6.3 Optimization Problems and BSDEs
6.3.1 Optimization Problem
in the case of the exponential utility. In a last step, for the determination of Hodges’ price, we shall change v into v − p.
6.3.1 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 strategy π ∈ Π(G) that maximizes EP(u(VTv(π) + X)), where the wealth process (Vt= Vtv(π), t ≥ 0) (for simplicity, we shall frequently skip v and π from 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(VTv(π) + X)ª
= 1 − inf
π∈Π(G)EP
¡e−%VTv(π)e−%X¢ .
We shall give three different methods to solve infπ∈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) with ZT = e−%X, which depends only on the claim X and parameters %, σ, ν, and such that the process (e−%Vtv(π)Zt, t ≥ 0) is a (P, G)-submartingale for any admissible strategy π, and is a martingale under P for some admissible strategy π∗∈ Π(G). Then, we would have
EP(e−%VTv(π)ZT) ≥ e−%V0v(π)Z0= e−%vZ0
for any π ∈ Π(G), with equality for some strategy π∗∈ Π(G). Consequently, we would obtain and thus we would be in the position to conclude that π∗ 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 of G-admissible trading strategies to such strategies that the ”martingale part” in (6.16) 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 the volume on Indifference prices and to the papers of Barles (1997), Rong [165] and the thesis of Royer [166] for BSDE with jumps.
We shall search the process Z in the class of all processes satisfying the following BSDE dZt= ztdt + bztdWt+ eztdMt, t ∈ [0, T [, ZT = e−%X, (6.15) where the process z = (zt, t ≥ 0) will be determined later (see equation (6.18) below). By applying Itˆo’s formula, we obtain It is easily seen that, assuming that the process Z is strictly positive, we have
π∗t = νZt+ σbzt
Now, let us choose the process z as follows zt= Zt
Note that with the above choice of the process z the drift term in (6.16) is positive for any admissible strategy π, and it is zero for π = π∗.
Given the above, it appears that we have reduced our problem to the problem of solving the BSDE (6.15) with the process z given by (6.18), i.e.,
in (6.16) is a true martingale part rather than a local-martingale part, then the process πt∗= 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 [156, 155] study the family of processes
Vt(v) = max
φ EP(U (v + Z T
t
φsdSs)|Gt)
where v is 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 BSDE
dV(t, v) = 1 2
1
Vvv(t, v)(ϕv(t, v) + λtVv(t, v))2dhµit+ ϕ(t, v)dµt+ dNt(v),
V(T, v) = U (v), (6.20)
where N is a martingale orthogonal to µ, and the optimal portfolio is proved to be φ∗t = −Stϕv(t, Vt∗) − λtVv(t, Vt∗)
Vvv(t, Vt∗) .
Analysis of the proof of the equation (1.4) in Mania and Tevzadze [156] 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 (Vt(v), t ≥ 0) satisfies the BSDE (6.20) with terminal condition V(T, v) = U (v + X). We note however that there are several technical conditions postulated in Mania and Tevzadze [156] 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 have dµt= StσdWtand λt= ν/(Stσ2), and the BSDE (6.20) reads
dV(t, v) = St2σ2
2Vvv(t, v)(ϕ(t, v) + ν σ2St
Vv(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
where N is a martingale orthogonal to W (hence, in our setting a martingale of the formRt
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 process V as V(t, v) = e−%vZt, and to factorize process ϕ as ϕ(t, v) = bϕ(t)e−%v. It follows that Z satisfies
dZt=
( bϕ(t) + ν σ2St
Zt)2
2Zt St2σ2dt + bϕ(t)StσdWt+ dNt, ZT = e−%X. Setting bzt= bϕ(t)σSt, we get
dZt= 1
2Zt(bzt+ν
σZt)2dt + bztdWt+ dNt, ZT = e−%X,
which is exactly equation (6.18), where N is a stochastic integral w.r.t. the martingale M , orthogonal to W . Thus, it appears that a solution to equation (6.18) is given as
Zt= e%vV(t, v), bzt= bϕ(t)σSt, and ezt= dNt
dMt
. The optimal portfolio is
σbzt+ Ztν
%σ2Zt which is exactly (6.17).
Remark 6.3.1 Analogous results follow from by Mania and Tevzadze [156] where a more general case of utility function is studied.
Duality Approach
We present now the duality approach (See for example Delbaen et al. [61], or Mania and Tevzadze [155]). In the case dSt= St(νdt + σdWt), the set of equivalent martingale measure (emm) is the set of probability measures Qψ defined as
dQψ|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 [140], 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 filtration F is equal to Q.) Let us denote by WtQ= Wt+ θt and cMt= Mt−Rt
0ψsξsds. The processes WQ and cM are Qψ martingales. Then,
Lt = exp µ
−θWt−1 2θ2t +
Z t
0
ln(1 + ψs)dHs− Z t
0
ψsξsds
¶
= exp µ
−θWtQ+θ2t 2 +
Z t
0
ln(1 + ψs)d cMs+ Z t
0
[(1 + ψs) ln(1 + ψs) − ψs]ξsds
¶
Hence, the relative entropy of Qψ with respect to P is
H(Qψ|P) = EQψ(ln LT) = EQψ Ã1
2θ2T + Z T
0
[(1 + ψs) ln(1 + ψs) − ψs]ξsds
! .
From duality theory, the optimization problem
π∈Π(G)inf EP
¡e−%VTv(π)e−%X¢
reduces to maximization over ψ of
EQψ(X − 1
%H(Qψ|P)), that is, maximization over ψ of
EQψ
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 + butdWtQ+ eutd cMt, UT = X − 1
2%θ2T.
Setting Yt= % Utwe obtain
dYt = ([(1 + ψt) ln(1 + ψt) − ψt]ξt) dt + bytdWtQ+ eytd cMt, YT = % X −1
2θ2T.
In terms of the martingale M , we get
dYt= ([(1 + ψt) ln(1 + ψt) − ψt(1 + eyt)]ξt) dt + bytdWtQ+ eytdMt,
The solution is obtained by maximization of the drift in the above equation w.r.t. ψ, which leads to 1 + ψs= eys. Consequently, the BSDE reads
dYt= −
³
eeyt− 1 − eyt
´
ξtdt + bytdWtQ+ eytdMt, YT = %X − 1 2θ2T, and setting Zt∗= exp(−Yt) we conclude that
dZt∗=1
2Zt∗byt2dt − Zt∗bytdWtQ+ Zt−∗ (eybt− 1)dMt, ZT∗ = exp(−%X + 1 2θ2T ), or, denoting bzt= −Zt∗ybt, ezt= Zt−∗ (ebyt− 1)
dZt∗= 1
2Zt∗zbt2dt + bztdWtQ+ bztdMt, ZT∗ = exp(−%X +1 2θ2T ), which is equivalent to (6.19). (Note that Zt= Zt∗e−12θ2(T −t).)