# Efficient Hedging of Options with Probabilistic Haar Wavelets

Authors: Pedro José Catuogno; Sebastián Esteban Ferrando; Alfredo Lázaro González
Journal: ISRN Probability and Statistics
 (2012)

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## Abstract

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.


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## Body

## 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 provideefficient 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 pathω, Π1 and Π2 two approximating hedging portfolios for a financial contract X, one will say that Π1 is more efficient than Π2 at ω if
(1.1)|Π1-X|≤|Π2-X|,VT(Π1)<VT(Π2),#Π1<#Π2
holds at the given ω, where VT(Πi)(ω) is the volume of transactions and #Πi(ω) the number of transactions, respectively, necessary to implement the portfolio Πi at ω.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 portfoliosΠi.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 X that initiates at t0 and expires at T with Vt0(X) denoting the risk neutral price (or value) of X. Consider a digital option with payoff X=1ST≥K and approximate this digital option with the following portfolio
(1.2)Π=1K2-K1(X1-X2),
where we go long on a European call X1=(ST-K1)+ with strike K1 and short on a European call X2=(ST-K2)+ with strike K2 and K1<K<K2. We obtain a better and better approximation to X by considering (K2-K1)→0. By no-arbitrage, it follows that Vt0(X)≈Vt0(Π) but the volume of transactions for Π (which in this static example is a constant) is equal to
(1.3)VT(Π)=1K2-K1[Vt0(X1)+Vt0(X2)],
which can be arbitrarily large as (K2-K1)→0. 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 X≈Π 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 formB=1[t1,T]1{St2∈(a,b)} (for given times t1<t2≤T), 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 [8–10] 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 〈X,uk〉 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. Section2 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(Ω,𝒜,P) denote an arbitrary probability space. The notation ∥∥22=〈,〉 stands for the inner product on L2(Ω,𝒜,P) and ∥∥1 denotes the norm in L1. The following Gundy's [12] definition is motivated by the standard Haar system on L2([0,1]).Definition 2.1.
An orthonormal system of functions{uk}k≥0 defined on Ω is called an H-system if and only if for any X∈L2(Ω,𝒜,P)(2.1)X𝒜m≡E(X∣u0,u1,…,um)=∑k=0m‍〈X,uk〉uk,∀m≥0,
where 𝒜m=σ(u0,…,um). The intended meaning of k≥0 in the above definition is to allow the system {uk}k≥0 to be finite or infinite. We also use the notation 𝒜∞=σ(∪m≥0𝒜m). In applications we will make use of the pointwise convergence of (2.1), which holds due to the martingale convergence theorem [13]. Moreover, if p∈[1,∞) is a given real number, then, for every X∈Lp, the sequence X𝒜m=E(X∣𝒜m) converges a.s. and in Lp to X∞=E(X∣𝒜∞). Convergence to X holds whenever σ(X)⊆𝒜∞.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{uk}k≥0 defined on Ω is an H -system if and only if the following three conditions hold.(1)
Eachuk assumes at most two nonzero values with positive probability.(2)
Theσ-algebra 𝒜m consists exactly of m+1 elements.(3)
E(uk+1u0,u1,…,uk)=0,k≥0. So the functions uk are martingale differences.Generating elements of discrete sigma algebras will be referred to asatoms.Corollary 2.3.
Assume that{uk}k≥0 is an H -system. Then, for each n≥0, un+1 takes two nonzero values (one positive and the other negative) only on one atom of 𝒜n (hence this atom becomes its support). Consequently, 𝒜n+1 consists of n atoms from 𝒜n and two more atoms obtained by splitting the remaining atom from 𝒜n.In view of the above proposition and its corollary, the functions in anH-system are natural generalizations of classic Haar functions, as the next definition states.Definition 2.4.
GivenA∈𝒜, P(A)>0, a function ψ is called a Haar function on A if there exist Ai∈𝒜, A0∩A1=∅, A=A0∪A1, ψ=a1A0+b1A1, and
(2.2)∫Ω‍ψ(ω)dP(ω)=0,∫Ω‍ψ2(ω)dP(ω)=1.
Occasionally ψ above will be denoted by ψA. We will attach the word Haar to several definitions and constructions related to H-systems.


### 2.1. Basic Properties ofH-Systems

It should be clear, from Corollary2.3, 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 ofΩ, ℛ≔{ℛj}j≥0, is called a binary sequence of partitions if, for j≥0, the members of ℛj have positive probability, ℛ0={Ω}, and, for j≥1, A∈ℛj if and only if it is also a member of ℛj-1 or, if not, there exists another member A' of ℛj, necessarily not in ℛj-1, such that A∪A'∈ℛj-1.
We setA0,0:=Ω, and hence ℛ0={A0,0}. For j≥1, if A∈ℛj and A=Ak,i∈ℛj-1 then A preserves its index. Otherwise (i.e., A∉ℛj-1, and not yet indexed) then there exists Ak,i∈ℛj-1 and A'∈ℛj such that
(2.3)Ak,i=A∪A′,
then set Ak+1,2i:=A and Ak+1,2i+1:=A'. In that case we will say “Ak,i splits in ℛj”.The indexk in Ak,i will be called the scale parameter (we will also call it the level), it indicates the number of times A0,0 has been split to obtain Ak,i. Notice that ℛj can have at most 2j members, and if Ak,i∈ℛj then k≤j and 0≤i≤2k-1. Observe also that ℛj refines ℛj-1. The tree shown in Figure 7 illustrates this indexation, it corresponds to an H-system with m=5.Observe thatA3,2,A3,3,A3,4,andA3,5 do not appear, the reason being that A2,1 and A2,2 do not split.Given a binary sequence of partitions{ℛj}j≥0, define the associated trees
(2.4)𝒯n≡∪j=0nℛj,𝒯≡∪n≥0𝒯n.
The parent-child relationship is given by the split relationship. For every splitting internal node A∈𝒯, we have, using the indexation introduced in Definition 2.5 (A=Ak,i=Ak+1,2i∪Ak+1,2i+1), a corresponding Haar function at that node
(2.5)ψk,i=ak,i1Ak+1,2i+bk,i1Ak+1,2i+1,
where, according to Definition 2.4,
(2.6)ak,i=P(Ak+1,2i+1)P(Ak,i)P(Ak+1,2i),bk,i=-P(Ak+1,2i)P(Ak,i)P(Ak+1,2i+1).
Given A∈𝒯, we have the natural associated tree 𝒯A with root A.The following sets of indexes will be used throughout the paper, and in AppendixA, consider j≥0 and let
(2.7)Hj≡{(k,i):Ak,i∈ℛj,Ak,i=Ak+1,2i∪Ak+1,2i+1withAk+1,2i,Ak+1,2i+1∈ℛj+1},Kj≡{(k,i):Ak,i∈ℛj}.Hj collects the indices of members of ℛj that split in ℛj+1, those are going to be the support of the Haar functions. In the tree illustration shown in Figure 7 we have K2={(2,0),(2,1),(2,2),(2,3)}, H2={(2,0),(2,3)}, K3={(3,0),(3,1),(2,1),(2,2),(3,6),(3,7)} and H3 not defined.For eachAk,i∈ℛj, let
(2.8)ϕk,i≡1Ak,iP(Ak,i).{ϕk,i}(k,i)∈Kj is an orthonormal basis of L2(Ω,σ(ℛj),P).  Under the assumption that all atoms in ℛj split, the present notation relates easily to the abstract definition in (2.1), namely, given a binary sequence of partitions {ℛj}j≥1, there is associated the H-system given by u0=ϕ0,0≡1Ω and {u2k+i≡ψk,i:(k,i)∈Hj,j≥1,}. In general the following expansion holds for any H-system:
(2.9)Xℛj≡E(X∣σ(ℛj))(ω)=〈X,ϕ0,0〉ϕ0,0+∑m=0j-1‍∑(k,i)∈Hm‍〈X,ψk,i〉ψk,i=∑(k,i)∈Kj‍〈X,ϕk,i〉ϕk,i.
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 useH-systems to approximate stochastic processes, a finite partition of the time interval [0,T] will provide a finite H-system, and each atom A in the associated tree 𝒯n 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.


## 2.1. Basic Properties ofH-Systems

It should be clear, from Corollary2.3, 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 ofΩ, ℛ≔{ℛj}j≥0, is called a binary sequence of partitions if, for j≥0, the members of ℛj have positive probability, ℛ0={Ω}, and, for j≥1, A∈ℛj if and only if it is also a member of ℛj-1 or, if not, there exists another member A' of ℛj, necessarily not in ℛj-1, such that A∪A'∈ℛj-1.
We setA0,0:=Ω, and hence ℛ0={A0,0}. For j≥1, if A∈ℛj and A=Ak,i∈ℛj-1 then A preserves its index. Otherwise (i.e., A∉ℛj-1, and not yet indexed) then there exists Ak,i∈ℛj-1 and A'∈ℛj such that
(2.3)Ak,i=A∪A′,
then set Ak+1,2i:=A and Ak+1,2i+1:=A'. In that case we will say “Ak,i splits in ℛj”.The indexk in Ak,i will be called the scale parameter (we will also call it the level), it indicates the number of times A0,0 has been split to obtain Ak,i. Notice that ℛj can have at most 2j members, and if Ak,i∈ℛj then k≤j and 0≤i≤2k-1. Observe also that ℛj refines ℛj-1. The tree shown in Figure 7 illustrates this indexation, it corresponds to an H-system with m=5.Observe thatA3,2,A3,3,A3,4,andA3,5 do not appear, the reason being that A2,1 and A2,2 do not split.Given a binary sequence of partitions{ℛj}j≥0, define the associated trees
(2.4)𝒯n≡∪j=0nℛj,𝒯≡∪n≥0𝒯n.
The parent-child relationship is given by the split relationship. For every splitting internal node A∈𝒯, we have, using the indexation introduced in Definition 2.5 (A=Ak,i=Ak+1,2i∪Ak+1,2i+1), a corresponding Haar function at that node
(2.5)ψk,i=ak,i1Ak+1,2i+bk,i1Ak+1,2i+1,
where, according to Definition 2.4,
(2.6)ak,i=P(Ak+1,2i+1)P(Ak,i)P(Ak+1,2i),bk,i=-P(Ak+1,2i)P(Ak,i)P(Ak+1,2i+1).
Given A∈𝒯, we have the natural associated tree 𝒯A with root A.The following sets of indexes will be used throughout the paper, and in AppendixA, consider j≥0 and let
(2.7)Hj≡{(k,i):Ak,i∈ℛj,Ak,i=Ak+1,2i∪Ak+1,2i+1withAk+1,2i,Ak+1,2i+1∈ℛj+1},Kj≡{(k,i):Ak,i∈ℛj}.Hj collects the indices of members of ℛj that split in ℛj+1, those are going to be the support of the Haar functions. In the tree illustration shown in Figure 7 we have K2={(2,0),(2,1),(2,2),(2,3)}, H2={(2,0),(2,3)}, K3={(3,0),(3,1),(2,1),(2,2),(3,6),(3,7)} and H3 not defined.For eachAk,i∈ℛj, let
(2.8)ϕk,i≡1Ak,iP(Ak,i).{ϕk,i}(k,i)∈Kj is an orthonormal basis of L2(Ω,σ(ℛj),P).  Under the assumption that all atoms in ℛj split, the present notation relates easily to the abstract definition in (2.1), namely, given a binary sequence of partitions {ℛj}j≥1, there is associated the H-system given by u0=ϕ0,0≡1Ω and {u2k+i≡ψk,i:(k,i)∈Hj,j≥1,}. In general the following expansion holds for any H-system:
(2.9)Xℛj≡E(X∣σ(ℛj))(ω)=〈X,ϕ0,0〉ϕ0,0+∑m=0j-1‍∑(k,i)∈Hm‍〈X,ψk,i〉ψk,i=∑(k,i)∈Kj‍〈X,ϕk,i〉ϕk,i.
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 useH-systems to approximate stochastic processes, a finite partition of the time interval [0,T] will provide a finite H-system, and each atom A in the associated tree 𝒯n 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 Definition2.1 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(Ω,𝒜,P) be a complete probability space and S=(St:0≤t≤T) be a continuous stochastic process defined on this probability space. Let ℱ={ℱt:0≤t≤T} be the filtration where ℱt is the completion of σ(Sr:0≤r≤t). Following Willinger and Taqqu [16, 17], we introduce the notion of skeleton-approach for stochastic processes.Definition 3.1.
Acontinuous-time skeleton approach of S is a triple (Iξ,ℱξ,ξ), consisting of an index-set Iξ, a filtration ℱξ={ℱtξ:0≤t≤T}, the skeleton filtration, and a ℱξ-adapted process ξ=(ξ:0≤t≤T) such that it verifies the following(1)
Iξ={0=t(ξ,0)<⋯<t(ξ,Nξ)=T}, where Nξ<∞.(2)
For eacht, ℱtξ is a finitely generated sub-σ-algebra of ℱt, with atomic partition 𝒫tξ.(3)
Fort∈[0,T]-Iξ, one set ℱtξ=ℱt(ξ,k)ξ if t∈[t(ξ,k),t(ξ,k+1)) for some 0≤k<Nξ.(4)
For each0≤t≤T, ξt=E(St∣ℱtξ).Definition 3.2.
A sequence(In,ℱn,ξn) of continuous time skeletons of S will be called a continuous-time skeleton approximation of S if the following three properties hold. The sequence In of indexes satisfies
(3.1)limn→∞|In|=0,
where |In|≡max{∣t(ξn,k)-t(ξn,k-1)∣:1≤k≤Nn}, and I≡∪nIn is a dense subset of [0,T]. For each 0≤t≤T,
(3.2)ℱtn↑ℱt,P({ω∈Ω:limn→∞sup0≤t≤T|St(ω)-ξtn(ω)|=0})=1.The fundamental result of Willinger ([16] pp 55, Lemma 4.3.1) 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.
AssumeS that is a continuous process, then there exists a continuous-time skeleton approximation for S.The atoms in Willinger's construction are given by intersections of sets of the formSti-1(a,b) corresponding to different times tj. Therefore, the atoms are natural events associated with level sets of the underlying process St.Each continuous time skeleton(Iξ,ℱξ,ξ) of S determines a sequence of nested finite partitions {𝒫tsξ}s=0Nξ. Clearly, there exists a binary sequence of partitions {ℛjξ}j≥0, and a finite sequence of indexes 0=j0<j1<⋯<jNξ such that
(3.3)ℛjsξ=𝒫tsξfors=0,…,Nξ.
Indeed, let ℛ0ξ≡{Ω}, and assume that ℛjξ, for j≤js, has been constructed and indexed as in Definition 2.5. then we may construct ℛjξ for j>js in the following way: first, include in Rjs+1ξ all those members of ℛjsξ(= 𝒫tsξ) that are also atoms of 𝒫ts+1ξ (these are events that do not undergo a split in the period [ts,ts+1).) Next, for each Ak,i∈ℛjsξ, which can be written as a union of B1,…,Bm∈𝒫ts+1ξ (m>1), set Ak+1,2i:=∪j=0[m/2]Bj and Ak+1,2i+1:=∪[m/2]+1mBj in ℛjs+1ξ. Continue this process, changing the role of ℛjsξ by ℛjs+1ξ, up to some js+1 satisfying Rjs+1ξ=𝒫ts+1ξ. Therefore, as an application of Theorem 2.6 to the binary sequence of partitions {ℛjξ}j≥0 described above, we obtain a finite H-system associated with each continuous time skeleton (Iξ,ℱξ,ξ) of S. Clearly, these H-systems are adapted to the filtration ℱtsξ, that is, ψk,i∈ℱtsξ for (k,i)∈Hj with j<js.Proposition 3.4.
Letℱ={ℱt:0≤t≤T} be the filtration where ℱt is the completion of σ(Sr:0≤r≤t). Then there exist a sequence of finite H -systems (ℋn={1Ω,ψk,in}) and two sequences of finite indexes (In={0=t0n<⋯<tNnn=T}) and (Jn={0=j0n<⋯<jNnn}) such that(1)
ψk,in∈ℱtsn for (k,i)∈Hj with j<jsn,(2)
for each0≤t≤T,
(3.4)limn→∞sup{St-ξtn:0≤t≤T}=0a.e.,whereξtn=E(St)1Ω+∑j=0jsn-1‍∑(k,i)∈Hj‍〈St,ψk,in〉ψk,in for t∈[ts-1n,tsn).Proof.
Let(In,ℱn,ξn) be a continuous-time skeleton approximation of S. Using Theorem 2.6 construct for each n an H-system (ℋn={1Ω,ψj,in}) associated with the sequence of partitions {ℛjn}j≥0 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), ξtn=E(St∣ℱtn)=E(St)1Ω+∑j=0jsn-1‍∑(k,i)∈Hj‍〈St,ψk,in〉ψk,in for t∈[ts-1n,tsn).To be clear, we briefly remark on the connection between Proposition3.4 and (2.1). The functions uk appearing in (2.1) are the functions ψk,in introduced in Proposition 3.4, the key remark being the extra parameter n that corresponds to the time interval discretizations In. Once more, we rewrite (2.1) in the notation of the present section,
(3.5)E(X∣σ(𝒫tsn))=E(X∣σ(ℛjsn))=E(X∣{1Ω,ψk,in}(k,i)∈Hj,0≤j<js)=E(X)1Ω+∑j=0js-1‍∑(k,i)∈Hj‍〈X,ψk,in〉ψk,in,∀s=0,…,Nn.
The sigma algebra 𝒜m, appearing in (2.1), in the present context equals 𝒜m=𝒜ms≡σ({ψk,in}(k,i)∈Hj,0≤j<js)=σ(𝒫tsn) (where ms is the cardinality of the set {ψk,in}(k,i)∈Hj,0≤j<js.) Notice also that
(3.6)E(X∣σ(ℛjsn))=E(X∣σ(ℛjs-1n))+∑j=js-1js-1‍∑(k,i)∈Hj‍〈X,ψk,in〉ψk,in,∀s=0,…,Nn.


## 4. Hedging Portfolios of Binary Options

This section illustrates howH-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 Section1. First note that u0=ϕ0,0=1Ω and, so, this function can be implemented by means of a bank account, the Haar functions are of the form ψk,i=ak,i1Ak+1,2i+bk,i1Ak+1,2i+1, which under natural conditions can be realized by means of binary options (involving short selling). Recall that ∫Ω‍ψk,idP=0 and the support of ψk,i is Ak,i, which is the disjoint union of Ak+1,2i and Ak+1,2i+1. It is clear that 〈X,ψk,i〉ψk,i approximates the oscillations of X-EAk,i(X) on Ak,i, where EAk,i(X) denotes the expectation on Ak,i. In general, the events Ak+1,2i and Ak+1,2i+1 will be level sets of financially relevant random variables, and hence the function ψk,i captures fluctuations in X 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 X and the use of binary options to capture the (local) oscillations of X 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 ψk,i is localized to its support, say the atom Ak,i, this atom will be localized in time to same interval [sa,ta] (essentially, this means that Ak,i is generated by the random variables {St}sa≤t≤ta) 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 w∈Ω only the Haar functions in (2.1) whose support contain this w 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 X a.e. In this way, we have a sequence of portfolios of binary options converging a.e. to X. 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 X. 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(Ω,ℱ,(ℱt)0≤t≤T,P) with the usual assumptions; we refer to [18] for background. Let B=(B(t)=ert) be the bond and a nonnegative adapted continuous stochastic process S=(St)0≤t≤T the price process. As in Section 3, ℱt is the completion of σ(Sr:0≤r≤t). We assume that P is the risk neutral measure, that is, the discounted price process (e-r(T-t)St) is a martingale. Let (In,ℱn,ξn) be a continuous-time skeleton approximation of S. For fixed n, let ℛn={ℛjn}j≥0 be a sequence of binary partitions associated with the sequence of partitions {𝒫tsn}s=0Nn as described in Section 3 (see (3.3)), associated, via Theorem 2.6, with the H-system {1Ω,ψk,in} defined on Ω. We also assume that a European derivative X∈L1(Ω,σ(∪n≥0ℱTn),P) is given. In the remaining of this section we will drop the index n as it will be fixed in all computations. The notation Vt(Z) will denote the risk neutral value process of an instrument Z, in our setup we have Vt(Z)=e-r(T-t)E(Z∣ℱt).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 theH-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.
LetE∈ℱT, we say that E is localized to the time interval [a,b], or that the time support of E, denoted by t-supp(E) is contained in [a,b] if E∈σ(St,t∈[a,b]). Also, let t-supp(E) denote the intersection of all such intervals [a,b].The following definition is an extension to partitions of the notion of time localization of events.Definition 4.2.
Let𝒫⊂ℱT be a partition of Ω. 𝒫 is said to be localized to the time interval [a,b] if each B∈𝒫 is localized to [a,b]. Moreover, define the t-supp(𝒫) as the intersection of the all intervals [a,b] such that 𝒫 is localized to that interval.The definition below is the cornerstone of our dynamic hedging strategy based onH-systems.Definition 4.3.
Letℛ={ℛj}0≤j≤J be a sequence of binary partitions, we say that ℛ is localized to the time sequence t0=0<⋯<tn=T if there exists a sequence j1<⋯<jn=J such that t-supp(ℛjs)⊂[t0,ts] for s=1,…,n. We call the sequence j1,…,jn the levels of localization of ℛ.The financial blocks underlyingℛ are the binary options
(4.1)Bk,i=(1Ak,i(t)≡1[ts,T](t)1Ak,i),(k,i)∈Kj,forjs<j≤js+1,
which are acquired at time ts-1 and reach their maturity at time ts. These binary options have payoff 1Ak,i at time ts.To have a financial realization of the hedging we are proposing, we need to assumeℛ to be admissible as defined in the next definition.Definition 4.4 (assumption on financial realization).
The binary sequence of partitionsℛ is called admissible if for any integer j and each atom Ak,i∈ℛj the binary options Bk,i 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 theHaar obligations as follows: Ψk,i, where Ψk,i(t)≡1[ts,T](t)ψk,i=1[ts,T](t)(ak,i1Ak+1,2i+bk,i1Ak+1,2i+1)=ak,iBk+1,2i+bk,iBk+1,2i+1, with (k,i)∈Hj, js≤j<js+1 which are obligations at time ts that are acquired at time ts-1, for each s=1,…,N. 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 usingH-systems. In fact, we introduce the strategy HΠ associated with Haar obligations and the strategy BΠ 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
H
Π
ℛ
(
X
)
=
(
H
Π
ℛ
(
X
)
t
) will be a predictable, vector-valued, stochastic processes constant on the intervals ts-1≤t<ts. The portfolio HΠℛ(X)t is rebalanced at times ts-1 replicating e-r(T-ts)E(X∣σ(ℛjs)) for s=1,…,N-1. As previously indicated, this portfolio approximates fluctuations of the option about its mean value by means of the Haar functions. Taking N=1 the construction gives, as a special case, an example of static hedging. At each time ts-1 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 HΠℛ(X)t. Here are the coordinates of HΠℛ(X)t for t∈[t0,t1)(4.2)e-r(T-t0)E(X)investedinthebondande-r(T-t1)〈X,ψk,i〉investedinΨk,ij=0,…j1-1,(k,i)∈Hj.The coefficients〈X,ψk,i〉 correspond to the expansion of E(X∣σ(𝒫ts)) given in (3.5).As a short hand notation we will use(4.3)xk[i]≡E(X∣σ(ℛj))(w)=1P(Ak,i)∫Ak,i‍XdP,ω∈Ak,i,for(k,i)∈Kj.
Observe that the purchasing value of this portfolio is Vt0(HΠℛ(X))=e-r(T-t0)E(X). The following (inductive) step will be to rebalance the portfolio at time ts-1. Assume that at this time we are in the event Ak0,i0 with (k0,i0)∈Kjs-1, and the value of this portfolio is e-r(T-ts-1)xk0[i0].The rebalanced coordinates ofHΠℛ(X)t for t∈[ts-1,ts) are
(4.4)e-r(T-ts-1)xk0[i0]investedinthebondande-r(T-ts)〈X,ψk,i〉investedinΨk,i,(k,i)∈Hjk0,i0,j=js-1,…js-1,
where Hjk0,i0≡{(k,i)∈Hj:Ak,i⊆Ak0,i0} (Hjk0,i0 could be empty if Ak0,i0 does not split at time ts). Recall that the obligations Ψk,i involved expire at time ts.Theorem 4.6.
Under the assumptionVts(Ψk,i)=0, for all (k,i) and all s=1,…,n-1, the portfolio HΠℛ(X)t is self-financing and replicates e-r(T-ts)E(X∣σ(ℛjs)) at s=1,…,N.Proof.
We proceed by induction ons. For s=1 the portfolio HΠℛ(X)t is given by (4.2) when t∈[t0,t1). It is clear from (3.5) that HΠℛ(X)t0 replicates e-r(T-t1)E(X∣σ(ℛj1)) and is self-financing because Vt0(HΠℛ(X)t0)=e-r(T-t0)E(X) since E(ψj,i)=0.
For convenience, we will use the notationt-=t-ϵ, ϵ>0. For the inductive step, at time ts-1 the process is in some event Ak0,i0 with (k0,i0)∈Kjs-1, and assume
(4.5)Vts-1(HΠℛ(X)ts-1-)(ω)=e-r(T-ts-1)E(X∣σ(ℛjs-1))(ω)=e-r(T-ts-1)xk0[i0]
for ω∈Ak0,i0. The re-balancing of HΠℛ(X)t at ts-1 is given by (4.4), for all t∈[ts-1,ts); the cases of Ak0,i0 splitting at the next level or not are simultaneously handled by the cases Hjk0,i0≠∅ or Hjk0,i0=∅, respectively. The purchasing of HΠℛ(X)ts-1 is self-financing since the value of the portfolio given by (4.4) is e-r(T-ts-1)xk0[i0], this follows from our assumption that Vts-1(Ψk,i)=0. Considering t=ts, by (3.6), (4.4), and (3.5), we compute
(4.6)Vts(HΠℛ(X)ts-)=e-r(T-ts-1)xk0[i0]er(ts-ts-1)1Ak0,i0+e-r(T-ts)∑j=js-1js-1‍∑(k,i)∈Hjk0,i0‍〈X,Ψk,i〉Vts(Ψk,i(ts))=e-r(T-ts)xk0[i0]1Ak0,i0+e-r(T-ts)∑j=js-1js-1‍∑(k,i)∈Hjk0,i0‍〈X,Ψk,i〉ψk,i=e-r(T-ts)E(X∣σ(ℛjs))a.e.onAk0,i0.The assumptionVts(Ψk,i)=0, needed in Theorem 4.6, will not hold for a general H-system. The following proposition provides a sufficient condition Vts(Ψk,i)=0 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 obligationsΨk,i satisfy ψk,i=a1A+b1B with A=E∩F0,  B=E∩F1, E∈ℱts, Fi∈ℱts+1, i=0,1 and
(4.7)aE(1F0∣ℱts)+bE(1F1∣ℱts)=0.
Then, Vts(Ψk,i)=0.Proof.
Notice thatE(ψk,i∣ℱts)=a1EE(1F0∣ℱts)+b1EE(1F1∣ℱts)=0.Corollary 4.8.
Consider the same setup as in Proposition4.7 but also E=A∪B, E∈ℱts, and instead of (4.7) assume the events Fi∈ℱts+1 to be independent with ℱts. Then, Vts(Ψk,i)=0.Proof.
We have(4.8)0=E(ψk,i∣ℱtsn)=a1EP(F0)+b1EP(F1).
This implies (4.7).Remark 4.9.
Corollary4.8 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 conditionVts(Ψk,i)=0 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 timet1 and a Haar obligation Ψ (starting at time t1 and maturing at time t2) such that Vt1(Ψ)≠0. Given w∈Ω assume that there exists A∈ℱt1 with P(A)>0 and w∈A such that EA(Ψ)=0. Then, the exact error (including the sign but excluding interest rates for simplicity) is given by
(4.9)E(X∣ℱt1n)-1P(A)∫A‍XdP.
We can then restart our approximations at t1 with a higher resolution n1 such that A∈ℱt1n1, so that E(X∣ℱt1n1)=(1/P(A))∫A‍XdP. In this way we accumulate the non-self-financing nature of the portfolio as errors in the following way (define n0≡n):
(4.10)∑i=0N-2‍E(X∣ℱti+1ni)-E(X∣ℱti+1ni+1).
A simple case where the errors (4.10) can be controlled a priori (i.e., at the start of hedging at t0) is when hedging a derivative X and there exist an integer K and times r0,…,rK such that X=X(Sr0,…,SrK).Portfolio of Characteristic Functions
This section shows how to construct a self-financing portfolioBΠℛ(X)t to hedge X. The portfolio BΠℛ(X)t will also be re-balanced at times t0,…,tN-1, replicating e-r(T-ts)E(X∣σ(ℛjs)) for s=1,…,N. Taking into account Remark 4.5, we will dispense with writing the analogue of Theorem 4.6 for BΠℛ(X)t, we mention that the sufficient conditions of Proposition 4.7 and Corollary 4.8 apply to this new portfolio as well.
We formalizeBΠℛ(X)t as a vector-valued process that is constant on the intervals ts-1≤t<ts. At time t0 it is defined, for t∈[t0,t1), by specifying its coordinates, namely how much to invest in each of the binary options,
(4.11)e-r(T-t0)E(X)investedinthebondande-r(T-t0)(xk[i]Bk,i-E(X))where(k,i)∈Kj1.
The cost of purchasing this portfolio is Vt0(BΠℛ(X))=e-r(T-t0)E(X)=e-r(T-t0)x0[0]. The inductive step will be to rebalance the portfolio at time ts-1. Assume that at this time the price process is in the event Ak0,i0 with (k0,i0)∈Kjs-1, and the value of this portfolio is e-r(T-ts-1)xk0[i0]. We need to specify the coordinates of BΠℛ(X)t, for ts-1≤t<ts, namely,
(4.12)e-r(T-ts-1)xk0[i0]investedinthebondande-r(T-ts-1)(xk[i]-xk0[i0])Bk,iwhere(k,i)∈Kjsk0,i0,
and Kjsk0,i0≡{(k,i)∈Kjs:Ak,i⊆Ak0,i0}.
K
j
s
k
0
,
i
0 reduces to (k0,i0) if Ak0,i0 does not split in 𝒫tsn.
In an analogous way to the done forHΠℛ(X) it is possible to prove that the strategy BΠℛ(X) is self-financing and replicates e-r(T-ts)E(X∣σ(ℛjs)) at s=1,…,N. It should be clear that the hedging strategies BΠℛ(X) and HΠℛ(X) can be intermixed at different time intervals [ts-1,ts).


### 4.2. Compressed Approximations

Letukii≥0 be a new indexing for our Haar system {uk}k≥0 that satisfies ∣〈X,uki+1〉∣≤∣〈X,uki〉∣, except for uk0≡u0. Define the R-term compressed approximation, which we will denote by XRc, by
(4.13)XRc≡∑i=0R‍〈X,uki〉uki.
As we will see, XRc provides a very good approximation with a small volume of transactions and using small values of R. This implies that few binary options are required to implement the portfolio HΠ. Notice that we have defined compression through the Haar representation, the same resulting approximating random variable (namely XRc) will also be used when computing volume of transactions for the portfolio BΠ. This implies that the number of binary options required to implement the portfolio BΠ is about half the number of binary options required to implement the portfolio HΠ. In practice we will apply the above-described compression at the beginning of each time interval [ts-1,ts) to the martingale differences as we explain next.For fixedn and a given sequence of times tsn, s=0,…,Nn, let ℱtsn represent the discrete skeleton atomic sigma algebra approximating that ℱts (see Definition 3.2). We also assume, t0n=t0 is the initial time, ℱt0 the trivial sigma algebra and tNn=T-the final time. For simplicity, we will not display the dependency on n in some of the notation, in particular we will write N=Nn and ts=tsn (we will keep using n to maintain the fundamental distinction between ℱtn and ℱt). Notice that by continuity of St we can uniformly approximate E(·∣ℱt)(w) with E(·∣ℱtn)(w) for n sufficiently large.For the sake of clarity we will add a superindexts to the notation introduced in Section 2.1, to indicate, for example, that {Ak,its}, (k,i)∈Kjs denotes the collection 𝒫tsn=ℛjs of atoms generating ℱtsn. We also add the super-index to other objects related to time ts. Recall that {ℛj}j≥0 is the binary sequence of partitions associated with the skeleton. Notice that Ak,it0=A0,0t0=Ω.Define(4.14)Dts-1,ts≡E(X∣ℱtsn)-E(X∣ℱts-1n),s=1,…,N.Considerψk,i∈ℱtsn, and then:
(4.15)〈Dts-1,ts,ψk,i〉≡〈E(X∣ℱtsn)-E(X∣ℱts-1n),ψk,i〉=〈E(X∣ℱtsn),ψk,i〉=〈X,ψk,i〉,
where the last two equalities follow from basic properties of the conditional expectation.Noticing that the disjoint atomsAj,lts-1, (j,l)∈ℛjs-1 are a partition of Ω and using (3.6), we obtain the decomposition
(4.16)Dts-1,ts=∑(j,l)∈Kjs-1‍1Aj,lts-1∑m=js-1js-1‍∑{(k,i)∈Hm:Ak,i⊂Aj,lts-1}‍〈Dts-1,ts,ψk,i〉ψk,i(4.17)=∑(j,l)∈Kjs-1‍1Aj,lts-1∑m=js-1js-1‍∑{(k,i)∈Hm:Ak,i⊂Aj,lts-1}‍〈X,ψk,i〉ψk,i.
Of course, ψk,i=ak,i1Ak+1,2i+bk,i1Ak+1,2i+1 is the Haar function at node Ak,i (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 Dts-1,tsc, it is the same function that is also expanded in the characteristic functions basis.Recalling the orthonormal system(4.18)ϕk,its≡1P(Ak,its)1Ak,its,for(k,i)∈Kjs,
we can then write, for w∈Aj,lts-1,
(4.19)Dts-1,tsc(w)=∑{(k,i)∈Kjs:Ak,its⊆Aj,lts-1}‍〈Dts-1,tsc,ϕk,its〉ϕk,its(w).To sum up, in practice both portfoliosBΠ and HΠ, at each interval [ts-1,ts), implement Dts-1,tsc 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.


## 4.1. Formal Developments

For the sake of simplicity, we will work in a market model(Ω,ℱ,(ℱt)0≤t≤T,P) with the usual assumptions; we refer to [18] for background. Let B=(B(t)=ert) be the bond and a nonnegative adapted continuous stochastic process S=(St)0≤t≤T the price process. As in Section 3, ℱt is the completion of σ(Sr:0≤r≤t). We assume that P is the risk neutral measure, that is, the discounted price process (e-r(T-t)St) is a martingale. Let (In,ℱn,ξn) be a continuous-time skeleton approximation of S. For fixed n, let ℛn={ℛjn}j≥0 be a sequence of binary partitions associated with the sequence of partitions {𝒫tsn}s=0Nn as described in Section 3 (see (3.3)), associated, via Theorem 2.6, with the H-system {1Ω,ψk,in} defined on Ω. We also assume that a European derivative X∈L1(Ω,σ(∪n≥0ℱTn),P) is given. In the remaining of this section we will drop the index n as it will be fixed in all computations. The notation Vt(Z) will denote the risk neutral value process of an instrument Z, in our setup we have Vt(Z)=e-r(T-t)E(Z∣ℱt).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 theH-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.
LetE∈ℱT, we say that E is localized to the time interval [a,b], or that the time support of E, denoted by t-supp(E) is contained in [a,b] if E∈σ(St,t∈[a,b]). Also, let t-supp(E) denote the intersection of all such intervals [a,b].The following definition is an extension to partitions of the notion of time localization of events.Definition 4.2.
Let𝒫⊂ℱT be a partition of Ω. 𝒫 is said to be localized to the time interval [a,b] if each B∈𝒫 is localized to [a,b]. Moreover, define the t-supp(𝒫) as the intersection of the all intervals [a,b] such that 𝒫 is localized to that interval.The definition below is the cornerstone of our dynamic hedging strategy based onH-systems.Definition 4.3.
Letℛ={ℛj}0≤j≤J be a sequence of binary partitions, we say that ℛ is localized to the time sequence t0=0<⋯<tn=T if there exists a sequence j1<⋯<jn=J such that t-supp(ℛjs)⊂[t0,ts] for s=1,…,n. We call the sequence j1,…,jn the levels of localization of ℛ.The financial blocks underlyingℛ are the binary options
(4.1)Bk,i=(1Ak,i(t)≡1[ts,T](t)1Ak,i),(k,i)∈Kj,forjs<j≤js+1,
which are acquired at time ts-1 and reach their maturity at time ts. These binary options have payoff 1Ak,i at time ts.To have a financial realization of the hedging we are proposing, we need to assumeℛ to be admissible as defined in the next definition.Definition 4.4 (assumption on financial realization).
The binary sequence of partitionsℛ is called admissible if for any integer j and each atom Ak,i∈ℛj the binary options Bk,i 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 theHaar obligations as follows: Ψk,i, where Ψk,i(t)≡1[ts,T](t)ψk,i=1[ts,T](t)(ak,i1Ak+1,2i+bk,i1Ak+1,2i+1)=ak,iBk+1,2i+bk,iBk+1,2i+1, with (k,i)∈Hj, js≤j<js+1 which are obligations at time ts that are acquired at time ts-1, for each s=1,…,N. 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 usingH-systems. In fact, we introduce the strategy HΠ associated with Haar obligations and the strategy BΠ 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
H
Π
ℛ
(
X
)
=
(
H
Π
ℛ
(
X
)
t
) will be a predictable, vector-valued, stochastic processes constant on the intervals ts-1≤t<ts. The portfolio HΠℛ(X)t is rebalanced at times ts-1 replicating e-r(T-ts)E(X∣σ(ℛjs)) for s=1,…,N-1. As previously indicated, this portfolio approximates fluctuations of the option about its mean value by means of the Haar functions. Taking N=1 the construction gives, as a special case, an example of static hedging. At each time ts-1 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 HΠℛ(X)t. Here are the coordinates of HΠℛ(X)t for t∈[t0,t1)(4.2)e-r(T-t0)E(X)investedinthebondande-r(T-t1)〈X,ψk,i〉investedinΨk,ij=0,…j1-1,(k,i)∈Hj.The coefficients〈X,ψk,i〉 correspond to the expansion of E(X∣σ(𝒫ts)) given in (3.5).As a short hand notation we will use(4.3)xk[i]≡E(X∣σ(ℛj))(w)=1P(Ak,i)∫Ak,i‍XdP,ω∈Ak,i,for(k,i)∈Kj.
Observe that the purchasing value of this portfolio is Vt0(HΠℛ(X))=e-r(T-t0)E(X). The following (inductive) step will be to rebalance the portfolio at time ts-1. Assume that at this time we are in the event Ak0,i0 with (k0,i0)∈Kjs-1, and the value of this portfolio is e-r(T-ts-1)xk0[i0].The rebalanced coordinates ofHΠℛ(X)t for t∈[ts-1,ts) are
(4.4)e-r(T-ts-1)xk0[i0]investedinthebondande-r(T-ts)〈X,ψk,i〉investedinΨk,i,(k,i)∈Hjk0,i0,j=js-1,…js-1,
where Hjk0,i0≡{(k,i)∈Hj:Ak,i⊆Ak0,i0} (Hjk0,i0 could be empty if Ak0,i0 does not split at time ts). Recall that the obligations Ψk,i involved expire at time ts.Theorem 4.6.
Under the assumptionVts(Ψk,i)=0, for all (k,i) and all s=1,…,n-1, the portfolio HΠℛ(X)t is self-financing and replicates e-r(T-ts)E(X∣σ(ℛjs)) at s=1,…,N.Proof.
We proceed by induction ons. For s=1 the portfolio HΠℛ(X)t is given by (4.2) when t∈[t0,t1). It is clear from (3.5) that HΠℛ(X)t0 replicates e-r(T-t1)E(X∣σ(ℛj1)) and is self-financing because Vt0(HΠℛ(X)t0)=e-r(T-t0)E(X) since E(ψj,i)=0.
For convenience, we will use the notationt-=t-ϵ, ϵ>0. For the inductive step, at time ts-1 the process is in some event Ak0,i0 with (k0,i0)∈Kjs-1, and assume
(4.5)Vts-1(HΠℛ(X)ts-1-)(ω)=e-r(T-ts-1)E(X∣σ(ℛjs-1))(ω)=e-r(T-ts-1)xk0[i0]
for ω∈Ak0,i0. The re-balancing of HΠℛ(X)t at ts-1 is given by (4.4), for all t∈[ts-1,ts); the cases of Ak0,i0 splitting at the next level or not are simultaneously handled by the cases Hjk0,i0≠∅ or Hjk0,i0=∅, respectively. The purchasing of HΠℛ(X)ts-1 is self-financing since the value of the portfolio given by (4.4) is e-r(T-ts-1)xk0[i0], this follows from our assumption that Vts-1(Ψk,i)=0. Considering t=ts, by (3.6), (4.4), and (3.5), we compute
(4.6)Vts(HΠℛ(X)ts-)=e-r(T-ts-1)xk0[i0]er(ts-ts-1)1Ak0,i0+e-r(T-ts)∑j=js-1js-1‍∑(k,i)∈Hjk0,i0‍〈X,Ψk,i〉Vts(Ψk,i(ts))=e-r(T-ts)xk0[i0]1Ak0,i0+e-r(T-ts)∑j=js-1js-1‍∑(k,i)∈Hjk0,i0‍〈X,Ψk,i〉ψk,i=e-r(T-ts)E(X∣σ(ℛjs))a.e.onAk0,i0.The assumptionVts(Ψk,i)=0, needed in Theorem 4.6, will not hold for a general H-system. The following proposition provides a sufficient condition Vts(Ψk,i)=0 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 obligationsΨk,i satisfy ψk,i=a1A+b1B with A=E∩F0,  B=E∩F1, E∈ℱts, Fi∈ℱts+1, i=0,1 and
(4.7)aE(1F0∣ℱts)+bE(1F1∣ℱts)=0.
Then, Vts(Ψk,i)=0.Proof.
Notice thatE(ψk,i∣ℱts)=a1EE(1F0∣ℱts)+b1EE(1F1∣ℱts)=0.Corollary 4.8.
Consider the same setup as in Proposition4.7 but also E=A∪B, E∈ℱts, and instead of (4.7) assume the events Fi∈ℱts+1 to be independent with ℱts. Then, Vts(Ψk,i)=0.Proof.
We have(4.8)0=E(ψk,i∣ℱtsn)=a1EP(F0)+b1EP(F1).
This implies (4.7).Remark 4.9.
Corollary4.8 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 conditionVts(Ψk,i)=0 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 timet1 and a Haar obligation Ψ (starting at time t1 and maturing at time t2) such that Vt1(Ψ)≠0. Given w∈Ω assume that there exists A∈ℱt1 with P(A)>0 and w∈A such that EA(Ψ)=0. Then, the exact error (including the sign but excluding interest rates for simplicity) is given by
(4.9)E(X∣ℱt1n)-1P(A)∫A‍XdP.
We can then restart our approximations at t1 with a higher resolution n1 such that A∈ℱt1n1, so that E(X∣ℱt1n1)=(1/P(A))∫A‍XdP. In this way we accumulate the non-self-financing nature of the portfolio as errors in the following way (define n0≡n):
(4.10)∑i=0N-2‍E(X∣ℱti+1ni)-E(X∣ℱti+1ni+1).
A simple case where the errors (4.10) can be controlled a priori (i.e., at the start of hedging at t0) is when hedging a derivative X and there exist an integer K and times r0,…,rK such that X=X(Sr0,…,SrK).Portfolio of Characteristic Functions
This section shows how to construct a self-financing portfolioBΠℛ(X)t to hedge X. The portfolio BΠℛ(X)t will also be re-balanced at times t0,…,tN-1, replicating e-r(T-ts)E(X∣σ(ℛjs)) for s=1,…,N. Taking into account Remark 4.5, we will dispense with writing the analogue of Theorem 4.6 for BΠℛ(X)t, we mention that the sufficient conditions of Proposition 4.7 and Corollary 4.8 apply to this new portfolio as well.
We formalizeBΠℛ(X)t as a vector-valued process that is constant on the intervals ts-1≤t<ts. At time t0 it is defined, for t∈[t0,t1), by specifying its coordinates, namely how much to invest in each of the binary options,
(4.11)e-r(T-t0)E(X)investedinthebondande-r(T-t0)(xk[i]Bk,i-E(X))where(k,i)∈Kj1.
The cost of purchasing this portfolio is Vt0(BΠℛ(X))=e-r(T-t0)E(X)=e-r(T-t0)x0[0]. The inductive step will be to rebalance the portfolio at time ts-1. Assume that at this time the price process is in the event Ak0,i0 with (k0,i0)∈Kjs-1, and the value of this portfolio is e-r(T-ts-1)xk0[i0]. We need to specify the coordinates of BΠℛ(X)t, for ts-1≤t<ts, namely,
(4.12)e-r(T-ts-1)xk0[i0]investedinthebondande-r(T-ts-1)(xk[i]-xk0[i0])Bk,iwhere(k,i)∈Kjsk0,i0,
and Kjsk0,i0≡{(k,i)∈Kjs:Ak,i⊆Ak0,i0}.
K
j
s
k
0
,
i
0 reduces to (k0,i0) if Ak0,i0 does not split in 𝒫tsn.
In an analogous way to the done forHΠℛ(X) it is possible to prove that the strategy BΠℛ(X) is self-financing and replicates e-r(T-ts)E(X∣σ(ℛjs)) at s=1,…,N. It should be clear that the hedging strategies BΠℛ(X) and HΠℛ(X) can be intermixed at different time intervals [ts-1,ts).


## 4.2. Compressed Approximations

Letukii≥0 be a new indexing for our Haar system {uk}k≥0 that satisfies ∣〈X,uki+1〉∣≤∣〈X,uki〉∣, except for uk0≡u0. Define the R-term compressed approximation, which we will denote by XRc, by
(4.13)XRc≡∑i=0R‍〈X,uki〉uki.
As we will see, XRc provides a very good approximation with a small volume of transactions and using small values of R. This implies that few binary options are required to implement the portfolio HΠ. Notice that we have defined compression through the Haar representation, the same resulting approximating random variable (namely XRc) will also be used when computing volume of transactions for the portfolio BΠ. This implies that the number of binary options required to implement the portfolio BΠ is about half the number of binary options required to implement the portfolio HΠ. In practice we will apply the above-described compression at the beginning of each time interval [ts-1,ts) to the martingale differences as we explain next.For fixedn and a given sequence of times tsn, s=0,…,Nn, let ℱtsn represent the discrete skeleton atomic sigma algebra approximating that ℱts (see Definition 3.2). We also assume, t0n=t0 is the initial time, ℱt0 the trivial sigma algebra and tNn=T-the final time. For simplicity, we will not display the dependency on n in some of the notation, in particular we will write N=Nn and ts=tsn (we will keep using n to maintain the fundamental distinction between ℱtn and ℱt). Notice that by continuity of St we can uniformly approximate E(·∣ℱt)(w) with E(·∣ℱtn)(w) for n sufficiently large.For the sake of clarity we will add a superindexts to the notation introduced in Section 2.1, to indicate, for example, that {Ak,its}, (k,i)∈Kjs denotes the collection 𝒫tsn=ℛjs of atoms generating ℱtsn. We also add the super-index to other objects related to time ts. Recall that {ℛj}j≥0 is the binary sequence of partitions associated with the skeleton. Notice that Ak,it0=A0,0t0=Ω.Define(4.14)Dts-1,ts≡E(X∣ℱtsn)-E(X∣ℱts-1n),s=1,…,N.Considerψk,i∈ℱtsn, and then:
(4.15)〈Dts-1,ts,ψk,i〉≡〈E(X∣ℱtsn)-E(X∣ℱts-1n),ψk,i〉=〈E(X∣ℱtsn),ψk,i〉=〈X,ψk,i〉,
where the last two equalities follow from basic properties of the conditional expectation.Noticing that the disjoint atomsAj,lts-1, (j,l)∈ℛjs-1 are a partition of Ω and using (3.6), we obtain the decomposition
(4.16)Dts-1,ts=∑(j,l)∈Kjs-1‍1Aj,lts-1∑m=js-1js-1‍∑{(k,i)∈Hm:Ak,i⊂Aj,lts-1}‍〈Dts-1,ts,ψk,i〉ψk,i(4.17)=∑(j,l)∈Kjs-1‍1Aj,lts-1∑m=js-1js-1‍∑{(k,i)∈Hm:Ak,i⊂Aj,lts-1}‍〈X,ψk,i〉ψk,i.
Of course, ψk,i=ak,i1Ak+1,2i+bk,i1Ak+1,2i+1 is the Haar function at node Ak,i (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 Dts-1,tsc, it is the same function that is also expanded in the characteristic functions basis.Recalling the orthonormal system(4.18)ϕk,its≡1P(Ak,its)1Ak,its,for(k,i)∈Kjs,
we can then write, for w∈Aj,lts-1,
(4.19)Dts-1,tsc(w)=∑{(k,i)∈Kjs:Ak,its⊆Aj,lts-1}‍〈Dts-1,tsc,ϕk,its〉ϕk,its(w).To sum up, in practice both portfoliosBΠ and HΠ, at each interval [ts-1,ts), implement Dts-1,tsc 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 portfoliosBΠ and HΠ (introduced in Section 4). Proposition 5.13 proves that the volume of transactions for the portfolio BΠ is smaller the one for HΠ; because of this result, a justification for our reliance on the portfolio HΠ is in order. It is only through the use of HΠ 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 BΠ. Moreover, the upper bounds for the volume of transactions rely on an analysis of the behavior of the inner products defining HΠ. 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 portfoliosBΠ and HΠ 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 ofSt 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 optionsBk,its=(1Ak,its(t)≡1[ts-1,T](t)1Ak,its), (k,i)∈Kjs as described in (4.1). The volume of transactions, at time ts-1, required for a financial implementation of such an option is denoted by VTts-1(Bk,its) and equals its value, that is,
(5.1)VTts-1(Bk,its)=e-r(ts-ts-1)E(1Ak,its∣ℱts-1).Definition 5.2.
The volume of transactions necessary to implement a portfolioΠ represented, in terms of the above binary options, by Π=∑‍ck,iBk,its, is given by
(5.2)VTts-1(Π)≡∑‍|ck,i|e-r(ts-ts-1)E(1Ak,its∣ℱts-1).Remark 5.3.
The binary options appearing in the representation of the random variableΠ are not assumed to constitute a fixed linearly independent set; therefore Π could be represented in different ways in terms of the linearly dependent set of characteristic functions 1Ak,its. This implies, in particular, that the volume of transactions as defined in (5.2) depends on the specific representation of Π in terms of binary options.Lemma 5.4.
Given the binary optionBk,its=1Ak,its, with Ak,its∈ℱtsn and Ak,its⊆Aj,lts-1∈ℱts-1n, for (k,i)∈Kjs, then
(5.3)E(1Ak,its∣ℱts-1n)=P(Ak,its)P(Aj,lts-1)1Aj,lts-1
and so
(5.4)E(VTts-1(Bk,its)-e-r(ts-ts-1)E(1Ak,its∣ℱts-1n))=0.Proof.
The proof follows by noticing thatAj,lts-1 is an atom of the atomic sigma algebra ℱts-1n.In order to obtain pointwise upper bounds we will need to substituteE(1Ak,its∣ℱts-1) by E(1Ak,its∣ℱts-1n) in (5.1) (and hence also in (5.2)). The resulting expressions will be called approximate values and denoted by VT^ts-1(Bk,its). By means of the same substitution, one can also obtain the approximate values VT^ts-1(Π) and VT^ts-1(Dts-1,ts) using (5.1) and (5.6), respectively. Notice that the approximate values converge pointwise to the corresponding exact values as n→∞. Therefore, we can concentrate on establishing pointwise error bounds for the approximated quantities.Remark 5.5.
Under the conditions of Corollary4.8, and in particular under the setting of Section 6, we have VTts-1(Dts-1,ts)=VT^ts-1(Dts-1,ts) and hence also VTts-1(Π)=VT^ts-1(Π). Equation (5.4) will be used in several instances to justify the equality E(VT(Dts-1,ts))=E(VT^(Dts-1,ts)), from which E(VTt0(Π))=E(VT^t0(Π)) 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 (BΠ) and the Haar portfolio (HΠ) 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 Π:Lemma 5.6.
The following expression provides the volume of transactions required for a financial implementation of the portfolio(s)Π.
(5.5)VTt0(Π)=∑s=1N‍∑(k,i)∈Kjs-1‍1Ak,its-1VTts-1(Dts-1,ts).Proof.
It is enough to establish (5.5) for any of the two portfolios. we consider HΠ for definiteness and argue by induction. Recall that A0,00=Ω, given that e-r(T-t0)E(X) is invested in the bond; at time t1 the Haar portfolio provides the approximation E(X∣ℱt1n). Therefore, only the difference Dt0,t1 require a financial realization using binary options; the associated volume of transactions is given by VTt0(Dt0,t1). Reasoning inductively, at time ts-1, the value of the Haar hedging strategy is E(X∣ℱts-1n). Therefore, in order to achieve the approximation E(X∣ℱtsn) at time ts, only the difference Dts-1,ts requires a financial realization using binary options; the associated volume of transactions is given by VTts-1(Dts-1,ts)(w) where w belongs to some Ak,its-1 with (k,i)∈Kjs-1. This completes the inductive argument and gives (5.5). An analogous reasoning provides the argument for the case of the BΠ portfolio.


### 5.1. Representation by Characteristic Functions

From (4.18), (4.19), and (5.2), we obtain the following proposition.Proposition 5.7.
Forw∈Aj,lts-1,
(5.6)VTts-1(Dts-1,ts)(w)=e-r(ts-ts-1)∑{(k,i)∈Kjs:Ak,its⊆Aj,lts-1}‍〈Dts-1,ts,ϕk,its〉P(Ak,its)E(1Ak,its∣ℱts-1)(w).(5.7)VT^ts-1(Dts-1,ts)(w)=e-r(ts-ts-1)1Aj,lts-1P(Aj,lts-1)∥1Aj,lts-1Dts-1,ts∥1.Corollary 5.8.
Consider an arbitrary path{Aks,ists}s=1,…,N, that is, Aks,ists⊆Aks-1,is-1ts-1. Then,
(5.8)VT^t0(BΠ)=∑s=1N‍e-r(ts-ts-1)1P(Aks,ists)∥1Aks,istsDts-1,ts∥1,a.e.on∩Aks,ists.Corollary 5.9.
(5.9)
E
(
V
T
t
0
(
B
Π
)
)
=
∑
s
=
1
N
‍
e
-
r
(
t
s
-
t
s
-
1
)
∥
D
t
s
-
1
,
t
s
∥
1
≤
∑
s
=
1
N
‍
e
r
(
t
s
-
t
s
-
1
)
∥
D
t
s
-
1
,
t
s
∥
2
.Proof.
Notice that(5.10)⋃{(k,i)∈Kjs:Ak,its⊆Aj,lts-1}‍Ak,its=Aj,lts-1,⋃(j,l)∈Kjs-1‍Aj,lts-1=Ω,
and also, Ak,its∩Ak',i'ts=∅. Using (5.5), (5.6), and Aj,lts-1∈ℱts-1, we obtain
(5.11)E(VTt0(BΠ))=∑s=1N‍e-r(ts-ts-1)∑(j,l)∈Kjs-1‍∑{(k,i)∈Kjs:Ak,its⊆Aj,lts-1}‍|〈Dts-1,ts,1Ak,its〉|P(Ak,its)E(1Aj,lts-1E(1Ak,its∣ℱts-1))=∑s=1N‍e-r(ts-ts-1)∥Dts-1,ts∥1.
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 portfolioHΠ, 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 Section2.1, we may set 𝒯Aj,lts-1 to denote the tree containing its root Aj,lts-1 and internal nodes from ∪m=js-1js-1{Ak,i∈Hm}. So the double sums appearing above could be written ∑m=js-1js-1‍∑{(k,i)∈Hm:Ak,i⊂Aj,lts-1}‍=∑Ak,i∈𝒯Aj,lts-1‍. Notice that the condition Ak,i⊂Aj,lts-1 includes the case Ak,i=Aj,lts-1.Using (5.2), the volume of transactions associated with decomposition (4.17) is given in the following Lemma.Lemma 5.11.
Forw∈Aj,lts-1,
(5.12)VTts-1H(Dts-1,ts)(w)=e-r(ts-ts-1)∑m=js-1js-1∑{(k,i)∈Hm:Ak,i⊂Aj,lts-1}‍|X,ψk,i|×(|ak,i|E(1Ak+1,2i∣ℱts-1)(w)+|bk,i|E(1Ak+1,2i+1∣ℱts-1)(w)).As previously indicated, the volume of transactions depends on the representation and we have used the notationVTH to highlight this fact. The approximate version of (5.12) is V^Tts-1H(Dts-1,ts) and is obtained by replacing ℱts-1 by ℱts-1n in (5.12).Lemma 5.12.
Forw∈Aj,lts-1,
(5.13)VT^ts-1H(Dts-1,ts)(w)=e-r(ts-ts-1)1Aj,lts-1P(Aj,lts-1)∑m=js-1js-1∑{(k,i)∈Hm:Ak,i⊂Aj,lts-1}‍|〈X,ψk,i〉|∥ψk,i∥1.The following proposition tells us that the portfolioBΠ has a smaller volume of transactions than the portfolio HΠ. Whenever appropriate we will use VTB to denote the volume of transactions for the portfolio BΠ.Proposition 5.13.
One has,(5.14)VT^ts-1B(Dts-1,ts)≤V^Tts-1H(Dts-1,ts).
Therefore,
(5.15)VT^t0(BΠ)≤V^Tt0(HΠ).Remark 5.14.
Of course, Remark5.5 applies and the above proposition implies the corresponding results after taking expectations. Moreover, Proposition 5.13 also holds if Dts-1,ts is replaced for any random variable Y∈ℱtsn, in particular Y=Dts-1,tsc, namely the compressed version of Dts-1,ts.Proof.
Making use of (4.16) and the triangle inequality,
(5.16)1Aj,lts-1P(Aj,lts-1)∥1Aj,lts-1Dts-1,ts∥1≤1Aj,lts-1P(Aj,lts-1)∑m=js-1js-1‍∑{(k,i)∈Hm:Ak,i⊂Aj,lts-1}‍|〈X,ψk,i〉|∥ψk,i∥1.
Then, (5.14) follows after observing (5.7) and (5.12).In the computations that follow we assume that our approximating martingale isregular [19], namely for all parents nodes Ak,i and its children we have,
(5.17)P(Ak+1,l)≤δP(Ak,i),l=2i,2i+1,
where δ≥1/2 is the same for all nodes. For simplicity we will take δ=1/2 as other choices will make the statements of our results more complex. Also, by means of Definition 5.15, nodes Ak,i belonging to a tree 𝒯n are classified according to their size relative to the inner products ∣〈X,ψk,i〉∣.Definition 5.15.
Consider a nodeAk,i belonging to a tree 𝒯n associated with a given H-system satisfying 〈X,ψk,i〉≠0. The node Ak,i is said to be of type I if
(5.18)|〈X,ψk,i〉|<P(Ak,i).
Furthermore, it is said to be of type II if ∣〈X,ψk,i〉∣≥P(Ak,i).
Also defineϵk,i by: ∣〈X,ψk,i〉∣=P(Ak,i)1/2+ϵk,i. Notice that ϵk,i>0 whenever Ak,i is of type I and that ϵk,i≤0 whenever Ak,i is of type II.For the following lemma, consider a generic nodeAj,lts-1 of a given H-system and its associated tree 𝒯Aj,lts-1. To simplify the notation, nodes Ak,i will always satisfy Ak,i∈𝒯Aj,lts-1. In Lemma 5.16 we will make use of the following definition
(5.19)ϵm≡min{ϵk,i:(k,i)∈Hm,Ak,ioftypeI}ϵts-1≡min{ϵm:js-1<m≤js}.
Notice that ϵts-1>0 and its value depend on the given node Aj,lts-1.Lemma 5.16.
Assume a given H -system is regular withδ=1/2. Let 0≤qmm be the number of instances for which nodes at level m of the tree 𝒯Aj,lts-1 are of type I and rs≡js-js-1. Then,
(5.20)∑Ak,i∈𝒯Aj,lts-1‍P(Ak,i)|〈X,ψk,i〉|≤(P(Aj,lts-1))1+ϵts-1∑m=0rs‍(qmδ1+ϵm)m+∑Ak,ioftypeII‍|〈X,ψk,i〉|2≤(P(Aj,lts-1))1+ϵts-111-2δ1+ϵts-1+∥1Aj,lts-1Dts-1,ts∥22.Proof.
Considering the fact thatℛm can have at most 2m members, hence qm≤2, we obtain
(5.21)∑Ak,ioftypeI‍P(Ak,i)|〈X,ψk,i〉|≤∑m=0rs‍(qmδ1+ϵm)m(P(Aj,lts-1))1+ϵm≤(P(Aj,lts-1))1+ϵts-1∑m=0rs‍(qmδ1+ϵm)m≤(P(Aj,lts-1))1+ϵts-111-2δ1+ϵts-1.
According to the decomposition,
(5.22)∑Ak,i∈𝒯Aj,lts-1‍P(Ak,i)|〈X,ψk,i〉|=∑Ak,ioftypeI‍P(Ak,i)∣〈X,ψk,i〉∣+∑Ak,ioftypeII‍P(Ak,i)|〈X,ψk,i〉|,
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, 〈X,ψk,i〉=〈Dts-1,ts,ψk,i〉, then
(5.23)∑Ak,ioftypeII‍P(Ak,i)|〈X,ψk,i〉|≤∑Ak,ioftypeII‍|X,ψk,i|2≤∥1Aj,lts-1Dts-1,ts∥22.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 Lemma5.16 and consider VTt0(HΠ) in (5.5) with N=1, that is, a static replication and so Aj,lt0=A0,0t0=Ω; then,
(5.24)VT^t0(HΠ)=e-r(T-t0)∑Ak,i∈𝒯Ω‍P(Ak,i)|〈X,ψk,i〉|(5.25)≤2δe-r(T-t0)×(∑m=0r1‍(qmδ1+ϵm)m+∑Ak,ioftypeII‍|〈X,ψk,i〉|2)(5.26)≤2δe-r(T-t0)×(11-2δ1+ϵt0+∥Dt0,T∥22).Proof.
Using (5.12) with s=1, the regularity assumption, and the constraints E(ψk,i)=0 and E(ψk,i2)=1, we obtain
(5.27)VT^t0(HΠ)=VTt0H(D0,T)≤2δe-r(T-t0)×(∑m=0j1-1‍∑{(k,i)∈Hm}‍P(Ak,i)|〈X,ψk,i〉|).
The result (5.24) then follows from Lemma 5.16.In Theorem5.18 we will make use of the following definition: consider a given path {Ajs,lsts}s=0N and
(5.28)ϵ≡mins=0,…,N-1ϵts.
Notice that ϵ>0 and it depends on the given path.The next theorem provides pointwise bounds for the volume of transactions for the portfolioHΠ in the general dynamic case (so N≥1). Corollary 5.17 is a special case of this theorem.Theorem 5.18.
Consider an arbitrary path{Ajs,lsts}s=0N with ω∈∩Ajs,lsts and the same hypothesis as in Lemma 5.16. At each node Ajs,lsts define the quantities qts-1=∑m=0rs‍(qmδ1+ϵm)m where the qm are as in Lemma 5.16. Then,
(5.29)VT^t0(HΠ)(w)=∑s=0N-1‍e-r(ts+1-ts)1P(Ajs,lsts)∑k,i∈𝒯Ajs,lsts‍P(Ak,i)|〈X,ψk,i〉|(5.30)≤2δ∑s=0N-1‍e-r(ts+1-ts)[(P(Ajs,lsts))ϵtsqts+1P(Ajs,lsts)∑Ak,ioftypeII‍|〈X,ψk,i〉|2](5.31)≤(1(1-2δ1+ϵ)1(1-δϵ)+∑s=0N-1‍1P(Ajs,lsts)∥1Ajs,lstsDts,ts+1∥22).Proof.
Using the approximated version of (5.12), the regularity assumption, and the constraints E(ψk,i)=0 and E(ψk,i2)=1, if ω∈∩Ajs,lsts, we obtain
(5.32)VT^tsH(Dts,ts+1)(w)≤2δe-r(ts+1-ts)×1P(Ajs,lsts)∑m=jsjs+1-1‍∑{(k,i)∈Hm:Ak,i⊂Ajs,lsts}‍P(Ak,i)|〈X,ψk,i〉|.
Using the first inequality in (5.20), inequality (5.32), and (5.5), we obtain for ω∈∩Ajs,lsts(5.33)VT^t0(HΠ)(w)≤2δ∑s=0N-1‍e-r(ts+1-ts)×[(P(Ajs,lsts))ϵtsqts+1P(Ajs,lsts)∑Ak,i∈𝒯Ajs,lstsoftypeII‍|〈X,ψk,i〉|2].
Notice that
(5.34)∑s=0N-1‍(P(Ajs,lsts))ϵts≤1+(δj1)ϵt1+⋯+(δjN-1)ϵtN-1,qts≤∑m=0rs+1‍(qmδ1+ϵ)m≤1(1-2δ1+ϵ).
Therefore,
(5.35)VT^t0(HΠ)(w)≤2δ(1(1-2δ1+ϵ)1(1-δϵ)+∑s=0N-1‍1P(Ajs,lsts)∥1Aj,ltsDts,ts+1∥22).Notice that Theorem5.18 provides two inequalities, the term after the first inequality represents afine bound and the term after the second inequality presents a coarser one. An analogous remark applies to Corollary 5.17. In the following Theorem we present an upper bound for the mean of the volume of transactions and make use only of the coarser versions of previous inequalities. The reason for this is that, at this point, we are mainly interested in showing how the upper bound for the mean value relates to the number of time steps N.Theorem 5.19.
Consider the same hypothesis as in Theorem5.18 and assume that ϵ is constant over all the tree paths. Then,
(5.36)E(VTt0(HΠ))≤2(1-2δ1+ϵ)+∥E(X∣ℱTn)-E(X)∥22.Proof.
Combining (5.5) with (5.12), we obtain
(5.37)E(VTt0(HΠ))=∑s=0N-1‍e-r(ts+1-ts)∑(j,l)∈Kjs‍∑m=jsjs+1-1‍∑{(k,i)∈Hm:Ak,i⊂Aj,lts}‍P(Ak,i)|〈X,ψk,i〉|.
Making use of the coarser inequality in (5.20), the fact that ∑(j,l)∈Kjs‍P(Aj,lts)=1 and j0≡0,
(5.38)E(VTt0(HΠ))=∑s=0N-1‍e-r(ts+1-ts)∑(j,l)∈Kjs‍((P(Aj,lts))1+ϵts1-2δ1+ϵts+∥1Aj,ltsDts,ts+1∥22)≤∑s=0N-1‍((δ1+ϵts)js1-2δ1+ϵts+∥Dts,ts+1∥22)≤1(1-2δ1+ϵ)1(1-δ1+ϵ)+∥E(X∣ℱTn)-E(X)∥22.Remark 5.20.
Notice how the summation over the time index is easily handled with a geometric sum resulting in1/(1-δ1+ϵ) and a telescopic sum for the squared norms of the martingale differences. As we indicated, one may substantially improve the factor 1/(1-2δ1+ϵ) by making use of finer bounds introduced in previous results. Numerical examples for volumes of transactions, their bounds and values of ϵ are displayed in Tables 1–6.Table 1
Relative errors, betweenX and XRc, in terms of number of transactions and scales. Single European call. Values of parameters as in Figure 1.
No. of transactions
J
=
6
J
=
8
J
=
10
J
=
12
J
=
14
J
=
16
R
=
5
0.1907
0.1907
0.1907
0.1907
0.1907
0.1907
R
=
15
0.0849
0.0631
0.0627
0.0627
0.0627
0.0627
R
=
30
0.0769
0.0421
0.0323
0.0312
0.0312
0.0312
R
=
64
0.0767
0.0361
0.02041
0.0157
0.0147
0.0147
R
=
128
NA
0.0351
0.0173
0.0101
0.0079
0.0074
R
=
256
NA
0.0351
0.0166
0.0084
0.0050
0.0040Table 2
Volume of transactions for single European call, that is, optionZ1. Values of parameters as in Figure 1, also, J=16 and Vt0(X)=0.809.
R
VT
^
(
Π
)
A
verage
VolTrBS
VT
^
(
HΠ
)
Fine bound
Coarse bound
Relative error
R
=
5
0.920208
53.32
1.6783
1.93614
22.9151
0.227333
R
=
10
0.95818
107.1
1.84845
2.36697
22.982
0.112873
R
=
15
0.95818
157.2
1.89718
2.5557
22.9943
0.0746597
R
=
20
0.959513
213.8
1.92295
2.70761
22.9983
0.0570624
R
=
25
0.959513
258.8
1.94195
2.84482
23.0004
0.0447283
R
=
30
0.959513
317
1.94998
2.90008
23.0015
0.0369233Table 3
Volume of transactions for portfolio composed of two calls and one put, that is, optionZ2. Values of parameters as in Figure 5, also, J=14 and Vt0(X)=2.3.
R
=
R
H
V
T
^
(
B
Π
)
Average
VolTrBS
V
T
^
(
H
Π
)
Fine bound
Coarse bound
Rel. Error
R
=
5
0.606359
27.1
1.04108
1.69357
7.42379
0.361129
R
=
10
0.652564
59.43
1.21478
2.0247
8.12219
0.167056
R
=
15
0.652564
90.75
1.26068
2.15734
8.89706
0.110117
R
=
20
0.658014
121.59
1.28732
2.28512
8.89993
0.0861122
R
=
25
0.658014
152.03
1.30466
2.40028
9.28114
0.0676239
R
=
30
0.659379
184.9
1.31901
2.61102
9.64969
0.056536Table 4
Dynamic hedging information for optionZ1. J=5 and RH=5 at each time step, remaining values of parameters as in Figure 1. The volume of transactions VT^(·) are averages over all the samples.
R (RH=5)
V
T
^
(
B
Π
)
V
T
^
(
H
Π
)
Fine bound
Coarse bound
Relative error
ϵ
R
=
10 (N=2)
1.26062
2.40553
3.73645
18.3088
0.0795891
0.0939396
R
=
15 (N=3)
1.47833
2.8358
4.46665
21.2139
0.0874086
0.0783706
R
=
20 (N=4)
1.63339
3.13998
5.08153
24.1135
0.09152
0.06748
R
=
25 (N=5)
1.8458
3.53484
5.50147
30.4139
0.0957869
0.0520525Table 5
Dynamic hedging information for the optionZ2. J=5 and RH=8 at each time step, remaining values of parameters as in Figure 5. The volume of transactions VT^(·) are averages over all the samples.
R (RH=8)
V
T
^
(
B
Π
)
V
T
^
(
H
Π
)
Fine bound
Coarse bound
Relative error
ϵ
R
=
16 (N=2)
0.593357
1.20214
3.85979
49.1631
0.128758
0.0305896
R
=
24 (N=3)
1.03076
1.96901
4.39251
55.9105
0.128939
0.0268424
R
=
32 (N=4)
1.15317
2.2223
7.59714
77.4299
0.125603
0.0192216
R
=
40 (N=5)
1.37302
2.74642
9.13719
121.156
0.1235
0.0121686Table 6
Dynamic hedging information for the optionZ3. Values of parameters: NA=2,…,5, KA=21.
R (RH=10)
V
T
^
(
B
Π
)
V
T
^
(
H
Π
)
Fine bound
Coarse bound
Relative error
ϵ
R
=
20 (N=2)
0.958413
1.66733
2.86227
8.78945
0.05
0.225927
R
=
30 (N=3)
1.01458
1.6126
2.93135
8.03383
0.0715974
0.245643
R
=
40 (N=4)
1.03078
1.53383
2.92057
7.8773
0.0740302
0.249264
R
=
50 (N=5)
1.22817
1.87943
3.1592
8.53391
0.0735656
0.225817


## 5.1. Representation by Characteristic Functions

From (4.18), (4.19), and (5.2), we obtain the following proposition.Proposition 5.7.
Forw∈Aj,lts-1,
(5.6)VTts-1(Dts-1,ts)(w)=e-r(ts-ts-1)∑{(k,i)∈Kjs:Ak,its⊆Aj,lts-1}‍〈Dts-1,ts,ϕk,its〉P(Ak,its)E(1Ak,its∣ℱts-1)(w).(5.7)VT^ts-1(Dts-1,ts)(w)=e-r(ts-ts-1)1Aj,lts-1P(Aj,lts-1)∥1Aj,lts-1Dts-1,ts∥1.Corollary 5.8.
Consider an arbitrary path{Aks,ists}s=1,…,N, that is, Aks,ists⊆Aks-1,is-1ts-1. Then,
(5.8)VT^t0(BΠ)=∑s=1N‍e-r(ts-ts-1)1P(Aks,ists)∥1Aks,istsDts-1,ts∥1,a.e.on∩Aks,ists.Corollary 5.9.
(5.9)
E
(
V
T
t
0
(
B
Π
)
)
=
∑
s
=
1
N
‍
e
-
r
(
t
s
-
t
s
-
1
)
∥
D
t
s
-
1
,
t
s
∥
1
≤
∑
s
=
1
N
‍
e
r
(
t
s
-
t
s
-
1
)
∥
D
t
s
-
1
,
t
s
∥
2
.Proof.
Notice that(5.10)⋃{(k,i)∈Kjs:Ak,its⊆Aj,lts-1}‍Ak,its=Aj,lts-1,⋃(j,l)∈Kjs-1‍Aj,lts-1=Ω,
and also, Ak,its∩Ak',i'ts=∅. Using (5.5), (5.6), and Aj,lts-1∈ℱts-1, we obtain
(5.11)E(VTt0(BΠ))=∑s=1N‍e-r(ts-ts-1)∑(j,l)∈Kjs-1‍∑{(k,i)∈Kjs:Ak,its⊆Aj,lts-1}‍|〈Dts-1,ts,1Ak,its〉|P(Ak,its)E(1Aj,lts-1E(1Ak,its∣ℱts-1))=∑s=1N‍e-r(ts-ts-1)∥Dts-1,ts∥1.
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 portfolioHΠ, 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 Section2.1, we may set 𝒯Aj,lts-1 to denote the tree containing its root Aj,lts-1 and internal nodes from ∪m=js-1js-1{Ak,i∈Hm}. So the double sums appearing above could be written ∑m=js-1js-1‍∑{(k,i)∈Hm:Ak,i⊂Aj,lts-1}‍=∑Ak,i∈𝒯Aj,lts-1‍. Notice that the condition Ak,i⊂Aj,lts-1 includes the case Ak,i=Aj,lts-1.Using (5.2), the volume of transactions associated with decomposition (4.17) is given in the following Lemma.Lemma 5.11.
Forw∈Aj,lts-1,
(5.12)VTts-1H(Dts-1,ts)(w)=e-r(ts-ts-1)∑m=js-1js-1∑{(k,i)∈Hm:Ak,i⊂Aj,lts-1}‍|X,ψk,i|×(|ak,i|E(1Ak+1,2i∣ℱts-1)(w)+|bk,i|E(1Ak+1,2i+1∣ℱts-1)(w)).As previously indicated, the volume of transactions depends on the representation and we have used the notationVTH to highlight this fact. The approximate version of (5.12) is V^Tts-1H(Dts-1,ts) and is obtained by replacing ℱts-1 by ℱts-1n in (5.12).Lemma 5.12.
Forw∈Aj,lts-1,
(5.13)VT^ts-1H(Dts-1,ts)(w)=e-r(ts-ts-1)1Aj,lts-1P(Aj,lts-1)∑m=js-1js-1∑{(k,i)∈Hm:Ak,i⊂Aj,lts-1}‍|〈X,ψk,i〉|∥ψk,i∥1.The following proposition tells us that the portfolioBΠ has a smaller volume of transactions than the portfolio HΠ. Whenever appropriate we will use VTB to denote the volume of transactions for the portfolio BΠ.Proposition 5.13.
One has,(5.14)VT^ts-1B(Dts-1,ts)≤V^Tts-1H(Dts-1,ts).
Therefore,
(5.15)VT^t0(BΠ)≤V^Tt0(HΠ).Remark 5.14.
Of course, Remark5.5 applies and the above proposition implies the corresponding results after taking expectations. Moreover, Proposition 5.13 also holds if Dts-1,ts is replaced for any random variable Y∈ℱtsn, in particular Y=Dts-1,tsc, namely the compressed version of Dts-1,ts.Proof.
Making use of (4.16) and the triangle inequality,
(5.16)1Aj,lts-1P(Aj,lts-1)∥1Aj,lts-1Dts-1,ts∥1≤1Aj,lts-1P(Aj,lts-1)∑m=js-1js-1‍∑{(k,i)∈Hm:Ak,i⊂Aj,lts-1}‍|〈X,ψk,i〉|∥ψk,i∥1.
Then, (5.14) follows after observing (5.7) and (5.12).In the computations that follow we assume that our approximating martingale isregular [19], namely for all parents nodes Ak,i and its children we have,
(5.17)P(Ak+1,l)≤δP(Ak,i),l=2i,2i+1,
where δ≥1/2 is the same for all nodes. For simplicity we will take δ=1/2 as other choices will make the statements of our results more complex. Also, by means of Definition 5.15, nodes Ak,i belonging to a tree 𝒯n are classified according to their size relative to the inner products ∣〈X,ψk,i〉∣.Definition 5.15.
Consider a nodeAk,i belonging to a tree 𝒯n associated with a given H-system satisfying 〈X,ψk,i〉≠0. The node Ak,i is said to be of type I if
(5.18)|〈X,ψk,i〉|<P(Ak,i).
Furthermore, it is said to be of type II if ∣〈X,ψk,i〉∣≥P(Ak,i).
Also defineϵk,i by: ∣〈X,ψk,i〉∣=P(Ak,i)1/2+ϵk,i. Notice that ϵk,i>0 whenever Ak,i is of type I and that ϵk,i≤0 whenever Ak,i is of type II.For the following lemma, consider a generic nodeAj,lts-1 of a given H-system and its associated tree 𝒯Aj,lts-1. To simplify the notation, nodes Ak,i will always satisfy Ak,i∈𝒯Aj,lts-1. In Lemma 5.16 we will make use of the following definition
(5.19)ϵm≡min{ϵk,i:(k,i)∈Hm,Ak,ioftypeI}ϵts-1≡min{ϵm:js-1<m≤js}.
Notice that ϵts-1>0 and its value depend on the given node Aj,lts-1.Lemma 5.16.
Assume a given H -system is regular withδ=1/2. Let 0≤qmm be the number of instances for which nodes at level m of the tree 𝒯Aj,lts-1 are of type I and rs≡js-js-1. Then,
(5.20)∑Ak,i∈𝒯Aj,lts-1‍P(Ak,i)|〈X,ψk,i〉|≤(P(Aj,lts-1))1+ϵts-1∑m=0rs‍(qmδ1+ϵm)m+∑Ak,ioftypeII‍|〈X,ψk,i〉|2≤(P(Aj,lts-1))1+ϵts-111-2δ1+ϵts-1+∥1Aj,lts-1Dts-1,ts∥22.Proof.
Considering the fact thatℛm can have at most 2m members, hence qm≤2, we obtain
(5.21)∑Ak,ioftypeI‍P(Ak,i)|〈X,ψk,i〉|≤∑m=0rs‍(qmδ1+ϵm)m(P(Aj,lts-1))1+ϵm≤(P(Aj,lts-1))1+ϵts-1∑m=0rs‍(qmδ1+ϵm)m≤(P(Aj,lts-1))1+ϵts-111-2δ1+ϵts-1.
According to the decomposition,
(5.22)∑Ak,i∈𝒯Aj,lts-1‍P(Ak,i)|〈X,ψk,i〉|=∑Ak,ioftypeI‍P(Ak,i)∣〈X,ψk,i〉∣+∑Ak,ioftypeII‍P(Ak,i)|〈X,ψk,i〉|,
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, 〈X,ψk,i〉=〈Dts-1,ts,ψk,i〉, then
(5.23)∑Ak,ioftypeII‍P(Ak,i)|〈X,ψk,i〉|≤∑Ak,ioftypeII‍|X,ψk,i|2≤∥1Aj,lts-1Dts-1,ts∥22.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 Lemma5.16 and consider VTt0(HΠ) in (5.5) with N=1, that is, a static replication and so Aj,lt0=A0,0t0=Ω; then,
(5.24)VT^t0(HΠ)=e-r(T-t0)∑Ak,i∈𝒯Ω‍P(Ak,i)|〈X,ψk,i〉|(5.25)≤2δe-r(T-t0)×(∑m=0r1‍(qmδ1+ϵm)m+∑Ak,ioftypeII‍|〈X,ψk,i〉|2)(5.26)≤2δe-r(T-t0)×(11-2δ1+ϵt0+∥Dt0,T∥22).Proof.
Using (5.12) with s=1, the regularity assumption, and the constraints E(ψk,i)=0 and E(ψk,i2)=1, we obtain
(5.27)VT^t0(HΠ)=VTt0H(D0,T)≤2δe-r(T-t0)×(∑m=0j1-1‍∑{(k,i)∈Hm}‍P(Ak,i)|〈X,ψk,i〉|).
The result (5.24) then follows from Lemma 5.16.In Theorem5.18 we will make use of the following definition: consider a given path {Ajs,lsts}s=0N and
(5.28)ϵ≡mins=0,…,N-1ϵts.
Notice that ϵ>0 and it depends on the given path.The next theorem provides pointwise bounds for the volume of transactions for the portfolioHΠ in the general dynamic case (so N≥1). Corollary 5.17 is a special case of this theorem.Theorem 5.18.
Consider an arbitrary path{Ajs,lsts}s=0N with ω∈∩Ajs,lsts and the same hypothesis as in Lemma 5.16. At each node Ajs,lsts define the quantities qts-1=∑m=0rs‍(qmδ1+ϵm)m where the qm are as in Lemma 5.16. Then,
(5.29)VT^t0(HΠ)(w)=∑s=0N-1‍e-r(ts+1-ts)1P(Ajs,lsts)∑k,i∈𝒯Ajs,lsts‍P(Ak,i)|〈X,ψk,i〉|(5.30)≤2δ∑s=0N-1‍e-r(ts+1-ts)[(P(Ajs,lsts))ϵtsqts+1P(Ajs,lsts)∑Ak,ioftypeII‍|〈X,ψk,i〉|2](5.31)≤(1(1-2δ1+ϵ)1(1-δϵ)+∑s=0N-1‍1P(Ajs,lsts)∥1Ajs,lstsDts,ts+1∥22).Proof.
Using the approximated version of (5.12), the regularity assumption, and the constraints E(ψk,i)=0 and E(ψk,i2)=1, if ω∈∩Ajs,lsts, we obtain
(5.32)VT^tsH(Dts,ts+1)(w)≤2δe-r(ts+1-ts)×1P(Ajs,lsts)∑m=jsjs+1-1‍∑{(k,i)∈Hm:Ak,i⊂Ajs,lsts}‍P(Ak,i)|〈X,ψk,i〉|.
Using the first inequality in (5.20), inequality (5.32), and (5.5), we obtain for ω∈∩Ajs,lsts(5.33)VT^t0(HΠ)(w)≤2δ∑s=0N-1‍e-r(ts+1-ts)×[(P(Ajs,lsts))ϵtsqts+1P(Ajs,lsts)∑Ak,i∈𝒯Ajs,lstsoftypeII‍|〈X,ψk,i〉|2].
Notice that
(5.34)∑s=0N-1‍(P(Ajs,lsts))ϵts≤1+(δj1)ϵt1+⋯+(δjN-1)ϵtN-1,qts≤∑m=0rs+1‍(qmδ1+ϵ)m≤1(1-2δ1+ϵ).
Therefore,
(5.35)VT^t0(HΠ)(w)≤2δ(1(1-2δ1+ϵ)1(1-δϵ)+∑s=0N-1‍1P(Ajs,lsts)∥1Aj,ltsDts,ts+1∥22).Notice that Theorem5.18 provides two inequalities, the term after the first inequality represents afine bound and the term after the second inequality presents a coarser one. An analogous remark applies to Corollary 5.17. In the following Theorem we present an upper bound for the mean of the volume of transactions and make use only of the coarser versions of previous inequalities. The reason for this is that, at this point, we are mainly interested in showing how the upper bound for the mean value relates to the number of time steps N.Theorem 5.19.
Consider the same hypothesis as in Theorem5.18 and assume that ϵ is constant over all the tree paths. Then,
(5.36)E(VTt0(HΠ))≤2(1-2δ1+ϵ)+∥E(X∣ℱTn)-E(X)∥22.Proof.
Combining (5.5) with (5.12), we obtain
(5.37)E(VTt0(HΠ))=∑s=0N-1‍e-r(ts+1-ts)∑(j,l)∈Kjs‍∑m=jsjs+1-1‍∑{(k,i)∈Hm:Ak,i⊂Aj,lts}‍P(Ak,i)|〈X,ψk,i〉|.
Making use of the coarser inequality in (5.20), the fact that ∑(j,l)∈Kjs‍P(Aj,lts)=1 and j0≡0,
(5.38)E(VTt0(HΠ))=∑s=0N-1‍e-r(ts+1-ts)∑(j,l)∈Kjs‍((P(Aj,lts))1+ϵts1-2δ1+ϵts+∥1Aj,ltsDts,ts+1∥22)≤∑s=0N-1‍((δ1+ϵts)js1-2δ1+ϵts+∥Dts,ts+1∥22)≤1(1-2δ1+ϵ)1(1-δ1+ϵ)+∥E(X∣ℱTn)-E(X)∥22.Remark 5.20.
Notice how the summation over the time index is easily handled with a geometric sum resulting in1/(1-δ1+ϵ) and a telescopic sum for the squared norms of the martingale differences. As we indicated, one may substantially improve the factor 1/(1-2δ1+ϵ) by making use of finer bounds introduced in previous results. Numerical examples for volumes of transactions, their bounds and values of ϵ are displayed in Tables 1–6.Table 1
Relative errors, betweenX and XRc, in terms of number of transactions and scales. Single European call. Values of parameters as in Figure 1.
No. of transactions
J
=
6
J
=
8
J
=
10
J
=
12
J
=
14
J
=
16
R
=
5
0.1907
0.1907
0.1907
0.1907
0.1907
0.1907
R
=
15
0.0849
0.0631
0.0627
0.0627
0.0627
0.0627
R
=
30
0.0769
0.0421
0.0323
0.0312
0.0312
0.0312
R
=
64
0.0767
0.0361
0.02041
0.0157
0.0147
0.0147
R
=
128
NA
0.0351
0.0173
0.0101
0.0079
0.0074
R
=
256
NA
0.0351
0.0166
0.0084
0.0050
0.0040Table 2
Volume of transactions for single European call, that is, optionZ1. Values of parameters as in Figure 1, also, J=16 and Vt0(X)=0.809.
R
VT
^
(
Π
)
A
verage
VolTrBS
VT
^
(
HΠ
)
Fine bound
Coarse bound
Relative error
R
=
5
0.920208
53.32
1.6783
1.93614
22.9151
0.227333
R
=
10
0.95818
107.1
1.84845
2.36697
22.982
0.112873
R
=
15
0.95818
157.2
1.89718
2.5557
22.9943
0.0746597
R
=
20
0.959513
213.8
1.92295
2.70761
22.9983
0.0570624
R
=
25
0.959513
258.8
1.94195
2.84482
23.0004
0.0447283
R
=
30
0.959513
317
1.94998
2.90008
23.0015
0.0369233Table 3
Volume of transactions for portfolio composed of two calls and one put, that is, optionZ2. Values of parameters as in Figure 5, also, J=14 and Vt0(X)=2.3.
R
=
R
H
V
T
^
(
B
Π
)
Average
VolTrBS
V
T
^
(
H
Π
)
Fine bound
Coarse bound
Rel. Error
R
=
5
0.606359
27.1
1.04108
1.69357
7.42379
0.361129
R
=
10
0.652564
59.43
1.21478
2.0247
8.12219
0.167056
R
=
15
0.652564
90.75
1.26068
2.15734
8.89706
0.110117
R
=
20
0.658014
121.59
1.28732
2.28512
8.89993
0.0861122
R
=
25
0.658014
152.03
1.30466
2.40028
9.28114
0.0676239
R
=
30
0.659379
184.9
1.31901
2.61102
9.64969
0.056536Table 4
Dynamic hedging information for optionZ1. J=5 and RH=5 at each time step, remaining values of parameters as in Figure 1. The volume of transactions VT^(·) are averages over all the samples.
R (RH=5)
V
T
^
(
B
Π
)
V
T
^
(
H
Π
)
Fine bound
Coarse bound
Relative error
ϵ
R
=
10 (N=2)
1.26062
2.40553
3.73645
18.3088
0.0795891
0.0939396
R
=
15 (N=3)
1.47833
2.8358
4.46665
21.2139
0.0874086
0.0783706
R
=
20 (N=4)
1.63339
3.13998
5.08153
24.1135
0.09152
0.06748
R
=
25 (N=5)
1.8458
3.53484
5.50147
30.4139
0.0957869
0.0520525Table 5
Dynamic hedging information for the optionZ2. J=5 and RH=8 at each time step, remaining values of parameters as in Figure 5. The volume of transactions VT^(·) are averages over all the samples.
R (RH=8)
V
T
^
(
B
Π
)
V
T
^
(
H
Π
)
Fine bound
Coarse bound
Relative error
ϵ
R
=
16 (N=2)
0.593357
1.20214
3.85979
49.1631
0.128758
0.0305896
R
=
24 (N=3)
1.03076
1.96901
4.39251
55.9105
0.128939
0.0268424
R
=
32 (N=4)
1.15317
2.2223
7.59714
77.4299
0.125603
0.0192216
R
=
40 (N=5)
1.37302
2.74642
9.13719
121.156
0.1235
0.0121686Table 6
Dynamic hedging information for the optionZ3. Values of parameters: NA=2,…,5, KA=21.
R (RH=10)
V
T
^
(
B
Π
)
V
T
^
(
H
Π
)
Fine bound
Coarse bound
Relative error
ϵ
R
=
20 (N=2)
0.958413
1.66733
2.86227
8.78945
0.05
0.225927
R
=
30 (N=3)
1.01458
1.6126
2.93135
8.03383
0.0715974
0.245643
R
=
40 (N=4)
1.03078
1.53383
2.92057
7.8773
0.0740302
0.249264
R
=
50 (N=5)
1.22817
1.87943
3.1592
8.53391
0.0735656
0.225817


## 6. Numerical Examples

This section presents output from a computer implementation based on the Brownian motion example from AppendixA. This type of H-system can be seen to correspond to an initial partition corresponding to the recursive construction of Willinger (referred to in our Section 3), we have chosen such an example for matters of simplicity. Also, the fact that we have chosen to discretize the geometric Brownian model is also a matter of convenience, any other model with continuous paths and independent increments will also be possible (see the discussion after Theorem 4.6). Figures and tables with numerical outputs are presented in Tables 1–6 and Figures 1–6.Figure 1
Approximations to single European call using delta hedging and a single Haar function (as explained in Section6). Values of the parameters used: N=1,  St0=20,r=0.05,σ=0.1,T-t0=1, K=21.Figure 2
Same as in Figure1 but with u0 added.Figure 3
Same as in Figure1 except R=20.Figure 4
L
2 norm of the (relative) errors between the option Z1 and delta hedging and Haar approximations respectively. The plot is in terms of the parameter R. Values of the parameters used: as in Figure 1.Figure 5
Approximations to portfolioZ2, constructed from two calls and one put, using delta hedging and the Haar system constructed via Brownian motion increments. Values of the parameters used: N=1, R=20,  St0=20,r=0.05,σ=0.1,T-t0=s1, K1=19.35,  K2=20.96, K3=22.7.Figure 6
L
2 norm of the (relative) errors between the option Z2 and delta hedging and Haar approximations, respectively. The plot is in terms of the parameter R. Values of the parameters used: as in Figure 5.Figure 7The simple discretization from AppendixA gives binary partitions ℛj={Ak,i} that are constructed via the increments of the Brownian motion and are characterized through the parameters N and j1,…,jN. In this example all elements of ℛj split in ℛj+1, therefore the indexation introduced in Section 2 gives for 0≤j≤jN(6.1)ℛj={Aj,i:0≤i≤2j-1},Kj={(j,i):0≤i≤2j-1},
and Hj=Kj for 0≤j<jN. Moreover,
(6.2)u0=1Ω,{u2j+i≡ψj,i:0≤j<jN,0≤i≤2j-1}.For the sake of simplicity we will takers≡js-js-1=J, for all s=1,…,N, where J is a fixed constant and j0≡0. Given the nature of our numerical experiments, there will be a natural way to select the value of N; the parameter J is chosen empirically in such a way that (consider n≡(N,J))
(6.3)∥X-E(X∣ℱtNn)∥2≤η,
for a negligible η. In our simulations we have taken J=14,16 for the static cases and J=5,8,10 (depending on the option under consideration) for each stage in the dynamic cases.We will make use of the notion ofcompression as described in Section 4.2 and some of the definitions and results introduced elsewhere in the paper.For dynamic hedging, at each nodeAj,lts-1 of an unfolding path w∈∩Ajs,lstss=0,…,N, we will keep a constant (i.e., independent of the node) number RH of inner products 〈X,ψk,i〉 with Ak,i∈𝒯Aj,lts-1. Therefore, RH is the number of Haar obligations (see Section 4), purchased at ts-1 and expiring at ts, which are used by the portfolio HΠ when constructing XRc. Notice that, with this in mind, we then require N×RH=R. Considering RH to be constant helps to easy the comparisons with delta hedging but it is not a necessary constraint. In practice one may choose RH adaptively depending on the node in order to make sure a certain level of desired approximation is reached. Notice that static hedging corresponds to N=1 and so RH=R.To summarize, the parameterR will be equal to the number of times the Black-Scholes portfolio is re-balanced when performing delta hedging and also equals the total number of Haar functions used in the final approximation when performing Haar hedging. R will be referred to as the number of transactions in the remaining of the paper. We re-balance the Black-Scholes portfolio at uniformly spaced time intervals. We report the errors in the approximations as well as the volume of transactions, both as a function of the number of transactions. Equivalently, one may report these quantities as functions of the approximation error.Whenever feasible, we report relative errors given by:(6.4)∥X-XRc∥2∥X∥2,
where, as usual, ∥∥2 is the L2 norm. Given the nature of our computations we will use E(X∣ℱtNn) instead of X in (6.4), this gives a negligible error (see (6.3)). In order to reduce computational times, and only when doing dynamic hedging, we will report a sampled version of (6.4) (see (6.15)).


### 6.1. Upper Bounds

The exact values for the volume of transactions for the portfolioHΠ are given by the equalities in (5.24) and (5.29) for the static and dynamic, case respectively. We will also report the fine upper bound given by (5.25) (for the static case) and (5.30) (for the dynamic case). Analogously, we provide the coarse upper bound given by (5.26) (for the static case) and the term (5.31) (for the dynamic case). Notice that all these quantities refer to the pointwise volume of transactions (which, in the static case, equals its mean value). Also the parameter ϵ will be computed.In practice, it is convenient to distinguish between two classes oftype I of Ak,i nodes (introduced in Definition 5.15), namely, the ones that satisfy the further constraint
(6.5)|〈X,ψk,i〉|<P(Ak,i)1/2+k.
For a relatively large k, we took k=3 in our simulations. This refinement reduces the values of the numbers qm and makes the values of the numbers ϵm and ϵ larger (and so makes the fine bounds tighter). These numbers appear in the inequalities of Corollary 5.17 and Theorem 5.18 The net effect on upper bounds of introducing the class of type I nodes satisfying (6.5) is an additive term with value no larger than 1.6. Our numerical simulations will report fine and coarse upper bounds including this modification.


### 6.2. Multiresolution Analysis Algorithm

The computations can be organized by means of the multiresolution analysis algorithm (MRA). The algorithm is an adaptation of the well-known algorithm from wavelet theory [20] to our probabilistic setting. This algorithm produces a relation between the samples of X, namely,
(6.6)xkts[i]=E(X∣ℱtsn)(w),ω∈Ak,its∈𝒫tsn,
and the Haar coefficient 〈X,ψk,i〉. This facilitates the computation and manipulation of these inner products. the algorithm for our setting is described in [14] (see also below).Given that we will be reporting pathwise quantities, we have to consider (6.6) relative to a given node Aj,lts-1, and therefore our algorithm consists of an unfolding path w∈∩Ajs,lsts and, at each time step ts, s=0,…,N-1, the quantities xkts[i] (given by (6.6)), conditioned to Ajs-1,ls-1ts-1, are inputs to the MRA algorithm described in [14]. Once the inner products 〈X,ψk,i〉 are available, we perform compression. After this, in order to compute the volume of transactions and the associated upper bounds, only the numbers 〈XRc,ψk,i〉 are needed.The input to the MRA is obtained by computing(6.7)xkts[i]=2js∫Ak,its‍X(ω)dP(ω),(k,i)∈Kjs,
where P(Ak,its)=1/2js. Recalling that rs=js-js-1=J, we observe that there are 2J numbers in the array xkts[·].For completeness, we include a sketch of the MRA, for the stages-1→s.Analysis Step
Setdk[i]≡〈X,ψk,i,〉(6.8)for1≤k≤J,xk[i]→{xk-1[i]=12(xk[2i+1]+xk[2i]),dk-1[i]=2-(js-1+k+1)/2(xk[2i]-xk[2i+1]),0≤i≤2k-1.Compression Step
For0≤k≤J-1, keep the greatest R coefficients dkcomp[i]≡dk[i], and set dkcomp[i]≡0 for the rest of the 2J-1-R ones.Reconstruction Step
Considerx0comp[0]≡x0[0], for k=1,…J,
(6.9)dk-1comp[i]→{xkcomp[2i]=xk-1comp[i]+2(js-1+k-1)/2dk-1comp[i],xkcomp[2i+1]=xk-1comp[i]-2(js-1+k-1)/2dk-1comp[i],0≤i≤2k-1.
Some details on how to computexkts[i] are provided later in this section.


### 6.3. Delta Hedging

Delta hedging will be performed overR time steps. Letting
(6.10)φti=∂Vti(X)∂Sti,
the volume of transactions for a Black-Scholes portfolio with rebalancing dates {ti},i=0,…,R-1 is given by
(6.11)∑i=0R-1‍[|φti-φti-1|Sti+(Bti-Bti-1er(ti-ti-1))+],
with φt-1=Bt-1≡0 (Bt is the bond used in Section 4.1). We have used equally spaced rebalancing dates. Given that (6.11) is a random quantity, we will report the average (AverageVolTrBS) over many samples.


### 6.4. Path Independent Options

For the path independent examples we will consider two cases:(i)N=1 which provides static hedging, in this case all the atomic sigma algebras 𝒜n are included in σ(WT) and 𝒜∞=σ(WT); (ii)N>1 which provides dynamic hedging. Mathematically, in the case of path independent options, already the case N=1 provides perfect replication in the limit of J→∞.(i) Static Case
The expressions for the pathwise volume of transactions for our portfolios in this case are(6.12)VT^t0(BΠ)=e-r(T-t0)∥XRc∥1=e-r(T-t0)∫Ω‍|XRc(ω)|dP(ω),VT^t0(HΠ)=e-r(T-t0)∑Ak,i∈𝒯Ω‍P(Ak,i)|〈XRc,ψk,i〉|.
Next we comment on the output displays (see Tables1–6 and Figures 1–6); consider first a single European call Z1=(ST-K)+, values of parameters are indicated in the text surrounding the figures. In Figures 1, 2, and 3 we present the Black-Scholes and Haar approximations with R=1,2,20, respectively. Notice how Figure 1 shows the Haar approximation with u1=1/2(1A1,0-1A1,1), which happens to give the largest inner product. Figure 2 shows the Haar approximation when u0 is added, giving the second largest inner product in this example. Figure 4 shows the estimation of the L2 norm of the errors as a function of R. Similar information is presented in Table 1; it illustrates the compression power of our H-system for the call option Z1 in the present static case. It shows that few Haar functions are needed to implement HΠ. That table provides the results for different values of J as this better illustrates the effect of the optimization carried out in (4.13).
As a second example we consider a portfolio built as a linear combination of European calls and puts as follows,Z2=(ST-K1)++(K2-ST)+-(ST-K3)+, values of parameters are indicated in the text surrounding Figure 5. Finally, Figure 6 shows the estimation of the L2 norm of the errors as a function of R.
Tables2 and 3 show the volume of transactions for the portfolio BΠ, for delta hedging and for the Haar portfolio HΠ. The tables also contain values of the fine and coarse upper bounds for the portfolio HΠ.
Notice the difference in magnitudes withAverageVolTrBS. The volume of transactions offers a clear numerical evidence of the different nature of Haar hedging and delta hedging, an analytical analysis was provided in Section 5.
It is expected, and it is confirmed by our experience with numerical examples, that the Haar approximation outperforms (in the sense of smaller error for equal value ofR) the Black-Scholes approximation whenever the payoff, or its derivative, contains discontinuities. Moreover, the Haar functions can be adapted to these discontinuities; for instance, we can choose u1 such that it is supported in the union of A1,0={ST<K} and A1,1={ST≥K} for the case of the European call. Our examples reflect these choices; for example, St0 was taken close to K so as the discontinuity in the first derivative of the European call becomes problematic for Black-Scholes approximation and can be reproduced efficiently by the Haar expansion. An extreme example of this kind will be the case of a digital option where, of course, the Haar expansions have no bearing as a hedging tool.
Naturally, it is easy to find situations where delta hedging outperforms Haar hedging as, for example, a position in a European call which is well in or out of the money. This is a situation where the linear approximation in delta hedging becomes very efficient. It is interesting to see under what conditions delta hedging and Haar hedging are complementary and to investigate how to combine both techniques.
Tables2 and 3 show that the behaviors of the portfolios BΠ and HΠ are similar in these static examples. It is also important to notice that we were able to make an effective use of the notion of compression because of the Haar representation; expansions on characteristic functions do not allow such a simple effective optimization procedure.(ii) Dynamic Case
Given a pathw∈∩s=0NAjs,lsts, the volume of transactions for our portfolios is given by:
(6.13)VT^t0(BΠ)(w)=∑s=0N-1‍e-r(ts+1-ts)1Ajs,lsts∑Ak,its+1∈𝒯Ajs,lsts‍P(Ak,its+1)P(Ajs,lsts)×|[1P(Ak,its+1)∫Ak,its+1‍X-1P(Ajs,lsts)∫Ajs,lsts‍X]|,(6.14)VT^t0(HΠ)=2δ∑s=0N-1‍e-r(ts+1-ts)1P(Ajs,lsts)∑Ak,i∈𝒯Ajs,lsts‍P(Ak,i)|〈X,ψk,i〉|.
In practice we do not reproduce Dts,ts+1 but its compressed version, and therefore the averages in (6.13) have to be replaced by the expressions described in the MRA above. Similarly the inner products in (6.14) have to be replaced by their compressed versions. An expression for the input to the MRA, at each dynamic stage, is provided by (6.18).
We illustrate the dynamic case for the path independent optionsZ1 and Z2 described above. Notice that our main point is to illustrate how the volume of transactions for the portfolios BΠ and HΠ behaves when transactions are performed over time, that is in a dynamic way. Naturally, we will report average values (over M random paths, we took M between 200 and 500) of all the quantities involved: volume of transactions, fine and coarse bounds, relative errors, and the values of ϵ. For example, if we sample w1,…,wM paths, the absolute error becomes
(6.15)1M∑m=1M‍|X(wi)-XRc(wi)|2.
Tables 4 and 5 display the resulting values averaged between M=200 and M=500 random paths for the options Z1 and Z2, respectively. The bounds correspond to the portfolio HΠ. Noticing the need to use the parameter RH, for simplicity we fixed this parameter to be RH=5 for Z1 and RH=8 for Z2 to reduce the error and increased N (and so also increased R=RH×N). One may also work with an adaptive value of RH. As indicated, we are reporting averages of VT^(·) over all the samples, the oscillations around these values seem to be negligible.
Another illustration is provided by apath dependent example, it necessarily requires dynamic hedging. Again, our main objective is to illustrate how the volume of transactions for both of our portfolios behaves when performing transactions over time. We consider the Asian option
(6.16)Z3=(1NA∑s=0NA-1‍StNA-s-KA)+.
It is natural to deploy the H-system along the required times, t0,t1,…,tNA and so we will take N=NA.
For each sampled path, the computation ofxkts[i] involves N iterated integrals. Indeed, the stock, discretized in time, follows an evolution S=F(Wt0,t1,…,WtN-1,tN) where {Wts-1,ts}s=1N are independent random variables with normal distribution, and then the atoms of ℱtsn contained in Aks-1,is-1ts-1 are given by
(6.17)Ak,its=Wts-1,ts-1((ciJ,ci+1J])∩Wts,ts+1-1((-∞,∞))∩…,∩WtN-1,tN-1((-∞,∞)),
where the ciJ are given in Appendix A. Consequently
(6.18)xkts[i]=2js∫-∞∞‍⋯∫-∞∞‍∫ciJci+1J‍∫cis-1Jcis-1+1J‍⋯∫ci1Jci1+1J‍(F(y1,…,yN)-KA)+e-y12/2⋯e-yN2/2dy1⋯dyN.
In order to avoid exceedingly long computational times we have takenNA=2,…,5. Notice that in order to reduce the error of approximation we are required to increase RH (we have taken RH=10). Table 6 displays the resulting values averaged over approximately M=500 random paths.


## 6.1. Upper Bounds

The exact values for the volume of transactions for the portfolioHΠ are given by the equalities in (5.24) and (5.29) for the static and dynamic, case respectively. We will also report the fine upper bound given by (5.25) (for the static case) and (5.30) (for the dynamic case). Analogously, we provide the coarse upper bound given by (5.26) (for the static case) and the term (5.31) (for the dynamic case). Notice that all these quantities refer to the pointwise volume of transactions (which, in the static case, equals its mean value). Also the parameter ϵ will be computed.In practice, it is convenient to distinguish between two classes oftype I of Ak,i nodes (introduced in Definition 5.15), namely, the ones that satisfy the further constraint
(6.5)|〈X,ψk,i〉|<P(Ak,i)1/2+k.
For a relatively large k, we took k=3 in our simulations. This refinement reduces the values of the numbers qm and makes the values of the numbers ϵm and ϵ larger (and so makes the fine bounds tighter). These numbers appear in the inequalities of Corollary 5.17 and Theorem 5.18 The net effect on upper bounds of introducing the class of type I nodes satisfying (6.5) is an additive term with value no larger than 1.6. Our numerical simulations will report fine and coarse upper bounds including this modification.


## 6.2. Multiresolution Analysis Algorithm

The computations can be organized by means of the multiresolution analysis algorithm (MRA). The algorithm is an adaptation of the well-known algorithm from wavelet theory [20] to our probabilistic setting. This algorithm produces a relation between the samples of X, namely,
(6.6)xkts[i]=E(X∣ℱtsn)(w),ω∈Ak,its∈𝒫tsn,
and the Haar coefficient 〈X,ψk,i〉. This facilitates the computation and manipulation of these inner products. the algorithm for our setting is described in [14] (see also below).Given that we will be reporting pathwise quantities, we have to consider (6.6) relative to a given node Aj,lts-1, and therefore our algorithm consists of an unfolding path w∈∩Ajs,lsts and, at each time step ts, s=0,…,N-1, the quantities xkts[i] (given by (6.6)), conditioned to Ajs-1,ls-1ts-1, are inputs to the MRA algorithm described in [14]. Once the inner products 〈X,ψk,i〉 are available, we perform compression. After this, in order to compute the volume of transactions and the associated upper bounds, only the numbers 〈XRc,ψk,i〉 are needed.The input to the MRA is obtained by computing(6.7)xkts[i]=2js∫Ak,its‍X(ω)dP(ω),(k,i)∈Kjs,
where P(Ak,its)=1/2js. Recalling that rs=js-js-1=J, we observe that there are 2J numbers in the array xkts[·].For completeness, we include a sketch of the MRA, for the stages-1→s.Analysis Step
Setdk[i]≡〈X,ψk,i,〉(6.8)for1≤k≤J,xk[i]→{xk-1[i]=12(xk[2i+1]+xk[2i]),dk-1[i]=2-(js-1+k+1)/2(xk[2i]-xk[2i+1]),0≤i≤2k-1.Compression Step
For0≤k≤J-1, keep the greatest R coefficients dkcomp[i]≡dk[i], and set dkcomp[i]≡0 for the rest of the 2J-1-R ones.Reconstruction Step
Considerx0comp[0]≡x0[0], for k=1,…J,
(6.9)dk-1comp[i]→{xkcomp[2i]=xk-1comp[i]+2(js-1+k-1)/2dk-1comp[i],xkcomp[2i+1]=xk-1comp[i]-2(js-1+k-1)/2dk-1comp[i],0≤i≤2k-1.
Some details on how to computexkts[i] are provided later in this section.


## 6.3. Delta Hedging

Delta hedging will be performed overR time steps. Letting
(6.10)φti=∂Vti(X)∂Sti,
the volume of transactions for a Black-Scholes portfolio with rebalancing dates {ti},i=0,…,R-1 is given by
(6.11)∑i=0R-1‍[|φti-φti-1|Sti+(Bti-Bti-1er(ti-ti-1))+],
with φt-1=Bt-1≡0 (Bt is the bond used in Section 4.1). We have used equally spaced rebalancing dates. Given that (6.11) is a random quantity, we will report the average (AverageVolTrBS) over many samples.


## 6.4. Path Independent Options

For the path independent examples we will consider two cases:(i)N=1 which provides static hedging, in this case all the atomic sigma algebras 𝒜n are included in σ(WT) and 𝒜∞=σ(WT); (ii)N>1 which provides dynamic hedging. Mathematically, in the case of path independent options, already the case N=1 provides perfect replication in the limit of J→∞.(i) Static Case
The expressions for the pathwise volume of transactions for our portfolios in this case are(6.12)VT^t0(BΠ)=e-r(T-t0)∥XRc∥1=e-r(T-t0)∫Ω‍|XRc(ω)|dP(ω),VT^t0(HΠ)=e-r(T-t0)∑Ak,i∈𝒯Ω‍P(Ak,i)|〈XRc,ψk,i〉|.
Next we comment on the output displays (see Tables1–6 and Figures 1–6); consider first a single European call Z1=(ST-K)+, values of parameters are indicated in the text surrounding the figures. In Figures 1, 2, and 3 we present the Black-Scholes and Haar approximations with R=1,2,20, respectively. Notice how Figure 1 shows the Haar approximation with u1=1/2(1A1,0-1A1,1), which happens to give the largest inner product. Figure 2 shows the Haar approximation when u0 is added, giving the second largest inner product in this example. Figure 4 shows the estimation of the L2 norm of the errors as a function of R. Similar information is presented in Table 1; it illustrates the compression power of our H-system for the call option Z1 in the present static case. It shows that few Haar functions are needed to implement HΠ. That table provides the results for different values of J as this better illustrates the effect of the optimization carried out in (4.13).
As a second example we consider a portfolio built as a linear combination of European calls and puts as follows,Z2=(ST-K1)++(K2-ST)+-(ST-K3)+, values of parameters are indicated in the text surrounding Figure 5. Finally, Figure 6 shows the estimation of the L2 norm of the errors as a function of R.
Tables2 and 3 show the volume of transactions for the portfolio BΠ, for delta hedging and for the Haar portfolio HΠ. The tables also contain values of the fine and coarse upper bounds for the portfolio HΠ.
Notice the difference in magnitudes withAverageVolTrBS. The volume of transactions offers a clear numerical evidence of the different nature of Haar hedging and delta hedging, an analytical analysis was provided in Section 5.
It is expected, and it is confirmed by our experience with numerical examples, that the Haar approximation outperforms (in the sense of smaller error for equal value ofR) the Black-Scholes approximation whenever the payoff, or its derivative, contains discontinuities. Moreover, the Haar functions can be adapted to these discontinuities; for instance, we can choose u1 such that it is supported in the union of A1,0={ST<K} and A1,1={ST≥K} for the case of the European call. Our examples reflect these choices; for example, St0 was taken close to K so as the discontinuity in the first derivative of the European call becomes problematic for Black-Scholes approximation and can be reproduced efficiently by the Haar expansion. An extreme example of this kind will be the case of a digital option where, of course, the Haar expansions have no bearing as a hedging tool.
Naturally, it is easy to find situations where delta hedging outperforms Haar hedging as, for example, a position in a European call which is well in or out of the money. This is a situation where the linear approximation in delta hedging becomes very efficient. It is interesting to see under what conditions delta hedging and Haar hedging are complementary and to investigate how to combine both techniques.
Tables2 and 3 show that the behaviors of the portfolios BΠ and HΠ are similar in these static examples. It is also important to notice that we were able to make an effective use of the notion of compression because of the Haar representation; expansions on characteristic functions do not allow such a simple effective optimization procedure.(ii) Dynamic Case
Given a pathw∈∩s=0NAjs,lsts, the volume of transactions for our portfolios is given by:
(6.13)VT^t0(BΠ)(w)=∑s=0N-1‍e-r(ts+1-ts)1Ajs,lsts∑Ak,its+1∈𝒯Ajs,lsts‍P(Ak,its+1)P(Ajs,lsts)×|[1P(Ak,its+1)∫Ak,its+1‍X-1P(Ajs,lsts)∫Ajs,lsts‍X]|,(6.14)VT^t0(HΠ)=2δ∑s=0N-1‍e-r(ts+1-ts)1P(Ajs,lsts)∑Ak,i∈𝒯Ajs,lsts‍P(Ak,i)|〈X,ψk,i〉|.
In practice we do not reproduce Dts,ts+1 but its compressed version, and therefore the averages in (6.13) have to be replaced by the expressions described in the MRA above. Similarly the inner products in (6.14) have to be replaced by their compressed versions. An expression for the input to the MRA, at each dynamic stage, is provided by (6.18).
We illustrate the dynamic case for the path independent optionsZ1 and Z2 described above. Notice that our main point is to illustrate how the volume of transactions for the portfolios BΠ and HΠ behaves when transactions are performed over time, that is in a dynamic way. Naturally, we will report average values (over M random paths, we took M between 200 and 500) of all the quantities involved: volume of transactions, fine and coarse bounds, relative errors, and the values of ϵ. For example, if we sample w1,…,wM paths, the absolute error becomes
(6.15)1M∑m=1M‍|X(wi)-XRc(wi)|2.
Tables 4 and 5 display the resulting values averaged between M=200 and M=500 random paths for the options Z1 and Z2, respectively. The bounds correspond to the portfolio HΠ. Noticing the need to use the parameter RH, for simplicity we fixed this parameter to be RH=5 for Z1 and RH=8 for Z2 to reduce the error and increased N (and so also increased R=RH×N). One may also work with an adaptive value of RH. As indicated, we are reporting averages of VT^(·) over all the samples, the oscillations around these values seem to be negligible.
Another illustration is provided by apath dependent example, it necessarily requires dynamic hedging. Again, our main objective is to illustrate how the volume of transactions for both of our portfolios behaves when performing transactions over time. We consider the Asian option
(6.16)Z3=(1NA∑s=0NA-1‍StNA-s-KA)+.
It is natural to deploy the H-system along the required times, t0,t1,…,tNA and so we will take N=NA.
For each sampled path, the computation ofxkts[i] involves N iterated integrals. Indeed, the stock, discretized in time, follows an evolution S=F(Wt0,t1,…,WtN-1,tN) where {Wts-1,ts}s=1N are independent random variables with normal distribution, and then the atoms of ℱtsn contained in Aks-1,is-1ts-1 are given by
(6.17)Ak,its=Wts-1,ts-1((ciJ,ci+1J])∩Wts,ts+1-1((-∞,∞))∩…,∩WtN-1,tN-1((-∞,∞)),
where the ciJ are given in Appendix A. Consequently
(6.18)xkts[i]=2js∫-∞∞‍⋯∫-∞∞‍∫ciJci+1J‍∫cis-1Jcis-1+1J‍⋯∫ci1Jci1+1J‍(F(y1,…,yN)-KA)+e-y12/2⋯e-yN2/2dy1⋯dyN.
In order to avoid exceedingly long computational times we have takenNA=2,…,5. Notice that in order to reduce the error of approximation we are required to increase RH (we have taken RH=10). Table 6 displays the resulting values averaged over approximately M=500 random paths.


## 7. Conclusions and Extensions

We have introduced a basic and general new framework to represent contingent claims. Key ingredients are the flexibility given by the possible space and time discretizations that can be adapted to a given class of options and the potential for financial realization of these discretizations. From a theoretical point of view, the approach is as fundamental as delta hedging and it is reasonable to think that it can be extended to other settings where this last technique is available. Some of the computational tools introduced could also be used even when an actual financial realization (of the approximation) is not available, pricing computations is an example. A main goal of the paper is to bring forward the importance of efficient hedging strategies, these alternative representations should be fully contrasted with delta hedging in order to compare trade-offs and limitations of this last technique. To this end, the paper emphasizes efficiency in relation to volume of transactions and gives theoretical as well as numerical evidence that the suggested new hedging strategy has several advantages over delta hedging.Further empirical and theoretical work is needed to assess the realm of applications where the new constructions offer a financial or computational advantage. The techniques could also be extended to the setting of higher-dimensional models.



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*Source: 946415-2012-09-18.xml*