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Local Risk-Minimization under the Benchmark Approach

We study the pricing and hedging of derivatives in incomplete financial markets by considering the local risk-minimization method in the context of the benchmark approach, which will be called benchmarked local risk-minimization. We show that the proposed benchmarked local risk-minimization allows to handle under extremely weak assumptions a much richer modeling world than the classical methodology.


Introduction
The valuation and hedging of derivatives in incomplete financial markets is a frequently studied problem in mathematical finance. The goal of this paper is to discuss the concept of local risk-minimization under the benchmark approach (see e.g. [8], [9], [15], [20] and [21]), a general modeling framework that only requires the existence of a benchmark, the numéraire portfolio. According to this approach, even under the absence of an equivalent local martingale measure (in short ELMM), contingent claims can be consistently evaluated by means of the so-called real world pricing formula, which generalizes standard valuation formulas, where the discounting factor is the numéraire portfolio and the pricing measure is the physical probability measure P. Local risk-minimization under the benchmark approach has been also studied in [7] in the case of jump-diffusion markets. In our paper the approach is more general, since we do not assume any specific market model for the primitive assets whose price processes may have jumps. We analyze the relationship between the classical local risk-minimization and the benchmark approach and revisit this quadratic hedging method in this modeling framework. This is rather different from [7], where the definition of cost process and optimal strategy are revisited in a Brownian setting and the square-integrability condition is dropped. In [7] the cost is then interpreted in a different sense in terms of cost condition. However, we should stress that the concept of risk (see Definition 3.3) associated to an admissible strategy is well-defined only if the cost process is assumed to be square-integrable. Another difference between the two papers is also that we are considering a general setting with no specification of the asset dynamics, while in [7] the Brownian setting of the underlying model plays a crucial role.
First of all, we study the local risk-minimization method in the case when the benchmarked asset prices are P-local martingales, which will correspond to benchmarked risk-minimization. This includes continuous market models (see Section 3.1) and a wide class of jump-diffusion models (see for example [21], Chapter 14, pages 513-549). This property implies several advantages since in market models, where the discounted asset prices are given by P-local martingales, the local risk-minimization method coincides with riskminimization, as introduced originally in [11]. In the local risk-minimization approach, the optimal strategy is often calculated by switching to a particular martingale measure P (the minimal martingale measure) and computing the Galtchouck-Kunita-Watanabe (in short GKW) decomposition of a benchmarked contingent claimĤ under P. However, this method has two main disadvantages: (i) the minimal measure P may not exist, as it is often the case in the presence of jumps affecting the asset price dynamics; (ii) if P exists, the GKW decomposition ofĤ under P must satisfy some particular integrability conditions under the real world probability measure P to give the Föllmer-Schweizer decomposition ofĤ. On the contrary, the risk-minimization approach that we discuss in this paper for the case of benchmarked market models, does not face the same technical difficulties as the local risk-minimization one. It formalizes in a straightforward mathematical way the economic intuition of risk and delivers always an optimal strategy for a given benchmarked contingent claimĤ ∈ L 2 (F T , P) 1 , obtained by computing the GKW decomposition ofĤ under P.
Furthermore, in this setting we establish a fundamental relationship between real world pricing and benchmarked risk-minimization. In market models, where the asset prices are given by P-local martingales, by Theorem 3.6 we will obtain the result that the benchmarked portfolio's value of the risk-minimizing strategy forĤ ∈ L 2 (F T , P) coincides with the real world pricing formula forĤ. The benchmarked contingent claimĤ can be written asĤ =Ĥ 0 + T 0 ξĤ u dŜ u + LĤ T P − a.s., (1.1) where LĤ is a square-integrable P-martingale with LĤ 0 = 0 strongly orthogonal 2 toŜ. Decomposition (1.1) allows us to decompose every squareintegrable benchmarked contingent claim as the sum of its hedgeable part H h and its unhedgeable partĤ u such that we can writê The space L 2 (FT , P) denotes the set of all FT -measurable random variables H such that E H 2 = H 2 dP < ∞. 2 Two P-local martingales M and N are called strongly orthogonal if their product M N is a P-local martingale.
Here the notation ξĤ · dŜ characterizes the integral of the vector process ξĤ with respect to the vector processŜ (see e.g. [19]). Note that the benchmarked hedgeable partĤ h can be replicated perfectly, i.e.
and ξĤ yields the fair strategy for the self-financing replication of the hedgeable part ofĤ. The remaining benchmarked unhedgeable part can be diversified and will be covered through the cost process C := LĤ −Ĥ 0 . The connection between risk-minimization and real world pricing is then an important insight, which gives a clear reasoning for the pricing and hedging of contingent claims via real world pricing also in incomplete markets. A natural question concerns indeed the invariance of the risk-minimizing strategy under a change of numéraire. By [3] this property always holds in the case of continuous assets prices. Here we show that this result is also true more generally: it is sufficient that the orthogonal martingale structure is generated by continuous P-(local) martingales.
Then we also study the case when the benchmarked processes are Psupermartingales. In particular we consider a market model, where incompleteness is due to incomplete information. In this setting we show that a benchmarked locally risk-minimizing strategy can be determined by computing the predictable projection of the strategy in the completed market without any specification of the asset price dynamics (see Theorem 3.15). The proof we provide holds when the discounted asset prices are special semimartingales in S 2 (P) 3 , hence in particular for all benchmarked underlying assets in S 2 (P) by the Doob's decomposition. This extends the results of [10], where they assume continuity of the underlying prices processes.
Finally, we provide some examples to illustrate how to compute the GKW decomposition in the minimal market model with random scaling, where there exists no ELMM, but the primitive assets are still P-local martingales if benchmarked.
The local risk-minimization method under the benchmark approach has acquired new importance for pricing and hedging in hybrid markets and insurance markets (see [1] and [4]). Since hybrid markets are intrinsically incomplete, perfect replication of contingent claims is not always possible and one has to apply one of the several methods for pricing and hedging in incomplete markets. Local risk-minimization appears to be one of the 3 Given the Doob-Meyer decomposition of a P-semimartingale X into a P-local martingale M = {Mt, t ∈ [0, T ]} and an Fpredictable process V = {Vt, t ∈ [0, T ]} of finite variation, we say that X ∈ S 2 (P) if the following integrability condition is satisfied Here |V | = {|V |t, t ∈ [0, T ]} denotes the total variation of the process V . most suitable methods when the market is affected by orthogonal sources of randomness, such as the ones represented by mortality risk and catastrophic risks. The results of this paper provide a new simplified framework for applying benchmarked local risk-minimization.

Financial Market
To describe a financial market in continuous time, we introduce a probability space (Ω, F, P), a time horizon T ∈ (0, ∞) and a filtration F := (F t ) 0≤t≤T that is assumed to satisfy F t ⊆ F for all t ∈ [0, T ], as well as the usual hypotheses of completeness and right-continuity and saturation by all P-null sets of F. In our market model we can find d adapted, nonnegative primary security account processes represented by (càdlàg) P-semimartingales Additionally, the 0-th security account S 0 t denotes the value of the adapted strictly positive savings account at time t ∈ [0, T ]. The j-th primary security account holds units of the j-th primary security plus its accumulated dividends or interest payments, j ∈ {1, 2, . . . , d}. In this setting, market participants can trade in order to reallocate their wealth.
Here δ j t , j ∈ {0, 1, . . . , d}, denotes the number of units of the j-th security account that are held at time t ≥ 0 in the corresponding portfolio S δ = {S δ t , t ∈ [0, T ]}. Following [3], we define the value S δ of this portfolio as given by a càdlàg optional process such that A strategy δ and the corresponding portfolioŜ δ are said to be self-financing if Note that the stochastic integral of the vector process δ with respect to S is well-defined because of our assumptions on δ. Furthermore, a ⊤ denotes the transpose of a. In general, we do not request strategies to be self-financing. Denote by V + x , (V x ), the set of all strictly positive, (nonnegative), finite, self-financing portfolios, with initial capital x > 0, (x ≥ 0). Definition 2.2. A portfolio S δ * ∈ V + 1 is called a numéraire portfolio, if any nonnegative portfolio S δ ∈ V + 1 , when denominated in units of S δ * , forms a P-supermartingale, that is,
To establish the modeling framework, we make the following (extremely weak) key assumption, which is satisfied for almost all models of practical interest, see e.g. [21] and [15].
There exists a numéraire portfolio S δ * ∈ V + 1 . From now on, let us choose the numéraire portfolio as benchmark. We call any security, when expressed in units of the numéraire portfolio, a benchmarked security and refer to this procedure as benchmarking. The benchmarked value of a portfolio S δ is of particular interest and is given by the ratioŜ If a benchmarked price process is a P-martingale, then we call it fair. In this case we would have equality in relationship (2.2) of Definition 2.2. The benchmark approach developed in [15], [17] and [21] uses the numéraire portfolio for derivative pricing without using equivalent martingale measures. In portfolio optimization the numéraire portfolio, which is also the growth optimal portfolio, is in many other ways the best performing self-financing portfolio, see [16] and [18]. As shown in [21], jump-diffusion and Itô process driven market models have a numéraire portfolio under very general assumptions, where benchmarked nonnegative portfolios turn out to be P-local martingales and, thus, P-supermartingales. In [15] the question on the existence of a numéraire portfolio in a general semimartingale market is studied. In order to guarantee the economic viability of our framework, we check whether obvious arbitrage opportunities are excluded. A strong form of arbitrage would arise when a market participant could generate strictly positive wealth from zero initial capital via his or her nonnegative portfolio of total wealth. Definition 2.4. A benchmarked nonnegative self-financing portfolioŜ δ is a strong arbitrage if it starts with zero initial capital, that isŜ δ 0 = 0, and generates some strictly positive wealth with strictly positive probability at a later time t ∈ (0, T ], that is P(Ŝ δ t > 0) > 0. Thanks to the supermartingale property (2.2), the existence of the numéraire portfolio guarantees that strong arbitrage is automatically excluded in the given general setting, see [21]. However, some weaker forms of arbitrage may still exist. These would require to allow for negative portfolios of total wealth of those market participants who fully focus on exploiting such weaker forms of arbitrage, which is not possible in reality due to bankruptcy laws. This emphasizes the fact that an economically motivated notion of arbitrage should rely on nonnegative portfolios. Within this paper, we consider a discounted European style contingent claim. Such a benchmarked claimĤ (expressed in units of the benchmark) is given by the F T -measurable, nonnegative random payoffĤ that is delivered at time T . We will here always assume that a benchmarked contingent claim H belongs to L 2 (F T , P). Given a benchmarked contingent claimĤ, there are at least two tasks that a potential seller ofĤ may want to accomplish: the pricing by assigning a value toĤ at times t < T ; and the hedging by covering as much as possible against potential losses arising from the uncertainty ofĤ. If the market is complete, then there exists a self-financing strategy δ whose terminal valuê S δ T equalsĤ with probability one, see [21]. More precisely, the real world pricing formulaŜ provides the description for the benchmarked fair portfolio at time t ∈ [0, T ], which is the least expensive P-supermartingale that replicates the benchmarked payoffĤ if it admits a replicating self-financing strategy δ H witĥ S δ H T =Ĥ. HereŜ δ H forms by definition a P-martingale. The benchmark approach allows other self-financing hedge portfolios to exist forĤ, see [21]. However, these nonnegative portfolios are not P-martingales and, as P-supermartingales, more expensive than the P-martingaleŜ δ H given in (2.3), see [21]. Completeness is a rather delicate property that does not cover a large class of realistic market models. Here we choose the (local) risk-minimization approach (see e.g. [10], [11] and [24]) to price non-hedgeable contingent claims. In this paper, we first investigate the case of benchmarked securities that represent P-local martingales and study risk-minimization as originally introduced in [11]. We will see that this covers many cases in the context of the benchmark approach including all continuous financial market models, a wide range of jump-diffusion driven market models and cases like the minimal market model that do not have an equivalent risk neutral probability measure. Then we will study the general case when benchmarked securities are P-supermartingales that are not necessarily P-local martingales. As indicated earlier, we will refer to local risk-minimization under the benchmark approach as benchmarked local risk-minimization.

Local Risk-Minimization with Benchmarked Assets
Our aim is to investigate a concept of local risk-minimization similar to the one in [13] and [24], which used the savings account as reference unit. Here we use the numéraire portfolio as discounting factor and benchmark. The main feature of a local risk-minimization concept is the fact that one insists on the replication requirementŜ δ T =Ĥ. IfĤ is not hedgeable, then this forces one to work with strategies that are not self-financing and the aim becomes to minimize the resulting intrinsic risk or cost under a suitable criterion. As we will see, rather natural and tractable are quadratic hedging criteria, where we refer to [24] and [13] for extensive surveys. Important is the fact that there are realistic situations that we will cover, which would be excluded because a minimal martingale measure may not exist for the respective models. For example, in the case of local riskminimization of financial derivatives based on insurance products, the minimal martingale measure may often not exist because of the presence of jumps in the underlying.
We recall that under Assumption 2.3, the benchmarked value of any nonnegative, self-financing portfolio forms a P-supermartingale, see (2.2). In particular, the vector of the d + 1 benchmarked primary security accountŝ S = (Ŝ 0 ,Ŝ 1 , . . . ,Ŝ d ) ⊤ forms with each of its components a nonnegative Psupermartingale. By Theorem VII.12 of [6], we know that the vector procesŝ S has a unique decomposition of the form where M is a vector P-local martingale and V is a right-continuous Fpredictable finite variation vector process with M 0 = V 0 = 0, with 0 denoting the (d + 1)-dimensional null vector. This expresses the fact that every right-continuous P-supermartingale is a special P-semimartingale.

Benchmarked Local Martingales.
We now discuss the case when benchmarked securities are P-local martingales. Let us assume that the vector of the d + 1 discounted primary security accounts continuous and null at 0. Moreover, M X is a vector P-local martingale and A X is an adapted, finite variation vector process. The bracket process Since Assumption 2.3 is in force, Theorem 3.4 of [14] ensures that the structure condition 4 is satisfied and the discounted numéraire portfolioS δ * t = S δ * t S 0 t at any time t is given byS where the processẐ corresponds to the stochastic exponential which is then well-defined and a strictly positive P-local martingale. Via Itô's product rule, it is easy to check that the vector processŜ of benchmarked primary security accounts is a P-local martingale, and thus, a P- 4 We say that X satisfies the structure condition if A X is absolutely continuous with respect to M X , in the sense that there exists an . . , d} and t ∈ [0, T ], and the mean-variance tradeoff processKt = t 0λ This implies that whenever we consider continuous primary security account processes, they are P-local martingales when expressed in units of the numéraire portfolio. In the general case when S t can have jumps, it is not possible to provide an analogous explicit description of the numéraire portfolio S δ * t or, more precisely, its generating strategy δ * . An implicit description can be found in [15], Theorem 3.15, or more generally in [12], Theorem 3.2 and Corollary 3.2. In both cases, δ * can be obtained by pointwise maximization of a function that is given explicitly in terms of semimartingale characteristics. If S is discontinuous, such a pointwise maximizer is only defined implicitly and neither of the above descriptions provides explicit expressions for δ * . However, a wide class of jump-diffusion market models is driven by primary security account processes that turn out to be, when expressed in units of the numéraire portfolio, P-local martingales, see e.g. [21], Chapter 14. For example, this is the case in jump-diffusion markets, that is, when security price processes exhibit intensity based jumps due to event risk, see [21], Chapter 14, page 513. These results allow us to consider below risk-minimization in the case when the benchmarked assets are given by P-local martingales.
3.1.1. Risk-Minimization with Benchmarked Assets. Since at this stage we refer to the case where benchmarked securities represent P-local martingales (i.e. we assume V ≡ 0 for all t ∈ [0, T ] in (3.1)), we study riskminimization as originally introduced in [11] under the benchmark approach, that is, benchmarked risk-minimization. In particular, since we are considering a (general) discounting factor (different from the usual money market account), we follow the approach of [3] for local risk-minimization under a given numéraire.
T ]} such that (i) the associated portfolioŜ δ is a square-integrable stochastic process whose left-limit is equal toŜ δ t− = δ t ·Ŝ t , (ii) the stochastic integral δ · dŜ is such that Here ..,d denotes the matrix-valued optional covariance process ofŜ.
Recall that the market may be not complete. We also admit strategies that are not self-financing and may generate benchmarked profits or losses over time.
Definition 3.2. For any L 2 -admissible strategy δ, the benchmarked cost processĈ δ is defined bŷ HereĈ δ t describes the total costs incurred by δ over the interval [0, t].
Definition 3.3. For an L 2 -admissible strategy δ, the corresponding risk at time t is defined bŷ where the benchmarked cost processĈ δ , given in (3.3), is assumed to be square-integrable.
IfĈ δ is constant, then it equals zero and the strategy is self-financing. Our goal is to find an L 2 -admissible strategy δ, which minimizes the associated risk measured by the fluctuations of its benchmarked cost process in a suitable sense.
Definition 3.4. Given a benchmarked contingent claimĤ ∈ L 2 (F T , P), an L 2 -admissible strategy δ is said to be benchmarked risk-minimizing if the following conditions hold: For the proof of Lemma 3.5, we refer to Section A in the Appendix. Hence benchmarked risk-minimizing strategies are "self-financing on average". We will see, to find a benchmarked risk-minimizing strategy corresponds to finding a suitable decomposition of the benchmarked claim adapted to this setting. Let M 2 0 (P) be the space of all square-integrable P-martingales starting at null at the initial time.
Proof. The proof follows from Theorem 2.4 of [24] and Lemma 3.5.
Thus, the problem of minimizing risk is reduced to finding the representation (3.4). A natural question is whether the benchmarked risk-minimizing strategy is invariant under a change of numéraire. We address this issue to Section 3.3.1.

3.2.
Relationship to Real World Pricing. Definition 3.7. We say that a nonnegative benchmarked contingent claim Decomposition (3.4) and Definition 3.7 allow us to decompose every nonnegative, square-integrable benchmarked contingent claim as the sum of its hedgeable partĤ h and its unhedgeable partĤ u such that we can writê , strongly orthogonal to each component ofŜ. There is a close relationship between benchmarked risk-minimization and real world pricing, as we will see now. Let us apply the real world pricing formula (2.3) to the benchmarked contingent claimĤ in order to get its benchmarked fair priceÛ H (t) at time t. Recall, by its martingale property that the benchmarked fair price is the best forecast of its future benchmarked prices. Due to the supermartingale property (2.2), it follows that we characterize, when using the real world pricing formula (2.3) for obtaining the fair price of the hedgeable part, the least expensive replicating portfolio forĤ h by taking the conditional expectation E Ĥ h F t under the real world probability measure P. Then by (3.5) we haveÛ for every t ∈ [0, T ]. Note that the benchmarked hedgeable partĤ h can be replicated perfectly, i.e.
In particular, for t = 0 one has for the benchmarked hedgeable part On the other hand, we have for the benchmarked unhedgeable part Consequently, for the nonnegative benchmarked payoffĤ, its benchmarked fair priceÛ H (0) at time t = 0, is given bŷ The real world pricing formula (2.3) appears in the form of a conditional expectation and, thus, as a projection in a least squares sense. More precisely, the benchmarked fair priceÛ H (0) can be interpreted as the least squares projection ofĤ into the space of F 0 -measurable benchmarked values. Note that the benchmarked fair priceÛ H u (0) of the unhedgeable partĤ u = LĤ T is zero at time t = 0. Recall that the benchmarked hedgeable part is priced at time t = 0 such that the minimal possible price, the fair price, results. Viewed from time t = 0 the benchmarked costĈ δ T =Ĥ 0 +LĤ T , see Theorem 3.6, has then minimal variance Var LĤ T . This means that the application of the real world pricing formula to a benchmarked payoff at time t = 0 leaves its benchmarked unhedgeable part totally untouched. This is reasonable because any extra trading could only create unnecessary uncertainty and potential additional benchmarked costs. Of course, once the benchmarked fair price is used to establish a hedge portfolio, a benchmarked cost emerges according to Theorem 3.6 if there was an unhedgeable part in the benchmarked contingent claim. The following practically important insight is worth mentioning: Remark 3.8. From a large financial institution's point of view, the benchmarked profits & losses due to the optimal costs in its derivative book have minimal variance when evaluated under real world pricing and viewed at time t = 0. If they are large in number and independent, then the Law of Large Numbers reduces asymptotically the variance of the benchmarked pooled profit & loss to zero and, thus, its value to zero. Obviously, requesting from clients higher prices than fair prices would make the bank less competitive. On the other hand, charging lower prices than fair prices would make it unsustainable in the long run because it would suffer on average a loss. In this sense fair pricing of unhedgeable claims is most natural and yields economically correct prices. Accordingly, benchmarked riskminimization is a very natural risk management strategy for pricing and hedging. Moreover, it is mathematically convenient and for many models rather tractable when using the GKW-decomposition.
With the above notation, we obtain by Theorem 3.6 and (3.4) for the benchmarked payoffĤ ∈ L 2 (F T , P) the following decomposition: This allows us to summarize the relationship between benchmarked riskminimization and real world pricing. In our settingĤ ∈ L 2 (F T , P) admits a benchmarked risk-minimizing strategy and the decomposition forĤ, provided by the real world pricing formula, coincides with the decomposition (3.4), where ξĤ yields the fair strategy for the self-financing replication of the hedgeable part ofĤ. The remaining benchmarked unhedgeable part, given by the benchmarked cost process LĤ , can be diversified. Note that diversification takes place under the real world probability measure and not under some putative risk neutral measure. This is an important insight, which gives a clear reasoning for the pricing and hedging of contingent claims via real world pricing in incomplete markets.

Local Risk-Minimization with Benchmarked Assets.
We now consider the general situation, where the vector of the d+1 benchmarked primary security accountsŜ = (Ŝ 0 ,Ŝ 1 , . . . ,Ŝ d ) ⊤ forms with each of its components a locally square-integrable nonnegative P-supermartingale with V = 0 in decomposition (3.1), and hence a special P-semimartingale. In view of Proposition 3.1 in [24], Definition 3.3 does not hold in this non-martingale case due to a compatibility problem. Indeed as observed in [24], at any time t we minimizeR δ t over all admissible continuations from t on and obtain a continuation which is optimal when viewed in t only. But for s < t, the s-optimal continuation from s onward highlights what to do on the whole interval (s, T ] ⊃ (t, T ] and this may be different from what the t-optimal continuation from t on prescribes. However, it is possible to characterize benchmarked pseudo-locally risk-minimizing strategies 5 through the following well-known result, see [24]. Proposition 3.9. A benchmarked contingent claimĤ ∈ L 2 (F T , P) admits a benchmarked pseudo-locally risk-minimizing strategy δ withŜ δ T =Ĥ P-a.s. if and only ifĤ can be written aŝ withĤ 0 ∈ L 2 (F 0 , P), ξĤ is an F-predictable vector process satisfying the following integrability condition where for ω ∈ Ω, dV s (ω) denotes the (signed) Lebesgue-Stieltjes measure corresponding to the finite variation function s → V s (ω) and |dV s |(ω) the 5 The original definition of a locally risk-minimizing strategy is given in [24] and formalizes the intuitive idea that changing an optimal strategy over a small time interval increases the risk, at least asymptotically. Since it is a rather technical definition, it has been introduced the concept of a pseudo-locally risk-minimizing strategy that is both easier to find and to characterize, as Proposition 3.9 will show in the following. Moreover, in the one-dimensional case and ifŜ is sufficiently well-behaved, pseudo-optimal and locally risk-minimizing strategies are the same. associated total variation measure, and LĤ ∈ M 2 0 (P) is strongly orthogonal to M . The strategy δ is then given by its benchmarked value process iŝ and the benchmarked cost process equalŝ Proof. This result extends Lemma 3.1 in [3]. For the reader's convenience we provide here briefly the proof of (3.7). It is formally analogous to the one of Lemma 3.1 in [3]. By Itô's formula, we have By Lemma 3.10, the cost process of a risk-minimizing strategy (with respect to a given discounting factor) is given by a P-martingale. This property provides a fundamental characterization of (local) risk-minimizing strategies (with respect to a given discounting factor). Here we show that they are invariant under a change of numéraire.
Proposition 3.11. Under the same hypotheses of the previous lemma, if the processC δ is a continuous P-local martingale strongly orthogonal to the martingale part ofS, thenĈ δ is also a (continuous) P-local martingale strongly orthogonal to the martingale part ofŜ.
Proof. This result generalizes Proposition 3.1 of [3]. The proof essentially follows from Itô's formula and Lemma 3.10. From integration by parts formula, we have that From Lemma 3.10, we have where we have used the fact thatC δ is a continuous P-local martingale strongly orthogonal to the martingale part ofS. Furthermore, If nowC δ is a P-local martingale strongly orthogonal toS, then By (3.8) and (3.9), we have that also d Ĉ δ , S S δ * t = 0, henceĈ δ is strongly orthogonal to the martingale part of S. This concludes the proof. Now it is possible to state the main result that guarantees that invariance under change of numéraire is kept in the case of right-continuous asset price processes if we assume that the cost processC δ is continuous. Theorem 3.12. Let δ be an L 2 -admissible strategy with respect to the numéraires S 0 and S δ * and assume thatC δ is continuous. If δ is locally risk-minimizing under the numéraire S 0 , then δ is locally risk-minimizing also with respect to the numéraire S δ * , i.e. it is benchmarked locally riskminimizing.
Proof. The proof is an immediate consequence of Proposition 3.11: if δ is a locally risk-minimizing strategy under S 0 , the costĈ δ is a P-local martingale strongly orthogonal to the martingale part ofŜ. But since the strategy is L 2 -admissible with respect to S δ * , the cost processĈ δ is actually a square integrable P-martingale.

3.4.
Benchmarked local risk-minimization under incomplete information. Here we show an example of benchmarked local risk-minimization that works under general assumptions onŜ. Similarly to [10], we consider a situation where the financial market would be complete if we had more information. The available information is described by the filtration F. We suppose that the benchmarked claimĤ is attainable with respect to some larger filtration. Only at the terminal time T , but not at times t < T , all the information relevant for a perfect hedging of a claim will be available to us. So letF := (F t ) 0≤t≤T be a right-continuous filtration such that We now show how the results of [10] hold without assuming that the underlying asset price processes are continuous, if we consider a general benchmarked market. Note furthermore that we are not going to assume that the benchmarked assets are P-local martingales. Consider now the benchmarked asset price processŜ. ThenŜ is a Psupermartingale and admits the Doob-Meyer's decomposition where A is an F-predictable increasing finite variation process and M is a P-local martingale.
Assumption 3. 13. Suppose now that the vector processŜ belongs to S 2 (P), that is, the space of P-semimartingales satisfying the integrability condition where |A| = {|A| t : t ∈ [0, T ]} is the total variation of A. In addition, the decomposition (3.1) ofŜ with respect to F is still valid with respect toF. In other words we assume that M is a P-martingale with respect toF, although it is adapted to the smaller filtration F.
Step 1. First we need to check that all components in (3.12) are square-integrable. Denote by p X the (dual) F-predictable projection of a process X. By (3.11) and by the properties of the F-predictable projection (see [6], VI.57), where the last equality holds since M is the F-predictable dual projection (see e.g. [6], VI.73) of [M ]. By Jensen's inequality we have for every predictable F-stopping time τ , on the set {τ < ∞}. Hence if we consider τ = s, again by the properties of the F-predictable dual projection, it follows that In order to show that T 0 ξĤ s dA s ∈ L 2 (F T , P), we prove that for any bounded F T -measurable random variable Z T with L 2 -norm Z T 2 . Let Z = {Z t , t ∈ [0, T ]} denote a right-continuous version with left-limits of the (F, P)-martingale E[Z T |F t ], t ∈ [0, T ], and put Z * = sup 0≤t≤T |Z t |.
Since A is F-predictable, we can use some properties of the F-predictable projection (see VI.45 and VI.57 in [6]), and obtain where in the last inequality we have used (3.11) and Doob's inequality for the supremum of a square-integrable P-martingale.
Step 2. Clearly,H 0 −Ĥ 0 ∈ L 2 (Ω,F 0 , P) is orthogonal to all squareintegrable stochastic integrals of M with respect to the filtrationF, hence in particular with respect to the filtration F. Then, it only remains to show that for all bounded F-predictable processes µ = {µ t , t ∈ [0, T ]}. This will imply that the (F, P)-martingale LĤ is orthogonal to M . First we note that (3.13) is equivalent to the following (3.14) Then we decompose the left-side of (3.14) into We have and 16) by the property of the F-predictable projection (see VI.45 in [6]). Since M is the F-predictable dual projection of [M ], we can rewrite (3.15) as follows where the second equality (3.17) follows from Remark 44(e) in [6]. Now, from the properties of the F-predictable projection it is clear thatξĤ can be replaced by ξĤ in (3.16), and this yields (3.13).

Applications
In the remaining part of the paper we discuss some examples that illustrate how classical (local) risk-minimization is generalized to benchmarked (local) risk-minimization in a market when there is no equivalent risk-neutral probability measure. Finally, we will demonstrate that the presence of jumps does not create a major problem, which is not easily resolved under classical (local) risk-minimization.

Benchmarked Risk-Minimization in the Minimal Market Model
with Random Scaling. The notion of a minimal market model (in short MMM) is due to E. Platen and has been introduced in a series of papers with various co-authors; see Chapter 13 of [21] for a recent textbook account. The version of the MMM described here, which generalizes the stylized version derived in Section 13.2 of [21], is governed by a particular choice of the discounted numéraire portfolio drift. The MMM generates stochastic volatilities that involve transformations of squared Bessel processes. We begin by describing a continuous financial market model almost similarly as in Chapter 10 of [21]. More precisely, in this framework uncertainty is modeled by d independent standard Wiener processes We assume that the value at time t of the savings account S 0 is given by T ]} denotes the F-adapted short term interest rate. For simplicity, we suppose r t = r ≥ 0, for every t ∈ [0, T ]. Furthermore, we assume that the dynamics of the primary security account processes S j = {S j t , t ∈ [0, T ]}, j ∈ 1, 2, . . . , d, are given by the SDE for t ∈ [0, T ] with S j 0 > 0. The j-th appreciation rate a j = {a j t , t ∈ [0, T ]} and the (j, k)-th volatility b j,k = {b j,k t , t ∈ [0, T ]} are F-predictable According to Section 13.4 of [21], we now assume that the discounted numéraire portfolio drift is given by where Z is a a squared Bessel process with a general dimension δ > 2 satisfying the SDE for t ∈ [0, T ] with Z 0 > 0. Note that with this choice of α, we have that for t ∈ [0, T ] and δ ∈ (2, ∞). We should stress that for the standard choice δ = 4 and γ t = 1 for every t ∈ [0, T ], we recover the stylized MMM, see Sect.13.2 of [21]. Note also that for the standard case with δ = 4 the discounted numéraire portfolio drift does not depend on Z t . According to [21], we assume here that the scaling process γ is a nonnegative, F-adapted stochastic process that satisfies a SDE of the form with a random initial value γ 0 > 0. HereW is a Wiener process that models some uncertainty in trading activity and is assumed to be independent of W . The scaling correlation ρ is, for simplicity, assumed to be constant. Under this formulation the dynamics of the diffusion process γ can be chosen to match empirical evidence, see Section 13.4 of [21] for some examples. Note that an equivalent risk neutral probability measure does not exist for the above model. The benchmarked savings accountŜ 0 and, thus, the candidate Radon-Nikodym derivative process Λ = {Λ t , t ∈ [0, T ]} with are by (8.7.24) of [21] strict F-local martingales when we assume no correlation, that is ρ = 0. Note also thatŜ 0 is not square-integrable, see formula (8.7.14) in [21]. We now consider an application of risk-minimization in the MMM with random scaling. For the sake of simplicity, we refer to a generalized MMM where we can find two primary security account processes S 1 and S 2 , whose dynamics are described at each time t ∈ [0, T ] by the SDE (4.1), for j = 1, 2.
We recall that the primary security account processes S j , j = 1, 2, are driven by the two independent standard Wiener processes W 1 and W 2 , i.e. their behavior is described by the following SDE: with S j 0 > 0. By applying Itô's formula, it is easy to compute the following SDE: dŜ j Then,Ŝ j is an F-(local) martingale for both j = 1, 2. We assume that E Ŝ j T < ∞, for j ∈ {1, 2}. Then by Corollary 4 on page 74 of Chapter II of [22], we obtain that E (Ŝ j T ) 2 = E [Ŝ j ] T < ∞ and thatŜ j is a square-integrable F-martingale for both j = 1, 2.
If we now restrict our attention to the market given by the numéraire portfolio S δ * and the savings account S 0 , the primary security account processes S 1 and S 2 cannot be perfectly replicated by investing in a portfolio containing only these two tradeable assets.
and let ϕ = (ξ, η) = {(ξ t , η t ), t ∈ [0, T ]} be an L 2 -admissible strategy such that At time t < T , we have that is the benchmarked fair price at time t of a zero coupon bond with maturity T and F W = (F W t ) 0≤t≤T denotes the natural filtration of W . We replicate S j by using S δ * and the money market account, or equivalently, we replicate the benchmarked primary security accountŜ j by investing in the benchmarked zero coupon bond and in 1. In general, we do not have an explicit joint density of (Z T , γ T ), which we would need to calculate the conditional expectation in (4.13). However, it is possible to characterize the benchmarked fair zero coupon bond pricing functionP T (·, ·, ·) as the solution of a Kolmogorov backward equation and provide its description by using numerical methods for solving partial differential equations (PDEs), see Section 15.7 in [21]. SinceP T (·, ·, ·) is of the form (4.13), then it will admit a Brownian martingale representation, i.e.

Benchmarked Risk-Minimization for a Defaultable Put on an
Index under the Stylized Minimal Market Model. The numéraire portfolio S δ * can be realistically interpreted as a diversified equity index, see [21]. Index linked variable annuities or puts on the numéraire portfolio are products that are of particular interest to pension plans. The recent financial crisis made rather clear that the event of a potential default of the issuing bank has to be taken into account. Set as before F W = (F W t ) 0≤t≤T . We now study the problem of pricing and hedging a defaultable put on the numéraire portfolio with strike K ∈ R + and maturity T ∈ (0, ∞) in the stylized MMM. Since the benchmarked payoffĤ of a put option is of the form F (Ŝ 0 T ), for a bounded function F , thenĤ ∈ L 2 (F W T , P) even ifŜ 0 t is not square-integrable for any t ∈ [0, T ]. The fair default free benchmarked pricep T,K (t) at time t is given by [21] in the form where Z 2 (x; ν, l) denotes the non-central chi-square distribution function with ν degrees of freedom, non-centrality parameter l and which is taken at the level x. Here we have Now, we extend the stylized MMM to include default risk. Beyond the traded uncertainty given by the standard F W -Wiener process W , there is also an additional source of randomness due to the presence of a possible default that, according to intensity based modeling, shall be modeled via a compensated jump process. More precisely, we assume that the random time of default τ is represented by a stopping time in the given filtration F. Let D be the default process, defined as D t = 1 {τ ≤t} , for t ∈ [0, T ]. We assume that τ admits an F W -intensity, that is, there exists an F W -adapted, nonnegative, (integrable) process λ such that the process is a P-martingale. Notice that for the sake of brevity we have writtenλ t = λ t 1 {τ ≥t} . In particular, we obtain that the existence of the intensity implies that τ is a totally inaccessible F W -stopping time, see [6], so that P(τ =τ ) = 0 for any F W -predictable stopping timeτ . Furthermore, we suppose that the default time τ and the underlying Wiener process W , are independent. When λ is constant, τ is the moment of the first jump of a Poisson process. Then, the benchmarked payoff of the defaultable put can be represented as follows:Ĥ whereδ is supposed to be the random recovery rate. In particular, we assume thatδ is a random variable in L 2 (F W T , P) depending only on T and τ , i.e. δ = h(τ ∧ T ), (4.20) for some Borel function h : (R, B(R)) → (R, B(R)), 0 ≤ h ≤ 1. Here we focus on the case when an agent recovers a random part of the promised claim at maturity. Moreover, we obtain thatĤ ∈ L 2 (F W T , P). Thus, we can apply the results of Section 3 to compute the decomposition (3.4) forĤ, i.e. the GKW decomposition ofĤ with respect to the P-local martingaleP (·, T ). By applying the real world pricing formula, we obtain the relationship for all t ∈ [0, T ]. Now it only remains to compute Ψ t . First we note that for each t ∈ [0, T ], we have where F τ stands for the cumulative distribution function of τ . We assume that F τ t < 1, for every t ∈ [0, T ], so thatQ is well-defined. We note that the second equality in the above derivation follows from Corollary 4.1.2 of [5]. By using the same arguments as in [2], we obtain for every t ∈ [0, T ] the equation where the functiong : R + → R is given by the formulã Here h is the function introduced in (4.20). Moreover, we have used the relationship that follows from Lemma 5.1 of [5]. Consequently, by applying Itô's formula, we get dÛ H (t) =p T,K (t)dΨ t + Ψ t− dp T,K (t), t ∈ [0, T ], and, thusĤ ,δ and the process Ψ, the benchmarked hedgeable part of the contingent claim forms a square-integrable P-martingale and the resulting strategy is L 2 -admissible.
since the residual optimal costĈδ T −Ĉδ t 0 must be orthogonal to all integrals of the form