# Arbitrage without borrowing or short selling?

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DOI: 10.1007/s11579-016-0180-x

- Cite this article as:
- Lukkarinen, J. & Pakkanen, M.S. Math Finan Econ (2017) 11: 263. doi:10.1007/s11579-016-0180-x

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## Abstract

We show that a trader, who starts with no initial wealth and is not allowed to borrow money or short sell assets, is theoretically able to attain positive wealth by continuous trading, provided that she has perfect foresight of future asset prices, given by a continuous semimartingale. Such an arbitrage strategy can be constructed as a process of finite variation that satisfies a seemingly innocuous self-financing condition, formulated using a pathwise Riemann–Stieltjes integral. Our result exemplifies the potential intricacies of formulating economically meaningful self-financing conditions in continuous time, when one leaves the conventional arbitrage-free framework.

### Keywords

Short selling Self-financing condition Arbitrage Riemann–Stieltjes integral Stochastic integral Semimartingale### Mathematics Subject Classification

60H05 90G10 60G44### JEL Classification

C22 G11 G14## 1 Introduction

Common sense suggest that arbitrage strategies—in the sense of mathematical finance, involving no initial wealth—should require short selling or an access to credit—an obvious *budget constraint*. Indeed, in the real world, and in discrete-time models as well, we can distinguish the first position in the risky asset prescribed by the strategy. If this position were not short, it would have to be funded by borrowed money. However, in the realm of continuous trading, there might not be any “first position”, as the composition of the portfolio can vary rather freely as a function of time, so it is not a priori clear if arbitrage strategies without short selling or borrowing are impossible.

*Self-financing conditions* are an important aspect of dynamic trading strategies. They should be seen as a means to enforce coherent accounting: All profits from trading must be credited to, and all trading costs debited from the money market account. In continuous time, self-financing conditions are formulated using stochastic integrals; see, e.g., Björk [2, Sects. 6.1 and 6.2]. In particular, for adapted strategies, *Itô integrals* can be used when the price process is a semimartingale. However, the choice of the integral is a rather delicate matter, as not all stochastic integrals lend themselves to economically meaningful self-financing conditions. (For example, the paper by Björk and Hult [3] documents some interpretability issues that arise from the use of *Skorohod integrals* and *Wick products* in self-financing conditions.) In any case, any sound self-financing condition should at the very least rule out arbitrage strategies without short selling or borrowing. After all, such trading strategies, which are able to generate wealth literally *ex nihilo*, should definitely not be self-financing.

Besides Itô integration, *pathwise Riemann–Stieltjes integrals* (see, e.g., Riga [10], Salopek [11], or Sottinen and Valkeila [12]) have often been seen as a “safe” way to formulate reasonable self-financing conditions. The reasons are manifold: Like Itô integrals, Riemann–Stieltjes integrals can, of course, be obtained transparently as limits of Riemann sums that reflect the natural self-financing condition for simple trading strategies. Also, a pathwise Riemann–Stieltjes integral coincides with the corresponding Itô integral whenever the latter exists. Recall that a Riemann–Stieltjes integral is guaranteed to exist for example when the integrator is continuous and the integrand is of *finite variation*. While it typically rules out the Markovian trading strategies that arise in dynamic hedging and utility maximisation, say, the finite variation assumption is economically justified as it amounts to keeping the trading volume of the strategy finite (which is an essential requirement under transaction costs); see, e.g., Longstaff [7].

However, it transpires that self-financing conditions based on pathwise Riemann–Stieltjes integrals alone do not necessarily prohibit pathological trading strategies (even of finite variation). We show in this note that, quite surprisingly, a Riemann–Stieltjes-based self-financing condition may in fact admit arbitrage strategies that require neither borrowing nor short selling if the trader has *perfect foresight* of the future prices of the risky asset.^{1} Our existence result for such strategies (Theorem 2.2, below) is valid provided that the price process is a continuous semimartingale with an equivalent local martingale measure and non-degenerate quadratic variation. While the requirement of perfect foresight is admittedly unusual, a sound self-financing condition should nevertheless prevent even a perfectly informed trader from executing such an egregious arbitrage strategy. More importantly, from a mathematical perspective, this result illustrates how stochastic integrals, even when defined pathwise, may not always behave as financial intuition would suggest. We additionally show that these arbitrage strategies would in fact not be possible if also the price process were of finite variation (Proposition 2.4, below). This indicates that the phenomenon documented in this note is intricately linked with the fine properties and “roughness” of the price process.

## 2 Model and main results

Let us consider a continuous-time market model with a risky asset and a risk-free money market account, where trading is possible up to a finite time horizon \(T \in (0,\infty )\). The price of the risky asset follows a continuous, positive-valued semimartingale \(S=(S_t)_{t \in [0,T]}\), defined on a complete probability space \((\Omega ,\mathscr {F},\mathbb {P})\). For simplicity, the interest rate of the money market account is zero. Additionally, we denote by \((\mathscr {F}^S_t)_{t \in [0,T]}\) the natural filtration of the price process *S*, augmented the usual way to make it complete and right-continuous, and by \(\langle S \rangle \) the quadratic variation process of *S*. Throughout the paper, we use the interpretation \(\inf \varnothing = \infty \).

*S*is understood as an Itô integral. Under the self-financing condition (2.2), the process \(\psi \) becomes redundant as, by plugging (2.2) into (2.1), we can solve for \(\psi _t\), to wit,

*S*is arbitrage-free. Indeed, if there exists a probability measure \(\mathbb {Q}\) on \((\Omega ,\mathscr {F})\) such that \(\mathbb {Q} \sim \mathbb {P}\) (where “\(\sim \)” denotes mutual absolute continuity of measures, as usual) and that

*S*is a local \(\mathbb {Q}\)-martingale, then a suitable version of the

*fundamental theorem of asset pricing*(e.g., [4, Corollary 1.2]) implies that there are no non-negative (adapted) processes \(\phi \) that would satisfy (2.4) and \(\mathbb {P}(V_t> 0)>0\) for some \(t \in (0,T]\).

*S*that appears in (2.2), (2.3) and (2.4) may not exist as an Itô integral. But if we assume that \(\phi \) is of finite variation, then the integral does exist as a Riemann–Stieltjes integral, see [13, Theorems 1.2.3 and 1.2.13], defined path-by-path for any \(t \in [0,T]\) by

*simple*, that is, piecewise constant.

### Remark 2.1

While trading strategies in mathematical finance literature are conventionally assumed to be adapted to the natural filtration of the price process, non-adapted strategies do appear in literature on insider trading; see, e.g., [1, 9]. More recently, it has also been suggested that (imprecise) prior information of future price changes at very short time scales may be available to high-frequency traders and market makers [5].

Our main result shows that, in this alternative framework, there are in fact non-trivial, non-negative processes \(\phi \) that satisfy the inequality (2.4). The proof of this result is carried out in Sect. 3, below.

### Theorem 2.2

*S*is a local \(\mathbb {Q}\)-martingale. Then there exists a non-negative process \(\phi =(\phi _t)_{t \in [0,T]}\), with càglàd sample paths of finite variation, such that \(\phi _0=0\), and

### Remark 2.3

- (i)While not explicitly stated above, the process \(\phi \) of Theorem 2.2 is indeed not (and could not be) adapted to \((\mathscr {F}^S_t)_{t \in [0,T]}\). The specification of \(\phi _t\) for any \(t\in (0,T]\) requires full knowledge of the path of
*S*until time*T*. However, the process \(\phi \) is adapted to the filtrationcorresponding to perfect foresight on$$\begin{aligned} \tilde{\mathscr {F}}^S_t \mathrel {\mathop :}=\mathscr {F}^S_T, \quad t \in [0,T]\,, \end{aligned}$$*S*, which also ensures that \(\phi \) does not depend on any (external) randomness beyond*S*. - (ii)
It is also worth stressing that the time horizon \(T\in (0,\infty )\) can be chosen freely, as long as \(\mathbb {P}(\langle S \rangle _T> 0)>0\) is satisfied. In particular, if

*S*has strictly increasing quadratic variation, then we can choose*T*to be arbitrarily small—that is, prior knowledge of the fluctuations of*S*is required only on a very short time interval. - (iii)
In mathematical finance literature, it is common to restrict trading strategies to be

*admissible*; see, e.g., [4, Definition 2.7]. While there are actually several slightly differing definitions of admissibility, they have the commonality that the value process of an admissible strategy is bounded from below (in some sense). The purpose of admissibility conditions is to preclude some outright pathological trading strategies, such as*doubling strategies*[4, p. 467]. It is worth stressing that the process \(\phi \) of Theorem 2.2 would*not*violate the typical admissibility conditions as the corresponding value process \(V_t = \int _0^t \phi _u \mathrm {d}S_u\), \(t \in [0,T]\), is non-negative due to the property (2.6a).

Curiously, the assumption about positive quadratic variation in Theorem 2.2—that is, *S* exhibits “enough” fluctuation—is rather crucial: Using a result [8, Theorem 3.1] on the positivity of Riemann–Stieltjes integrals, we can show that arbitrage without borrowing or short selling is in fact eliminated in this setting if also the price process *S* is of finite variation:

### Proposition 2.4

### Proof

*S*is a continuous semimartingale, the assumption \(\langle S \rangle _T=0\) implies that the sample paths of

*S*are of finite variation. Now if \(\phi _t >0\) for some \(t \in [0,T]\), then it follows

^{2}from [8, Theorem 3.1] that

### Remark 2.5

In some way, Theorem 2.2 and Proposition 2.4 defy the usual mathematical finance intuition that “smooth” price processes are easier to arbitrage than “rough” ones. Here the “roughness” of *S* is the very property that makes it possible to construct the process \(\phi \) in Theorem 2.2.

In Theorem 2.2, we assume that the price process *S* is arbitrage-free whilst the strategy \(\phi \) may not be adapted. This is, of course, only one of the possible departures from the standard arbitrage-free setting. Alternatively, one could also consider a scenario where the process *S* is a very general continuous process that may admit arbitrage and \(\phi \) is an adapted strategy of finite variation and ask, how the stronger form of arbitrage without short selling and borrowing can be excluded. This looks less straightforward and may require some new techniques and estimates for Riemann–Stieltjes integrals, so we leave the question open:

### Open Question 2.6

When *S* is a general positive, continuous process (not necessarily a semimartingale), under which conditions on *S* is arbitrage without borrowing or short selling excluded in the context of strategies of finite variation? We remark that, to this end, the process *S* should satisfy some kind of a non-degeneracy condition, as integrands similar to \(\phi \) of Theorem 2.2 can be constructed for deterministic continuous paths that exhibit enough variation; see [8, Theorem 2.1].

## 3 Proof of Theorem 2.2

Before proving Theorem 2.2 rigorously, we describe intuitively how the process \(\phi \) is constructed. The idea is to structure \(\phi \) from a sequence of non-overlapping static positions in the risky asset, so that they have an “accumulation point” at \(\rho \), see Fig. 1, bottom-right panel, for an illustration. The sizes of these static positions are chosen so that they are gradually increasing (from zero) and the positions are timed, using the quadratic variation of *S* and perfect foresight, so that the price of the asset is known to increase during each holding period.

each position can be fully funded using the profits from the preceding positions (without needing to borrow money),

the cumulative trading volume remains finite, which is equivalent to \(\phi \) being of finite variation.

We introduce now some additional notation that are needed in the sequel. For all \(x,y \in \mathbb {R}\), we denote \(x \vee y \mathrel {\mathop :}=\max \{ x,y\}\) and \(x^+ \mathrel {\mathop :}=x \vee 0\). If *X* and *Y* are identically distributed random variables, we write \(X\mathop {=}\limits ^{d}Y\). Suppose that \(A \in \mathscr {F}\). Then we say that a property \(\mathscr {P}\) (provided that it is “\(\mathscr {F}\)-measurable”) holds \(\mathbb {P}\)-a.s. on *A*, if \(\mathbb {P}(\{\mathscr {P} \} \cap A) = \mathbb {P}(A)\). We use the convention that \(\mathbb {N}\mathrel {\mathop :}=\{1,2,\ldots \}\).

As a preparation, we prove now two technical lemmata, which will be instrumental in the proof of Theorem 2.2.

### Lemma 3.1

### Proof

Consider a sequence \((y_n)_{n=1}^\infty \) and \(\alpha \in (0,1)\) that satisfy the assumptions given above. Define then \(\beta \mathrel {\mathop :}=\frac{2 \alpha }{1+\alpha }\) and a sequence \((x_n)_{n=1}^\infty \) through (3.2). Clearly, then \(0<\alpha<\beta <1\) and \(0<x_n\leqslant 1\) for all \(n \in \mathbb {N}\).

### Lemma 3.2

### Proof

The proof of Theorem 2.2 is based on the observation that the properties (2.6a) and (2.6b) the process \(\phi \) is expected to satisfy are robust to time changes and equivalent changes of the probability measure. Under the assumptions of Theorem 2.2, we can represent the process *S* as a time-changed Brownian motion under an equivalent local martingale measure. Therefore we can verify (2.6a) and (2.6b) relying on the properties of Brownian motion via Lemmata 3.1 and 3.2.

### Proof of Theorem 2.2

*S*by a positive constant. By rescaling

*S*, the probability of the event \(\{\sup _{t \in [0,T]} S_t \leqslant 1\}\) can be made to be arbitrarily close to one. In particular, we may assume, without loss of generality, that

*S*has been rescaled. This implies that \(\mathbb {P}(\sup _{t \in [0,T]} S_t \leqslant 1,\, \langle S \rangle _T> 0)>0\), so we can find a constant \(c>0\) such that the event \(A_c \mathrel {\mathop :}=\{\sup _{t \in [0,T]} S_t \leqslant 1\} \cap \{ \langle S \rangle _T > c\}\) satisfies \(\mathbb {P}(A_c)>0\).

*S*is continuous, also its quadratic variation process \(\langle S \rangle \) is \(\mathbb {P}\)-a.s. continuous [6, Theorem 17.5]. Thus we have \(\mathbb {P}\)-a.s. on \(A_c\),

*S*is a local \(\mathbb {Q}\)-martingale. Then, clearly, \(\mathbb {Q}(A_c)>0\). By the Dambis–Dubins–Schwarz theorem [6, Theorem 18.4], there exists a standard Brownian motion \(B=(B_t)_{t \geqslant 0}\) defined on an extension \(\big (\bar{\Omega },\bar{\mathscr {F}},\bar{\mathbb {Q}}\big )\) of \((\Omega ,\mathscr {F},\mathbb {Q})\), such that the scaled Brownian motion \(B'_t \mathrel {\mathop :}=\sqrt{c} B_t\), \(t \geqslant 0\), satisfies

*t*, which dispels any concerns about convergence of the random sum.) Since (3.11) holds \(\mathbb {P}\)-a.s. on \(A_c\), Lemma 3.1 with \(\alpha = \frac{1}{2}\) ensures that \(\sum _{n=1}^\infty H_n < \infty \)\(\mathbb {P}\)-a.s. on \(A_c\), which in turn implies that the process \(\phi \) is \(\mathbb {P}\)-a.s. càglàd and of finite variation with \(\phi _0=0\). Thus, by [13, Theorems 1.2.3 and 1.2.13], the stochastic integral \(\int _0^t \phi _u \mathrm {d}S_u\) exists as a pathwise Riemann–Stieltjes integral for any \(t \in [0,T]\) and is given by

In many cases, it is actually sufficient to have perfect foresight only on an arbitrarily short time interval, as is pointed out in Remark 2.3.

The term *non-vanishing* in the statement of [8, Theorem 3.1] is potentially misleading. The appropriate interpretation is that *the integrand g should not be identically zero*. It is also worth mentioning that the assumption \(g(a)=0\) therein can be trivially weakened to \(g(a)\geqslant 0\); see [8, p. 401].

## Acknowledgments

The research of J. Lukkarinen has been partially supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (Project 271983) and from an Academy Project (Project 258302). M. S. Pakkanen acknowledges partial support from CREATES (DNRF78), funded by the Danish National Research Foundation, from the Aarhus University Research Foundation (project “Stochastic and Econometric Analysis of Commodity Markets”), and from the Academy of Finland (Project 258042).

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