Table of Contents
ISRN Probability and Statistics
Volume 2012, Article ID 946415, 37 pages
Research Article

Efficient Hedging of Options with Probabilistic Haar Wavelets

1Departamento de Matemática, Universidade Estadual de Campinas, 13.081-97 Campinas, SP, Brazil
2Department of Mathematics, Ryerson University, 350 Victoria Street, Toronto, ON, Canada M5B 2K3
3Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, 7600 Mar del Plata, Argentina

Received 29 March 2012; Accepted 19 April 2012

Academic Editors: M. Galea, J. Hu, and P. E. Jorgensen

Copyright © 2012 Pedro José Catuogno et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The paper brings forward the issue of efficient representations of financial claims; in particular it addresses the problem of large transaction costs in hedging replications. Inspired by the localized properties of wavelets basis, Haar systems associated with space-time discretizations of continuous stochastic processes are proposed as a means to address the issue of efficient pathwise approximation. Theoretical developments are presented that justify the use of these approximations to construct self-financing portfolios by means of binary options. Upper bounds on the volume of transactions required to implement these portfolios are then established to illustrate the quality of the proposed approximations. The approach is applicable to general financial claims of European type, including path-dependent ones, for continuous underlying processes. Several numerical results and comparisons with delta hedging are also presented.

1. Introduction

From a mathematical perspective, hedging of financial contracts is a pathwise approximation that can also be realized by means of a financial portfolio. Constraints such as transactions costs and the discrete nature of hedging times, bring the question of efficient approximations upon us. In particular, existence of such approximations could make the markets more efficient. To the best of our knowledge, the issue of efficiency has, so far, not been addressed directly in the mathematical finance literature.

In delta hedging, the underlying is used to construct the portfolio replication; this involves an implicit linear spatial approximation, resulting from the Taylor expansion used in Ito's formula, for the value of the option. This approximate hedging gives a pointwise error the quality of which depends on the efficiency of this space-time approximation.

The aim of the paper is to introduce new expansions that provide efficient hedging portfolio approximations to financial instruments. These approximations are naturally realizable via binary options and the notions of efficiency that we single out in our study are the number of transactions and the volume of transactions. This latter notion is defined as the total money exchanged during transactions required to set up the approximating hedging portfolio. These notions relate to the problem of transactions costs under the assumption that costs are proportional to the volume of transactions and/or there are costs related to the number of transactions (e.g., a fix cost per transaction). We express this symbolically below.

Definition 1.1. Given a path , and two approximating hedging portfolios for a financial contract , one will say that is more efficient than at if holds at the given , where is the volume of transactions and the number of transactions, respectively, necessary to implement the portfolio at .

Clearly, the above definition can be easily modified to require the inequalities to hold with large probability or in the mean. It should be noted that the notion of volume of transactions is coordinate dependent, that is, it depends on the explicit approximation used to define the portfolios .

Our approach also allows to minimize the number of transactions required to implement the approximating portfolio, a small number makes the implementation of the portfolio realistic and reduces transaction costs when fixed costs are charged per transactions or they depend on the number of transactions in a laddered way.

Therefore, we aim to reduce the number and volume of transactions for a given approximation error; a key ingredient of our approach is the use of localized basic functions. The construction of our approximations is inspired by wavelet theory and nonlinear approximation theory (see [1, 2]), which studies efficient representations of functional classes. In the setup of the paper, we use the notion of H-system, an orthonormal set adapted to the given stochastic processes.

Reference [3] provides evidence of the high transaction costs incurred when performing frequent rebalancing in continuous models. Moreover, the volume of transactions required to implement delta hedging can be proved to diverge in a Black and Scholes model [4]. On the other hand, we describe next a related problem that occurs when European call options are used for hedging. For simplicity, consider an option that initiates at and expires at with denoting the risk neutral price (or value) of . Consider a digital option with payoff and approximate this digital option with the following portfolio where we go long on a European call with strike and short on a European call with strike and . We obtain a better and better approximation to by considering . By no-arbitrage, it follows that but the volume of transactions for (which in this static example is a constant) is equal to which can be arbitrarily large as . Therefore, when decreasing the error of approximation we have the undesirable effect of increasing the volume of transactions. This is due to the fact that the approximation is obtained by cancelation of (unbounded) terms and each term entering in this approximation will contribute separately to the volume of transactions. The example could be generalized to other payoffs containing a steep profile; the wavelet literature [5] points to the need to use localized approximating functions to handle singularities.

It is our contention that, in the two cases described above, the problem of large volume of transactions is created because both approximations cannot provide efficient approximations to some type of discontinuity present in the problem. By making use of binary options of the form (for given times ), conveniently structured in an H-system, we prove that there exist financial portfolios built from these basic building blocks that provide small errors of approximation making use of few binary options and incurring in a low volume of transactions. In order to indicate the quality of our approximations, and as a contrast to existing hedging strategies, we provide bounds for the volume of transactions required to implement our approximating portfolios. We also provide several simulation examples; both the upper bounds and the numerical simulations provide excellent results. The upper bounds rely on the property of localization of our basis functions, this property implies that in order to approximate discontinuities only few localized basis functions are needed (at this point our approach is motivated by wavelet theory, [6, 7]). This is a crucial ingredient in order to achieve a small volume of transactions while using few transactions and achieving a small approximating error, see our Theorems 5.18 and 5.19 Some of these characteristics are in contrast with other approaches to hedging [810] which, similarly to our approach, use portfolios of simple options to hedge complex portfolios.

To indicate the essence of our approach, we point to (2.1), its right hand side is just a rewrite of the left hand-side in terms of the martingale differences that always form an orthogonal set. A novelty is in the writing of the conditional expectation as a Fourier expansion; the inner products are a set of new coordinates with useful properties and information which can be used to achieve efficient approximations. Our expansions can be implemented by means of binary options; the idea of using binary options as providing a general approximation tool has been previously treated in [11]. Our approach relies on the use of linear combinations of binary options that have value zero at the time of entering the contracts. It is then important to limit their number and analyze the necessary volume of transactions to implement them so as to keep the transaction costs low. A more detailed explanation of the meaning of our approximations is presented at the opening of Section 4.

The rest of the paper is organized as follows. Section 2 defines H-systems and develops some of the relationships between H-systems and sequences of partitions. Section 3 summarizes Willinger's main result on existence of discretization of stochastic processes and connects them to our setup of H-systems. Section 4 motivates and develops our main application to hedging a given European option. Section 5 presents several upper bounds for the volume of transactions required to implement the approximating portfolios described in Section 4. Section 6 presents several numerical examples and comparisons with delta hedging. Section 7 summarizes the main results of the paper. Appendix A introduces a simple example of an H-system in a basic financial setting. All the figures and tables mentioned in the paper present results from the numerical experiments. As a technical note, and for matters of convenience, we will suppress writing a.e. (almost everywhere) from many statements.

2. H-Systems

Let denote an arbitrary probability space. The notation stands for the inner product on and denotes the norm in . The following Gundy's [12] definition is motivated by the standard Haar system on .

Definition 2.1. An orthonormal system of functions defined on is called an H-system if and only if for any where . The intended meaning of in the above definition is to allow the system to be finite or infinite. We also use the notation . In applications we will make use of the pointwise convergence of (2.1), which holds due to the martingale convergence theorem [13]. Moreover, if is a given real number, then, for every , the sequence converges a.s. and in to . Convergence to holds whenever .

The following proposition, which is proven in [12], gives an alternative characterization of H-systems equivalent to Definition 2.1.

Proposition 2.2. An orthonormal system defined on is an H -system if and only if the following three conditions hold.(1) Each assumes at most two nonzero values with positive probability.(2) The -algebra consists exactly of elements.(3). So the functions are martingale differences.

Generating elements of discrete sigma algebras will be referred to as atoms.

Corollary 2.3. Assume that is an H -system. Then, for each , takes two nonzero values (one positive and the other negative) only on one atom of (hence this atom becomes its support). Consequently, consists of atoms from and two more atoms obtained by splitting the remaining atom from .

In view of the above proposition and its corollary, the functions in an H-system are natural generalizations of classic Haar functions, as the next definition states.

Definition 2.4. Given , , a function is called a Haar function on if there exist , , , , and Occasionally above will be denoted by . We will attach the word Haar to several definitions and constructions related to H-systems.

2.1. Basic Properties of H-Systems

It should be clear, from Corollary 2.3, that an H-system naturally defines a binary tree of partitions, these are formally introduced in the next definition.

Definition 2.5. A sequence of partitions of , , is called a binary sequence of partitions if, for , the members of have positive probability, , and, for , if and only if it is also a member of or, if not, there exists another member of , necessarily not in , such that .
We set , and hence . For , if and then preserves its index. Otherwise (i.e., , and not yet indexed) then there exists and such that then set and . In that case we will say “ splits in ”.

The index in will be called the scale parameter (we will also call it the level), it indicates the number of times has been split to obtain . Notice that can have at most members, and if then and . Observe also that refines . The tree shown in Figure 7 illustrates this indexation, it corresponds to an H-system with .

Observe that do not appear, the reason being that and do not split.

Given a binary sequence of partitions , define the associated trees The parent-child relationship is given by the split relationship. For every splitting internal node , we have, using the indexation introduced in Definition 2.5 (), a corresponding Haar function at that node where, according to Definition 2.4, Given , we have the natural associated tree with root .

The following sets of indexes will be used throughout the paper, and in Appendix A, consider and let collects the indices of members of that split in , those are going to be the support of the Haar functions. In the tree illustration shown in Figure 7 we have , , and not defined.

For each , let is an orthonormal basis of . Under the assumption that all atoms in split, the present notation relates easily to the abstract definition in (2.1), namely, given a binary sequence of partitions , there is associated the H-system given by and . In general the following expansion holds for any H-system: Therefore, one has the following result (the formal proof and more details are provided in [14].)

Theorem 2.6. Every H-system induces naturally a binary sequence of partitions and reciprocally.

We will use H-systems to approximate stochastic processes, a finite partition of the time interval will provide a finite H-system, and each atom in the associated tree will correspond to an event in the space-time discretization of the process. Section 3 describes general results that construct these space-time approximations that provide pointwise convergence to a given continuous stochastic process. These results will guarantee the availability of H-systems for further developments in the paper.

3. Discretizations of Stochastic Processes

In order to make use of Definition 2.1 and Theorem 2.6 we will need results establishing the existence of discrete approximations of continuous stochastic processes. The main result is presented in Proposition 3.4, it gives an existence result that can be employed in our applications. Alternative constructions of H-systems are presented in [14]. Reference [15] describes ways to construct H-systems associated with nested partitions.

Let be a complete probability space and be a continuous stochastic process defined on this probability space. Let be the filtration where is the completion of . Following Willinger and Taqqu [16, 17], we introduce the notion of skeleton-approach for stochastic processes.

Definition 3.1. A continuous-time skeleton approach of is a triple , consisting of an index-set , a filtration , the skeleton filtration, and a -adapted process such that it verifies the following(1), where .(2)For each , is a finitely generated sub--algebra of , with atomic partition .(3)For , one set if for some .(4)For each , .

Definition 3.2. A sequence of continuous time skeletons of will be called a continuous-time skeleton approximation of if the following three properties hold. The sequence of indexes satisfies where , and is a dense subset of . For each ,

The fundamental result of Willinger ([16] pp 55, Lemma ) is stated next, it guarantees the existence of continuous-time skeleton approximations for continuous processes. These discrete pathwise approximations are finite in space and time.

Lemma 3.3. Assume that is a continuous process, then there exists a continuous-time skeleton approximation for .

The atoms in Willinger's construction are given by intersections of sets of the form corresponding to different times . Therefore, the atoms are natural events associated with level sets of the underlying process .

Each continuous time skeleton of determines a sequence of nested finite partitions . Clearly, there exists a binary sequence of partitions , and a finite sequence of indexes such that Indeed, let , and assume that , for , has been constructed and indexed as in Definition 2.5. then we may construct for in the following way: first, include in all those members of (= ) that are also atoms of (these are events that do not undergo a split in the period ).) Next, for each , which can be written as a union of (), set and in . Continue this process, changing the role of by , up to some satisfying . Therefore, as an application of Theorem 2.6 to the binary sequence of partitions described above, we obtain a finite H-system associated with each continuous time skeleton of . Clearly, these H-systems are adapted to the filtration , that is, for with .

Proposition 3.4. Let be the filtration where is the completion of . Then there exist a sequence of finite H -systems and two sequences of finite indexes and such that(1) for with ,(2)for each ,

where for .

Proof. Let be a continuous-time skeleton approximation of . Using Theorem 2.6 construct for each an H-system associated with the sequence of partitions satisfying (3.3). In order to conclude the proof it is sufficient to observe that, in virtue of the main property of H-systems, (2.1), for .

To be clear, we briefly remark on the connection between Proposition 3.4 and (2.1). The functions appearing in (2.1) are the functions introduced in Proposition 3.4, the key remark being the extra parameter that corresponds to the time interval discretizations . Once more, we rewrite (2.1) in the notation of the present section, The sigma algebra , appearing in (2.1), in the present context equals (where is the cardinality of the set .) Notice also that

4. Hedging Portfolios of Binary Options

This section illustrates how H-systems can be applied in financial mathematics. It develops briefly a theory of hedging based on binary options, the martingale property of the H-system is put to use in this theory.

Let us first discuss the main idea of how the developments in the present section address the problem of high volume of transactions described in Section 1. First note that and, so, this function can be implemented by means of a bank account, the Haar functions are of the form , which under natural conditions can be realized by means of binary options (involving short selling). Recall that and the support of is , which is the disjoint union of and . It is clear that approximates the oscillations of on , where denotes the expectation on . In general, the events and will be level sets of financially relevant random variables, and hence the function captures fluctuations in due to these two financial events. In short, the financial meaning of (2.1) is the use of the bank account to capture the mean value of and the use of binary options to capture the (local) oscillations of about this (local) mean value. Even though Haar hedging uses (binary) options to build the replicating portfolio, it will be misleading to call it a static [9] type of hedging as we explain next. In general, each is localized to its support, say the atom , this atom will be localized in time to same interval (essentially, this means that is generated by the random variables ) and will also be localized in space (it will be the level set of some appropriate random variable). This localization of the Haar functions, and hence of the binary options, has the effect that for a given unfolding path only the Haar functions in (2.1) whose support contain this have to be implemented by the Haar hedging portfolio. This is the essence of dynamic hedging.

In short, the left-hand side of (2.1) is a martingale that, under appropriate conditions, approximates a.e. In this way, we have a sequence of portfolios of binary options converging a.e. to . Moreover these portfolios can be implemented dynamically, via financial transactions, in a self-financing way due to the martingale property. Therefore, we have available a discrete, self-financing, hedging strategy to approximate . This hedging strategy will be referred to as Haar hedging next.

4.1. Formal Developments

For the sake of simplicity, we will work in a market model with the usual assumptions; we refer to [18] for background. Let be the bond and a nonnegative adapted continuous stochastic process the price process. As in Section 3, is the completion of . We assume that is the risk neutral measure, that is, the discounted price process is a martingale. Let be a continuous-time skeleton approximation of . For fixed , let be a sequence of binary partitions associated with the sequence of partitions as described in Section 3 (see (3.3)), associated, via Theorem 2.6, with the H-system defined on . We also assume that a European derivative is given. In the remaining of this section we will drop the index as it will be fixed in all computations. The notation will denote the risk neutral value process of an instrument , in our setup we have .

As a sufficient condition for the atoms in a binary sequence of partitions to be used in a dynamic hedging portfolio we will impose a natural association between the martingale property of the H-system and a sequence of rebalancing times. In particular, in order to define dynamic hedging strategies, we will use the concept of time support of events.

Definition 4.1. Let , we say that is localized to the time interval , or that the time support of , denoted by is contained in if . Also, let denote the intersection of all such intervals .

The following definition is an extension to partitions of the notion of time localization of events.

Definition 4.2. Let be a partition of . is said to be localized to the time interval if each is localized to . Moreover, define the as the intersection of the all intervals such that is localized to that interval.

The definition below is the cornerstone of our dynamic hedging strategy based on H-systems.

Definition 4.3. Let be a sequence of binary partitions, we say that is localized to the time sequence if there exists a sequence such that for . We call the sequence the levels of localization of .

The financial blocks underlying are the binary options which are acquired at time and reach their maturity at time . These binary options have payoff at time .

To have a financial realization of the hedging we are proposing, we need to assume to be admissible as defined in the next definition.

Definition 4.4 (assumption on financial realization). The binary sequence of partitions is called admissible if for any integer and each atom the binary options are available for trading, in particular, short selling is possible.

In order to simplify the exposition, when defining the Haar hedging portfolio below, we will further define the Haar obligations as follows: , where , with , which are obligations at time that are acquired at time , for each . At no point in the paper we require that the Haar obligations exist as financial instruments, they are financially realized only in terms of the binary options.

Remark 4.5. Next we will define two hedging strategies via self-financing portfolios that approximate a European option using H-systems. In fact, we introduce the strategy associated with Haar obligations and the strategy associated to binary options. These two portfolios will be used in such a way that the two will give exactly the same approximations at all times (see Section 4.2), The difference being that they will be implemented through different binary options and hence will have different volume of transactions.

Haar Hedging Portfolio
will be a predictable, vector-valued, stochastic processes constant on the intervals . The portfolio is rebalanced at times replicating for . As previously indicated, this portfolio approximates fluctuations of the option about its mean value by means of the Haar functions. Taking the construction gives, as a special case, an example of static hedging. At each time we will specify how much to invest in the bond and how much to invest in the Haar obligations available at that rebalancing time, this will specify the coordinates of the vector . Here are the coordinates of for

The coefficients correspond to the expansion of given in (3.5).

As a short hand notation we will use Observe that the purchasing value of this portfolio is . The following (inductive) step will be to rebalance the portfolio at time . Assume that at this time we are in the event with , and the value of this portfolio is .

The rebalanced coordinates of for are where ( could be empty if does not split at time ). Recall that the obligations involved expire at time .

Theorem 4.6. Under the assumption , for all and all , the portfolio is self-financing and replicates at .

Proof. We proceed by induction on . For the portfolio is given by (4.2) when . It is clear from (3.5) that replicates and is self-financing because since .
For convenience, we will use the notation , . For the inductive step, at time the process is in some event with , and assume for . The re-balancing of at is given by (4.4), for all ; the cases of splitting at the next level or not are simultaneously handled by the cases or , respectively. The purchasing of is self-financing since the value of the portfolio given by (4.4) is , this follows from our assumption that . Considering , by (3.6), (4.4), and (3.5), we compute

The assumption , needed in Theorem 4.6, will not hold for a general H-system. The following proposition provides a sufficient condition to be valid, a useful form of the result appears in Corollary 4.8.

Proposition 4.7. Consider an H -system where the associated Haar obligations satisfy with , , , , and Then, .

Proof. Notice that .

Corollary 4.8. Consider the same setup as in Proposition 4.7 but also , , and instead of (4.7) assume the events to be independent with . Then, .

Proof. We have This implies (4.7).

Remark 4.9. Corollary 4.8 applies when the atoms from H-systems are constructed by splitting a given event by means of an independent random variable. In particular it can be applied to processes with independent increments as in the Black and Scholes model in Section 6.
If the condition does not hold, we need to absorb the non self-financing nature of the portfolio as an additional approximating error. We do not analyze this error here but only provide the following informal argument.
Consider a re-balancing time and a Haar obligation (starting at time and maturing at time ) such that . Given assume that there exists with and such that . Then, the exact error (including the sign but excluding interest rates for simplicity) is given by We can then restart our approximations at with a higher resolution such that , so that . In this way we accumulate the non-self-financing nature of the portfolio as errors in the following way (define ): A simple case where the errors (4.10) can be controlled a priori (i.e., at the start of hedging at ) is when hedging a derivative and there exist an integer and times such that .

Portfolio of Characteristic Functions
This section shows how to construct a self-financing portfolio to hedge . The portfolio will also be re-balanced at times , replicating for . Taking into account Remark 4.5, we will dispense with writing the analogue of Theorem 4.6 for , we mention that the sufficient conditions of Proposition 4.7 and Corollary 4.8 apply to this new portfolio as well.
We formalize as a vector-valued process that is constant on the intervals . At time it is defined, for , by specifying its coordinates, namely how much to invest in each of the binary options, The cost of purchasing this portfolio is . The inductive step will be to rebalance the portfolio at time . Assume that at this time the price process is in the event with , and the value of this portfolio is . We need to specify the coordinates of , for , namely, and .
reduces to if does not split in .
In an analogous way to the done for it is possible to prove that the strategy is self-financing and replicates at . It should be clear that the hedging strategies and can be intermixed at different time intervals .

4.2. Compressed Approximations

Let be a new indexing for our Haar system that satisfies , except for . Define the -term compressed approximation, which we will denote by , by As we will see, provides a very good approximation with a small volume of transactions and using small values of . This implies that few binary options are required to implement the portfolio . Notice that we have defined compression through the Haar representation, the same resulting approximating random variable (namely ) will also be used when computing volume of transactions for the portfolio . This implies that the number of binary options required to implement the portfolio is about half the number of binary options required to implement the portfolio . In practice we will apply the above-described compression at the beginning of each time interval to the martingale differences as we explain next.

For fixed and a given sequence of times , , let represent the discrete skeleton atomic sigma algebra approximating that (see Definition 3.2). We also assume, is the initial time, the trivial sigma algebra and -the final time. For simplicity, we will not display the dependency on in some of the notation, in particular we will write and (we will keep using to maintain the fundamental distinction between and ). Notice that by continuity of we can uniformly approximate with for sufficiently large.

For the sake of clarity we will add a superindex to the notation introduced in Section 2.1, to indicate, for example, that , denotes the collection of atoms generating . We also add the super-index to other objects related to time . Recall that is the binary sequence of partitions associated with the skeleton. Notice that .


Consider , and then: where the last two equalities follow from basic properties of the conditional expectation.

Noticing that the disjoint atoms , are a partition of and using (3.6), we obtain the decomposition Of course, is the Haar function at node (see (2.5)). In applications (see Section 6) we will perform compression by ordering the Fourier coefficients appearing in (4.17). This results in a compressed martingale difference , it is the same function that is also expanded in the characteristic functions basis.

Recalling the orthonormal system we can then write, for ,

To sum up, in practice both portfolios and , at each interval , implement and hence provide the same approximation. Their implementation in terms of binary options is different and, hence, the resulting volume of transactions differ. This last topic is studied in Section 5.

5. Volume of Transactions

The results in the present section provide upper bounds, in the mean and pointwise, for the volume of transactions required to implement the portfolios and (introduced in Section 4). Proposition 5.13 proves that the volume of transactions for the portfolio is smaller the one for ; because of this result, a justification for our reliance on the portfolio is in order. It is only through the use of that we can perform compression as described in Section 4.2; in particular, this allows to use few binary options while retaining a small error of approximation. As indicated above, the compressed approximation is also the one to be implemented by . Moreover, the upper bounds for the volume of transactions rely on an analysis of the behavior of the inner products defining . Other aspects of our approximations used to obtain the upper bounds are the following: they approximate the relevant conditional expectations, the martingale property is used in the self-financing nature of the portfolios, the orthogonality of the martingale differences holds, and the localized property of the basis functions is used throughout (as in (5.10)).

We believe that the quality of the upper bounds described in the present section provides evidence of the efficiency of our approximations as far as volume of transactions is concerned. We have written the results for the portfolios and but they remain valid for their compressed versions as well. Numerical results are presented in Section 6.

The results assume the existence of a skeleton approximation (and, hence, continuity of will be assumed) as described in Section 3; this assumption is to make sure we have available a family of finite H-systems that approximate the given process.

Definition 5.1. Consider the binary options , as described in (4.1). The volume of transactions, at time , required for a financial implementation of such an option is denoted by and equals its value, that is,

Definition 5.2. The volume of transactions necessary to implement a portfolio represented, in terms of the above binary options, by , is given by

Remark 5.3. The binary options appearing in the representation of the random variable are not assumed to constitute a fixed linearly independent set; therefore could be represented in different ways in terms of the linearly dependent set of characteristic functions . This implies, in particular, that the volume of transactions as defined in (5.2) depends on the specific representation of in terms of binary options.

Lemma 5.4. Given the binary option , with and , for , then and so

Proof. The proof follows by noticing that is an atom of the atomic sigma algebra .

In order to obtain pointwise upper bounds we will need to substitute by in (5.1) (and hence also in (5.2)). The resulting expressions will be called approximate values and denoted by . By means of the same substitution, one can also obtain the approximate values and using (5.1) and (5.6), respectively. Notice that the approximate values converge pointwise to the corresponding exact values as . Therefore, we can concentrate on establishing pointwise error bounds for the approximated quantities.

Remark 5.5. Under the conditions of Corollary 4.8, and in particular under the setting of Section 6, we have and hence also . Equation (5.4) will be used in several instances to justify the equality , from which follows (these expectation results follow in general, that is independently of the validity of Corollary 4.8).

We will analyze the volume of transactions needed to implement the binary portfolio () and the Haar portfolio () introduced in Section 4.1. The notation will be designed, as much as possible, so the developments cover both cases simultaneously. When stating results that apply to both portfolios we will designate either of them by :

Lemma 5.6. The following expression provides the volume of transactions required for a financial implementation of the portfolio(s) .

Proof. It is enough to establish (5.5) for any of the two portfolios. we consider for definiteness and argue by induction. Recall that , given that is invested in the bond; at time the Haar portfolio provides the approximation . Therefore, only the difference require a financial realization using binary options; the associated volume of transactions is given by . Reasoning inductively, at time , the value of the Haar hedging strategy is . Therefore, in order to achieve the approximation at time , only the difference requires a financial realization using binary options; the associated volume of transactions is given by where belongs to some with . This completes the inductive argument and gives (5.5). An analogous reasoning provides the argument for the case of the portfolio.

5.1. Representation by Characteristic Functions

From (4.18), (4.19), and (5.2), we obtain the following proposition.

Proposition 5.7. For ,

Corollary 5.8. Consider an arbitrary path , that is, . Then,

Corollary 5.9.

Proof. Notice that and also, . Using (5.5), (5.6), and , we obtain The inequality in (5.9) follows from Cauchy-Schwarz.

5.2. Representation by Haar Functions

The previous bounds for the volume of transactions did not use the Haar coefficients. In this section we deal with the Haar portfolio , this amounts to representing the martingale differences using the Haar functions instead of the characteristic functions.

Remark 5.10. Recalling the tree notation introduced in Section 2.1, we may set to denote the tree containing its root and internal nodes from . So the double sums appearing above could be written . Notice that the condition includes the case .

Using (5.2), the volume of transactions associated with decomposition (4.17) is given in the following Lemma.

Lemma 5.11. For ,

As previously indicated, the volume of transactions depends on the representation and we have used the notation to highlight this fact. The approximate version of (5.12) is and is obtained by replacing by in (5.12).

Lemma 5.12. For ,

The following proposition tells us that the portfolio has a smaller volume of transactions than the portfolio . Whenever appropriate we will use to denote the volume of transactions for the portfolio .

Proposition 5.13. One has, Therefore,

Remark 5.14. Of course, Remark 5.5 applies and the above proposition implies the corresponding results after taking expectations. Moreover, Proposition 5.13 also holds if is replaced for any random variable , in particular , namely the compressed version of .

Proof. Making use of (4.16) and the triangle inequality, Then, (5.14) follows after observing (5.7) and (5.12).

In the computations that follow we assume that our approximating martingale is regular [19], namely for all parents nodes and its children we have, where is the same for all nodes. For simplicity we will take as other choices will make the statements of our results more complex. Also, by means of Definition 5.15, nodes belonging to a tree are classified according to their size relative to the inner products .

Definition 5.15. Consider a node belonging to a tree associated with a given H-system satisfying . The node is said to be of type I if Furthermore, it is said to be of type II if .
Also define by: . Notice that whenever is of type I and that whenever is of type II.

For the following lemma, consider a generic node of a given H-system and its associated tree . To simplify the notation, nodes will always satisfy . In Lemma 5.16 we will make use of the following definition Notice that and its value depend on the given node .

Lemma 5.16. Assume a given H -system is regular with . Let be the number of instances for which nodes at level of the tree are of type I and . Then,

Proof. Considering the fact that can have at most members, hence , we obtain According to the decomposition, In order to complete the proof, it only remains to bound the second summation in the right-hand side of (5.22). As we remarked before, , then

From the above lemma we deduce the following result on static Hedging for the Haar portfolio.

Corollary 5.17. Assume the same hypothesis as in Lemma 5.16 and consider in (5.5) with , that is, a static replication and so ; then,