Optimization Problem

in the case of the exponential utility. In a last step, for the determination of Hodges’ price, we shall changev intov−p.

1.3.1 Optimization Problem

Our first goal is to solve an optimization problem for an agent who sells a claim X. To this end, it suffices to find a strategyπ∈ Π(G) that maximizes EP(u(V_T^v(π)+X)), where the wealth process (Vt=V_t^v(π), t≥0) (for simplicity, we shall frequently skipv andπfrom the notation) satisfies

dVt=φtdSt=πt(νdt+σdWt), V0=v.

We consider the exponential utility function u(x) = 1−e^−%x, with % > 0.

Therefore, sup

π∈Π(G)

EP

©u(V_T^v(π) +X)ª

= 1− inf

π∈Π(G)EP

¡e^−%VTv(π)e^−%X¢ . We shall give three different methods to solve inf_π∈Π(G)EP

¡e^−%VTv(π)e^−%X¢ .

Direct method

We describe the idea of a solution; the idea follows the dynamic programming principle.

Suppose that we can find a G-adapted process (Zt, t≥0) withZT =e^−%X, which depends only on the claimX and parameters%, σ, ν, and such that the process (e^−%Vtv(π)Zt, t≥0) is a (P,G)-submartingale for any admissible strat-egy π, and is a martingale under Pfor some admissible strategy π^∗ ∈ Π(G).

Then, we would have

EP(e^−%VTv(π)ZT)≥e^−%V0v(π)Z0=e^−%vZ0

for anyπ∈ Π(G), with equality for some strategy π^∗ ∈Π(G). Consequently, we would obtain

π∈Π(G)inf EP

¡e^−%VTv(π)e^−%X¢

=EP

¡e^{−%VTv(π∗)}e^−%X¢

=e^−%vZ0, (1.15) and thus we would be in the position to conclude thatπ^∗is an optimal strategy.

In fact, it will turn out that in order to implement the above idea we shall need to restrict further the class ofG-admissible trading strategies to such strategies that the ”martingale part” in (1.17) determines a true martingale rather than a local-martingale.

In what follows, we shall use the BSDE framework. We refer the reader to the chapter by ElKaroui and Hamad´ene in this volume and to the papers of Barles (1997), Rong (1997) and the thesis of Royer (2002) for BSDE with jumps.

We shall search the process Z in the class of all processes satisfying the following BSDE

dZt=ztdt+bztdWt+zetdMt, t∈[0, T[, ZT =e^−%X, (1.16) where the processz = (zt, t≥0) will be determined later (see equation (1.19) below). By applying Itˆo’s formula, we obtain

d(e^−%Vt) =e^−%Vt¡¡₁

2%²π_t²σ²−%πtν¢

dt−%πtσ dWt

¢, so that

d(e^−%VtZt) = e^−%Vt¡

zt+Zt(¹₂%²π²_tσ²−%πtν)−%πtσbzt

¢dt +e^−%Vt¡

(bz_t−%π_tσZ_t)dW_t+ez_tdM_t¢

. (1.17)

Let us chooseπ^∗= (π_t^∗, t≥0) such that it minimizes, for everyt, the following expression

Zt

¡₁

2%²π_t²σ²−%πtν¢

−%πtσbzt=−%πt(νZt+σbzt) +¹₂%²π_t²σ²Zt. It is easily seen that, assuming that the processZ is strictly positive, we have

π^∗_t = νZt+σbzt

%σ²Zt = 1

%σ

³ θ+ bzt

Zt

´

. (1.18)

1.3. OPTIMIZATION PROBLEMS AND BSDES 19 Now, let us choose the processz as follows

z_t=Z_t¡

Note that with the above choice of the process z the drift term in (1.17) is positive for any admissible strategyπ, and it is zero forπ=π^∗.

Given the above, it appears that we have reduced our problem to the problem of solving the BSDE (1.16) with the processzgiven by (1.19), i.e.,

 the ”martingale part” in (1.17) is a true martingale part rather than a local-martingale part, then the process

π^∗_t = 1 will be an optimal portfolio, i.e.,

π∈Π(G)inf EP

¡e^−%VTv(π)e^−%X¢

=EP

¡e^{−%VTv(π∗)}e^−%X¢ .

However, this BSDE is not of standard. This is a BSDE with jumps, and existence theorems and comparison theorems are known only if the driver is Lipschitz. Hence, we shall establish the existence using another approach, an approach due to Mania and Tevzadze.

Mania and Tevzadze approach

In a very general setting, when the underlying asset is of the form dSt=dµt+λtdhµit

where µ is a continuous local martingale, Mania and Tevzadze (2003a) study the family of processes

wherevis a real-valued deterministic parameter. They establish that the process (V(t, v) = Vt(v), t ≥ 0) (which depends on the parameter v) is solution of a

whereN is a martingale orthogonal toµ, and the optimal portfolio is proved to be

φ^∗_t =−Stϕv(t, V_t^∗)−λtVv(t, V_t^∗) Vvv(t, V_t^∗) .

Analysis of the proof of the equation (1.4) in Mania and Tevzadze (2003a) reveals that their results carry to the case when

Vt(v) = max

φ E(U(v+ Z _T

t

φsdSs+X)|Gt)

for a claim X satisfying appropriate integrability conditions, in which case the process (V_t(v), t ≥ 0) satisfies the BSDE (1.21) with terminal condition V(T, v) = U(v+X). We note however that there are several technical condi-tions postulated in Mania and Tevzadze (2003a) that need to be verified before their results can be adopted.

In the particular case when the dynamics of the underlying asset follows dSt=St(νdt+σdWt)

we havedµt=StσdWtandλt=ν/(Stσ²), and the BSDE (1.21) reads dV(t, v) = S_t²σ²

2Vvv(t, v)(ϕ(t, v) + ν

σ²StVv(t, v))²dt+ϕ(t, v)StσdWt+dNt

= 1

2σ²Vvv(t, v)(ϕ(t, v)σ²St+νVv(t, v))²dt+ϕ(t, v)StσdWt+dNt

whereN is a martingale orthogonal toW (hence, in our setting a martingale of the formR_t

0ψsdMs). The terminal condition is V(T, v) =U(v+X). and the optimal portfolio is

φ^∗_t =−Stϕv+Vvν/(σ²St) Vvv .

Here,U is an exponential function. Thus, it is convenient to factorize processV asV(t, v) =e^−%vZt, and to factorize processϕasϕ(t, v) =ϕ(t)eb ^−%v. It follows thatZ satisfies

dZt=

(ϕ(t) +b ν σ²StZt)²

2Zt S_t²σ²dt+ϕ(t)Sb tσdWt+dNt, ZT =e^−%X. Settingzbt=ϕ(t)σSb t, we get

dZt= 1 2Zt

(bzt+ ν

σZt)²dt+zbtdWt+dNt, ZT =e^−%X,

1.3. OPTIMIZATION PROBLEMS AND BSDES 21 which is exactly equation (1.19), where N is a stochastic integral w.r.t. the martingale M, orthogonal to W. Thus, it appears that a solution to equation (1.19) is given as

Remark. Analogous results follow from by Mania and Tevzadze (2003b) where a more general case of utility function is studied.

Duality Approach

We present now the duality approach (See for example Delbaen et al. (2002), or Mania and Tevzadze (2003b)). In the case dSt = St(νdt+σdWt), the set of equivalent martingale measure (emm) is the set of probability measures Q^ψ defined as

dQ^ψ|Gt =LtdP|Gt

where

dLt=Lt−(−θdWt+ψtdMt)

where ψ is a G-predictable process, with ψ > −1 and θ is the risk premium θ=ν/σ. Indeed, using Kusuoka representation theorem (1999), we know that any strictly positive martingale can be written of the form

dLt=Lt−(`tdWt+ψtdMt).

The discounted price of the default-free asset is a martingale under the change of probability, hence, it is easy to check that`t=−θ. (We have already noticed that the restriction of any emm to the filtrationFis equal toQ.) Let us denote by W_t^Q =Wt+θt andMct =Mt−R_t

Hence, the relative entropy ofQ^ψ with respect toPis H(Q^ψ|P) =E_Q^ψ(lnLT) =E_Q^ψ

From duality theory, the optimization problem

π∈Π(G)inf EP

¡e^−%VTv(π)e^−%X¢

reduces to maximization overψof E_Q^ψ(X−1

%H(Q^ψ|P)), that is, maximization overψof

E_Qψ

à X− 1

2%θ²T−1

% Z _T

0

[(1 +ψs) ln(1 +ψs)−ψs]ξsds

! .

We solve this latter problem by operating dUt =

µ1

%[(1 +ψt) ln(1 +ψt)−ψt]ξt

¶

dt+butdW_t^Q+eutdMct, UT = X− 1

2%θ²T.

SettingYt=% Utwe obtain

dYt = ([(1 +ψt) ln(1 +ψt)−ψt]ξt)dt+bytdW_t^Q+yetdMct, YT = % X−1

2θ²T.

In terms of the martingaleM, we get

dYt= ([(1 +ψt) ln(1 +ψt)−ψt(1 +yet)]ξt)dt+ybtdW_t^Q+eytdMt, The solution is obtained by maximization of the drift in the above equation w.r.t. ψ, which leads to 1 +ψ_s=ye_s. Consequently, the BSDE reads

dYt=−³

e^yet−1−yet

´

ξtdt+ybtdW_t^Q+yetdMt, YT =%X−1 2θ²T, and settingZ_t^∗= exp(−Yt) we conclude that

dZ_t^∗= 1

2Z_t^∗yb²_tdt−Z_t^∗ybtdW_t^Q+Z_t−^∗ (e^byt−1)dMt, Z_T^∗ = exp(−%X+1 2θ²T), or, denotingbzt=−Z_t^∗byt,zet=Z_t−^∗ (e^byt−1)

dZ_t^∗= 1

2Z_t^∗zb_t²dt+bztdW_t^Q+zbtdMt, Z_T^∗ = exp(−%X+1 2θ²T), which is equivalent to (1.20). (Note thatZt=Z_t^∗e^{−12θ2(T−t)}.)

1.3. OPTIMIZATION PROBLEMS AND BSDES 23