime-switching Esscher transform as in 137

Elliott et al. [11] to determine an equivalent martingale measure.

138

For doing so, we define the process Y by 139

Yt= Z t

0

µr−1

2σ_r²

dr+ Z t

0

σrdWr+ Z t

0

Z

R\{0}

zNe^X(dr;dz)− Z t

0

Z

R\{0}

(e^z−1−z)ρ^X(dz)dr (2.17) As in [35], let consider the following set

140

Θ :=

(

(θt)t∈T |θt:=

N

X

i=1

θihX_t−|e_iiwith (θ1, θ2, ..., θN)∈R^Nsuch thatE^P h

e⁻

Rt 0θrdYr

F_T^Xi

<∞ )

.

Forθ:= (θ_t)t∈T ∈Θ, the generalized Laplace transform of a G-adapted processY is defined 141

as 142

M_Y(θ)t:=E^P h

e⁻

Rt 0θrdYr

F_T^Xi

. (2.18)

Notice that contrary to the usual Esscher transform, the expectation involved here is taken 143

conditionally on the information of all the future of the Markov chainX. With this extended 144

definition of a Laplace transform, we can now define the generalized Esscher transform (with 145

respect to the parameterθ calledEsscher parameter).

146

Let Λ^θ ={Λ^θ_t}t∈T denote aG-adapted stochastic process defined as 147

Λ^θ_t := e^−R0tθrdYr

M_Y(θ)_t , t∈ T; θ∈Θ. (2.19)

It can be shown that (see for example, [11]) 148

Λ^θ_t = exp

"

− Z t

0

θ_rσ_rdW_r−1 2

Z t 0

θ²_rσ_r²dr− Z t

0

Z

R\{0}

θ_r⁻zNe^X(dr;dz)

− Z t

0

Z

R\{0}

e^−zθr −1 +θrz

ρ^X(dz)dr

#

. (2.20)

Moreover, as proven in [35], the stochastic process Λ^θ ={Λ^θ_t}_t∈T defined by (2.19) is a positive 149

(G,P)-martingale and 150

E^P[Λ^θ_t] = 1, ∀t∈ T. (2.21)

From Equation 2.21, we deduce that the process Λ^θ ={Λ^θ_t}_t∈T given by Equation (2.20) is 151

a density process inducing a change of measure in the probability space (Ω,G_T). Indeed, by 152

setting 153

dQ^θ dP

_G

t

= Λ^θ_t t∈ T, (2.22)

we define for each processθin Θ a new probability measureQ^θ equivalent toP. Actually,Q^θ 154

is just an equivalent probability measure, to transform it into a martingale equivalent measure 155

we need to impose some conditions generally known as martingale condition. It stipulates 156

that the discounted stock price{ξ_t}_t∈T would be a G-martingale underQ^θ. Then, 157

E^Qθ h

ξt|G₀i

=ξ(0), ∀t∈ T. (2.23)

Hence, we have 158

Proposition 2.2. An equivalent probability measureQ^θ defined through (2.22) is an equiva- 159

lent martingale measure on(Ω,G_T), i.e. it satisfies condition (2.23), if and only if the process 160

θ satisfies the following equation 161

µt−rt−θtσ²_t + Z

R\{0}

(e^z−1)(e^−zθt−1)ρ^X(dz) = 0, ∀t∈ T. (2.24) Proof. The proof is a straightforward adaptation of that of Proposition 2.2 in Elliott et al.

162

[11]. The main ingredient is an explicit computation of the generalized Laplace transform 163

defined by (2.18).

164

However, the process θis completely determined by the vector (θ1, θ2, . . . , θM) solution of 165

the system of equations 166

µ_i−r_i−θ_iσ_i²+ Z

R

(e^z−1)(e^−zθi−1)ρ_i(z)dz= 0, (2.25) fori= 1,2, . . . , N.

167 168

For pricing purposes, we need to know the dynamics of the discounted stock price under the 169

martingale probability measureQ^θ.The following proposition states a result in this direction.

170

Proposition 2.3. Under risk-neutral probability measure Q^θ, the discounted stock price pro- 171

cessξ is solution to the following stochastic differential equation 172

(

dξt =Fσ(t, ξ_t⁻, Xt)dW_t^θ+R

R\{0}Fγ(t, ξ_t⁻, Xt)Ne^θ(dt;dz)

ξ(0) =S₀ >0 P-a.s. f or 0≤t≤T, (2.26)

where 173

• W^θ defined by 174

W_t^θ :=Wt+ Z t

0

θrσrdr, (2.27)

is the standard Brownian motion under Q^θ; 175

• Ne^θ defined by 176

Ne^θ(dr;dz) =N^X(dr;dz)−ρ^θX(dz)dr, (2.28) is the compensated measure ofN^X under Q^θ withρ^θX(dz) :=e^−θzρ^X(dz).

177 178

Proof. This follows easily from Equation (2.8) by the application of Girsanov-Meyer Theorem 179

(See Øksendal and Sulem [27], Protter [30]).

180

3. The locally risk-minimizing hedging problem 181

In this section, we recall some terminology on local risk minimization. We shall simply give 182

necessary results; for further informations, the reader is referred to the survey of Schweizer 183

[34] from which our presentation owes much.

184

3.1. Review of some notions on the risk-minimization approach.

185 186

This concept has been introduced by F¨ollmer and Sondermann [20] for nonredundant (or non-attainable) contingent claim written on a one-dimensional, square-integrable discounted risky assetξwhich is a martingale under the original measureP. Concretely, given a stochastic basis as above the goal consist to minimize the conditional remaining risk : R_t:=E^P[(C_T − C_t)²|F_t] for all t ∈ T. Here C_t stands for the cost process and is defined as the difference between the value of the (portfolio) strategy detained by the investor at timetand the gains made from trading in the financial market up to timet. Let L²(ξ) the space of all R-valued predictable processφsuch that

||φ||_L2(ξ):=

E^P hZ T

0

φ²_ud[ξ, ξ]_ui¹₂

<∞,

A trading strategy is a pair of processes ϕ = (φ, ψ) where ψ is an adapted process and 187

φ ∈ L²(ξ) is a F-predictable process, such that the value process V := φξ +ψ has right 188

continuous sample paths andE^P[V_t²]<∞for every t∈ T(i.eVt∈ L²(Ω,P) for everyt∈ T).

189

For a trading strategy ϕ = (φ, ψ), where φ = (φ_t)t∈T denotes at time t, the number of 190

stocks held andψ= (ψt)t∈T the amount invested in the money market account.

191

LetHbe a claim which isF_T-measurable and square-integrable. Consider a strategies that replicate the contingent claimH at timeT; that is the strategies with the assumption

V_T =H P-a.s.

Such strategies are calledH-admissible.

192

A trading strategy ϕ such that Ct(ϕ) = C0(ϕ) for all t ∈ T is called self-financing. Fur- 193

thermore, if the cost processC_t(ϕ) is aP-martingale thenϕis said to bemean self-financing.

194

Definition 3.1. Let (φ, ψ) and (φ,e ψ)e be H-admissible strategies. Then (φ,e ψ)e is called a 195

H-admissible strategy continuation of (φ, ψ) at time t ∈ [0, T) if φes = φs for s ∈ [0, t] and 196

ψes=ψs for s∈[0, t).

197

The following result obtained by F¨ollmer and Sondermann [20] is based on the Galtchouk- 198

Kunita-Watanabe (GKW) decomposition (see Kunita-Watanabe [25]) ofH and gives a risk- 199

minimizing hedging strategy under full information.

200

Theorem 3.2. Assume the GKW decomposition of the claim H∈ L²(Ω,P) given by H =H₀+

Z _T

0

φ^H_s dξ_s+L^H_T,

withφ^H ∈ L²(ξ),L^H a square-integrableP-martingale orthogonal toξwithH₀ =E^P[H]P-a.s.

201

Then, the trading strategyϕ^⊗= (φ^⊗, ψ^⊗) defined by 202

(φ^⊗_t , ψ^⊗_t ) := (φ^H_t , H₀+ Z t

0

φ^H_s dξ_s−φ^H_t ξ_t+L^H_t ), ∀t∈[0, T] (3.1) isH-admissible and risk-minimizing. Its associated risk processR^⊗ is given by

203

R^⊗_t =E^P[(L^H_T −L^H_t )²|F_t], P−a.s. ∀t∈[0, T]. (3.2) Furthermore, this strategy is unique.

204

From now on, we assume that the one-dimensional discounted asset ξ is no longer a mar- 205

tingale under the measurePbut only a semimartingale with the following decomposition 206

ξ=ξ₀+Z+A (3.3)

whereZa square-integrable martingale for whichZ0 = 0, andAa predictable process of finite 207

variation |A|(i.e sup_τPNτ

i=1|A_ti−Ati−1|<∞) for every partitionτ ofT. In this situation, we 208

cannot longer apply the preceding result of F¨ollmer and Sondermann [20]. To deal with such 209

a case, Schweizer [33, 34] introduced the concept of locally risk-minimizing strategy where 210

the conditional variances are kept as small as possible but now in a local manner. Now, 211

to adapt the definition of a trading strategy in this case we need that φ ∈ L²(Z) and that 212

RT

0 |φ_udAu| ∈ L²(Ω,P).

213

Definition 3.3. (small perturbation). A trading strategy ∆ = (δ, ) is called a small pertur- 214

bation if it satisfies the following conditions:

215

• δ is bounded;

216

• RT

0 |δ_u||dA_u| is bounded;

217

• δT =T = 0.

218

For any subinterval(s, T]⊂[0, T], we define the small perturbation∆

_(s,T] := (δ1_(s,T], 1_(s,T]).

219

Now we can define 220

Definition 3.4. (locally risk-minimizing strategy). For a trading strategy ϕ, a small pertur- 221

bation∆ and a partition τ of [0, T] the risk-quotient (R-quotient) r^τ[ϕ,∆] which is a sort of 222

relative local risk is defined as 223

r^τ[ϕ,∆] := X

ti,ti+1∈τ

R_ti(ϕ+ ∆ _(t

i,ti+1])− R_ti(ϕ)

E^P[hZi_ti+1− hZi_ti|F_ti] 1_(ti,ti+1]. (3.4) A trading strategy ϕ is called locally risk-minimizing if

lim inf

n→∞r^τn[ϕ,∆]≥0, P× hZi −a.s.

for every small perturbation∆and every increasing sequence (τ_n) of partitions ofT such that 224

||τ_n|| →0.

225

To present the main results, we need the following technical assumptions:

226

Assumption 3.5.

227 228

• (A1)ForP-almost allωthe measure on[0, T]induced byhZi(ω)has the whole interval 229

[0, T]as its support, i.ehZi should be P-almost surely strictly increasing on the whole 230

interval [0, T].

231

• (A2) A is continuous.

232

• (A3) A is absolutely continuous with respect to hZi with a density α satisfying

E^P hZ T

0

|α_u|max(log|α_u|,0)dhZi_ui

<∞.

A sufficient condition for (A3) is that E^P hRT

0 |α_u|²dhZi_ui

<∞ and one refers to that by 233

saying: ξ satisfiesthe Structure Condition (SC). We can remark that with assumption(A2), 234

ξ is aspecial semimartingale. We can now state the optimality result.

235

Theorem 3.6. A contingent claim H ∈ L²(Ω,P) admits a (pseudo-optimal) locally risk- 236

minimizing strategy ϕ = (φ, ψ) with VT(ϕ) =H P a.s. if and only if H can be written 237

as 238

H =H₀+ Z T

0

φ^H_s dξ_s+L^H_T P a.s. (3.5)

with H0 ∈ L²(Ω,P), φ^H ∈ L²(ξ), L^H a square-integrable P-martingale null at the origin and 239

P-strongly orthogonal toM. The strategy ϕ is then given by 240

φ_t =φ^H_t , t∈[0, T] and

241

C_t(ϕ) =H₀+L^H_t , t∈[0, T];

its value process is 242

V_t(ϕ) =C_t(ϕ) + Z t

0

φ_sdξ_s=H₀+ Z t

0

φ^H_s dξ_s+L^H_t , t∈[0, T]. (3.6)

Proof. See Proposition 3.4 of Schweizer [34].

243

Equation (3.5) is called F¨ollmer-Schweizer decomposition (FS) for the contingent claim 244

H. In practice, to obtain this decomposition is very difficult so the more natural approach 245

introduced by F¨ollmer and Schweizer [19] consist to use a Girsanov transformaton to shift 246

the problem back to a martingale measure where standard techniques as Galchouk-Kunita- 247

Watanabe projection is available.

248

4. Main results 249

4.1. A martingale representation property.

250 251

In this section, we give an explicit representation of a martingale which is useful for the 252

problem of hedging in the context of a Markov-modulated L´evy model. The proof of the result 253

is similar to the one given by Elliottet al. [15]. We give an explicit martingale representation 254

of the wealth process which will be useful later on in the finding of an optimal strategy the 255

proof of our main result.

256

First, it is easy to see that the Esscher transform change of measure Λ^θ introduced in 257

Section 2.3 is solution to this following SDE 258







Λ_{t, u}(x) = 1 +Ru

t Λ_{t, r}⁻(x)(−θ_rσ_r)(r, ξ_{t, r}⁻(x), X_r)dW_r +Ru

t

R

R\{0}Λ_{t, r}⁻(x)(e^{−zθr(r,ξt, r−(x),Xr)}−1)Ne^X(dr;dz) Λt, t(x) = 1 P a.s. f or 0≤t < u≤T.

(4.1)

Indeed, for allt∈[0, T], Λ^θ_t = Λ_{0, t}(x).

259 260

Now, consider a function c(·) : (0,+∞) →R such that c(·) is twice differentiable and c(·) 261

and ^∂c(·)_∂x are at most linear growth in x. We shall determine the current price at time tof a 262

contingent claim of the formc(S_T), which is the payoff of the claim at maturityT > t. In the 263

sequel, we have to work with the discounted claim as function of the discounted stock price, 264

that is:

265

ˆ

c(ξ0,T) :=R⁻¹_T c(RTξ0,T(x0)) =R⁻¹_T c(ST). (4.2) So, we assume that the process θ is chosen such that E^Q

θ[ˆc²(ξ0,T(x0))] < ∞ and then we 266

define the square-integrable (G,Q^θ)-martingale{V_t}_t∈[0,T] as:

267

V_t:=E^Q

θ[ˆc(ξ_0,T(x₀))|G_t], t∈[0, T]. (4.3) As (X, ξ) and (X,Λ) are Markov additive processes (See C¸ inlar [4]) we have that they verify 268

the Markov property with respect to the large filtration G. Hence, we obtain by using 269

Bayes’rule 270

Vt:=E^Q

θ[ˆc(ξ_{0, T}(x0))|G_t]

= E^P[Λ_{0, T}(x₀)ˆc(ξ_{0, T}(x₀))|G_t] E^P[Λ0, T(x0)|G_t]

=E^P

hΛ_{0, t}(x₀)Λ_{t, T}(x)ˆc(ξ_{t, T}(x)) Λ0, t(x0)

G_ti

, because E^P[Λt, T(x)|G_t] = 1;

=E^P[Λ_{t, T}(x)ˆc(ξ_{t, T}(x))|G_t]

=E^P[Λt, T(x)ˆc(ξt, T(x))|X_t=e, ξ0, t(x0) =x]. (4.4) Thus, we define for each x∈(0,+∞) and e∈S,

271

V(t, x,e) :=E^P[Λ_{t, T}(x)ˆc(ξ_{t, T}(x))|X_t=e, ξ0, t(x0) =x] (4.5) (=E^Q

θ[ˆc(ξ_{t, T}(x))|X_t=e, ξ_{0, t}(x₀) =x]).

For each (t, u) such that 0≤t < u≤T, let introduce the following processes:

272

(1) L defined by 273

Lt, u:=

Z u t

∂(−θ_rσr)

∂ξ (r, ξt, r(x), Xr)×∂ξt, r

∂x dW_r^θ +

Z u t

Z

R\{0}

h

e^{zθr(r, ξt, r−(x), Xr)}∂e^{−zθr(r, ξt, r−(x), Xr)}

∂ξ × ∂ξ_{t, r}⁻

∂x (x) i

Ne^θ(dr, dz), (2) K defined by

274

K_{t, u} :=

Z u t

Λ_{t, r}(x+ζ_t(y)) Λt, r(x)

h(−θ_rσ_r)(r, ξ_{t, r}(x+ζ_t(y)), X_r) + (θ_rσ_r)(r, ξ_{t, r}(x), X_r)i dW_r^θ

+ Z u

t

Z

R\{0}

Λ_{t, r}⁻(x+ζ_t(y)) Λ_{t, r}⁻(x)

he^{−zθr(r, ξt, r−(x+ζt(y)), Xr)}−e^{−zθr(r, ξt, r−(x), Xr)} e^{−zθr(r, ξt, r−(x), Xr)}

i

Ne^θ(dr, dz) withξ_t⁻ =x, ζt(y) :=ζ(t, x, y),

275

(3) V the vector process defined by 276

V(t, ξ_{0, t}(x₀)) :=

V(t, ξ_{0, t}(x₀),e₁), V(t, ξ_{0, t}(x₀),e₂), . . . , V(t, ξ_{0, t}(x₀),e_M) . Now, we are able to give an martingale representation for the {V_t}_t∈T.

277

Proposition 4.1. The(G,Q^θ)-martingale {V_t}_t∈T has the representation 278

Vt=V0+ Z t

0

φ^c_r(ξr, Xr)dW_r^θ+ Z t

0

Z

R\{0}

φ^d_r(z, ξ_r⁻, X_r⁻)Ne^θ(dr, dz) + Z t

0

hα_r, dΓri, (4.6) where φ^c, φ^d and α are such that,

279 E^Q

θh RT

0 (Φ^c_r)²dri

< ∞, E^Q

θh RT

0 ||α_r||²dri

< ∞ and E^Q

θh RT

0

R

R\{0}(φ^d_r(z))²ρ^X(dz)dri

<∞, 280

with the following explicit expressions 281

φ^c_r(ξ_r, X_r) =E^Q

θh

L_{r, T}ˆc(ξ_{r, T}(x)) + ∂ˆc

∂ξ(ξ_{r, T}(x))∂ξ_{r, T}

∂x (x)

X_r=e, ξ_{0, r}(x₀) =xi

σ_r(r, ξ_r, X_r);

(4.7) φ^d_r(y, ξ_r⁻, X_r) =E^Q

θh

(K_{r, T} + 1)ˆc(ξ_{r, T}(x−+ζ_r(z)))−ˆc(ξ_{r, T}(x))

X_r=e, ξ_{0, r}(x₀) =xi

; (4.8)

α_t=V(t, ξ_{0, t}(x₀))∈R^M. (4.9)

withx=ξ_{0, r}(x₀) and x−=ξ_{0, r}⁻(x₀).

282

In order to prove Proposition 4.1, we need the subsequent result 283

Lemma 4.2. The following identities hold 284

∂Λ_{t, T}

∂x (x) = Λt, T(x)×Lt, T (4.10)

and 285

Λ_{t, T}(x+ζ(z))−Λ(x) = Λ_{t, T}(x)×K_{t, T}. (4.11)

Proof. See Appendix.

286

Now, we give the proof of the Proposition 4.1.

287

Proof. (Proposition 4.1) 288

289

Noting that 290

V(t, ξ_t, X_t) =hV(t, ξ_t)|X_ti, (4.12) we obtain dy differentiation

291

dV(t, ξt, Xt) =hdV(t, ξ_t)|X_ti+hV(t, ξ_t)|dX_ti, (4.13) and from Itˆo differentiation rule

292

dV(t, ξ_t, X_t) =

V(t, ξ_t)

dX_t

+ ∂V

∂tdt+ ∂V

∂ξdξ_t+1 2

∂²V

∂ξ² d[ξ, ξ]^c_t (4.14) +

Z

R\{0}

h

V(t, ξ_t⁻e^z)−V(t, ξ_t⁻)−∆ξt

∂V

∂ξ i

N^X(dt, dz)

Xt

From (3.3), we deduce that 293

dXt=AX_t⁻dt+dΓt. (4.15)

By replacing this last expression in (4.14), we obtain 294

dV(t, ξ_t, X_t)

=

*"

∂V

∂t + 1 2σ_t²ξ_t²−

∂²V

∂ξ² + Z

R\{0}

h

V(t, ξ_t⁻e^z)−V(t, ξ_t⁻)−ξ_t⁻(e^z−1)∂V

∂ξ i

ρ^θX(dz)

# dt

Xt

+

+

*

V(t, ξ_t)

AX_t⁻ +

dt+

*

V(t, ξ_t)

dΓ_t +

+

*

σtξ_t⁻∂V

∂ξ dW_t^θ+ Z

R\{0}

h

V(t, ξ_t⁻e^z)−V(t, ξ_t⁻)

iN˜^θ(dt, dz)

Xt

+

(4.16)

As{V_t=V(t, ξt, Xt)}t∈T is a (G,Q^θ)-martingale, his continuous finite variation part would 295

be identically equal to zeroQ^θ a.s, thus 296

*∂V

∂t +1

2σ²_tξ²_t∂²V

∂ξ² + Z

R\{0}

h

V(t, ξ_t⁻e^z)−V(t, ξ_t⁻)−ξ_t⁻(e^z−1)∂V

∂ξ i

ρ^θX(dz)

X_t +

+

*

V(t, ξ_t)

AX_t⁻ +

= 0 (4.17)

which is equivalent with X_t=e to:

297

∂V

∂t(t, ξt,e) +1

2σ²_tξ²_t∂²V

∂ξ² (t, ξt,e) + D

V(t, ξt) AX_t⁻

E

+ Z

R\{0}

h

V(t, ξ_t⁻e^z,e)−V(t, ξ_t⁻,e)−ξ_t⁻(e^z−1)∂V

∂ξ(t, ξ_t,e)i

ρ^θX(dz) = 0. (4.18) Hence, back to Equation (4.16), we deduce that

298

V(t, ξt,e) =V(0, ξ0, X0) + Z t

0

σsξs

∂V

∂ξ(s, ξs, Xs)dW_s^θ +

Z t 0

Z

R\{0}

h

V(s, ξ_s⁻e^z, Xs)−V(s, ξ_s⁻, Xs)

iN˜^θ(ds, dz) + Z t

0

D

V(s, ξs) dΓs

E . (4.19) We deduce from the uniqueness of the decomposition of the special semimartingale V that 299

• Φ^c_t(ξ_t) =σ_tξ_t^∂V_∂ξ(t, ξ_t,e);

300

• Φ^d_t(z, ξ_t⁻) =V(t, ξ_t⁻e^z,e)−V(t, ξ_t,e);

301

• αt=V(t, ξt).

302

To obtain a more explicit expressions for these quantities, we write by noting that ξ0, t =x 303

and ξ_{0, t}⁻ =x−

304

Φ^c_t(ξt) = xσt(t, x,e)∂V

∂x(t, x,e)

= xσt(t, x,e) ∂

∂xE^P[Λt, T(x)ˆc(ξt, T(x))|X_t=e, ξ0, t(x0) =x] by (4.5)

= xσt(t, x,e)E^P

h∂Λt, T

∂x (x)ˆc(ξ_{t, T}(x)) + Λ_{t, T}(x)∂ˆc

∂ξ(ξ_{t, T}(x))∂ξt, T

∂x (x)

Xt=e, ξ0, t(x0) =x i

= xσt(t, x,e)E^P h

Λt, T(x)Lt, Tˆc(ξt, T(x)) + Λ_{t, T}(x)∂ˆc

∂ξ(ξ_{t, T}(x))∂ξt, T

∂x (x)

X_t=e, ξ_{0, t}(x₀) =xi

by Lemma 4.2

= xσ_t(t, x,e)E^Q

θh

L_{t, T}c(ξˆ _{t, T}(x)) + ∂ˆc

∂ξ(ξ_{t, T}(x))∂ξ_{t, T}

∂x (x)

X_t=e, ξ_{0, t}(x₀) =xi

. (4.20) In the same way,

305

Φ^d_t(z, ξ_t⁻) = V(t, ξ_t⁻e^z,e)−V(t, ξ_t⁻,e)

= E^P h

Λ_{t, T}(x−+ζ_r(z))ˆc(ξ_{t, T}(x−+ζ_r(z)))

X_t=e, ξ_{0, t}(x₀) =xi

− E^P h

Λ_{t, T}(x)ˆc(ξ_{t, T}(x))

X_t=e, ξ_{0, t}(x₀) =xi

= E^P h

Λ_{t, T}(x−+ζ_r(z))−Λ_{t, T}(x) ˆ

c(ξ_{t, T}(x−+ζ_r(z)))

X_t=e, ξ_{0, t}(x₀) =xi + E^P

h

Λ_{t, T}(x)

ˆ

c(ξ_{t, T}(x−+ζr(z)))−ˆc(ξ_{t, T}(x))

Xt=e, ξ0, t(x0) =x i

= E^P h

Λt, T(x)Kt,T

ˆ

c(ξt, T(x−+ζr(z))

+ Λt, T(x)

ˆ

c(ξt, T(x−+ζr(z)))−c(ξˆ t, T(x))

Xt=e, ξ0, t(x0) =x i

by Lemma 4.2

= E^Q

θh

(K_t,T+ 1)ˆc(ξ_{t, T}(x−+ζ_r(z)))−c(ξˆ _{t, T}(x))

X_t=e, ξ_{0, t}(x₀) =xi

. (4.21) Finally, we have to show that the different component involved in (4.19) are mutually or- 306

thogonal (G,Q^θ)-local martingale, that is, the different product W^θ·N˜^θ(·, dz), W^θ ·Γ and 307

Γ·N˜^θ(·, dz) are (G,Q^θ)-local martingale. The claim is easy verified for the first ones by noting 308

thatW^θis an continuous (G,Q^θ) local-martingale such thatW₀^θ = 0 whereas ˜N^θ(·, dz) and Γ 309

are pure jump (G,Q^θ) local-martingales. For the last, we have∀t∈ T and ∀i∈ {1,2, ..., M}

310

[Γⁱ,N˜^θ(·, dz)]_t = X

0≤s≤t

∆Γⁱ_s∆ ˜N^θ(s, dz)

= 0. (4.22)

This result comes from Assumption 2.1 and the decomposition theorem of the (additive) 311

component of the MAP (X, S) given in C¸ inlar [5], theorem 2.23.

312

4.2. The locally risk-minimizing hedging Problem under full information for the 313

model (2.4)-(2.2).

314 315

In this section, we consider the problem of hedging a contingent claim H in the Markov- 316

modulated exponential L´evy model given by (2.2)-(2.4) given that the information set is G.

317

In general, in such a market the claim H cannot be perfectly hedged. Therefore, we need 318

to take into account the market participant’s attitude toward risk in the search of the viable 319

market transactions. One way of doing this in the literature consists to optimize a given 320

criterion based or not on the preference of the market participant. In particular, the choice 321

of quadratic criterion is quite natural and pertinent because it leads to a linear pricing rule 322

which is very meaningful in financial economics.

323

LetB be a contingent claim with a discounted payoffH = ˆc(ξ0, T(x0))∈ L²(Ω,P). Follow- 324

ing Schweizer [32], a locally risk-minimizing strategyϕ= (φ, ψ) which generates ˆc(ξ_{0, T}(x₀)) 325

must be such that 326

(1) V_T = ˆc(ξ_{0, T}(x₀))P-a.s.;

327

(2) V_t(ϕ) =V₀(ϕ) +Rt

0φ_rdξ_r+ Υ_t, for allt∈[0, T];

328

(3) Υ is a martingale underPand Υ is orthogonal to the martingale partZ ofξ underP. 329

We shall require that (Vt(ϕ))0≤t≤T is a (G,Q^θ)-martingale. With this assumption and Equa- 330

tion (4.5), we have 331

V_t(ϕ) = E^Q

θ[V_T(ϕ)|G_t]

= E^Q

θ[ˆc(ξ0, T(x0))|X_t=e, ξ0, t=x]

= V(t, x,e).

Now we can state the main proposition if this section.

332

Proposition 4.3. Assume σ_t > 0 for all t ∈ [0, T]. If there exists a process θ^∗ satisfying 333

(2.24) and such that 334

θ_t^∗= µt−rt

σ²_t +R

R\{0}(e^x−1)²ρ^X(dx), (4.23) 335

e^−zθt∗−1 =− (µt−rt)(e^z−1) σ_t²+R

R\{0}(e^x−1)²ρ^X(dx), ∀z∈R (4.24) then there exists a minimal martingale measure defined by the Esscher transform Λ^θ∗. Fur- 336

thermore, the locally risk-minimizing strategy for the contingent claimH is given by 337

φ^∗_t = 1

ξ_t⁻ ×σ_tφ^c_t(ξ_t, X_t) +R

R\{0}(e^z−1)φ^d_t(y, ξ_t⁻, X_t⁻)ρ^X(dz) σ_t²+R

R\{0}(e^x−1)²ρ^X(dx) , (4.25) and

338

ψ^∗_t := Vt(ϕ)−φ^∗_tξt

= E^Q

θ∗

[ˆc(ξ_{0, T}(x₀))|X_t=e, ξ_{0, t}=x]−φ^∗_tξ_t. (4.26) Proof.

339

1- We have to show that if there exists a processθ^∗ satisfies the Equations (2.24), (4.23) and 340

(4.24) then the process Λ^θ∗ defines a minimal martingale measure in the sense of Schweizer 341

[31].

342