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Non Quadratic Local Risk-Minimization for Hedging Contingent Claims in the Presence of Transaction Costs
Frédéric Abergel, Nicolas Millot
To cite this version:
Frédéric Abergel, Nicolas Millot. Non Quadratic Local Risk-Minimization for Hedging Contingent
Claims in the Presence of Transaction Costs. 2011. �hal-00621256v2�
Non quadratic local risk-minimization for hedging contingent claims in the presence of transaction costs
Frederic Abergel and Nicolas Millot ∗ December 5, 2011
Abstract
This paper is devoted to the study of derivative hedging in incomplete markets when frictions are considered. We extend the general local risk minimisation approach introduced in [1] to account for liquidity costs, and derive the corresponding optimal strategies in both the discrete- and continuous-time settings. We examplify our method in the case of stochastic volatility and/or jump-diusion models.
Introduction
The optimal hedging of derivatives in uncomplete markets is a subject of the utmost importance from a application-driven point of view, as well as an introduction to a host of many challenging mathematical problems. Originating in the pioneering work of [9], the local risk minimization method has been identied as one of the most intuitive and practical way of dening realistic hedging strategies and related option prices.
In the original approach developped in [9], the local risk was dened as the second moment of the incremental cost between two consecutive re-hedging periods. In a recent work [1], this approach was revisited so as to extend it to general, convex local risk functionals, and the corresponding optimality conditions were derived in the discrete- and continuous-time settings. This article is devoted to the task of extending our previous results to the realistically important case of transaction costs. More precisely, and contrarily to one of the early and important contributions in that direction, see [14], we do not introduce a bid-ask spread, as this would lead to innite costs in the continuous limit, but rather, consider as in [5] a supply curve corresponding to the existence of nite liquidity at a given price. Such a paradigm is especially well-suited to the situation of a trader hedging a large book or trading in an illiquid market. It also connects with recent researches on orderbook modelling and market impact, when the supply curve is seen as a smoothed-out version of a stochastic orderbook prole.
The main result of this paper is twofold : on one hand, in a discrete-time setting, the optimality system is fully characterized, and admits a natural interpretation in terms of a non-linear martingale transform orthogonal to the martingale part of a modied price process. Then, extensions to a continuous-time setting are considered, for which only the case of It o processes can be understood in full generality. For processes having discontinuous paths, pseudo-optimality can be considered, but the connection with the original minimization problem is an open question.
The paper is organized as follows: Section 1 presents, in the discrete-time case, the basic denitions of the cost of a strategy and its associated risk. Section 2 contains the optimality and pseudo-optimality conditions in discrete time, and provide an interpretation of the optimal strategy in terms of a non-linear martingale transform orthogonal to the cost-adjusted price process (the supply price). In Section 3, we extend those results to the continuous-time setting, while Section 4 and 5 are respectively devoted to applications to stochastic volatility and jump-diusion models.
∗
Laboratory of Mathematics Applied to Systems, École Centrale Paris, 92290 Châtenay Malabry, France
1 Liquidity costs and risk process
1.1 Cost process
In this section, we investigate the discrete time setting, studying the existence and uniqueness of solutions to the minimization problem. The market will classically be represented, see e.g. [9][1], by a multi-period model where the risky asset is a strictly positive semimartingale S k , (k = 0, · · · , T ) on some probabil- ity space (Ω, F , P ) . Let F k denote the σ− eld of events which are observable up to and including time k . We assume that S k is adapted and square-integrable and that the conditional variance of its returns E (S k+1 − S k ) 2 |F k
− E (S k+1 − S k |F k ) 2 is strictly positive P -almost surely. In order to simplify the expo- sition, the risk-free rate is supposed to be deterministic and therefore, using discounted stock prices, one can assume that it is zero.
The task of interest is that of hedging a contingent claim associated to a square-integrable random variable H ∈ L 2 (P ) of the following form H = δ H S T + β H , δ H and β H being F T − measurable random variables.
We thus consider a general trading strategy Φ represented by two stochastic processes δ k , (k = 0, · · · , T ) and β k , (k = 0, · · · , T ) , both adapted to F k and in L 2 (P ) . δ k is the amount of stock held during the k th period = [t k , t k+1 ) , and has to be xed at the beginning of that period. That is to say, δ k is F k − measurable (k = 0, · · · , T ) , and likewise for β k , the amount held in the cash account during the k th period.
The theoretical value of the portfolio at time k is its value right after applying the strategy and is given by V k = δ k S k + β k , k = 1, · · · , T.
We admit only strategies such that each V k is square-integrable and which replicate the contingent claim H , i.e. we require V T = H , which for instance is the case upon choosing δ T = δ H T and β T = β T H .
Denote by ∆C k the incremental cost of applying strategy Φ at time t k , k > 0 . In the presence of liquidity costs on the stock, ∆C k is given by
∆C k (Φ) = L ((δ k − δ k−1 ), S k , t k ) + (β k − β k−1 ) ∀k ∈ {k = 1, · · · , T }
where the function L measures the costs of adjusting the stock part, thereby accounting for the liquidity eect, namely:
- If (δ k − δ k−1 ) > 0 , meaning that the strategy requires to buy stock, it might not necessarily be possible at the theoretical price S k but rather, at a higher price, so that the bigger the quantity to acquire, the greater the marginal costs.
- If on the contrary (δ k − δ k−1 ) < 0 , meaning that the strategy now requires to sell, again it might not necessarily be possible at the theoretical price S k but rather, at a lower price, so that, once again, the bigger the quantity to sell, the greater the marginal costs.
1.2 Liquidity costs
As a consequence of the nite liquidity observed on real markets and described above, it is legitimate to assume that L : ( R , R + , R + ) → R is a strictly increasing, convex function of its rst argument satisfying L(0, ., .) = 0 . Let us make the further assumption that it is dierentiable with respect to its rst argument, with ∂L ∂x (0, S, .) = S . At this stage, it is noteworthy to point out that the bid/ask spread is not taken into account, nor is the potential market impact of a transaction. This last assumption amounts to assuming that the period of trading is much greater than the relaxation time of the market impact function.
If there exists an adapted function g , i.e. g = g(x, t, ω) with ω ∈ F k , such that the liquidity costs can be written as L((δ k − δ k−1 ), S k , t k ) = (δ k − δ k−1 )g((δ k − δ k−1 ), t k ) , then g is called the supply curve. We refer to [5] for more details on the self-nancing approach in the case of a supply curve. Here, a more general assumption is made, namely, that there exists an increasing density function l : ( R , R + , R + ) → R, l ∈ C 1 representing the cost of buying a marginal amount of stock. That is to say, L has the following form:
L (u, S k , t k ) = Z u
0
l(x, S k , t k )dx. (1.1)
In particular, l(0, S k , t k ) is equal to S k in the absence of bid/ask spread. Assumption (1.1) corresponds to a smoothing of the realistic orderbook prole giving the quantity available at a given price.
In order to make the extension to continuous-time more tractable whilst not narrowing the scope of the paper, we shall further assume that the marginal costs can be written as a stationary function times the theoretical spot price S , i.e. l(x, S, t) = l(x)S t .
1.3 Local risk
The local risk is naturally dened, see e.g. [1], as the conditional expectation given information up to time k of the functional associated to the risk function f , of the costs incurred at time k + 1 . This reads
∆R k (Φ) = E (f (∆C k+1 )|F k ) or, with obvious notation,
∆R k (Φ) = E k (f (∆C k+1 )) .
Note that, contrarily to the case of innite liquidity, these assumptions on the liquidity costs together with the convexity of the risk function f do not ensure that (x, y) 7→ f (L(x)S + y) is a convex function. This lack of convexity will obviously make uniqueness results dicult to obtain.
2 Optimal and pseudo-optimal strategies
The dention of an optimal strategy is now addressed. Classically, as in [9][1], such a strategy sequentially minimizes the incremental risk process backward in time, and indeed solves the following problem
Problem (*) Given a contingent claim H , nd an admissible strategy Φ
∗such that
∀k ∈ (0, · · · , T − 1) , ∀Φ admissible with δ k+1 = δ k+1
∗and β k+1 = β k+1
∗, ∆R k (Φ) ≥ ∆R k (Φ
∗)
Standard regularity and convexity conditions on f , L
1, as well as the assumptions on S k and β k , provide the existence of an optimal strategy solution to the rst-order optimality equations
E k (f
0(∆C k+1 (Φ
∗))) = 0 E k (f
0(∆C k+1 (Φ
∗)) L
0((δ k+1 − δ k ), S k+1 , t k+1 )) = 0 or equivalently
E k (f
0(∆C k+1 (Φ
∗))) = 0
E k (f
0(∆C k+1 (Φ
∗)) l (δ k+1 − δ k ) S k+1 ) = 0 (2.1) where L
0stands for the partial derivatives of L with respect to its rst argument.
We now prove the existence of a locally risk-minimizing strategy.
Theorem 2.1 Problem (*) has a at least one solution Φ
∗whose components δ
∗and β
∗solve the set of equations ( (2.1)).
Proof Let h(x, y, ω) ≡ E k (f (L((U − x), S, t k+1 ) + (V − y))) (ω) with U , V and S ∈ L 2 (P ) . We rst observe that, thanks to the structure hypotheses on liquidity costs, for a xed ω , h is a continuous and dierentiable function of (x, y) and therefore reaches its minimum at (x
∗, y
∗) only if (x
∗, y
∗) is a critical point of h , i.e.
∇h(x
∗, y
∗) = 0 . Secondly, there holds that lim
||(x,y)||→∞h(x, y, ω) = +∞ P − a.e. , so that h has a global
1
We refer the reader to [1] for the detailed statement of these conditions.
minimum P − almost surely. There remains to prove that (x
∗, y
∗) is F k − measurable: let D n = {j2
−n|j ∈ Z } be the set of dyadic rational of order n and dene
(x n (ω), y n (ω)) = inf{(x, y) ∈ D n × D n , h(x, y, ω) ≤ h(x
0, y
0, ω)∀(x
0, y
0) ∈ D n × D n }.
Since ω 7→ h(x, y, ω) is F k − measurable, (x n , y n ) is also F k − measurable. As (x n , y n ) is bounded in n P − a.e.
and h is continuous in (x, y) , (˜ x, y) = lim inf ˜ n→∞ (x n , y n ) is a F k − measurable minimizer of h , satisfying
∇h(˜ x, y) = 0 ˜ . This ends the proof of Theorem 2.1
The set of equations (2.1) can be given a natural interpretation after the introduction of the two processes C k f = P k
i=1 f
0(∆C i ) and S S k = P k
i=1 (l (∆δ i ) S i − l(0)S i−1 ) = P k
i=1 (l (∆δ i ) S i − S i−1 ) with initial conditions C 0 f = 0 and S 0 S = S 0 : (2.1 simply states that C k f is a martingale strongly orthogonal to the martingale part of S k S
k . The rst process will be referred to as the f − costs process as in [1], while the new process S S will be referred to as the supply price process. Following [?], this property of C k f will be termed "pseudo- optimality". Let us also mention that, in the case of "innite" liquidity l(.) = 1 , the supply price process is just the stock price S , and one recovers the results of [1].
3 Continuous time setting
Let now (Ω, F , P ) be a probability space with a ltration (F t ) 0≤t≤T satisfying the usual conditions of right- continuity and completeness. T ∈ R
∗+denotes a xed and nite time horizon. Furthermore, we assume that F 0 is trivial and that F T = F . The risky asset S = (S t ) 0≤t≤T is supposed to be a strictly positive semimartingale
S = S 0 + M + A
such that M = (M t ) 0≤t≤T is a square-integrable martingale with M 0 = 0 , and A = (A t ) 0≤t≤T is a con- tinuous and adapted process of nite variation |A| with A 0 = 0 . Throughout this article, we shall use a right-continuous version of S .
The aim of this section is to dene and characterize the f -cost process and the supply process, so as to exend the notions of pseudo-optimality to the continuous-time case. In order to do so, we need to introduce some denitions that will extend the rather intuitive notions of the discrete-time setting to the more intricate continuous-time models.
3.1 Trading strategies and local risk
A general trading strategy Φ is then a pair of càdlàg and adapted processes δ = (δ t ) 0≤t≤T , β = (β t ) 0≤t≤T while a contingent claim is described by a random variable H ∈ L 2 (P ) , with H = δ H S T + β H , δ H and β H being F T − measurable random variables.
In order to dene the processes which are the basic ingredients of pseudo-optimality in continuous time, we need to restrict the set of trading strategies to those we call H -admissible according to the
Denition 3.1.1 A trading strategy will be called H -admissible if it meets the following requirements
δ T = δ H P − a.s.
β T = β H P − a.s.
δ has nite and integrable quadratic variation β has nite and integrable quadratic variation
δ and β have nite and integrable quadratic covariation .
We now recap the denitions (see [1]) required to extend the concept of local risk-minimization to the
continuous time framework.
Small perturbations
Denition 3.1.2 A small perturbation is a bounded admissible
2strategy φ = (β, δ) such that β T = 0 and δ T = 0 .
Local risk along a partition
Given an H − admissible strategy Φ , a partition τ of [0, T ] , where τ = {0 = t 0 , t 1 , · · · , t k = T } and a small perturbation φ , one denes the process
r f τ [Φ, φ](t, ω) = X
t
i,t
i+1∈τ∆R t
i(Φ + φ| [t
i,t
i+1( )(ω) − ∆R t
i(Φ)(ω) t i+1 − t i
1 [t
i,t
i+1( (t) with ∆R t
i(Φ) = E f (∆C t
i+1)|F t
iand (as in the previous sections) ∆C t
i+1is the incremental cost associ- ated to a strategy.
The concept of local risk minimization can now be specied:
Denition 3.1.3 An H− admissible strategy Φ is called locally risk-minimizing for the contingent claim H if for every small perturbation φ and every increasing sequence of partitions (τ n ) n∈
Ntending to the identity, there holds
lim inf
n→∞ r τ
n[Φ, φ] ≥ 0 P − a.e.
3.2 The f − costs process
Given a general trading strategy Φ , its f − cost process C t (Φ)
3is dened as the limit, whenever it exists, of
l
nX
k=1
f
0(L(δ τ
kn− δ τ
k−1n, S τ
kn) + β τ
kn− β τ
k−1n),
where convergence takes place in ucp topology, for any sequences P n of Riemann partitions of [0, T ] of length l n ( X T stands for the process stopped at T ).
Below is a result showing that the f − cost process of an H − admissible strategy is well dened.
Proposition 3.1 The f − cost process of an H− admissible strategy Φ is well-dened and is given by the formula
C t (Φ) = f
00(0)
V t − V 0 − R t
0+ δ s− dS s + 1 2 l
0(0) R t
0+ S s− d[δ, δ] c s + f
(3)2 (0)
[β, β] c t + 2 R t
0+ S s− d[β, δ] c s + R t
0+ S s− 2 d[δ, δ] c t + P
0<s≤t f
0(∆β s + L(∆δ s , S s )) − f
00(0)(∆β s + ∆δ s S s ) (3.1) with notation [X, Y ] c standing for the continuous part of the (càdlàg) quadratic covariation process.
Proof The proof relies on exactly the same ingredients as in Theorem 2 of [1], where the case without transac- tion costs is thoroughly studied along the lines of the proof of It o formula's proof for general semimartingales in e.g. [16]. The only (minor) dierence lies in the use of Taylor's theorem, which we apply to f
0(L(x)S + y) rather than f
0(S) .
2
Admissible means that it satises the same regularity requirements as an H− admissible strategy without the equality constraints on the terminal conditions.
3
The superscript f is dropped as the cost function is xed once and for all and no confusion can occur.
Corollary 3.2 The f − cost process of an H − admissible strategy Φ can also be expressed in terms of the portfolio value V
C t (Φ) = f
00(0)
V t − V 0 − R t
0+ δ s− dS s
+f
00(0)l
0(0)
1 2
R t
0+ S s− d[δ, δ] c s + f
(3)2 (0)
[V, V ] c t − 2 R t
0+ δ s− d[V, S] c s + R t
0+ δ 2 s− d[S, S] c t + P
0<s≤t f
0(∆V s − δ s− ∆S s + L(∆δ s , S s ) − ∆δ s S s )
− P
0<s≤t f
00(0)(∆V s − δ s− ∆S s ). (3.2)
Proof A straightforward application of the properties of quadratic variations, when viewing β as a function of V in formula (3.1).
Compared to the case of innite liquidity, the additional term in the expression of C t (Φ) f
00(0)l
0(0)
1 2
R t
0+ S s− d[δ, δ] c s , and it is non-decreasing given the convexity of both f and L .
3.3 The supply price process
Exactly as in the previous paragraph, for an H − admissible trading strategy Φ , one can dene the supply price process S t S (Φ) as the limit in ucp topology, whenever it exists, of
l
nX
k=1
l(δ τ
kn− δ τ
k−1n)S τ
kn− S τ
k−1nfor any sequences P n of Riemann partitions of [0, T ] of length l n . Given an H − admissible strategy Φ , the existence of the associated supply price is ensured by the following result.
Proposition 3.3 The supply price process S S of an H − admissible strategy Φ is well-dened and given by the formula
S t S (Φ) = S t + l
0(0)
δ t S t − δ 0 S 0 − R t
0+ δ s− dS s
+ 1 2 l
00(0) R t
0+ S s− d[δ, δ] c s + P
0<s≤t (l((∆δ s ) − 1)S s − l
0(0)∆δ s S s ) . (3.3) Proof The proof follows the same lines as that of Proposition 3.1 and is omitted.
4 Application to stochastic volatility models
Of great interest is the particularization of the general concepts previously dened to some specic asset dynamics. In this section, the case of stochastic volatility is considered. In order to derive an explicit formula for the f− cost and supply price processes, and completely characterize pseudo-optimal strategies, we then let ( S , σ ) be a solution of the following set of SDE's
dS t = µ t dt + σ t dW t 1 (4.1)
dσ t = γ t dt + Σ t dW t 2 (4.2)
where (W 1 , W 2 ) is a two-dimensional Wiener process under P with correlation ρ , i.e. d < W 1 , W 2 > t = ρdt . Under appropriate conditions hold for the functions µ t , γ t and Σ t , see e.g. [?], (4.1, 4.2) admits a unique strong continuous solution with S t > 0 and σ t > 0 . We will from now on assume that such conditions hold true and restrict our study to a Markovian framework, thereby looking for the optimal strategy Φ as a smooth function of the state variables
δ t = δ(t, S t , σ t )
V t = V (t, S t , σ t ).
4.1 PDE formulation
In order to derive a set of PDE's satised by pseudo-optimal strategies, one rst has to rewrite the cost process as a function of the diusion parameters and the strategy. A straightforward calculation using (3.2) yields
C t (Φ) = Z t
0
f
00(0)
∂V
∂u + ∂V
∂S µ u + ∂V
∂σ γ u + 1 2
∂ 2 V
∂S 2 σ u 2 + 1 2
∂ 2 V
∂σ 2 Σ 2 u + ∂ 2 V
∂S∂σ ρσ u Σ u − δ u µ u
+ f
00(0)l
0(0) S 2
u∂δ
∂S
2
σ 2 u + ∂σ ∂δ 2
Σ 2 u + 2 ∂S ∂δ ∂σ ∂δ ρσ u Σ u
+ f
(3)2 (0)
∂V
∂S
2
σ u 2 + ∂V ∂σ 2
Σ 2 u + 2 ∂V ∂S ∂V ∂σ ρσ u Σ u
− f (3) (0)δ u ∂V
∂S σ u 2 + ∂V ∂σ ρσ u Σ u
+ f
(3)2 (0) δ 2 u σ 2 u i du +
Z t 0
f
00(0) ∂V
∂S − δ u
σ u dW u 1 + Z t
0
f
00(0) ∂V
∂σ Σ u dW u 2 . Likewise, using 3.3), there holds for the supply price process
S S t (Φ) = S t + l
0(0)
δ t S t − δ 0 S 0 − Z t
0
δ u µ u du − Z t
0
δ u σ u dW u 1
+ 1 2 l
00(0) Z t
0
∂δ
∂S 2
σ u 2 + ∂δ
∂σ 2
Σ 2 u + 2 ∂δ
∂S
∂δ
∂σ ρσ u Σ u
! du.
Now, applying to the strategy Φ the rst pseudo-optimality criterion, i.e. that C must be a martingale under the measure P , we nd a rst fully non-linear PDE satised by the strategy (V, δ)
f
00(0) ∂V
∂u + ∂V
∂S µ u + ∂V
∂σ γ u + 1 2
∂ 2 V
∂S 2 σ u 2 + 1 2
∂ 2 V
∂σ 2 Σ 2 u + ∂ 2 V
∂S∂σ ρσ u Σ u − δ u µ u
+f
00(0)l
0(0) S u
2
∂δ
∂S 2
σ 2 u + ∂δ
∂σ 2
Σ 2 u + 2 ∂δ
∂S
∂δ
∂σ ρσ u Σ u
!
+ f (3) (0) 2
∂V
∂S 2
σ u 2 + ∂V
∂σ 2
Σ 2 u + 2 ∂V
∂S
∂V
∂σ ρσ u Σ u
!
−f (3) (0)δ u
∂V
∂S σ 2 u + ∂V
∂σ ρσ u Σ u
+ f (3) (0)
2 δ 2 u σ 2 u = 0.
with terminal condition corresponding V T = δ H S T + β H .
In order to apply the second pseudo-optimality criterion, i.e. that the martingale C must be orthogonal to the martingale part of the supply price process S S , we rst identify the martingale part of the latter
S t S (Φ) − E S S t (Φ)
= Z t
0
1 + l
0(0)S ∂δ
∂S
σ u dW u 1 + Z t
0
l
0(0)S ∂δ
∂σ Σ u dW u 2 , so that the second PDE satised by (V, δ) is
∂V
∂S − δ 1 + l
0(0)S ∂δ
∂S
σ 2 + ∂V
∂σ
1 + l
0(0)S ∂δ
∂S
ρσΣ +
∂V
∂S − δ ∂δ
∂σ l
0(0)SρσΣ + ∂V
∂σ
∂δ
∂σ l
0(0)SΣ 2 = 0.
With some rearrangements, the pseudo-optimal strategy Φ is shown to solve the following coupled system of nonlinear PDEs
∂V
∂u + ∂V ∂S µ + ∂V ∂σ γ + 1 2 ∂ ∂S
2V
2σ 2 + 1 2 ∂ ∂σ
2V
2Σ 2 + ∂S∂σ ∂
2V ρσΣ = δµ + α
∂V
∂S σ + ∂V ∂σ ρΣ − δσ 2
+ (1 − ρ 2 ) ∂V ∂σ 2 Σ 2 +l
0(0) S 2
∂δ
∂S σ + ∂σ ∂δ ρΣ 2
+ (1 − ρ 2 ) ∂δ ∂σ 2 Σ 2
∂V
∂S − δ
1 + l
0(0)S ∂S ∂δ
σ 2 + ∂V ∂σ 1 + l
0(0)S ∂S ∂δ ρσΣ + ∂V ∂S − δ ∂δ
∂σ l
0(0)SρσΣ + ∂V ∂σ ∂δ ∂σ l
0(0)SΣ 2 = 0
(4.3)
with terminal condition V T = δ H S T + β H .
A system such as (4.3) is quite challenging: one can see it as a parabolic equation coupled with a nonlinear, stationary hyperbolic equation which can be viewed as a constraint. The study of (4.3) will be the subject of another work.
The case of a complete market
The case of a complete market corresponds to Σ , the volatility of volatility, equal to zero. The equation for the hedge ratio δ then reduces to
∂V
∂S − δ 1 + l
0(0)S ∂δ
∂S
= 0.
so that a sucient condition is that V, δ is a solution to
δ = ∂V
∂S (4.4)
∂V
∂u + 1 2
∂ 2 V
∂S 2 σ 2
1 + l
0(0)S ∂ 2 V
∂S 2
= 0. (4.5)
Upon the generalized Black and Scholes PDE (4.5) having a solution, a property easily shown to hold when the contingent claim has a convex payo, equation (4.4) gives a perfect hedge. As already holds in the innite liquidity case, the solution does not depend on the risk function f . In fact, the f − cost process is identically zero, which amounts to having a self-nancing strategy incorporating liquidity costs that perfectly replicates the contingent claim H .
4.2 The minimization problem
Despite the fact that, in discrete time, a pseudo-optimal strategy satisfying ((2.1)) might not be optimal, in continuous time, and when working with continuous path processes, there exists a correspondence between the two concepts
4. As a matter of fact, we now prove that a strategy solving system of equations (4.3) is locally risk-minimizing for the function f .
Given the smoothness of the risk function f and the liquidity costs function L , one can write a Taylor expansion around the unperturbed strategy Φ . Given a partition τ and t ∈ [0, T ] , and assuming without loss of generality that t is one of the t i , there holds that
r f τ [Φ, φ](t, ω) = ∆R t
i(Φ + φ| [t
i,t
i+1( )(ω) − ∆R t
i(Φ)(ω) t i+1 − t i
= E t
if (∆C t
i+1(Φ + φ| [t
i,t
i+1( ))
(ω) − E t
if (∆C t
i+1(Φ)) (ω) t i+1 − t i
4
We refer the reader to paragraph 3.2 of [1] for the precise denition of local risk-minimization in continuous time.
Taylor's formula applied to g : (x, y) 7→ f (L(x) + y) then yields for some φ ˜ ∈ []
f (∆C t
i+1(Φ + φ| [t
i,t
i+1() )) = f (∆C t
i+1(Φ)) − β t
if
0(∆C t
i+1(Φ)) − δ t
iL
0(Φ)f
0(∆C t
i+1(Φ)) + 1
2 δ t 2
ih( ˜ φ) + 1
2 (β t
i+ δ t
iL
0( ˜ φ)) 2 g( ˜ φ)
where g( ˜ φ) = f
00(∆C t
i+1( ˜ φ)) and h( ˜ φ) = L
00( ˜ φ)f
0(∆C t
i+1( ˜ φ)) with φ ˜ = ( ˜ β, δ) ˜ such that | β| ≤ ˜ β and | δ| ≤ ˜ δ . Using the standing assumptions on f , namely f
0(0) = 0, f
00(0) > 0 , the remainder term δ t 2
ih( ˜ φ) + (β t
i+ δ t
iL
0( ˜ φ)) 2 g( ˜ φ) will remain strictly positive in a neighborhood of t i for δ t
iand β t
ismall enough. Rearranging and simplifying yields
r τ f [Φ, φ](t, ω) = β t
iE t
if
0(∆C t
i+1(Φ)) (ω)
t i+1 − t i + δ t
iE t
iL
0(Φ)f
0(∆C t
i+1(Φ)) (ω)
t i+1 − t i (4.6)
+ 1 2
E t
iδ t 2
i
h( ˜ φ) (ω)
t i+1 − t i (4.7)
+ 1 2
E t
i(β t
i+ δ t
iL
0( ˜ φ)) 2 g( ˜ φ) (ω)
t i+1 − t i . (4.8)
Thanks to the pathwise conitnuity of It o processes, there holds
t
i+1lim
→tiE t
if
0(∆C t
i+1(Φ)) (ω) t i+1 − t i
= Λ (f
0◦ ∆C) t
i
t
i+1lim
→tiE t
iL
0(Φ)f
0(∆C t
i+1(Φ)) (ω)
t i+1 − t i = Λ (L
0· f
0◦ ∆C) t
i
and
t
i+1lim
→tiE t
ih( ˜ φ)
(ω) t i+1 − t i
= Λh t
it
i+1lim
→tiE t
ig( ˜ φ)
(ω)
t i+1 − t i = Λg t
it
i+1lim
→tiE t
iL
0g( ˜ φ) (ω) t i+1 − t i
= Λ (L
0· g) t
i
t
i+1lim
→tiE t
iL
02g( ˜ φ) (ω)
t i+1 − t i = Λ L
02· g
t
iwhere Λ is the innitesimal generator associated with the diusion:
Λh = ∂h
∂S µ + ∂h
∂σ γ + 1 2
∂ 2 h
∂S 2 σ 2 + 1 2
∂ 2 h
∂σ 2 Σ 2 + ∂ 2 h
∂S∂σ ρσΣ.
Finally, one obtains that the process r f τ in (4.6) is given by
r f τ [φ, ∆](t, ω) = β t Λ (f
0◦ ∆C) t + δ t Λ (L
0· f
0◦ ∆C) t (4.9) + 1 2 β t 2 Λg t + 2β t δ t Λ (L
0· g) t + δ t 2 Λ L
02· g + h
t
. (4.10)
Upon setting the rst component δ of the perturbation equal to zero (that is, we perturb only β ), a rst condition for a strategy φ to be locally risk-minimizing is derived:
β t Λ (f
0◦ ∆C) t + 1
2 β 2 t Λg t ≥ 0 P − a.e. ∀β t .
Hence, there holds Λ (f
0◦ ∆C) t = 0 . Similarly, upon setting now β = 0 , the following second condition for the strategy φ follows:
δ t Λ (L
0· f
0◦ ∆C) t + 1
2 δ t 2 Λ L
02· g + h
t ≥ 0 P − a.e. ∀δ t .
As a consequence, Λ (L
0· f
0◦ ∆C) t = 0 , and one can easily workout the equivalence below Λ (f
0◦ ∆C) t = 0,
Λ (L
0· f
0◦ ∆C) t = 0
⇔
f
00(0)
∂V
∂u + ∂V ∂S µ u + ∂V ∂σ γ u + 1 2 ∂ ∂S
2V
2σ 2 u + 1 2 ∂ ∂σ
2V
2Σ 2 u + ∂S∂σ ∂
2V ρσ u Σ u − δ u µ u +f
00(0)l
0(0) S 2
u∂δ
∂S
2
σ 2 u + ∂δ ∂σ 2
Σ 2 u + 2 ∂S ∂δ ∂σ ∂δ ρσ u Σ u + f
(3)2 (0)
∂V
∂S
2
σ u 2 + ∂V ∂σ 2
Σ 2 u + 2 ∂V ∂S ∂V ∂σ ρσ u Σ u
−f (3) (0)δ u ∂V ∂S σ u 2 + ∂V ∂σ ρσ u Σ u
+ f
(3)2 (0) δ 2 u σ u 2 = 0,
∂V
∂S − δ
1 + l
0(0)S ∂S ∂δ
σ 2 + ∂V ∂σ 1 + l
0(0)S ∂S ∂δ ρσΣ + ∂V ∂S − δ ∂δ
∂σ l
0(0)SρσΣ + ∂V ∂σ ∂σ ∂δ l
0(0)SΣ 2 = 0.
As claimed in the beginning of this section, one can see that the optimal strategies with respect to local risk-minimization are the same as the pseudo-optimal strategies. This result is similar to that obtained in [1] in the case of "innite" liquidity, the only requirement being that the innitesimal generator is a local operator.
5 Application to stochastic volatility/jump diusion models
This section is devoted to a situation where non-quadratic risk denitely implies a dierent hedging strategy.
The evolution of S is modelled by an SDE with stochastic volatility and Poisson jumps in the vein of the Bates model [?]
dS t = µ t dt + σ t dW t 1 + kdN t
dσ t = γ t dt + Σ t dW t 2 ,
where as before W 1 and W 2 are Wiener processes under P and d < W 1 , W 2 > t = ρdt , N t is a Poisson process with intensity λ , and the amplitude of the jumps k has probability distribution K . We also assume that W t , N t and k are independent. Similarly to the case of stochastic volatility, standard assumptions are made to ensure that the set of SDE has a unique strong solution.
With these assumptions, one can look for the optimal strategy Φ , in a Markovian framework, as a function of the state variables
δ t = δ(t, S t , σ t )
V t = V (t, S t , σ t )
and derive an equation for these quantities.
5.1 PIDE formulation
The PIDE's corresponding to respectively the portfolio value and optimal strategy follow as in Section ??
fro a rewriteing of the cost process C t (Φ) = R t
0
f
00(0)
∂V
∂u + ∂V ∂S µ u + ∂V ∂σ γ u + 1 2 ∂ ∂S
2V
2σ 2 u + 1 2 ∂ ∂σ
2V
2Σ 2 u + ∂S∂σ ∂
2V ρσ u Σ u − δ u µ u
+ f
(3)2 (0)
∂V
∂S
2
σ u 2 + ∂V ∂σ 2
Σ 2 u + 2 ∂V ∂S ∂V ∂σ ρσ u Σ u
+ l
0(0)S 2
u∂δ
∂S
2
σ u 2 + ∂σ ∂δ 2
Σ 2 u + 2 ∂S ∂δ ∂σ ∂δ ρσ u Σ u
− f (3) (0)δ u− ∂V ∂S σ 2 u + ∂V ∂σ ρσ u Σ u
+ f
(3)2 (0) δ u 2 σ 2 u du
+ R t
0 f
00(0) ∂V ∂S − δ u−
σ u dW u 1
+ R t
0 f
00(0) ∂V ∂σ Σ u dW u 2
+ R t
0
R
R
f
0(∆V u − δ u− ∆S u + L(∆δ u , S u ) − ∆δ u S u )K(k)dkdN u
which follows from equation (3.2). Note that ∆V u , the jump of V when there S has a jump ∆S u of size k at time u , is equal to V (u−, S u− + k, σ u− ) − V (u−, S u− , σ u− ) , and similarly for ∆δ u .
Applying the rst pseudo-optimality criterion to the strategy Φ , i.e. that C is a martingale under the measure P , yields the PIDE satised by the portfolio value V
f
00(0) ∂V
∂u + ∂V
∂S µ u + ∂V
∂σ γ u + 1 2
∂ 2 V
∂S 2 σ 2 u + 1 2
∂ 2 V
∂σ 2 Σ 2 u + ∂ 2 V
∂S∂σ ρσ u Σ u − δ u− µ u
+ f (3) (0) 2
∂V
∂S 2
σ u 2 + ∂V
∂σ 2
Σ 2 u + 2 ∂V
∂S
∂V
∂σ ρσ u Σ u
!
+ l
0(0)S u
2
∂δ
∂S 2
σ u 2 + ∂δ
∂σ 2
Σ 2 u + 2 ∂δ
∂S
∂V
∂σ ρσ u Σ u
!
−f (3) (0)δ u−
∂V
∂S σ u 2 + ∂V
∂σ ρσ u Σ u
+ f (3) (0) 2 δ u− 2 σ 2 u +
Z
R
f
0(∆V u − δ u− ∆S u )K(k)dkλ u = 0 with terminal conditions
V T = δ H S T + β H .
In order to apply the second pseudo-optimality criterion, i.e. that the martingale C be orthogonal to the martingale part of the supply price process S S , we rst identify the martingale part
S t S (Φ) − E S S t (Φ)
= Z t
0
1 + l
0(0)S ∂δ
∂S
σ u dW u 1 + Z t
0
l
0(0)S ∂δ
∂σ Σ u dW u 2 + R t
0
R
R
((l(∆δ u ) − 1)S u + k) K(k)dkd N ˜ u
with N ˜ , the compensated Poisson process of N . Therefore the second PDE satised by the strategy (V, δ) is ∂V
∂S − δ 1 + l
0(0)S ∂δ
∂S
σ 2 + ∂V
∂σ
1 + l
0(0)S ∂δ
∂S
ρσΣ +
∂V
∂S − δ ∂δ
∂σ l
0(0)SρσΣ + ∂V
∂σ
∂δ
∂σ l
0(0)SΣ 2 +
Z
R
