Hedging Expected Losses on Derivatives in Electricity Futures Markets

1.6 MB·

by  Adrien Nguyen Huu

Hedging Expected Losses on Derivatives
in Electricity Futures Markets
Adrien Nguyen Huu∗1 and Nadia Oudjane2
1IMPA, Rio de Janeiro, Brasil
2EDF R&D, Clamart, France
February 3, 2014
4
1
0
2
1 Introduction

n
a
J Weproposeinthiscontributionamethodinvolvingnumericalimplementationforpartiallyhedging

1 financial derivatives on electricity futures contracts. Electricity futures markets present specific
3 features. As a non-storable commodity, electrical energy is delivered as a power over time periods.

Similarly, futures contracts exchange a present power price for delivery over a fixed future period
]
R against the future quoted price on that period: they refer explicitly to swap contracts, see [2].
P Electricity being non-storable, arbitrage arguments do not hold, preventing anyone to construct a
.
n term structure via usual tools. As a natural result of liquidity restriction and uncertainty in the
i
f future, a limited number of contracts are quoted and the length of their delivery period increases
-
q withtheirtimetomaturity. Thisphenomenonwecallthegranularity of the term structure implies
[ that the most desired and flexible contracts (weekly or monthly-period contracts) are only quoted

1 a short time before their maturity. If one desires a fixed price for power delivered on a given
v distant month in the future, she shall get a contract covering a wider period, e.g. a quarter or a
1
year contract. However, the risk remains if one is endowed with a derivative upon such a short
7
2 period contract, before this contract is even quoted.
8 This is the explored situation: we consider here an agent endowed with a derivative upon a
.
1 non-yetquotedasset. Inpractice, theunhedgeableriskofsuchapositionismanagedbydeploying
0
a cross-hedging strategy with a correlated quoted asset, see [22], [13] and [18]. The risk cannot be
4
1 completely eliminated without prohibitive cost: the market is incomplete and the methodology
: requires a pricing criterion, as proposed in the above references. In what follows, we develop a
v
i partial hedging procedure in order to satisfy a loss constraint in expectation. We aim at providing
X
aready-to-implementmethodtoattainsuchobjectives. Thisnecessitatesrecenttoolsofstochastic
r
a control [5] and numerical approximations of coupled forward backward SDEs. However, we relate
constantly the obtained results to well-known formulas and concepts, so that the method is easy
to assimilate. We also provide strong and sufficient assumptions in order to avoid involved proofs
in difficult cases. We believe that the specific method proposed hereafter can be understood and
applied without much effort, and be profitable to a numerous variety of hedging problems.
The approach we adopt has been originated by Föllmer and Leukert [14, 15] but we develop
the problem in the framework of control theory. The minimal initial portfolio value needed to
satisfy the constraint of expectation of losses can be expressed as a value function of a stochastic
∗Corresponding author: [email protected]
1
targetproblem. ThestochastictargetapproachhasbeenintroducedbySonerandTouzi[20,21]to
formulatethepricingprobleminacontrolfashion. Ithasbeenextendedtoexpectationcriteriaby
Bouchard, Elie and Touzi [5], and applied for quantile hedging [5], loss constraints in liquidation
problems [3], or loss constraints with small transaction costs [7]. However, the general approach
is rather technical and necessitates to solve a non-linear Partial Differential Equations (PDE),
when one has first proved a comparison theorem to resolve the uniqueness problem for the value
function.
Here, we use the application of [5], where nice results can be provided in complete market
without appealing to comparison arguments. However, our initial problem cannot be tackled
directly with the stochastic target formulation of [5] or [19]. The unhedgeable risk coming from
the apparition of the desired asset price generates a non-trivial extension to the usual framework.
Instead, we proceed in three steps in a backward fashion:
1. We first formulate in Section 3 the stochastic target problem in complete market, using an
easy extension of the application in [5]. Our approach relies strongly on the convexity of
the loss function. By convex duality arguments, the expected loss target can be expressed
in order to provide a formula which is very close to a usual risk-neutral pricing formulation.
This allows to express the value function V at the precise moment of the apparition of the
missing contract.
2. To treat the random apparition of a non-yet traded asset, we use a face-lifting procedure
that provides a new (intermediary) target for the period before quotation of the underlying
asset. This is done in Section 4, in coherence with the initial hedging criterion, and allows
to retrieve a complete market setting where results of step 1 can be partially used.
3. When the complete market setting cannot provide analytical formulas, as it happens after
step 2, we shall make use of numerical approximations. We propose such an algorithm in
Section 5, and illustrate the efficiency of the method in Section 6.
Asitwillbeunderstood,thismethodcanbeappliedrecursivelybyproceedingrepeatedlytosteps1
and 2along the numericalalgorithm provided instep 3. Theremaining of the contribution follows
the above order, preluded by the introduction of the problem in section 2, where we develop the
archetypal situation encountered by a financial agent on the market.
2 Description of the control problem
Let 0 < T < T∗ < ∞. We consider an agent endowed with a financial option with expiration
date T∗, payoff g on a futures covering a monthly period. However, this asset is not yet quoted
at initial time t = 0, and appears on the market at time T. The month is covered by a futures
with delivery over a larger period, e.g. a quarter futures. We denote by X := (X ) the
t t∈[0,T∗]
discounted price of this instrument, avoiding the introduction of an interest rate hereafter. We
assume it is available over the whole period of interest [0,T∗], whereas the monthly-period futures
upon which the agent has a position is only available on the interval [T,T∗].
The heart of our approach is to assume a structural relation between the two instruments
above, namely that the return of the two assets are perfectly correlated. More precisely, the
monthly futures price is supposed to be given as the product, ΛX , of the quarterly futures price
t
X and a given shaping factor Λ > 0, assumed to be a bounded random variable revealed at
t
time T, completely independent of the asset price X. The boundness condition follows from a
structural relation between the two futures contracts, assuming implicitly that the underlying
spot price is non-negative.
2
Weconsideraprobabilityspace(Ω,F,P)supportingaBrownianmotionW andthevariableΛ.
ThefiltrationFisgivenbyF := σ(W , 0 ≤ s ≤ t)fort < T,andbyF := σ(W , 0 ≤ s ≤ t)∧σ(Λ)
t s t s
for t ≥ T.
Assumption 1. We denote by L the support of Λ, which is supposed to be a bounded subset of
R+, and ρ its probability measure on L.
On the period [0,T], the agent trades with the asset Xt,x, which is assumed to be solution to
the SDE:
dXt,x = µ(s,Xt,x)ds+σ(s,Xt,x)dW , for s ≥ t, and Xt,x = x . (1)
s s s s t
We assume the following for Xt,x to be well and uniquely defined.
Assumption 2. The functions (µ,σ) : [0,T∗]×R → R×R∗ are assumed to be such that the
+ +
following properties are verified:
1. µ and σ are uniformly Lipschitz and verify |µ(t,x)|+|σ(t,x)| ≤ K(1+|x|) for any (t,x) ∈
[0,T∗]×R .
+
2. for any x > 0, we have Xt,x > 0 P−a.s. for all s ∈ [0,T∗]
s
3. σ(t,x) > 0 for any (t,x) ∈ [0,T∗]×R∗;
+
4. if x = 0 then σ(t,x) = 0 for all t ∈ [0,T∗];
5. σ is continuous in the time variable on [0,T∗]×R ;
+
6. µ and σ are such that
µ(t,x)
θ(t,x) := < ∞ uniformly in (t,x) ∈ [0,T∗]×R∗ . (2)
σ(t,x) +
Eq. (2) implies the so-called Novikov condition.The submarket composed of only (X )
t t∈[0,T∗]
is then a complete market associated to the Brownian subfiltration. The filtration F relates to an
incomplete market because of Λ, which is unknown on the interval [0,T) and cannot be hedged by
a self-financed admissible portfolio defined as follows. In that manner, the market can be labelled
as semi-complete in the sense of Becherer [1].
Definition 1. An admissible self-financed portfolio is a F-adapted process Yt,x,y,ν defined by
Yt,x,y,ν = y ≥ −κ and
t
(cid:90) s
Yt,x,y,ν := y+ ν dXt,x , s ∈ [t,T∗] , (3)
s u u
t
where ν ∈ U denotes a strategy and U the set of admissible strategies at time t, which is the set
t,y t,y
of R-valued F-progressively measurable, square-integrable processes such that Yt,x,y,ν ≥ −κ P−a.s.
s
for all s ∈ [t,T∗] and some κ ≥ 0 representing a finite credit line for the agent.
According to the asset model (1), it is strictly equivalent to consider a hedging portfolio with
Xt,x on [0,T∗] or a switch at any time r ∈ [T,T∗] for the newly appeared asset ΛXt,x:
(cid:90) s∧r (cid:90) t
Yt,x,y,ν = y+ ν dXt,x+ ν(cid:48)d(ΛXt,x) ,r ≥ T and t ≥ 0 ,
s u u u u
t s∧r
if ν(cid:48) = ν/Λ for u ≥ r. We thus assume that the agent trades with X on [0,T∗].
Assumption 3. The final payoff of the option is given by g(ΛX ), where the function g is
T∗
assumed to be Lipschitz continuous.
3
As it is well-known in the literature, the superhedging price of such an option can be pro-
hibitive, even with Assumption 1 where Λ is bounded. To circumvent this problem, we propose to
controltheexpectedlossesonpartialhedging. Thatmeansthattheagentgivesherselfathreshold
p < 0 and a loss function (cid:96) to evaluate the loss over her terminal position g(ΛXt,x)−Yt,x,y,ν.
T∗ T∗
Definition 2. A loss function (cid:96) : R → R is assumed to be continuous, strictly convex and
+ +
strictly increasing on R with polynomial growth. We normalize the function so that (cid:96)(0) = 0.
+
The agent’s objective at time t is to find the minimal value y and a portfolio strategy ν ∈ U
t,y
such that
(cid:20) (cid:18) (cid:19)(cid:21)
(cid:16) (cid:16) (cid:17) (cid:17)+
E −(cid:96) g ΛXt,x −Yt,x,y,ν ≥ p . (4)
T∗ T∗
More generally, it will be useful to measure the deviation between the payoff and the hedging
portfolio through a general map Ψ : R ×R → R . The previous example corresponds to the
+ −
specific case where
Ψ(x,y) = −(cid:96)((g(x)−y)+). (5)
Assumption 3 and Definition 2 imply that Ψ defined above satisfies the following assumption:
Assumption 4. The function Ψ : R ×R → R is assumed to be
+ −
• continuous with polynomial growth in (x,y).
• concave and increasing in y on cl{y ∈ [−κ,∞) : Ψ(x,y) < 0} for any x ∈ R .
+
Notice that Ψ(x,y) = 0 in (5) for any y ≥ g(x), which makes Ψ not invertible on the domain
R for any fixed x. Under Assumption 4, we can define the y inverse Ψ−1(x,p) on R ×R as a
+ −
convex increasing function of p, where
Ψ−1(x,0) = inf{y ≥ −κ : Ψ(x,y) = 0} . (6)
As explained above, the valuation approach of (4) has been introduced in [15, 16]. It can be
written under the stochastic control form of [5] allowed by the Markovian framework (1)-(3).
Definition 3. Let (t,x,p) ∈ S := [0,T∗]×R∗ ×R∗ be given. Then we define the value function
+ −
V on S as
(cid:110) (cid:104) (cid:105) (cid:111)
V(t,x,p) := inf y ≥ −κ : E Ψ(ΛXt,x,Yt,x,y,ν) ≥ p for ν ∈ U . (7)
T∗ T∗ t,y
Advanced technicalities and details on the general setting for the stochastic target problem
with controlled loss can be found in [5] and [19]. In particular, we introduce the value function V
onlyontheopendomainof(x,p),andavoidtreatingthespecificcasex = 0orp = 0. Furthermore,
the function V is implicitely bounded by the superhedging price of g(ΛXt,x) as we will see in the
T∗
next section. Notice that the expectation in (7) involves an integration w.r.t. the law of Λ for
t ∈ [0,T), before the shaping factor Λ is revealed. Thus, the present problem is not standard
due to the presence of Λ: the filtration F is not only due to the Brownian motion, and dynamic
programming arguments of [5] do not apply directly. The approach we undertake is to separate
the complete and incomplete market intervals in a piecewise problem.
Example 1. A particular example of loss function we will use in sections 5 and 6 is the special
case of lower partial moment
(cid:96)(x) := xk1 /k , with k > 1 . (8)
{x≥0}
4
In the case of k = 1, (which is not considered here) we obtain a criterion close to the expected
shortfall, independently studied in [12], whereas allowing k = 0 allows to retrieve precisely the
quantile hedging problem [5], although (cid:96) is not a loss function as in Definition 2 in that case. The
case k = 2 gets closer to the quadratic hedging (or mean-variance hedging) objective, but with the
advantage of considering only losses, and not gains. Notice that (cid:96) as in equation (8) for k > 1
follows Definition 2 and that Assumption 4 holds with (5) if Assumption 3 holds.
We finish this section with additional notations. In the sequel, we will denote S := S ∪S
1 2
whereS := [0,T]×R∗ ×R∗ andS := [T,T∗]×R∗ ×R∗ arethetwodomainsofthevaluefunction.
1 + − 2 + −
For any real valued function ϕ, defined on S, we denote by ϕ or ϕ the partial derivatives with
t x
respect to t or x. Partial derivatives of other variables, or second order partial derivatives are
written in the same manner.
3 Solution in complete market
3.1 General solution and risk-neutral expectation
On the interval [T,T∗], the variable Λ is known and the asset ΛX is tradable. We assume that Λ
takes the value λ ∈ L. On this interval, λ can be seen as a coefficient affecting the loss function Ψ,
andtheproblem(7)followsthestandardformulationof[5]: thefiltrationisgeneratedbythepaths
of the Brownian motion. In section 4, we reduce the incomplete market setting of period [0,T)
to complete market problem of the form (7) on [T,T∗], in a similar Brownian framework, with a
function Ψ which is no more derived from a loss function as in (5), motivating the introduction
of the general function Ψ.
Consequently and without loss of generality, we temporarily assume L = {1}, and omit λ in
the notation. On [0,T∗] then, we can work on problem (7) in the Brownian filtration and apply
most results of [5]. The filtration F for t ≥ T in that case is given by F := σ(W : T ≤ s ≤ t)
t s
on a complete probability space (Ω,F,P) with P[Λ = 1] = 1.
Proposition 1. Under Assumptions 2 and 4, the function V is given on S by
(cid:104) (cid:16) (cid:16) (cid:17)(cid:17)(cid:105)
V(t,x,p) = EQt Ψ−1 Xt,x,J Xt,x,Qt,q (9)
T∗ T∗ T∗
where J(x,q) := argsup(cid:8)pq−Ψ−1(x,p) : p ≤ 0(cid:9) and
(cid:90) s (cid:90) s
Xt,x = x+ σ(u,Xt,x)dWQt and Qt,q = q+ Qt,qθ(u,Xt,x)dWQt , (10)
s u u s u u u
t t
withWQt beingtheBrownianmotionundertheprobabilityQt,thelatterbeingdefinedbydPt/dQt =
(cid:104) (cid:16) (cid:17)(cid:105)
Qt,1 and used for expectation (9). Finally q is given such that E J Xt,x,Qt,q = p.
T∗ T∗
To obtain such a result, we proceed in several steps. We first use Proposition 3.1 in [5]
to express problem (7) in the standard form of [20], see Lemma 1 below. We then apply the
Geometric Dynamic Programming Principle (GDP) Principle of [20] in order to obtain a PDE
characterization of V in the viscosity sense. These results are recalled in Appendix. We then
use properties of the convex conjugate of V to obtain V as the risk-neutral expectation (9) of
Proposition 1.
Lemma 1. Let Pt,p,α be a F-adapted stochastic process defined by
(cid:90) s
Pt,p,α = p+ α Pt,p,αdW , 0 ≤ t ≤ s ≤ T∗ , (11)
s u u u
t
5
where α is a F-progressively measurable process taking values in R. Let us denote A the set of
t,p
such processes such that Pt,p,α and αPt,p,α are square-integrable processes. Let Assumptions 2 and
4 hold. Then on S
(cid:110) (cid:111)
V(t,x,p) = inf y ≥ −κ : Yt,x,y,ν ≥ Ψ−1(Xt,x,Pt,p,α) for (ν,α) ∈ U ×A . (12)
T∗ T∗ T∗ t,y t,p
Moreover, for a given triplet (t,y,p), if there exists ν ∈ U and a F-progressively measurable
t,y
process α taking values in R such that
Yt,x,y,ν ≥ Ψ−1(Xt,x,Pt,p,α) P−a.s. , (13)
T∗ T∗ T∗
then there exists α(cid:48) ∈ A such that Yt,x,y,ν ≥ Ψ−1(Xt,x,Pt,p,α(cid:48)) P−a.s.
t,p T∗ T∗ T∗
Proof. Under Assumption 4, Ψ−1(x,p) is well defined with (6). According to the polynomial
growth of Ψ combined with Assumption 2, the stochastic integral representation theorem can be
applied. The first result (12) is then a reformulation of Proposition 3.1 in [5].
The second result echoes Assumption 4 and Remark 6 in [3], as the statement is missing in
[5]. According to (11), for p < 0, the process Pt,p,α is a negative local martingale, and therefore
(cid:104) (cid:105)
a bounded submartingale. Thus, E Ψ(Xt,x,Yt,x,y,ν) ≥ p. According to the growth condition of
T∗ T∗
Ψ, the martingale representation theorem implies the existence of a square-integrable martingale
Pt,p,α(cid:48), with Pt,p,α(cid:48) = p and according to (13),
t
(cid:104) (cid:105) (cid:104) (cid:105)
E Ψ(Xt,x,Yt,x,y,ν)−Pt,p,α(cid:48) = E Ψ(Xt,x,Yt,x,y,ν) −p ≥ 0 .
T∗ T∗ T∗ T∗ T∗
Since Ψ(x,y) ∈ R for any (x,y) ∈ R × R, we can choose Pt,p,α(cid:48) to follow dynamics (11).
− +
This implies that α(cid:48) is a real-valued (F )-progressively measurable process such that (α(cid:48)Pt,p,α(cid:48)) ∈
t
L2([0,T∗]×Ω), and α(cid:48) ∈ A .
t,p
We can now turn to the proof of Proposition 1 by using the GDP recalled in Appendix. The
proof follows closely developments of section 4 in [5], and is given for the sake of clarity, as for the
introduction of important objects for section 4.3.
Proof. (Proposition 1) We divide the proof in three steps
1. We introduce conjugate and local test functions. LetV bethelowersemi-continuousversion
∗
of V, as defined in Appendix. According to Assumption 4, dynamics (3) and definition (7), V
and V are increasing functions of p on R . The boundedness of V implies also the finitness of
∗ −
V . For (t,x,q) ∈ [0,T∗]×R∗ ×R∗, we introduce the convex conjugate of V in p < 0, i.e.,
∗ + + ∗
V˜(t,x,q) := sup{pq−V (t,x,p) : p ≤ 0} .
∗
The map q (cid:55)→ V˜(.,q) is convex and upper-semi-continuous on R∗.
+
Letϕ˜beasmoothfunctionwithboundedderivatives, suchthat(t ,x ,q ) ∈ [0,T∗)×R∗ ×R∗
0 0 0 + +
isalocalmaximizerofV˜−ϕ˜with(V˜−ϕ˜)(t ,x ,q ) = 0. Themapq (cid:55)→ ϕ˜(.,q)isconvex. Without
0 0 0
lossofgenerality, wecanassumethatϕ˜isstrictlyconvexwithquadraticgrowthinq, seetheproof
in Section 4 of [5].
The convex conjugate of ϕ˜ with respect to q is a strictly convex function of p defined by
ϕ(t,x,p) := sup{qp−ϕ˜(t,x,q) : q ≥ 0}. We can then properly define the map (t,x,q) (cid:55)→
(ϕ )−1(t,x,q) on [0,T∗] × R∗ × R∗, where the inverse is taken in the p variable. According
p + +
to the definition of V˜ and the quadratic growth of ϕ˜, there exists p ≤ 0 such that for the fixed
0
q ,
0
p q −V (t ,x ,p ) = V˜(t ,x ,q ) = ϕ˜(t ,x ,q ) = sup{pq −ϕ(t ,x ,p)}
0 0 ∗ 0 0 0 0 0 0 0 0 0 0 0 0
p≤0
6
which, by taking the left and right sides of the above equation, implies that (t ,x ,p ) is a local
0 0 0
minimizer of V −ϕ and (V −ϕ)(t ,x ,p ) = 0. The first order condition in the definition of ϕ
∗ ∗ 0 0 0
implies that p = (ϕ )−1(t ,x ,q ).
0 p 0 0 0
2. We prove that V˜ is subsolution to a linear PDE. In our case, the control ν takes unbounded
values by definition of U . It implies, together with Assumption 2, that
t,y
N (t,x,p,d ,d ) := (cid:8)(u,a) ∈ R2 : |σ(t,x)(u−d )−apd | = 0(cid:9) (cid:54)= ∅ (14)
0 x p x p
for any (t,x,p,d ,d ) ∈ [0,T∗]×R ×R ×R2. This holds in particular for the set
x p + −
N (t ,x ,p ,ϕ (t ,x ,p ),ϕ (t ,x ,p ))
0 0 0 0 x 0 0 0 p 0 0 0
whichisthencomposedofelementsoftheform((ϕ +apϕ /σ)(t ,x ,p ),a)fora ∈ R. According
x p 0 0 0
to Theorem 2 in Appendix and changing ν for its new expression, ϕ in (t ,x ,p ) verifies
0 0 0
(cid:26) (cid:27)
1 1
−ϕ − σ2(t ,x )ϕ − inf (ap )2ϕ −ap (θ(t ,x )ϕ −σϕ ) ≥ 0 . (15)
t 0 0 xx 0 pp 0 0 0 p xp
2 a∈R 2
Since ϕ (t ,x ,p ) > 0, the infimum in the above equation is reached for
pp 0 0 0
(cid:18) (cid:19)
σϕ −θϕ
a := − xp p (t ,x ,p ) ∈ R , (16)
0 0 0
p ϕ
0 pp
which can be plugged back into (15) to obtain a new inequality at (t ,x ,p ):
0 0 0
−ϕ − 1σ2(t ,x )ϕ + 1(ϕ )−1(cid:0)θ(t ,x )ϕ −σ(t ,x )ϕ (cid:1)2 ≥ 0 . (17)
t 0 0 xx pp 0 0 p 0 0 xp
2 2
Recall that p = (ϕ )−1(t ,x ,q ). According to the Fenchel-Moreau theorem, ϕ˜ is its own
0 p 0 0 0
biconjugate, ϕ˜(t,x,q) = sup{pq−ϕ(t,x,p) : p ≤ 0}, and ϕ˜(t ,x ,q ) = p q −ϕ(t ,x ,p ). By
0 0 0 0 0 0 0 0
differentiating in p, we get ϕ (t ,x ,p ) = q . It follows by differentiating again that at point
p 0 0 0 0
(t ,x ,p ) we have the following correspondence:
0 0 0
(cid:32) (cid:33)
ϕ˜2 1 ϕ˜
xq xq
(ϕ ,ϕ ,ϕ ,ϕ ,ϕ ) = −ϕ˜ ,−ϕ˜ ,−ϕ˜ + , ,− . (18)
t x xx pp xp t x xx
ϕ˜ ϕ˜ ϕ˜
qq qq qq
Plugging (18) into (17), we get that ϕ˜ satisfies at (t ,x ,q )
0 0 0
−ϕ˜ − 1 (cid:0)σ2ϕ˜ +|θ|2q2ϕ˜ +2µϕ˜ (cid:1)(t ,x ,q ) ≤ 0 . (19)
t 2 xx 0 qq xq 0 0 0
By arbitrariness of (t ,x ,q ) ∈ [0,T∗)×R∗ ×R∗, this implies that V˜ is a subsolution of (19)
0 0 0 + +
on [0,T∗)×R∗ ×R∗. The terminal condition is given by the definition of V˜ and Theorem 2 in
+ +
Appendix:
V˜(T∗,x,q) = sup{pq−V (T∗,x,p)} = sup(cid:8)pq−Ψ−1(x,p)(cid:9) . (20)
∗
p≤0 p≤0
3. We compare V to a conditional expectation. Let V¯ be the function defined by V¯(t,x,q) =
(cid:104) (cid:105)
EQt V˜(T∗,Xt,x,Qt,q) for (t,x,q) ∈ [0,T∗]×(0,∞)2, with dynamics for s ∈ [0,T∗] given by
T∗ T∗
(cid:90) s (cid:90) s
Xt,x = x+ σ(u,Xt,x)dWQt and Qt,q = q+ Qt,qθ(u,Xt,x)dWQt ,
s u u s u u u
t t
where Q is a P-equivalent measure such that dP/dQ = Qt,1. According to the Feynman-Kac
t t
formula, V¯ is a supersolution to equation (19), and thus V¯ ≥ V˜. According to Assumption 4,
7
p (cid:55)→ Ψ−1(.,p) is convex increasing on R . Thus, for sufficiently large values of q, J(x,q) :=
−
argsup(cid:8)pq−Ψ−1(x,p) : p ≤ 0(cid:9) is well-defined and can take any value in R . By the implicit
−
(cid:104) (cid:16) (cid:17)(cid:105)
function theorem, there exists a function q of (t,x,p) such that EQt Qt,1J Xt,x,Qt,q(t,x,p) = p.
T∗ T∗ T∗
Therefore,
V(t,x,p) ≥ V (t,x,p) ≥ sup(cid:8)qp−V¯(t,x,q) : q ≥ 0(cid:9)
∗
(cid:104) (cid:16) (cid:17)(cid:105)
≥ pq(t,x,p)−EQt V˜ T∗,Xt,x,Qt,q(t,x,p)
T∗ T∗
(cid:16) (cid:104) (cid:16) (cid:17)(cid:105)(cid:17)
≥ q(t,x,p) p−EQt Qt,1J Xt,x,Qt,q(t,x,p)
T∗ T∗ T∗
(cid:104) (cid:16) (cid:16) (cid:17)(cid:17)(cid:105)
+EQx Ψ−1 Xt,x,J Xt,x,Qt,q(t,x,p)
T∗ T∗ T∗
(cid:104) (cid:16) (cid:16) (cid:17)(cid:17)(cid:105)
≥ EQt Ψ−1 Xt,x,J Xt,x,Qt,q(t,x,p) =: y(t,x,p) .
T∗ T∗ T∗
By the martingale representation theorem, there exists ν ∈ U such that
t,y
(cid:16) (cid:16) (cid:17)(cid:17)
Yt,y(t,x,p),ν = Ψ−1 Xt,x,J Xt,x,Qt,q(t,x,p)
T∗ T∗ T∗ T∗
which implies that for p ≤ 0
(cid:104) (cid:16) (cid:17)(cid:105) (cid:104) (cid:16) (cid:17)(cid:105) (cid:104) (cid:16) (cid:17)(cid:105)
E Ψ Xt,x,Yt,y(t,x,p),ν ≥ E J Xt,x,Qt,q(t,x,p) = EQt Qt,1J Xt,x,Qt,q(t,x,p)
T∗ T∗ T∗ T∗ T∗ T∗ T∗
≥ p
and therefore, by definition of the value function y(t,x,p) ≥ V(t,x,p) which provides the equality
and (9) for (t,x,p) ∈ S.
3.2 Application to the interval [T,T∗]
We now come back to [T,T∗], where the shaping factor Λ, is assumed to take the known value
λ at time T. To highlight the effect of λ as a given state parameter for t ≥ T, we denote by
V(t,x,p,λ) the value function defined similarly as (7) where Ψ is given by (5).
Definition 4. Let (t,x,p,λ) ∈ S ×L. We can define the value function on S ×L as
2 2
(cid:20) (cid:21)
(cid:16) (cid:16) (cid:17) (cid:17)+
V(t,x,p,λ) := inf{y ≥ −κ : E (cid:96) g λXt,x −Yt,x,y,ν ≤ −p for ν ∈ U } , (21)
T∗ T∗ t,y
Notice that if X is an exponential process, then we can explicitely change V(t,x,p,λ) for
V(t,λx,p,1), recalling the assumption P[Λ = 1] = 1 of the previous section. Let us recall the
standard pricing concepts in complete market. Under Assumption 2, we can define Q the P-
equivalent martingale measure defined by
ddQP(cid:12)(cid:12)(cid:12)(cid:12) = exp(cid:26)−(cid:90) tθ(s,Xs)dWs− 12 (cid:90) t|θ(s,Xs)|2ds(cid:27) ,t ≥ T . (22)
Ft T T
In the present setting, Q is the unique risk-neutral measure with the drifted Brownian motion
Q (cid:82)t
W = W + θ(s,X )ds, and we can provide a unique no-arbitrage price for g(λX ).
t t T s T∗
Definition 5. We define for (t,x,λ) ∈ [T,T∗]×R∗ ×L the function
+
C(t,x,λ) := EQ[g(λXt,x)] for t ≥ T . (23)
T∗
8
According to Assumption 2 and Assumption 3, we can apply Proposition 6.2, in [17], which
implies that for any λ ∈ L, (t,x) (cid:55)→ C(t,x,λ) is Lipschitz continuous in the spatial variable
with the same Lipschitz constant K as g. Moreover, C(·,·,λ) is the unique classical solution of
polynomial growth to the Black-Scholes equation
1
−C − σ2(t,x)C = 0 on [T,T∗)×(0,+∞) (24)
t xx
2
with terminal condition C(T∗,x,λ) = g(λx). Now Proposition 1 applies to provide the following
when we use a loss function as in Definition 2.
Corollary 1. Let Assumptions 2 and 3 hold. Then V is given on S ×L by
2
V(t,x,p,λ) = EQ[g(λXt,x)−(cid:96)−1(−Pt,p)] (25)
T∗ T∗
where Pt,p is a F -measurable random variable defined by
T∗ T∗
(cid:32) (cid:32) (cid:33)(cid:33)
(cid:90) T∗ 1 (cid:90) T∗
Pt,p = j qexp θ(s,Xt,x)dWQ− θ2(s,Xt,x)ds (26)
T∗ s s 2 s
t t
(cid:104) (cid:105)
with j(q) := −(((cid:96)−1)(cid:48))−1(q) and q in (26) such that EQ Pt,p = p . Moreover, V is convex and
T∗
increasing in p and is C1,2 in (t,x) on [T,T∗]×R∗.
+
The solution (25)-(26) can be explicitly computed in simple cases, see section 5 below. Notice
that V is bounded on [T,T∗]×R∗ ×R∗ ×L by C. Since (cid:96) is convex increasing on R , −(cid:96)−1 is
+ − +
convex. A look at (25)-(26) then convinces that V is continuous in p on R∗, but Proposition 3.3
−
in [5] can be used in our setting to assert that V is also convex in p. According to what was said
about the function C, V is C1,2 in (t,x) on [T,T∗]×R∗.
+
Remark 1. It is noticeable that the value function (25) is composed of the Black-Scholes price
of the claim g(λXt,x) minus a term that corresponds to a penalty in the dual expression of the
T∗
acceptance set if(cid:96) is a risk measure, see [16]. The hedging strategy is to be modified in consequence.
First note that (25) is a conditional expectation of a function of two Markov processes (Xt,x,Qt,q),
for q well chosen, so that we can write Yt,V(t,x,p),ν := y(Xt,x,Qt,q) := V(s,Xt,x,Pt,p). According
s s s s s
to Corollary 1, y is a regular function, and Xt,x,Qt,q are martingales under the probability Q, as
well as Yt,V(t,x,p),ν. Therefore Itô formula provides
dYt,V(t,x,p),ν = y (Xt,x,Qt,q)dXt,x+y (Xt,x,Qt,q)dQt,q .
s x s s s q s s s
Now expressing dWQ with respect to dXt,x, we obtain
s s
dYt,V(t,x,p),ν = (cid:0)y (Xt,x,Qt,q)+µ(s,Xt,x)Qt,qy (Xt,x,Qt,q)(cid:1)dXt,x (27)
s x s s s s q s s s
which allows to deduce the optimal dynamic strategy ν.
Corollary 1 retrieves solutions of [15]. The original problem in the latter is to minimize the
expectedlossgivenaninitialportfoliovalue,buttheauthorsprovethatourversionoftheproblem
isequivalent. ThestochastictargetproblemofDefinition3isactuallylinkedtoanoptimalcontrol
problem in a similar way, as noticed in the introduction of [5] and developped in [4]. We can
introduce the value function giving the minimal loss that can be achieved at time T∗, with at time
t, the initial capital y, Xt,x = x and the shaping factor Λ = λ.
t
9
Definition 6. For (t,x,y,λ) ∈ [T,T∗]×R∗ ×R×L we define
+
(cid:26) (cid:20) (cid:18) (cid:19)(cid:21) (cid:27)
(cid:16) (cid:16) (cid:17) (cid:17)+
U(t,x,y,λ) := sup E −(cid:96) g λXt,x −Yt,y,ν : ν ∈ U . (28)
T∗ T∗ t,y
Thiscorrespondstotheproblemoffindingthebestreachablethresholdpiftheinitialportfolio
valueisgivenbyy attimet. Thisresultisofgreatuseintheforthcomingresolutionoftheproblem
before T, by the following connection with V.
Lemma 2. For (t,x,y,λ) ∈ [T,T∗]×R∗ ×[−κ,∞)×L we define
+
V−1(t,x,y,λ) := sup{p ≤ 0 : V(t,x,p,λ) ≤ y} .
Then we have
U(t,x,y,λ) = V−1(t,x,y,λ) . (29)
Moreover the function U is a concave increasing function of y bounded by 0 from above.
Equality (29) is a direct application of Lemma 2.1 in [4], whereas the properties of U fall from
the properties of V and Definition 6.
4 Solution in incomplete market
We now turn to the solution of the problem before T, i.e., when Λ is unkown. We first provide
the face-lifting procedure that allows to reduce the new situation to the one handled in Section 3.
This procedure can be done according to two model paradigms:
1. the law ρ of Λ is supposed to be known;
2. only the support L of the law of Λ is supposed to be known.
The first approach is a probabilistic approach with a prior distribution, whereas the second one
is a robust approach, in connection with robust finance with parameter uncertainty, see [6] for a
control theory version. Finally, the numerical complexity of the face-lifting procedure pushes us
to give up explicit formulas for a numerical approach. We thus modify results of Section 3 to
provide a convenient formulation of the problem to be numerically approximated in Section 5.
4.1 Faced-lifted intermediary condition with prior on the shaping factor
When t < T, the problem (7) cannot be treated with the methodology developped in [5]. To solve
the problem, we are guided by the following argument. Considering (t,y) ∈ [0,T)×[−κ,∞) and
a strategy ν ∈ U , we arrive at time T to the wealth Yt,x,y,ν ≥ −κ, at the apparition of the
t,y T
exogenous risk factor Λ. Assume that the agent wants to control the expected level of risk p < 0,
at time T, by using the portfolio Yt,x,y,ν. It is obviously not possible with certainty if Λ takes
T
a value λ such that Yt,y,ν < V(T,Xt,x,p,λ) . However, the optimal strategy after T consists in
T T
optimizingtheportfoliobytryingtoachievetheoptimalexpectedleveloflossU(T,Xt,x,Yt,y,ν,λ)
T T
given by Definition 6. In the complete market setting, this achievement is possible.
Lemma 3. Under Assumptions 2 and 4, there exists a map (x,y,λ) (cid:55)→ ν(x,y,λ) ∈ U on
T,y
R∗ ×[−κ,∞)×L such that
+
(cid:104) (cid:16) (cid:17)(cid:105)
E Ψ λXT,x,YT,y,ν(x,y,λ) ≥ U(T,x,y,λ) . (30)
T∗ T∗
10