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arXiv:1103.1006v1 [q-fin.GN] 5 Mar 2011

ARBITRAGE AND HEDGING IN A NON PROBABILISTIC

FRAMEWORK

A. ALVAREZ, S. FERRANDO AND P. OLIVARES

DEPARTMENT OF MATHEMATICS, RYERSON UNIVERSITY

Abstract. The paper studies the concepts of hedging and arbitrage in a non

probabilistic framework. It provides conditions for non probabilistic arbitrage

based on the topological structure of the trajectory space and makes connec-

tions with the usual notion of arbitrage. Several examples illustrate the non

probabilistic arbitrage as well perfect replication of options under continuous

and discontinuous trajectories, the results can then be applied in probabilis-

tic models path by path. The approach is related to recent financial models

that go beyond semimartingales, we remark on some of these connections and

provide applications of our results to some of these models.

1. Introduction

Modern mathematical finance relies on the notions of arbitrage and hedging repli-

cation; generally, these ideas are exclusively cast in probabilistic frameworks. The

possibility of dispensing with probabilities in mathematical finance, wherever pos-

sible, may have already occurred to several researchers. A reason for this is that

hedging results do not depend on the actual probability distributions but on the

support of the probability measure. Actually, hedging is clearly a pathwise notion

and a simple view of arbitrage is that there is a portfolio that will no produce any

loss for all possible paths and there exists at least one path that will provide a

profit. This informal reasoning suggests that there is no need to use probabilities

to define the concepts even though probability has been traditionally used to do

so. The paper makes an attempt to study these two notions without probabilities

in a direct and simple way.

From a technical point of view, we rely on a simple calculus for non differentiable

functions introduced in [12] (see also [20]). This calculus is available for a fairly

large class of functions. Hedging results that only depend on this pathwise calculus

can be considered independently of probabilistic assumptions. In [4] the authors

take this point of view and develop a discrete framework, and its associated limit,

to hedge continuous paths with a prescribed 2-variation. Reference [4] is mostly

devoted to payoff replication for continuous trajectories and does not address the

issue of non probabilistic arbitrage. The present paper formally defines this last

notion in a context that allows for trajectories with jumps, develops some of the

basic consequences and presents some simple applications including novel results

to probabilistic frameworks.

Results on the existence of arbitrage in a non standard framework (i.e. a non

semimartingale price process) leads to interesting and challenging problems. In

order to gain a perspective on this issue, recall that a consequence of the funda-

mental theorem of asset pricing of Delbaen and Schachermayer in [10] is that under

1

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2ALVAREZ, FERRANDO AND OLIVARES

the NFLVR condition and considering simple predictable portfolio strategies, the

price process of the risky asset necessarily must be a semi-martingale. Recent lit-

erature ([7], [16], [9], [3]) describes pricing results in non semi-martingale settings;

the restriction of the possible portfolio strategies has been a central issue in these

works. In [7] the permissible portfolio strategies are restricted to those for which

the time between consecutive transactions is bounded below by some number h. In

[16] these results are extended, also considering portfolios having a minimal fixed

time between successive trades. In [9] the notion of A-martingale is introduced in

order to have the no-arbitrage property for a given class A of admissible strategies.

We also treat this problem but with a different perspective, our main object of

study is a class of trajectories J = J(x0) starting at x0. To this set of deterministic

trajectories we associate a class of admissible portfolios A = AJ(x0)which, under

some conditions, is free of arbitrage and allow for perfect replication to take place.

These two notions, arbitrage and hedging, are defined without probabilities. Once

no arbitrage and hedging have been established for the non probabilistic market

model (J(x0),AJ(x0)), these results could be used to provide a fair price for the

option being hedged. These pricing results will not be stated explicitly in the paper

and will be left implicit.

Some technical aspects from our approach relate to the presentation in [3], sim-

ilarities with [3] are expressed mainly in the use of a continuity argument which

is also related to a small ball property. Our approach eliminates the probability

structure altogether and replaces it with appropriate classes of trajectories; the new

framework also allows to accommodate continuous and discontinuous trajectories.

In summary, our work intends to develop a probability-free framework that al-

lows us to price by hedging and no-arbitrage. The results, obtained under no

probabilistic assumptions, will depend however, on the topological structure of the

possible trajectory space and a restriction on the admissible portfolios by requiring

a certain type of continuity property. We connect our non probabilistic models

with stochastic models in a way that arbitrage results can be translated from the

non probabilistic models to stochastic models, even if these models are not semi-

martingales. The framework handles naturally general subsets of the given trajec-

tory space, this is not the case in probabilistic frameworks that rely in incorporating

subsets of measure zero in the formalism.

The paper is organized as follows.Section 2 briefly introduces some of the

technical tools we need to perform integration with respect to functions of finite

quadratic variation and defines the basic notions of the non probabilistic framework.

Section 3 proves two theorems, they are key technical results used throughout the

rest of the paper. The theorems provide a tool connecting the usual notion of

arbitrage and non probabilistic arbitrage. Section 4 introduces classes of modeling

trajectories, we prove a variety of non probabilistic hedging and no arbitrage results

for these classes. Section 5 presents several examples in which we apply the non

probabilistic results to obtain new pricing results in several non standard stochastic

models. Appendix A provides some information on the analytical version of Ito

formula that we rely upon. Appendix B presents some technical results needed in

our developments.

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ARBITRAGE AND HEDGING IN A NON PROBABILISTIC FRAMEWORK3

2. Non Probabilistic Framework

We make use of the definition of integral with respect to functions with un-

bounded variation but with finite quadratic variation given in [12].

Let T > 0 be a fixed real number and let T ≡ {τn}∞

n=0where

τn=?0 = tn,0< ··· < tn,K(n)= T?

mesh(τn) = max

are partitions of [0,T] such that:

tn,k∈τn|tn,k− tn,k−1| → 0.

Let x be a real function on [0,T] which is right continuous and has left limits

(RCLL for short), the space of such functions will be denoted by D[0,T]. The

following notation will be used, ∆xt= xt− xt−and ∆x2

Financial transactions will take place at times belonging to the above discrete

grid but, otherwise, time will be treated continuously, in particular, the values x(t)

could be observed in a continuous way.

A real valued RCLL function x is of quadratic variation along T if the discrete

measures

ξn=

?

converge weakly to a Radon measure ξ on [0,T] whose atomic part is given by the

quadratic jumps of x:

t= (∆xt)2.

ti∈τn

(xti+1− xti)2ǫti

(1)[x]T

t= ?x?T

t+

?

s≤t

∆x2

s,

where [x]Tdenotes the distribution function of ξ and ?x?Tits continuous part.

Considering x as above and y to be a function on [0,T] × D, we will formally

define the F¨ ollmer’s integral of y respect to x along τ over the interval [0,t] for

every 0 < t ≤ T. We should note that while the integral over [0,t] for t < T will

be defined in a proper sense, the integral over [0,T] will be defined as an improper

F¨ ollmer’s integral.

Definition 1. Let 0 < t < T and x and y as above, the F¨ ollmer’s integral of y with

respect to x along T is given by

?t

τn∋tn,i≤t

provided the limit in the right-hand side of (2) exists. The F¨ ollmer’s integral over

the whole interval [0,T] is defined in an improper sense:

(2)

0

y(s,x) dxs= lim

n→∞

?

y(tn,i,x) (x(tn,i+1) − x(tn,i)),

?T

0

y(s,x) dxs= lim

t→T

?t

0

y(s,x) dxs,

provided the limit exists.

Consider φ ∈ C1(R) (i.e. a function with domain R and first derivative con-

tinuous), take y(s,x) ≡ φ(x(s−)) then (2) exists. More generally, if y(s,x) =

φ(s,x(s−),g1(t,x−),...,gm(t,x−)) where the gi(·,x) are functions of bounded vari-

ation (that may depend on the past values of the trajectory x up to time t) then

(2) exists. Moreover, in these two instances an Ito formula also holds, we refer to

Appendix A for some details. Several of our non probabilistic arbitrage arguments

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4ALVAREZ, FERRANDO AND OLIVARES

will depend only on assuming the existence of integrals of the form?t

x(t), 0 ≤ t < s, in these instances, and for the sake of generality, we will work under

this general assumption.

Next, we introduce the concepts of predictability, admissibility and self-financing

in a non probabilistic setting. The NP prefix will be used throughout the paper

indicating some non probabilistic concept. For a given real number x0, the central

modeling object is a set of trajectories x starting at x0, so x:[0,T] → R with

x(0) = x0. We will assume that these functions are RCLL and belong to a given set

of trajectories J(x0). In order to easy the notation, this last class may be written

as J when the point x0is clear from the context.

Some of our results apply to rather general trajectory classes, particular trajec-

tory classes will be needed to deal with hedging results and applications to classical

models and will be introduced at due time.

We assume the existence of a non risky asset with interest rates r ≥ 0 which,

for simplicity, we will assume constant, and a risky asset whose price trajectory

belongs to a function class J(x0). For convenience, in several occasions, we will

restrict our arguments to the case r = 0.

0φ(s,x)dxsfor

a given generic integrand φ(s,x) that (potentially) depends on all the path values

A NP-portfolio Φ is a function Φ: [0,T] × J(x0) → R2, Φ = (ψ,φ), satisfying

Φ(0,x) = Φ(0,x′) for all x,x′∈ J(x0). We will also consider the associated projec-

tion functions Φx:[0,T] → R2and Φt:J(x0) → R2, for fixed x and t respectively.

The value of a NP-portfolio Φ is the function VΦ:[0,T] × J(x0) → R given by:

VΦ(t,x) ≡ ψ(t,x) + φ(t,x) x(t).

Definition 2. Consider a class J(x0) of trajectories starting at x0:

i) A portfolio Φ is said to be NP-predictable if Φt(x) = Φt(x′) for all x,x′∈

J(x0) such that x(s) = x′(s) for all 0 ≤ s < t and Φx(·) is a left continuous

function with right limits (LCRL functions for short) for all x ∈ J(x0).

ii) A portfolio Φ is said to be NP-self-financing if the integrals

and?t

?t

where V0= V (0,x) = ψ(0,x) + φ(0,x) x(0) for any x ∈ J(x0).

iii) A portfolio Φ is said to be NP-admissible if Φ is NP-predictable, NP-self-

financing and VΦ(t,x) ≥ −A, for a constant A = A(Φ) ≥ 0, for all t ∈ [0,T]

and all x ∈ J(x0).

?t

0ψ(s,x) ds

0φ(s,x) dxsexist for all x ∈ J(x0) as a Stieljes and F¨ ollmer integrals

respectively, and

?t

VΦ(t,x) = V0+

0

ψ(s,x) r ds +

0

φ(s,x)dxs, ∀x ∈ J(x0),

Remark 1.

(1) Two identical trajectories up to time t will lead to identical portfolio strate-

gies up to time t.

(2) Notice that the notion of NP-admissible portfolio is relative to a given class

of trajectories J, classes of NP portfolios will be denoted AJ or A for

simplicity .

The following definition is central to our approach.

Definition 3. A NP-market is a pair (J,A) where J represents a class of possible

trajectories for a risky asset and A is an admissible class of portfolios .

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ARBITRAGE AND HEDGING IN A NON PROBABILISTIC FRAMEWORK5

The following definition provides the notion of arbitrage in a non probabilistic

framework.

Definition 4. We will say that there exists NP-arbitrage in the NP-market (J,A)

if there exists a portfolio Φ ∈ A such that VΦ(0,x) = 0 and VΦ(T,x) ≥ 0 for all

x ∈ J, and there exists at least one trajectory x∗∈ J such that VΦ(T,x∗) > 0. If

no NP-arbitrage exists then we will say that the NP-market (J,A) is NP-arbitrage-

free.

The notion of probabilistic market that we use through the paper is similar to

the one in [3]. Assume a filtered probability space (Ω,F,(Ft)0≤t≤T,P) is given.

Let Z be an adapted stochastic process modeling asset prices defined on this space.

A portfolio strategy Φzis a pair of stochastic processes Φz= (ψz,φz). The value

of a portfolio Φzat time t is a random variable given by:

VΦz(t) = ψz

A portfolio Φzis self-financing if the integrals?t

and

?t

A portfolio Φzis admissible if Φzis self-financing, predictable (i.e. measurable with

respect to Ft−) and there exists Az= Az(Φz) ≥ 0 such that VΦz(t) ≥ − AzP -a.s.

∀ t ∈ [0,T].

t+ φz

tZt.

0ψz

s(ω)ds and?t

0φz

s(ω)dZs(ω)

exist P -a.s. as a Stieltjes integral and a F¨ ollmer stochastic integral respectively

VΦz(t) = VΦz(0) + r

0

ψz

sds +

?t

0

φz

sdZs, P − a.s.

Definition 5. A stochastic market defined on a filtered probability space (Ω,F,(Ft)t≥0,P)

is a pair (Z,AZ) where Z is an adapted stochastic process modeling asset prices and

AZis a class of admissible portfolio strategies.

Remark 2. We assume F0is the trivial sigma algebra, furthermore, without loss

of generality, we will assume that the constant z0= Z(0,w) is fixed, i.e. we assume

the same initial value for all paths. The constant VΦz(0,w) will also be denoted

VΦz(0,z0).

The notion of arbitrage in a probabilistic market is the classical notion of arbitrage

(which in this paper will be referred simply as arbitrage). The market (Z,AZ)

defined over (Ω,F,(Ft)t≥0,P) has arbitrage opportunities if there exists Φz∈ AZ

such that VΦz(0) = 0 and VΦz(T) ≥ 0 P -a.s., and P(VΦz(T) > 0) > 0.

3. Non Probabilistic Arbitrage Results

Our technical approach to establish NP arbitrage results is to link them to clas-

sical arbitrage results. Somehow surprisingly, this connection will allow us to apply

the so obtained NP results to prove non existence of arbitrage results in new prob-

abilistic markets. This section provides two basic theorems, Theorem 1 allows to

construct NP markets free of arbitrage from a given arbitrage free probabilistic

market. Applications of this theorem are given in Section 4. Theorem 2 presents a

dual result allowing to construct probabilistic, arbitrage free, markets from a given

NP market which is arbitrage free. Applications of this theorem are given in Section

5.

In order to avoid repetition we will make the following standing assumption for

the rest of the section: for all the set of trajectories J and price processes Z to be

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6ALVAREZ, FERRANDO AND OLIVARES

considered, there exists a metric space (S,d) satisfying J ⊆ S and Z(Ω) ⊆ S up to

a set of measure zero. All topological notions considered in the paper are relative

to this metric space. Examples in later sections will use the uniform distance

d(x,y) = ||x − y||∞ where ||x||∞ ≡ sups∈[0,T]|x(s)| and S the set of continuous

functions x with x(0) = x0= z0. For trajectories with jumps, later sections will

use the Skorohod distance, denoted by ds, and S the set of RCLL functions x with

x(0) = x0= z0.

While the main results in this section can be formulated in terms of isomorphic

and V-continuous portfolios (see Definitions 10 and 11), the presentation makes use

the following weaker notion of connected portfolios; this approach provides stronger

results.

Definition 6. Let (J,A) and (Z,AZ) be respectively NP and stochastic markets.

Φ ∈ A is said to be connected to Φz∈ AZif the following holds in a set of full

measure:

VΦz(0,z0) = VΦ(0,x0)

and for any fixed x ∈ J and arbitrary ρ > 0 there exists δ = δ(x,ρ) > 0 such that

(3)if d(Z(w),x) < δ then VΦz(T,ω) ≥ VΦ(T,x) − ρ.

Given a class of stochastic portfolios AZ, Section 4 constructs NP-admissible

portfolios Φ ∈ A with the goal of obtaining NP arbitrage free markets (J,A).

Each such collection of portfolios A is defined as the largest class of NP admissible

portfolios connected to an element from AZ. Here is the required definition.

Definition 7. Let (Z,AZ) be a stochastic market on (Ω,F,Ft,P), and J a set of

trajectories. Define:

[AZ] ≡ {Φ : Φ is NP-admissible, ∃ Φz∈ AZs.t. Φ is connected to Φz}.

Theorem 1. Let (Z,AZ) be a stochastic market and J a set of trajectories. Fur-

thermore, assume the following conditions are satisfied:

C0: Z(ω) ⊆ J a.s.

C1: Z satisfies a small ball property with respect to the metric d and the space J,

namely for all ǫ > 0:

P (d(Z,x) < ǫ) > 0, ∀x ∈ J.

Then, the following statement holds.

If (Z,AZ) is arbitrage free then (J,[AZ]) is NP-arbitrage free.

Proof. We proceed to prove the statement by contradiction. Suppose then, that

there exists a NP-arbitrageportfolio Φ ∈ [Az]; therefore VΦ(0,x) = 0 and VΦ(T,x) ≥

0 for all x ∈ J and there is also x∗∈ J satisfying VΦ(T,x∗) > 0. From the def-

inition of [Az], it follows that there exists Φz∈ Azconnected to Φ satisfying

VΦz(0,z0) = VΦ(0,x0) = 0. Using C0, consider the case when there existˆΩ, a

measurable set of full measure, such that Z(ω) ⊆ J hods for all w ∈ˆΩ. As-

sume further, there exists a measurable setˆΩ1 ⊆ˆΩ with P(ˆΩ1) > 0 such that

VΦz(T,ω) < 0 holds for all ω ∈ˆΩ1. The relation “Φ is connected to Φz” holds in

a set of full measure which is independent on any given x, then, we may assume

without loss of generality that (3) holds for all w ∈ˆΩ1. Consider an arbitrary

ˆ ω ∈ˆΩ1and use the notation x ≡ Z(ˆ ω) ∈ J; for an arbitrary ρ > 0 we then have:

VΦz(T, ˆ ω) ≥ VΦ(T,x) − ρ; ρ being arbitrary, this gives a contradiction. Therefore,

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ARBITRAGE AND HEDGING IN A NON PROBABILISTIC FRAMEWORK7

VΦz(T,ω) ≥ 0 a.s. holds. Consider now x∗fixed as above, an arbitrary ρ > 0 and

δ > 0 given by the fact that Φ is connected to Φz. Condition C1implies that the

set Bρ= {w : d(Z(w),x∗) < δ} satisfies P(Bρ) > 0 for any ρ > 0, then using (3) we

obtain VΦz(T,ω) ≥ VΦ(T,x∗) − ρ, which we may assume without loss of generality

holds for all w ∈ Bρ. Clearly, VΦ(T,x∗) > 0 being fixed, there exist a small ρ∗> 0

such that VΦz(T,ω) > 0 for all w ∈ Bρ∗ and P(Bρ∗) > 0. This concludes the

proof.

?

In Section 4 we are faced with the following problem: given Az, we need to prove

that a given NP admissible portfolio Φ belongs to [Az]. The following proposition

provides a sufficient condition to check that a given Φ is connected to a certain

Φz. The stronger setting of the proposition also allows to see the condition (3) as

a weak form of lower semi continuity of the value of the NP portfolio.

Proposition 1. Let (Z,AZ) be a stochastic market and J a set of trajectories and

assume that C0from Theorem 1 holds. Then, if a NP-admissible portfolio Φ is such

that VΦ(T,·):J → R is lower semi-continuous with respect to metric d and there

exist Φz∈ AZsuch that VΦz(0,z0) = VΦ(0,x0) and VΦz(T,w) = VΦ(T,Z(w)) then

Φ is connected to Φz(so Φ ∈ [AZ].)

Proof. Consider Φ and Φzsatisfying the hypothesis of the proposition. The lower

semi continuity means that for a given x ∈ J and any ρ > 0 there exists δ > 0

satisfying: if d(x′,x) < δ, with x′∈ J then

VΦ(T,x′) ≥ VΦ(T,x) − ρ. (4)

Consider now w to be in the set of full measure where Z(Ω) ⊆ J holds; fix x ∈

J and ρ > 0 arbitrary. Consider now δ as given by the lower semi continuity

assumption, then, if d(Z(w),x) < δ, taking x′≡ Z(w) we obtain VΦz(T,w) =

VΦ(T,x′) ≥ VΦ(T,x) − ρ.

?

In order to construct arbitrage free probabilistic markets from NP markets free

of arbitrage we will make use of the following notion.

Definition 8. Let (J,A) and (Z,AZ) be respectively NP and stochastic markets.

Φz∈ AZis said to be connected to Φ ∈ A if the following holds in a set of full

measure:

VΦz(0,z0) = VΦ(0,x0)

and for any fixed x ∈ J and arbitrary ρ > 0 there exists δ = δ(x,ρ) > 0 such that

(5) if d(Z(w),x) < δ then VΦ(T,x) ≥ VΦz(T,ω) − ρ.

In order to apply results obtained for NP-markets to stochastic markets, in

particular non-semimartingale processes, Section 5 makes use of the following con-

struction: starting from a class of NP admissible portfolios A, a class of portfolios

AZis defined as the largest collection of admissible portfolios which are connected

to elements from A. This construction will give an arbitrage free stochastic market

(Z,AZ). Here is the required definition.

Definition 9. Let (J,A) be a NP market and (Ω,F,(Ft)t≥0,P) a filtered proba-

bility space. Let Z be an adapted stochastic process defined on this space. Define:

[A]Z≡ {Φz: Φzis admissible, ∃ Φ ∈ A s.t. Φzis connected to Φ}.

The following theorem is the dual version of Theorem 1.

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8ALVAREZ, FERRANDO AND OLIVARES

Theorem 2. Let (J,A) be a NP market and (Ω,F,(Ft)t≥0,P) a filtered probability

space. Let Z be an adapted stochastic process defined on this space. Furthermore,

assume C0and C1from Theorem 1 hold.

Then the following statement holds:

If (J,A) is NP-arbitrage free then (Z,[A]z) is arbitrage free.

Proof. We argue by contradiction, suppose there exists an arbitrage portfolio Φz∈

[A]z; therefore, VΦz(0,w) = 0 and VΦz(T,w) ≥ 0 a.s. Moreover, there exists a

measurable set D ⊆ Ω satisfying

(6)VΦz(T,w) > 0 for all w ∈ D and P(D) > 0.

Because Φz∈ [A]z, we know that Φzis connected to some Φ ∈ A. Then, 0 =

VΦz(0,z0) = VΦ(0,x0) = VΦ(0,x) for all x ∈ J. Assume now there exists ˜ x ∈ J

and VΦ(T, ˜ x) < 0, by C1and (5), given ρ > 0, we obtain

(7) VΦ(T, ˜ x) ≥ VΦz(T,ω) − ρ

a.s. for w ∈ Bρ ≡ {ω : d(Z(ω), ˜ x) < δ} and δ > 0 is as in (5).

fact that VΦz(T,ω) ≥ 0 we arrive at a contradiction and conclude VΦ(T,x) ≥ 0

for all x ∈ J. Assume now VΦ(T,x) = 0 for all x ∈ J, because C′

there exists ω∗∈ D and x∗≡ Z(ω∗) ∈ J. The relation “Φzis connected to Φ”

holds in a set of full measure which is independent on any given x, then, we may

assume without loss of generality that (5) holds for ω∗. Then, using C1we obtain:

VΦ(T,x∗) ≥ VΦz(T,ω∗) − ρ for all ρ > 0. This implies VΦ(T,x∗) > 0.

Using the

0and (6),

?

The following proposition provides sufficient conditions to check that a certain

Φzis connected to a NP portfolio Φ.

Proposition 2. Let (J,A) be a NP market, Z an adapted stochastic process defined

on the filtered probability space (Ω,F,(Ft)t≥0,P) and assume C0from Theorem 1

holds. Then, if Φzis an admissible portfolio such that there exists Φ ∈ A satisfying:

VΦ(T,·) : J → R is upper semi-continuous with respect to metric d, VΦz(0,z0) =

VΦ(0,x0) and VΦz(T,w) = VΦ(T,Z(w)) a.s., then Φzis connected to Φ (so Φz∈

[A]Z.)

Proof. Consider Φ and Φzsatisfying the hypothesis of the proposition. The upper

semi continuity means that for a given x ∈ J and any ρ > 0 there exists δ > 0

satisfying: if d(x′,x) < δ, with x′∈ J then

VΦ(T,x) ≥ VΦ(T,x′) − ρ.

Consider now w to be in the set of full measure where Z(Ω) ⊆ J holds; fix x ∈

J and ρ > 0 arbitrary. Consider now δ as given by the upper semi continuity

assumption, then, if d(Z(w),x) < δ, taking x′≡ Z(w) we obtain VΦ(T,x) ≥

VΦ(T,x′) − ρ = VΦz(T,w) − ρ.

?

For simplicity, in most of our further developments, we will make use of stronger

notions than connected and lower and upper semi-continuous portfolios. Namely,

isomorphic and V-continuous portfolios, here are the definitions.

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ARBITRAGE AND HEDGING IN A NON PROBABILISTIC FRAMEWORK9

Definition 10. Let (J,A) and (Z,AZ) be respectively NP and stochastic markets

and assume the condition C0 from Theorem 1 holds. A NP portfolio Φ ∈ A and

Φz∈ AZare said to be isomorphic if P-a.s.:

Φz(t,ω) = Φ(t,Z(ω))

for all 0 ≤ t ≤ T.

Definition 11. Let (J,A) be a NP market. A NP portfolio Φ ∈ A is said to be

V-continuous with respect to d if the functional VΦ(T,·):J → R is continuous with

respect to the topology induced on J by distance d.

Whenever the distance d is understood from the context we will only refer to the

portfolio as V-continuous. The intuitive notion of a V-continuous portfolio is that

small changes in the asset price trajectory will lead to small changes to the final

value of the portfolio.

Remark 3. Clearly, if ΦT:J → R2is continuous then Φ is V-continuous.

Propositions 1 and 2 plus the definition of V-continuity give the following corol-

lary.

Corollary 1. Consider the setup of Definition 10. In particular, consider Φ and

Φzto be isomorphic, furthermore, assume Φ to be V-continous, then:

• Φ is connected to Φzand so Φ ∈ [AZ].

• Φzis connected to Φ and so Φz∈ [A]Z.

In many of our examples, we will rely on Corollary 1 to check if given portfolios

belong to [AZ] or [A]Z. In each of our examples, introduced in later sections, it

will arise the question on how large are the classes of portfolios [AZ] and [A]Zas

the Definitions 7 and 9 do not provide a direct characterization of its elements. For

each of our examples we will prove that specific classes of portfolios do belong to

[AZ] and [A]Z, answering the question in general is left to future research.

Arbitrage in subsets

Consider J∗⊆ J, it is natural to look for conditions that provide a relation-

ship between the arbitrage opportunities of these two sets. The NP framework

allows a simple result, Proposition 3 below, which provides a clear contrast with

the probabilistic framework which, in particular, is not able to provide an answer

when J∗is a subset of measure zero (see Example 1). There exist cases, for exam-

ple if J∗= {x ∈ J : xT > x0erT}, for which there exists an obvious NP-arbitrage

portfolio by borrowing money from the bank and investing on the asset. Proposi-

tion 3 shows that the no-arbitrage property for a NP-market (J,A) is inherited by

NP-markets whose trajectories J∗are dense on J.

Proposition 3. Consider the NP-market (J,A) where A is some class of NP-

admissible, V-continuous portfolio strategies (with respect to metric d). Let J∗⊂ J

be a subclass of trajectories such that J∗is dense in J with respect to the metric d

and consider AJ∗ to be the restriction of portfolio strategies in A to the subclass J∗.

Then the NP-Market (J∗,AJ∗) is NP-arbitrage-free if (J,A) is NP-arbitrage-free.

Proof. Assuming that there exists Φ ∈ AJ∗, an arbitrageopportunity on (J∗,AJ∗),

we will derive an arbitrage strategy in (J,A). To achieve this end, it is enough

Page 10

10 ALVAREZ, FERRANDO AND OLIVARES

to prove that VΦ(0,x) = 0 and VΦ(T,x) ≥ 0, both relations valid for all x ∈ J.

The first relation should hold because of the density assumption and the fact that

VΦ(0,·) is a continuous function on J. The second relationship follows similarly

using continuity of VΦ(T,·) on J.

?

4. Examples: Arbitrage and Hedging in Trajectory Classes

This section provides examples of NP markets (J,A) which are free of arbitrage

and in which general classes of payoffs can be hedged. Several of the results from

Section 3 are applied in order to gain a more complete understanding of these

examples, in particular, we provide several details about the characterizations of

the portfolios Φ ∈ A.

A main example deals with continuous trajectories (this set is denoted by Jσ

other examples deal with trajectories containing jumps. Several aspects of these

different examples are treated in a uniform way illustrating the flexibility of using

different topologies in the trajectory space. The classes of trajectories to be intro-

duced could be considerably enlarged by allowing the parameter σ to be a function

of t (obeying some regularity conditions). Our results apply to such (extended)

classes as well, in the present paper we will restrict σ to be a constant for simplic-

ity. We also restrict to hedging results to path independent derivatives but expect

the results can be extended to path independent derivatives as well.

The replicating portfolio strategies that we will obtain in a NP-market are es-

sentially the same that in the corresponding stochastic frameworks, for example,

to replicate a payoff when prices lie in our example Jσ

trajectories, we use the well known delta-hedging as in the Black-Scholes model. In

the available literature there exist several results related to the robustness of delta

hedging, see for example: [4], [19], [3] and [9]. A point to emphasize is the fact

that the replication results, being valid in a different sense (probability-free), are

valid also when considering subclasses of trajectories J∗⊂ J. Formally, this fact

is not available in probabilistic frameworks due to the technical reliance on sets of

measure zero or non-measurable sets.

τ),

τ, comprised of continuous

4.1. Non Probabilistic Black-Scholes Model. Denote by ZT([0,T]) the collec-

tion of all continuous functions z(t) such that [z]T

Notice that ZT([0,T]) includes a.s. paths of Brownian motion if T is a refining

sequence of partitions ([17].)

For a given sequence of subdivisions T define,

• Given constants σ > 0 and x0 > 0, let Jσ

valued functions x for which there exists z ∈ ZT([0,T]) such that:

t= t for 0 ≤ t ≤ T and z(0) = 0.

τ(x0) to be the class of all real

x(t) = x0eσz(t).

According to (35), the class Jσ

x(0) = x0 and quadratic variation satisfying d?x?τ

class Jσ

τ(x0) will be considered as a subset of the continuous functions with the

uniform topology induced by the uniform distance.

τ(x0) is the class of continuous functions x with

t= σ2x(t)2dt. The trajectory

Remark 4. The class Jσ

geometric Brownian motion, as an illustration we indicate that if z = B+y, with B

τ(x0) includes trajectories of processes different than the

Page 11

ARBITRAGE AND HEDGING IN A NON PROBABILISTIC FRAMEWORK11

a Brownian motion and y a process with zero quadratic variation, then the trajec-

tories of z belongs to Zτ([0,T]), hence the trajectories of the process x0eσzbelong

to Jσ

τ(x0).

As previously suggested, the hedging results in this class have been already

obtained, more or less explicitly, in several papers (see [4] and [19] for example).

Theorem 3. Let J∗be a class of possible trajectories, J∗⊂ Jσ

[0,T] × R+→ R be the solution of the PDE

τand let v(·,·) :

(8)

∂v

∂t(t,x) + r x∂v

∂x(t,x) +σ2x2

2

∂2v

∂x2(t,x) − r v(t,x) = 0

with terminal condition v(T,x) = h(x) where h(·) is Lipschitz. Then, the delta hedg-

ing NP-portfolio Φt≡ (v(t,x(t)) − ̺(t,x(t))x(t),̺(t,x(t))) where ̺(t,x) ≡∂v

replicates the payoff h at maturity time T for all x ∈ J∗.

∂x(t,x)

Proof. The existence and uniqueness of the solution of (8) is guaranteed because h

is Lipschitz. If x ∈ J∗⊂ Jσ

τ(x0) then we know that x is of quadratic variations

and d?x?τ

using (8) and noticing that the integral?T

(9) h(x(T)) = v(T,x(T)) = lim

t= σ2x(t)2dt, so applying Itˆ o-F¨ ollmer formula, taking ̺(t,x) =∂v

0r(v(s,x(s))−̺(s,(s,x(s))x(s))ds exists,

we obtain:

∂x(t,x),

u→Tv(u,x(u)) =

lim

u→T

?

v(0,x(0)) +

?u

0

∂v

∂t(s,x(s))ds +1

2

?u

0

∂2v

∂x2(s,x(s))d?x?τ

s+

?u

0

̺(s,x(s))dx(s)

?

=

lim

u→T

?

v(0,x(0)) +

?u

0

r[v(s,x(s)) − ̺(s,x(s))x(s)]ds +

?u

0

̺(s,x(s))dx(s)

?

=

v(0,x(0)) +

?T

0

r[v(s,x(s)) − ̺(s,x(s))x(s)]ds + lim

u→T

?u

0

̺(s,x(s)) dx(s) =

v(0,x(0)) +

?T

0

r[v(s,x(s)) − ̺(s,x(s))x(s)]ds +

?T

0

̺(s,x(s))dx(s).

The analysis in (9) implies that the NP-portfolio

(10)Φt= (v(t,x(t)) − ̺(t,x(t))x(t),̺(t,x(t)))

replicates the payoff h at maturity time T.

?

Corollary 2. The delta hedging portfolio given by (10) is NP-admissible and V-

continuous relative to the uniform topology.

Proof. The self-financing and predictable properties follow from the definition and

constructions in Theorem 3 by noticing that x(t) = x(t−). The portfolio is admis-

sible, with A = 0, by the known property v(t,x(t)) ≥ 0. V-continuity follows from

VΦ(T,x) = h(x(T)) and the fact that h is continuous.

?

Page 12

12 ALVAREZ, FERRANDO AND OLIVARES

4.1.1. Arbitrage in Jσ

where possible trajectories are in Jσ

Black and Scholes model (Z,AZ); in particular, AZ= AZ

the admissible portfolios in the Black-Scholes stochastic market. Corollary 1 will

be used to show that a large class of portfolios belong to [AZ

we incorporate portfolio strategies that depend on past values of the trajectory and

not just on the spot value ([3]).

τ. We analyze next the problem of arbitrage in a market

τ. We will make use Theorem 1 applied to the

BS, where AZ

BSdenotes

BS]; towards this end,

Definition 12. A hindsight factor g over some class of trajectories J is a mapping

g : [0,T] × J → R satisfying:

i) g(t,η) = g(t, ˜ η) whenever η(s) = ˜ η(s) for all 0 ≤ s ≤ t.

ii) g(·,η) is of bounded variation and continuous for every η ∈ J.

iii) There is a constant K such that for every continuous function f.

????

?t

0

f(s)dg(s,η) −

?t

0

f(s)dg(s, ˜ η)

????≤ K max

0≤r≤tf(r)?η − ˜ η?∞

Another definition of [3] are the smooth strategies introduced next.

Definition 13. A portfolio strategy Φ = (ψt,φt)0≤t≤T over the class of trajectories

J is called smooth if:

i) The number of assets held at time t, φt, has the form

(11)φt(x) = φ(t,x) = G(t,xt,g1(t,x),...,gm(t,x))

for all t ∈ [0,T] and for all x ∈ J where G ∈ C1([0,T] × R × Rm) and the

gi’s are hindsight factors

ii) There exists A > 0 such that VΦ(t,x) ≥ −A ∀t ∈ [0,T] and ∀x ∈ J.

Given the notation and assumptions from Definition 13, an application of Itˆ o-

F¨ ollmer formula (34) proves that the integrals?t

B) relate the smoothness condition in (11) with the admissibility conditions of both,

stochastic and NP portfolios respectively.

0φ(s,x)dx(s) exist for all t ∈ [0,T]

if Φ = (ψt,φt) is smooth. Propositions 11 and 12 (stated and proven in Appendix

Proposition 4. If Φ is a smooth portfolio strategy over Jσ

distance then Φ is V-continuous.

τ(x0) and d is the uniform

Proposition 4 follows immediately from Lemma 4.5 in [3].

The following result is well known ([14]), we present a proof for completeness.

Lemma 1. Let y be a continuous function y : [0,T] → R with y(0) = 0. If W is a

Brownian motion defined on a probability space (Ω,F,P) then for all ǫ > 0,

(12)P

?

ω : sup

s∈[0,T]

|Ws(ω) − y(s)| < ǫ

?

> 0.

Proof. Function y is continuous on [0,T], therefore is uniformly continuous so for

all ǫ > 0, there exists δ > 0 such that

|t2− t1| < δ =⇒ |y(t2) − y(t1)| < ǫ/3

Let M be an integer, M > T/δ, and define points si= iT/M, for i = 0,...,M. By

definition |si+1− si| < δ, so |y(si+1) − y(si)| < ǫ/3.

Page 13

ARBITRAGE AND HEDGING IN A NON PROBABILISTIC FRAMEWORK13

Define for all 1 ≤ i ≤ M

Ai=

?

ω :sup

si−1≤t≤si

??Wt− Wsi−1

??< ǫ/2

?

Bi=?ω :??(Wsi− Wsi−1) − (y(si) − y(si−1))??< ǫ/6M?

hand, it is obvious that events Ωiand Ωjare independent for i ?= j since increments

of Brownian motions on disjoint intervals are independent. So

and Ωi= AiBi. It is immediate, from results in [15], that P(Ωi) > 0. On the other

(13) P

?M

i=1

?

Ωi

?

=

M

?

i=1

P(Ωi) > 0

We will prove now that

(14)

M

?

i=1

Ωi⊂

?

ω : sup

s∈[0,T]

|Ws(ω) − y(s)| < ǫ

?

Let ω ∈?M

(15)

i=1Ωi, then ω ∈?M

i=1Biso for all k = 1,...,M −1, applying triangular

inequality:

|Wsk(ω) − y(sk)| ≤

k

?

i=1

??(Wsi(ω) − Wsi−1(ω)) − (y(si) − y(si−1))??< kǫ/6M ≤ ǫ/6

Also ω ∈ Ak+1, so |Wt(ω) − Wsk(ω)| < ǫ/2 for all t ∈ [sk,sk+1]. We also know that

|y(sk) − y(t)| ≤ ǫ/3 for all t ∈ [sk,sk+1]. Using again triangular inequality:

(16)

|Wt(ω) − y(t)| ≤ |Wt(ω) − Wsk(ω)|+|Wsk(ω) − y(sk)|+|y(sk) − y(t)| < ǫ/2+ǫ/6+ǫ/3 = ǫ

is valid for all t ∈ [sk,sk+1] for all k so (16) is valid for all t ∈ [0,T] and (14) is

true. From (14) and (13) we obtain (12) and the Lemma is proved

?

A main consequence of Theorem 1 and the previous definitions and results is the

following Theorem.

Theorem 4. Let (Z,AZ

BS) be the Black-Scholes stochastic market defined by

Zt= x0e(µ−σ2/2)t+σWt,

where µ and σ > 0 are constant real numbers, W is a Brownian Motion, and AZ

is the class of all admissible strategies for Z. Consider the class of trajectories Jσ

with the uniform topology. We have:

i) The NP market (Jσ

BS]) is NP arbitrage-free.

ii) [AZ

BS] contains:

a) the smooth strategies such that the hindsight factors gisatisfy that gi(t,X)

are (Ft−)-measurable,

b) delta hedging strategies.

BS

τ

τ,[AZ

Proof. i) By the definition of Z and Jσ

satisfied. Also, condition C1from Theorem 1 follows from Lemma 1. As (Z,AZ

is arbitrage-free (see for example [10]) then the NP market (Jσ

arbitrage-free according to Theorem 1.

τ clearly condition C0 in Theorem 1 is

BS)

τ,[AZ

BS]) is NP

Page 14

14ALVAREZ, FERRANDO AND OLIVARES

ii) Let Φ be a smooth strategy over Jσ

condition (39) in Proposition 12 holds, therefore Φ is NP-admissible; Φ is also V-

continuous as consequence of Proposition 4. Define a.s. Φzas Φz(t,ω) = Φ(t,Z(ω));

Proposition 11 shows that the stochastic portfolio Φzis predictable, LCRL and self-

financing. The admissibility of Φzresults from ii) in Definition 13, hence Φz∈ AZ

As Φ and Φzare isomorphic and Φ is V-continuous, Corollary 1 applies so Φ is

connected to Φzand Φ ∈ [AZ

BS]. For the hedging strategies the same arguments

apply and the V-continuity and admissibility follow by an application of Corollary

2.

τAs the trajectories in Jσ

τare continuous,

BS.

?

Remark 5. In the framework of Theorem 4, where trajectories in J are continuous

it is not difficult to see that ˜ g(t,x) = min0≤s≤tx(s), as well as the maximum and

the average, are hindsight factors over J (see [3]), moreover ˜ g(t,X) is a (Ft−)-

measurable random variable.

Remark 6. It can be proved that [AZ

portfolio strategies satisfying

BS] also contains simple (piece-wise constant)

φt=

L

?

l=1

1(sl−1,sl](t)G(t,x(sl−1))

where 0 = s0< s1< ··· < sL= T, the siare deterministic and G is C1. This is

consequence of Remark 4.6 of [3].

Theorem 4 is the analogous in our framework of the known absence of arbitrage

in the Black-Scholes model, a property that in fact we use in the above proof.

In a classical stochastic framework, the absence of arbitrage is equivalent to the

existence of at least one risk neutral probability measure, the next example shows

a possible trajectory class which has no obvious probabilistic counterpart.

Example 1. Define the class

Jσ

τ,Q= {x ∈ Jσ

τ: x(T) ∈ Q}

where Q is the set of rational numbers.

Consider [AZ

V -continuous portfolios in [AZ

portfolios which also satisfies that the market (Jσ

AV

Jσ

Jσ

(Jσ

Jσ

BS] as defined in Theorem 4. Let AV⊂ [AZ

BS]. Item ii) in Theorem 4, AVis a large class of

BS] be the class of all

τ,AV) is NP-arbitrage free. Let

τ,Qbe the restriction of portfolio strategies in AVto the subclass of trajectories

τ,Q. As Jσ

τ,Q,AV

τ,Qis dense on Jσ

τ,Q) is NP-arbitrage-free.

τ, applying Proposition 3 we conclude that the market

The absence of arbitrage for model in Example 1 and replicating portfolio in

Theorem 3 imply that it is possible to price derivatives using the Black-Scholes

formula also for this model, even if there is no obvious intuitive measure over the

possible set of trajectories. In fact, the set Jσ

Black-Scholes model, therefore if a measure is defined over this set, it will not be

absolutely continuous with respect to the Wiener measure. Hence, it is not clear

how to price derivatives under a stochastic model following a risk neutral approach,

if the trajectories of the asset price process belong to Jσ

τ,Qhas null probability under the

τ,Q.

Page 15

ARBITRAGE AND HEDGING IN A NON PROBABILISTIC FRAMEWORK15

4.2. Non Probabilistic Geometric Poisson Model. This section studies hedg-

ing and arbitrage in specific examples of trajectory classes with jumps. Denote by

N([0,T]) the collection of all functions n(t) such that there exists a non nega-

tive integer m and positive numbers 0 < s1 < ... < sm < T such that n(t) =

?

The following class of real valued functions will be another example of possible

trajectories for the asset price.

• Given constants µ,a ∈ R and x0 > 0, let Ja,µ(x0) to be the class of all

functions x for which exists n(t) ∈ N([0,T]) such that:

si≤t1[0,t](si). The function n(t) is considered as identically zero on [0,T] when-

ever m = 0.

(17)x(t) = x0eµt(1 + a)n(t).

The function n(t) counts the number of jumps present in the path x until, and

including, time t. Note also that the definition of Ja,µ(x0) does not depend on the

particular subdivision T used elsewhere in the paper.

The natural probabilistic counterpart for this model is the Geometric Poisson

model

Zt= x0eµt(1 + a)Np

where NP= (NP

t) is a Poisson process on a filtered probability space (Ω,F,Ft,P).

Notice that P(Z(w) ∈ Ja,µ(x0)) = 1. Even if this stochastic model has limited

practical use in finance, it has theoretical importance because, together with the

Black-Scholes model, they are the only exponential L´ evy models leading to complete

markets, see [8].

t,

Remark 7. The class Ja,µ(x0) includes trajectories of processes different than

the Geometric Poisson model, in fact if N is a renewal process, trajectories of the

process Z defined as Zt= x0eµt(1 + a)Ntare also in Ja,µ(x0).

A replicating portfolio for trajectories in Ja,µcorresponds to the probabilistic-

free version of the hedging strategy associated to the Geometric Poisson model, see

[6].

Suppose we have an European type derivative with payoff h(x(T)). For simplicity

we consider interest rate r = 0. We are looking for a NP-admissible portfolio

strategy that perfectly replicates the payoff h(x(T)). The next Theorem provides

the answer to this NP hedging question.

Theorem 5. Let J∗be a class of possible trajectories for the asset price, J∗⊂

Ja,µ(x0). Consider that aµ < 0 and let λ = −µ/a. Define˜F(s,t) by:

˜F(t,s) = e−λ(T−t)

∞

?

k=0

h?seµ(T−t)(1 + a)k?(T − t)k

k!

.

Then, the portfolio Φt= (ψt,φt) where

φt=

˜F(t,(a + 1)x(t−)) −˜F(t,x(t−))

a x(t−)

,

and ψt=˜F(t,x(t−)) −φtx(t−), whose initial value is˜F(0,x0), replicates the Lips-

chitz payoff h(x(T)) at time T for every x ∈ J∗.

We will not provide a proof of Theorem 5 as it can be easily extracted from

[6] even though that reference obtains a probabilistic result considering n(t) (as