Computer and Information Science » Numerical Analysis and Scientific Computing » "MATLAB - A Fundamental Tool for Scientific Computing and Engineering Applications - Volume 3", book edited by Vasilios N. Katsikis, ISBN 978-953-51-0752-1, Published: September 26, 2012 under CC BY 3.0 license. © The Author(s).

MATLAB Aided Option Replication

By Vasilios N. Katsikis
DOI: 10.5772/45723

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MATLAB Aided Option Replication

Vasilios N. Katsikis

1. Introduction

In reality markets are incomplete in the sense that perfect replication of contingent claims using only the underlying asset and a riskless bond is impossible. In other words, that is perfect risk transfer is not possible since some payoffs cannot be replicated by trading in marketed securities. From the work of Ross in [21], it is evident that whenever the payoff of every call or put option can be replicated then the securities market is complete. In addition, an important implication of the aforementioned work of Ross, is the existence of options that cannot be replicated by the primitive securities when markets are incomplete. In [7], the authors came to the conclusion that Ross’s result is, in fact, a negative result since it asserts that in an incomplete market one cannot expect to replicate the payoff of each option even if the underlying asset is traded. In the same paper, it is proved that if the number of securities is less than half the number of states of the world, then (generically) not a single option can be replicated by traded securities. In [10], the author extended the aforementioned result in [7], to accommodate cases where the condition on the number of primitive securities is not imposed. In particular, it is proved that if there exists no binary payoff vector in the asset span, then for each portfolio there exists a set of nontrivial exercise prices of full measure such that any option on the portfolio with an exercise price in this set is non-replicated. Furthermore, note that the results of Ross for two-date security markets with finitely many states holds for security markets with more than two dates, see [8, 9].

It is well accepted that the lattice theoretic ideas are the most important technical contributions of the large literature on infinite-dimensional modern mathematical finance (for example lattice theoretic ideas in general equilibrium theory). However, ordered vector spaces that are not lattice ordered arise naturally in models of portfolio trading. Moreover, if available securities have smooth payoffs, then the portfolio space is never a vector lattice. It should be pointed out that since call and put options are vector lattice operations in the space of contingent claims, their replication by available securities requires a vector lattice structure in the portfolio space. There is a large literature on vector lattice theory related to mathematical economics; see for instance [1, 2, 3, 4, 5, 6, 7, 17, 18, 19, 20, 22, 23].

On the other hand, there is an obvious need for properly structured high performance computational methods on vector lattices. Moreover, the main concern is to describe, in computational terms, and then solve problems arising from mathematical economics such as portfolio insurance and option replication. A lot of work in this area has been done in [11, 12, 14, 15, 16].

In this chapter, we focus to the option replication problem. We consider an incomplete market of primitive securities, meaning that some call and put options need not be marketed and our objective is to describe an efficient method for computing maximal submarkets that replicate any option. Even though, there are several important results on option replication they cannot provide a method for the determination of the replicated options. By using the theory of vector-lattices and positive bases it is provided a procedure in order to determine the set of securities with replicated options. In particular, it is shown that the union of all maximal replicated submarkets (i.e., submarketsY, such that any option written on the elements of Ycan be replicated and Yis as large as possible) defines a set of elements such that any option written on these elements is replicated.

In [11, 12, 13, 14, 16], it was shown that it is possible to construct computational methods in order to efficiently compute vector sublattices and lattice-subspaces of Rm as well as in the general case ofC[a,b]. In addition, these methods has been successfully applied in portfolio insurance and in completion of security markets.

Here we consider a two-period security market Xwith a finite number mof states and a finite number of primitive securities with payoffs in Rm and we construct computational methods in order to determine maximal replicated submarkets of Xby using the theory of vector sublattices and lattice-subspaces. Moreover, in the theory of security markets it is a usual practice to take call and put options with respect to the riskless bond1=(1,1,...,1). Then, the completion F1(X) of Xby options is the subspace of Rm generated by all options written on the elements ofX{1}. Since the payoff space isRm, which is a vector lattice, in the case where 1Xthen F1(X) is exactly the vector sublattice generated byX. If, in addition, Xis a vector sublattice of Rm then F1(X)=Xtherefore any option is replicated, unfortunately this situation is extremely rare.

A recent article, [15], provided a computationally efficient method for computing maximal submarkets that replicate any option. In particular, by using computational methods and techniques from [11, 12, 13, 14] in order to determine vector sublattices and their positive bases, it is presented a procedure in order to calculate the set of securities with replicated options. The aforementioned article emphasizes the most important interrelationship between the theory of vector lattices, positive bases, projection bases and the problem of option replication.

The material in this chapter is spread out in 5 sections. Section 2 is divided in two subsections; in the first the fundamental properties of lattice-subspaces and vector sublattices are presented, whereas in the second we introduce the basic results for vector sublattices, positive bases and projection bases of Rk together with the solution to the problem of whether a finite collection of linearly independent, positive vectors of Rk generates a lattice-subspace or a vector sublattice. In section 3, there are three subsections where it is discussed the theoretical background for option replication. Also, section 3 emphasis the most important interrelationship between positive bases, projection bases and the problem of option replication. Section 4 presents an algorithm for calculating maximal submarkets that replicate any option followed by the corresponding computational approach. Conclusions and research directions are provided in Section 5.

In this chapter, all the numerical tasks have been performed using the Matlab 7.8 (R2009a) environment on an Intel(R) Pentium(R) Dual CPU T2310 @ 1.46 GHz 1.47 GHz 32-bit system with 2 GB of RAM memory running on the Windows Vista Home Premium Operating System.

2. Basic results in the theory of positive bases and projection bases of Rm

In this section, a brief introduction is provided to the theory of vector lattices ofRm. Furthermore, we present some basic results related to the theory of positive bases and projection bases ofRm.

2.1. Preliminaries and notation

Initially, we recall some definitions and notation from the vector lattice theory. Let Rm={x=(x(1),x(2),...,x(m))|x(i)R,foreachi}, where we view Rm as an ordered space. The pointwise order relation in Rm is defined by

 x≤y  if  and  only  if  xi≤yi, for  each   i=1,...,m. (1)

The positive cone of Rm is defined by R+m={xRm|x(i)0,foreachi} and if we suppose that Xis a vector subspace of Rm then Xordered by the pointwise ordering is an ordered subspace ofRm, with positive cone X+=XR+m. By {e1,e2,...,em} we shall denote the usual basis ofRm. A point xRmis an upper bound, (resp. lower bound) of a subset SRmif and only ifyx   (resp.xy), for all yS.For a two-point set S={x,y},we denote by xy   (resp.xy)the supremum of Si.e., its least upper bound (resp. the infimum of Si.e., its greatest lower bound). Thus, xy   (resp.xy)is the componentwise maximum (resp. minimum) of xand ydefined by

 x∨yi=maxxi,yix∧yi=minxi,yi,   for all  i=1,...,m. (2)

For anyx=(x(1),x(2),...,x(m))Rm, the set supp(x)={i|x(i)0}is the support ofx. The vectors x,yRmhave disjoint supports ifsupp(x)supp(y)=.

An ordered subspace Zof Rm is a vector sublattice or a Riesz subspace of Rm if for any x,yZthe supremum and the infimum of the set {x,y} in Rm belong to Z.

Assume that Xis an ordered subspace of Rm and B={b1,b2,...,bμ}is a basis for X.Then Bis a positive basis of Xif for each xXit holds: xis positive if and only if its coefficients in the basis Bare positive. In other words, Bis a positive basis of Xif the positive cone X+ of Xhas the form,

 X+={x=∑i=1μ‍λibi|λi≥0,for  each  i}. (3)

Then, for any x=i=1μλibiand y=i=1μρibiwe have xyif and only if λiρi for each i=1,2,...,μ.A positive basis B={b1,b2,...,bμ}is a partition of the unit if the vectors bi have disjoint supports andi=1μbi=1.

Recall that a nonzero element x0 of X+ is an extremal point of X+ if, for any xX,0xx0impliesx=λx0, for a real numberλ. Since each element bi of the positive basis of Xis an extremal point ofX+, a positive basis of Xis unique in the sense of positive multiples.

The existence of positive bases is not always ensured, but in the case where Xis a vector sublattice of Rm then Xalways has a positive basis. If B={b1,b2,...,bμ}is a positive basis for a vector sublattice Xthe lattice operations inX, namely xyfor the supremum and xyfor the infimum of the set {x,y} inX, are given by

 x∨y=∑i=1μ‍maxλi,ρibi  and   x∧y=∑i=1μ‍min{λi,ρi}bi, (4)

for each

 x=∑i=1μ‍λibi,y=∑i=1μ‍ρibi∈X. (5)

Suppose that Lis a finite dimensional subspace of C(Ω)generated by a set {z1,z2,...,zr} of linearly independent positive vectors ofC(Ω). If Zis the sublattice of C(Ω)generated by Land {b1,...,bμ} is a positive basis for Z(μ=dim(Z)) then, a projection basis {b~1,b~2,...,b~r} of Zis a basis for Lsuch that its elements are projections of the elements of the positive basis{b1,...,bμ}. In our current work we consider that Ω={1,2,...,m}henceC(Ω)=Rm.

For an extensive presentation of vector sublattices as well as for notation not defined here we refer to [11, 12, 13, 15, 16] and the references therein.

2.2. Theoretical background

In this section we present theoretical results for positive bases and projection bases inRm. Given a collection x1,x2,...,xn of linearly independent, positive vectors of Rm we define the function,

 h:1,2,...,m→Rn  such   that   h(i)=(x1(i),x2(i),...,xn(i)) (6)

and the function,

 β:1,2,...,m→Rn  such  that  β(i)=h(i)∥h(i)∥1 (7)

for each i{1,2,...,m}with h(i)10. We shall refer to βas the basic function of the vectors x1,x2,...,xn. The set

 R(β)={β(i)|i=1,2,...,m,  with  ∥h(i)∥1≠0}, (8)

is the range of the basic function and the cardinal number, cardR(β),of R(β)is the number of different elements of R(β).

Suppose that Zdenotes the sublattice of Rm generated byX=[x1,x2,...,xn]. We shall denote by P1,P2,...,Pn,Pn+1,...,Pμ an enumeration of R(β)such that the first nvertices P1,P2,...,Pn are linearly independent andμ=dim(Z). Note that such an enumeration always exists. The notation, ATstands for the transpose of a matrix A.A procedure in order to construct the sublattice Zis given by the following theorem.

Theorem 1 Suppose that the above assumptions are satisfied. Then,

1. Xis a vector sublattice of Rm if and only if R(β)has exactly npoints (i.e.,μ=n). Then a positive basis b1,b2,...,bn for Xis defined by the formula
 (b1,b2,...,bn)T=A-1(x1,x2,...,xn)T, (9)

where Ais the n×nmatrix whose ith column is the vector Pi, for each i=1,2,...,n.It is clear that in such a case Zand Xcoincide.

1. Letμ>n. IfIs=β-1(Ps), and
 xs=∑i∈Is‍∥h(i)∥1ei,  s=n+1,n+2,...,μ, (10)

then

 Z=[x1,x2,...,xn,xn+1,...,xμ], (11)

is the vector sublattice generated by x1,x2,...,xn and dimZ=μ.

For a positive basis {b1,b2,...,bμ} ofZ, consider the basic function γof {x1,x2,...,xμ} with range,R(γ)={P'1,P'2,...,P'μ}. Then, the relation

 (b1,b2,...,bμ)T=B-1(x1,x2,...,xμ)T (12)

where Bis the μ×μmatrix with columns the vectorsP'1,P'2,...,P'μ, defines a positive basis forZ.

The notion of the projection basis is important for our study. Furthermore, in the following, we are interested for a projection basis that corresponds to a positive basis. Let {z1,z2,...,zr} be a set of linearly independent and positive vectors of Rm then by using Theorem 1 we construct the sublattice Zof Rm generated by these vectors. Ifdim(Z)=μ, by Theorem 1, a positive basis {b1,b2,...,bμ} of Zcan be determined. The basic result for calculating the projection basis that corresponds to the positive basis {b1,b2,...,bμ} of Zis the following theorem.

Theorem 2 Suppose that βis the basic function of the vectors {z1,z2,...,zr} and P1,P2,...,Pr, Pr+1,...,Pμ is an enumeration of the range of βsuch that the first rvectors P1,P2,...,Pr are linearly independent and suppose also that zr+1,...,zμ are the new vectors constructed in Theorem 1. If L=[z1,z2,...,zr]is the subspace of Rm generated by the vectors z1,z2,...,zr then,

1.  Z=L⊕[zr+1,...,zμ], (13)
2.  {br+1,br+2,...,bμ}={2zr+1,2zr+2,...,2zμ}, (14)
3. Ifbi=bi~+b'i, with bi~Landb'i[zr+1,...,zμ], for eachi=1,2,...,r, then {b1~,b2~,...,br~} is a basis for Lwhich is given by the formula
 (b1~,b2~,...,br~)T=A-1(z1,z2,...,zr)T, (15)

where Ais the r×rmatrix whose ith column is the vectorPi, fori=1,2,...,r. This basis, {b1~,b2~,...,br~}is called the projection basis of Land has the property: The rfirst coordinates of any element xLexpressed in terms of the basis {b1,b2,...,bμ} coincide with the corresponding coordinates of xin the projection basis, i.e.,

 x=∑i=1μ‍λibi∈L⇒x=∑i=1r‍λibi~ (16)

Suppose that Zis the sublattice generated by a collection z1,z2,...,zr of linearly independent, positive vectors of Rm and {d1,d2,...,dμ} is a positive basis forZ.

Then, by Theorem 2, if

 (d1~,d2~,...,dr~)T=A-1(z1,z2,...,zr)T, (17)

where Ais the r×rmatrix whose ith column is the vectorPi, for i=1,2,...,rthen {d1~,d2~,...,dr~} is the projection basis ofL=[z1,z2,...,zr]. The projection basis {d1~,d2~,...,dr~} is called the projection basis of Xcorresponding to the basis{d1,d2,...,dμ}. The following proposition allows us to determine a positive basis and its corresponding projection basis. Moreover, the calculated positive basis is a partition of the unit.

Proposition 1 Suppose that {di} is the basis of Zgiven by equation (2) of Theorem 1 and {di~} is the projection basis of L=[z1,z2,...,zr]corresponding to the basis {di}. Then {bi=didi|i=1,2,...,μ} is the positive basis of Zwhich is a partition of the unit and {bi~=di~di|i=1,2,...,n} is the projection basis of Lcorresponding to the basis {bi} ofZ.

In the following, we shall denote by 1the vector1=(1,1,...,1). A vector xis a binary vector ifx0=(0,0,...,0), x1and x(i)=0orx(i)=1, for anyi. Let {bi|i=1,2,...,μ} be a positive basis of Zwhich is a partition of the unit and let {bi~|i=1,2,...,n} be the projection basis of Lcorresponding to the basis{bi}. A partition δ={σi|i=1,2,...,k}of {1,2,...,n} is proper if for anyr=1,2,...,k, the vector wr=iσrbi~ is a binary vector withr=1kwr=1. If there is no proper partition of {1,2,...,n} strictly finer thanδ, then we say that δis a maximal proper partition of{1,2,...,n}.

Example 1 Let {bi|i=1,2,3} be a positive basis such that the corresponding projection basis is the following

 b~1=(12,1,0,1,0),  b~2=(12,0,0,0,1),  b~3=(0,0,1,0,0). (18)

We calculate the partitions of {1,2,3} which are the following:

 δ1={σ1,σ2},  where  σ1={1},  σ2={2  3} (19)
 δ2={σ1,σ2},  where  σ1={2},  σ2={3  4} (20)
 δ3={σ1,σ2},  where  σ1={3},  σ2={1  2} (21)
 δ4={σ1,σ2,σ3},  where  σ1={1},  σ2={2},  σ3={3}. (22)

Then δ1 is not a proper partition since w1=iσ1b~i=b~1 and b~1 is not a binary vector. Similarly, δ2is not a proper partition. On the other hand δ3 is a proper partition since w1=iσ1b~i=b~3 and b~3 is a binary vector, w2=iσ2b~i=b~1+b~2=(1,1,0,1,1)which is a binary vector andw1+w2=(1,1,1,1,1). Notice that δ3 is strictly finer thanδ4, hence δ3 is a maximal proper partition of{1,2,3}.

3. Option replication

In this section we shall discuss the economic model of our study. Moreover, first we discuss an inductive method for calculating the completion of security markets. So, if 1Xthen it is possible to determine a basic set of marketed securities i.e., a set of linearly independent and positive vectors of Xand the sublattice of Rm generated by a basic set of marketed securities isF1(X). Finally, F1(X)has a positive basis which is a partition of the unit. Under these observations we present an algorithmic procedure in order to determine maximal submarkets that replicate any option.

3.1. The economic model

In our economy there are two time periods, t=0,1, where t=0denotes the present and t=1denotes the future. We consider that at t=1we have a finite number of states indexed bys=1,2,...,m, while at t=0the state is known to bes=0.

Suppose that, agents trade x1,x2,...,xn non-redundant securities in periodt=0, then the future payoffs of x1,x2,...,xn are collected in a matrix

 A=[xi(j)]i=1,2,...nj=1,2,...,m∈Rm×n (23)

where xi(j) is the payoff of one unit of security iin statej. In other words, Ais the matrix whose columns are the non-redundant security vectorsx1,x2,...,xn. It is clear that the matrix Ais of full rank and the asset span is denoted byX=Span(A). So, Xis the vector subspace of Rm generated by the vectorsxi. That is, Xconsists of those income streams that can be generated by trading on the financial market. A portfolio is a column vector θ=(θ1,θ2,...,θn)Tof Rn and the payoff of a portfolio θis the vector x=AθRmwhich offers payoff x(i)in statei, wherei=1,...,m. A vector inRm, is said to be marketed or replicated if xis the payoff of some portfolio θ(called the replicating portfolio ofx), or equivalently ifxX. Ifm=n, then markets are said to be complete and the asset span coincides with the spaceRm. On the other hand, if n<m,the markets are incomplete and some state contingent claim cannot be replicated by a portfolio.

In the following, we shall denote by 1the riskless bond i.e., the vector1=(1,1,...,1). A vector xis a binary vector ifx0=(0,0,...,0), x1and x(i)=0orx(i)=1, for anyi. The call option written on the vector xRmwith exercise price αis the vectorc(x,a)=(x-α1)+=(x-α1)0. The put option written on the vector xRmwith exercise price αis the vectorp(x,a)=(α1-x)+=(α1-x)0. If yis an element of a Riesz space then the following lattice identities hold, y=y+-y-andy-=(-y)+. It is clear that x-α1=(x-α1)+-(x-α1)-=(x-α1)+-(α1-x)+=c(x,α)-p(x,α).Therefore we have the identity

 x-a1=c(x,a)-p(x,a), (24)

which is called put-call parity.

If both c(x,α)>0andp(x,α)>0, we say that the call option c(x,α)and the put option p(x,α)are non trivial and the exercise price αis a non trivial exercise price ofx. If c(x,α)and p(x,α)belong to Xthen we say that c(x,α)and p(x,α)are replicated. If we suppose that 1Xand at least one ofc(x,a), p(x,a)is replicated, then both of them are replicated since,x-α1=c(x,α)-p(x,α).

3.2. Completion of security markets

We shall discuss the problem of completion by options of a two-period security market in which the space of marketed securities is a subspace ofRm. The present study involves vector sublattices generated by a subset Bof Rm of positive, linearly independent vectors. A computational solution to this problem is provided by using the SUBlat Matlab function from [16].

Let us assume that in the beginning of a time period there are nsecurities traded in a market. Let S={1,...,m}denote a finite set of states and xjR+m be the payoff vector of security jin mstates. The payoffs x1,x2,...,xn are assumed linearly independent so that there are no redundant securities. If θ=(θ1,θ2,...,θn)Rnis a non-zero portfolio then its payoff is the vector

 T(θ)=∑i=1n‍θixi. (25)

The set of payoffs of all portfolios is referred as the space of marketed securities and it is the linear span of the payoffs vectors x1,x2,...,xn in Rm which we shall denote it byX, i.e.,

 X=[x1,x2,...xn]. (26)

For any x,uRmand any real number athe vector cu(x,a)=(x-au)+ is the call option and pu(x,a)=(au-x)+ is the put option of xwith respect to the strike vector uand exercise pricea.

Let Ube a fixed subspace of Rm which is called strike subspace and the elements of Uare the strike vectors. Then, the completion by options of the subspace Xwith respect to Uis the space FU(X) which is defined inductively as follows:

• X1is the subspace of Rm generated by O1, whereO1={cu(x,a)|xX,uU,aR}, denotes the set of call options written on the elements ofX,
• Xnis the subspace of Rm generated by On, whereOn={cu(x,a)|xXn-1,uU,aR}, denotes the set of call options written on the elements ofXn-1,
•  FU(X)=∪n=1∞Xn (27)
.

The completion by options FU(X) of Xwith respect to Uis the vector sublattice of Rm generated by the subspaceY=XU. The details are presented in the next theorem,

Theorem 3 In the above notation, we have

1.  Y⊆X1 (28)
,
2. FU(X)is the sublattice S(Y)of Rm generated byY, and
3. ifUX, then FU(X) is the sublattice of Rm generated by X.

Any set {y1,y2,,yr} of linearly independent positive vectors of Rm such that FU(X) is the sublattice of Rm generated by {y1,y2,,yr} is a basic set of the market.

Theorem 4 Any maximal subset {y1,y2,,yr} of linearly independent vectors of Ais a basic set of the market, whereA={x1+,x1-,,xn+,xn-}, if UXandA={x1+,x1-,,xn+,xn-,u1+,u1-,,ud+,ud-}, if UÖX

The space of marketed securities Xis complete by options with respect to UifX=FU(X).

From theorem 1 it follows,

Theorem 5 The space Xof marketed securities is complete by options with respect to Uif and only if UXandcardR(β)=n. In addition, the dimension of FU(X) is equal to the cardinal number ofR(β). Therefore, FU(X)=Rkif and only if cardR(β)=k.

Example 2 Suppose that in a security market, the payoff space is R12 and the primitive securities are:

 x1=(1,2,2,-1,1,-2,-1,-3,0,0,0,0) (29)
 x2=(0,2,0,0,1,2,0,3,-1,-1,-1,-2) (30)
 x3=(1,2,2,0,1,0,0,0,-1,-1,-1,-2) (31)

and that the strike subspace is the vector subspace Ugenerated by the vector

 u=(1,2,2,1,1,2,1,3,-1,-1,-1,-2). (32)

Then, a maximal subset of linearly independent vectors of {x1+,x1-,x2+,x2-,x3+,x3-,u1+,u1-} can be calculated by using the following code:

where Xdenotes a matrix whose rows are the vectors x1,x2,x3,u. We can determine the completion by options of Xi.e., the spaceFU(X), with the SUBlat function from [16] by using the following code:

The results then are as follows

3.3. Computation of maximal submarkets that replicate any option

We consider a two-period security market Xwith a finite number mof states and a finite number of primitive securities with payoffs in Rm and we construct computational methods in order to determine maximal submarkets of Xthat replicate any option by using the results provided in subsection 2.2. In particular, in the theory of security markets it is a usual practice to take call and put options with respect to the riskless bond1=(1,1,...,1). Then, the completion F1(X) of Xby options (see subsection 3.2) is the subspace of Rm generated by all options written on the elements ofX{1}. Since the payoff space isRm, which is a vector lattice, in the case where 1Xthen F1(X) is exactly the vector sublattice generated byX. If, in addition, Xis a vector sublattice of Rm then F1(X)=Xtherefore any option is replicated.

A basic set of marketed securities (i.e., a set of linearly independent and positive vectors) of Xalways exist and the sublattice of Rm generated by a basic set of marketed securities isF1(X). In addition, F1(X)has a positive basis which is a partition of the unit.

Let us assume that Xis generated by a basic set of marketed securities, then from Theorem 1 it is possible to determine a positive basis {b1,b2,...,bμ} ofF1(X).

The sublatticeZ, generated by a basic set of marketed securities, is exactly F1(X) and F1(X) has a positive basis which is a partition of the unit, i.e.,i=1μbi=1. This is possible since the notion of a positive basis is unique in the sense of positive multiples therefore we are able to extract from the positive basis {b1,b2,...,bμ} another positive basis {d1,d2,...,dμ} of F1(X) which is a partition of the unit. Therefore, let us denote by {d1,d2,...,dμ} a positive basis of F1(X) which is a partition of the unit. Then, by Theorem 2, if

 (d1~,d2~,...,dr~)T=A-1(z1,z2,...,zr)T, (33)

where Ais the r×rmatrix whose ith column is the vectorPi, for i=1,2,...,rthen {d1~,d2~,...,dr~} is a projection basis ofF1(X). The projection basis {d1~,d2~,...,dr~} is the projection basis of Xcorresponding to the basis{d1,d2,...,dμ}. For Z=F1(X)proposition 1 takes the following form.

Proposition 2 Suppose that {di} is the basis of F1(X) given by equation (2) of theorem 1 and {di~} is the projection basis of Xcorresponding to the basis {di}. Then {bi=didi|i=1,2,...,μ} is the positive basis of F1(X) which is a partition of the unit and {bi~=di~di|i=1,2,...,n} is the projection basis of Xcorresponding to the basis {bi} ofF1(X).

Suppose that Yis a subspace ofX, then if F1(Y)Xwe say that Yis replicated. If, in addition, for any subspace Zof Xwith YÜZwe have that XÜF1(Z)then Yis a maximal replicated subspace or a maximal replicated submarket ofX. Note that, the replicated kernel of the market, i.e., the union of all maximal replicated subspaces of the market is the set of all elements xof Xso that any option written on xis replicated.

Let {bi i=1,2,...,μ} be a positive basis of F1(X) which is a partition of the unit and let {bi~ i=1,2,...,n} be the projection basis of Xcorresponding to the basis{bi}. Recall that, a partition δ={σi|i=1,2,...,k}of {1,2,...,n} is proper if for anyr=1,2,...,k, the vector wr=iσrbi~ is a binary vector withr=1kwr=1. If there is no proper partition of {1,2,...,n} strictly finer thanδ, then we say that δis a maximal proper partition of{1,2,...,n}.

The following theorem provides the development of a method in order to determine the set of securities with replicated options by using the theory of positive bases and projection bases.

Theorem 6 Let {bi,i=1,2,...,μ} be the positive basis of F1(X) which is a partition of the unit and let {bi~,i=1,2,...,n} be the projection basis of Xcorresponding to the basis{bi}. If Yis a subspace ofX, the following are equivalent:

1. Yis a maximal replicated subspace ofX,
2. there exists a maximal proper partition δ={σi|i=1,2,...,k}of {1,2,...,n} so that Yis the sublattice of Rm generated byδ.

The set of maximal replicated submarkets of Xis nonempty.

4. The computational method

We present the proposed computational method that enables us to determine maximal submarkets that replicate any option. Our numerical method is based on the introduction of the mrsubspace function, from [15], that allow us to perform fast testing for a variety of dimensions and subspaces.

4.1. Algorithm for calculating maximal submarkets that replicate any option

Recall that Xis the security market generated by a collection {x1,x2,...,xn} of linearly independent vectors (not necessarily positive) ofRm. If 1Xthen it is possible to determine a basic set of marketed securities i.e., a set of linearly independent and positive vectors ofX. This is possible through the following easy proposition:

Proposition 3 Ifa=max{xi|i=1,2,...,n}, then at least one of the two sets of positive vectors of X

 {yi=a1-xi|i=1,2,...,n},  {zi=2a1-xi|i=1,2,...,n}, (34)

consists of linearly independent vectors.

The main steps of the underlying algorithmic procedure that enables us to determine maximal submarkets that replicate any option are the following:

1. Use proposition 3 in order to determine a basic set {y1,y2,...,yn} of marketed securities.

2. Use Equation (1) in order to determine the basic curve βof the vectorsyi.

3. Determine the range R(β)ofβ.

4. Use Theorem 1 in order to construct the vector sublattice generated byy1,y2,...,yn, which is exactly the completion by options F1(X) ofX. Then, determine a positive basis {d1,d2,...,dμ} forF1(X).

5. Use Theorem 2 in order to determine a projection basis {d1~,d2~,...,dn~} ofX.

6. Use Proposition 1 in order to determine a positive basis {b1,b2,...,bμ} of F1(X) which is a partition of the unit and the corresponding projection basis{b1~,b2~,...,bn~}.

7. Calculate all the possible proper partitions of the set{1,2,...,n}.

8. Decide which of the proper partitions created in step (7) are maximal proper partitions and determine the corresponding maximal replicated submarkets.

In [15], it is presented the translation followed by the implementation of the aforementioned algorithm in Rm within a Matlab-based function named mrsubspace. Moreover, in the same paper, computational experiments assessing the effectiveness of this function and lead us to the conclusion that the mrsubspace function provides an important tool in order to investigate replicated subspaces.

4.2. The computational approach - Code features

We shall discuss the proposed computational method that enables us to determine maximal submarkets that replicate any option. The standard method used currently to determine the maximal replicated submarkets is based on a manual processing. From section 3, it is evident that the required number of verifications for this process can be of significant size even in a relatively low-dimensional space, thus rendering the problem too difficult to solve. Our numerical method is based on the introduction of the mrsubspace function, from [15] that allow us to perform fast testing for a variety of dimensions and subspaces.

The structure of the code ensures flexibility, meaning that it is convenient for applications as well as for research and educational purposes. The given security marketX, generated by the linearly independent vectorsx1,x2,...,xn, must be given under a matrix notation with columns the vectorsx1,x2,...,xn. The mrsubspace function must be stored in a Matlab-accessible directory and then the input data, i.e., the matrixX, can be typed directly in the Matlab’s environment. Under the following command,

mrsubspace(X);

the program solves the problem of option replication and prints out the maximal proper partitions as well as the corresponding maximal replicated subspaces. If Xis a vector sublattice, then X=F1(X)and any option is replicated. In the case where the initial space Xis not a vector sublattice, it is possible to produce the normalized positive basis and the corresponding projection basis with the following code,

[Npb,Cprb] = mrsubspace(X)

Inside the code there are several explanations that indicate the implemented part of the algorithm. A user proficient in Matlab can easily use the code and modify it if needed. Especially, the user can isolate a part of the code according to his/her special needs to solve different problems like

• Determine a basic set of marketed securities.

• Find the completion F1(X) of Xby options in Rm or find the vector sublattice generated by a finite collection of linearly independent vectors ofRm.

• Calculate a positive basis and a projection basis for a finite dimensional vector sublattice.

In the last part of the code, entitled Maximal proper partitions - Maximal replicated subspaces, the user can change the way that the mrsubspace function understands the values 0 and 1, according to his/her knowledge and needs.

Example 3 Consider the following 5 vectors x1,x2,...,x5 in R10,

 x1x2x3x4x5=01011112211112111212111211121111111112212121111111 (35)
and X=[x1,x2,...,x5]is the marketed space.

Note that1=x5-x4+x1. In order to determine the maximal replicated subspaces for the above collection of vectors we use the following simple code:

as a result and after removing irrelevant Matlab output one gets

Therefore, the marketed space Xhas two maximal replicated subspaces, {1}  {2  3}  {4}  {5}and {1}  {2}  {3  4}  {5} are maximal proper partitions with corresponding maximal replicated subspaces the subspaces

 Y1=[(1,0,1,0,0,0,0,0,0,0),(0,1,0,1,1,1,1,0,0,0),(0,0,0,0,0,0,0,1,1,0), (0,0,0,0,0,0,0,0,0,1)] (36)

and

 Y2=[(1,0,1,0,0,0,0,0,0,0),(0,1,0,0,1,1,1,0,1,0),(0,0,0,1,0,0,0,1,0,0),(0,0,0,0,0,0,0,0,0,1)], (37)

respectively. The replicated kernel of the market isY=Y1Y2.

5. Conclusions

In this chapter, computational methods for option replication are presented based on vector lattice theory. It is well accepted that the lattice theoretic ideas are one of the most important technical contributions of the large literature on modern mathematical finance. In this chapter, we consider an incomplete market of primitive securities, meaning that some call and put options need not be marketed. Our objective is to describe an efficient method for computing maximal submarkets that replicate any option. Even though, there are several important results on option replication they cannot provide a method for the determination of the replicated options. By using the theory of vector-lattices and positive bases it is provided a procedure in order to determine the set of securities with replicated options. Moreover, we determine those subspaces of the marketed subspace that replicate any option by introducing a Matlab function, namely mrsubspace. The results of this work can give us an important tool in order to study the interesting problem of option replication of a two-period security market in which the space of marketed securities is a subspace ofRm. This work is based on a recent work, [Katsikis, 2011], regarding computational methods for option replication.

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