2.1. Setting

Let (Ω, ℱ, ℙ) be a probability space endowed with a filtration 𝔽 = (ℱ_t)_t≥0 satisfying the usual conditions of right-continuity and completeness and supporting a càdlàg (right-continuous with left limits) real-valued^{Footnote 2} adapted process X = (X_t)_t≥0. The filtration 𝔽 represents the flow of available information, while the process X represents the gains process of a risky asset, discounted with respect to some baseline security. In the case of a dividend paying asset, this corresponds to the sum of the discounted ex-dividend price and the cumulated discounted dividends. We do not assume that X is a semimartingale or that the initial sigma-field ℱ₀ is trivial. The results presented below apply to any model in a finite time horizon T < + ∞ by simply considering the stopped process X^T. We let ΔX = (ΔX_t)_t≥0 denote the jump process of X, with ΔX_t ≔ X_t − X_{t −}, for t ≥ 0. Following the convention of [Reference Jacod and Shiryaev20], we let ΔX ₀ = 0. We refer to [Reference Jacod and Shiryaev20] for all unexplained notions related to the general theory of stochastic processes.

A stopping time T is said to be a jump time of X if [[T]] ⊆ {ΔX ≠ 0} (up to an evanescent set).^{Footnote 3} We say that X exhibits predictable jumps if there exists at least one jump time T which is a predictable time and such that the random variable ${{\bf{1}}_{\{ T \lt + \infty ,{\kern 1pt} \Delta {X_T} > 0\} }}$ is ℱ _T−-measurable. Strengthening this definition, we say that X exhibits fully predictable jumps if there exists at least one predictable jump time T such that the random variable ΔX_T 1_{{T < +∞}} is ℱ_{T −}-measurable. In other words, X exhibits predictable jumps if there exists at least one predictable jump time at which the direction (and even the size, in the case of a fully predictable jump) of the jump is known just before the occurrence of the jump. We aim to relate the absence of predictable jumps (and of fully predictable jumps) to minimal and realistic no-arbitrage properties.

2.2. Buy-and-hold strategies and flash strategies

We describe the activity of trading in the financial market according to the following definition.

A flash strategy represents the possibility of investing at higher and higher frequencies. In the limit, the strategy converges to a (bounded) position ξ which is constructed and then immediately liquidated at some random time τ. If by doing so and starting from zero initial wealth an investor can reach a strictly positive amount of wealth (provided that the investor trades at all, i.e. from time τ onwards), then the flash strategy is said to generate a sure profit. In the case of a constant profit, the amount of wealth generated by the flash strategy is perfectly known in advance. The requirement that the positions (ξⁿ)_n∊ℕ are uniformly bounded means that an investor is not allowed to make larger and larger trades as the holding period τ_n − σ_n converges to zero. This makes flash strategies feasible by placing market orders in financial markets with finite liquidity. Observe also that no trading activity occurs in the limit on the event {τ = + ∞}.

In the limit, a sure profit does not involve any risk, since the gains from trading converge to a strictly positive random variable. Moreover, it turns out that the components (hⁿ)_n∊ℕ of a flash strategy generating a sure profit can be chosen in such a way that the potential losses incurred by each individual buy-and-hold strategy hⁿ are uniformly bounded, for all sufficiently large n (see Section 2.3). Note also that, if a flash strategy generates a constant profit for some c > 0, then there exist constant profits for every c > 0, since the flash strategy can be arbitrarily rescaled. A further important property of the notion of constant profit via flash strategies is its robustness with respect to small transaction costs (see Section 3.2 below).

2.3. Predictable jumps and sure profits via flash strategies

The following theorem shows that the absence of sure profits via flash strategies is equivalent to the absence of predictable jumps. This result relies on the fact that predictable jumps are anticipated by a sequence of precursory signals which can be used to construct a sequence of buy-and-hold strategies forming a flash strategy.

Theorem 2.1. The process X does not exhibit predictable (fully predictable, resp.) jumps if and only if there are no sure (constant, resp.) profits via flash strategies.

Proof. We first prove that if X exhibits predictable jumps, then there exist sure profits. To this effect, let T be a predictable time with [[T]] ⊆ {ΔX ≠ 0} such that the random variable ${{\bf{1}}_{\{ T \lt + \infty ,{\kern 1pt} \Delta {X_T} > 0\} }}$ is ℱ _T−-measurable. For simplicity of notation, we set ΔX_T = 0 on {T = +∞}. In view of [Reference Jacod and Shiryaev20, Theorem I.2.15], there exists an announcing sequence (ρ_n)_n∊ℕ of stopping times satisfying ρ_n < T and such that ρ_n increases to T for n → + ∞. For each n ∊ ℕ, let σ_n ≔ ρ_n ∧ n and τ_n ≔ T ∧ n and define the sequence (hⁿ)_n∊ℕ by

(2.1) $${h^n} = {\xi ^n}{{\bf{1}}_{]]{\sigma _n},{\tau _n}]]}},\quad {\rm{where}} {\xi ^n} : = 2 {\mathbb {P}}(\Delta {X_T} > 0|{{\cal F}_{{\sigma _n}}}) - 1\quad {\rm{for every}} n \in .$$

As a consequence of the martingale convergence theorem, the sequence (ξⁿ)_n∊ℕ converges a.s. to the random variable

$$\xi : = 2{\mathbb P} (\Delta {X_T} > 0|{{\cal F}_{T - }}) - 1 = {{\bf{1}}_{\{ \Delta {X_T} > 0\} }} - {{\bf{1}}_{\{ \Delta {X_T} \le 0\} }},$$

where we have used the fact that ${{\bf{1}}_{\{ \Delta {X_T} > 0\} }}$ is ℱ _T−-measurable. This shows that (hⁿ)_n∊ℕ is a flash strategy in the sense of Definition 2.2. To prove that it generates a sure profit, it suffices to remark that, for every t ≥ 0, it holds that $\mathop {\lim }\nolimits_{n \to + \infty } {X_{{\tau _n} \wedge t}} = {X_{T \wedge t}}$ and

$$\mathop {\lim }\limits_{n \to + \infty } {X_{{\sigma _n} \wedge t}} = {X_{T - }}{{\bf{1}}_{\{ T \le t\} }} + {X_t}{{\bf{1}}_{\{ T > t\} }},$$

so that

$$\mathop {\lim }\limits_{n \to + \infty } {({h^n} \cdot X)_t} = \mathop {\lim }\limits_{n \to + \infty } ({\xi ^n}({X_{{\tau _n} \wedge t}} - {X_{{\sigma _n} \wedge t}})) = \xi \Delta {X_T}{{\bf{1}}_{\{ T \le t\} }} = |\Delta {X_T}|{{\bf{1}}_{\{ T \le t\} }}\quad {\rm{a}}{\rm{.s}}.,$$

thus showing that (hⁿ)_n∊ℕ generates a sure profit.

We now turn to the converse implication. Let (hⁿ)_n∊ℕ be a flash strategy, composed of elements of the form ${h^n} = {\xi ^n}{{\bf{1}}_{]]{\sigma _n},{\tau _n}]]}}$, generating a sure profit with respect to a random variable ζ and a stopping time τ with {τ < + ∞} ⊆ {ζ > 0}. It can be checked that lim_k→+∞ (hⁿ · X)_t−1/k = (hⁿ · X)_t− uniformly over n ∈ ℕ. Indeed, defining $\bar \xi : = \mathop {\sup }\nolimits_{n \in {\mathbb N}} |{\xi ^n}|$ (which is a bounded random variable due to Definition 2.2), it holds that

$$\matrix{ {\mathop {\lim }\limits_{k \to + \infty } \mathop {\sup }\limits_{n \in {\mathbb N}} |({h^n} \cdot X{)_{t - 1/k}} - {{({h^n} \cdot X)}_{t - }}| = \mathop {\lim }\limits_{k \to + \infty } \mathop {\sup }\limits_{n \in {\mathbb N}} |{\xi ^n}(X_{t - 1/k}^{{\tau _n}} - X_{t - 1/k}^{{\sigma _n}}) - {\xi ^n}(X_{t - }^{{\tau _n}} - X_{t - }^{{\sigma _n}})|} \hfill \cr {\quad \quad \le \bar \xi \mathop {\lim }\limits_{k \to + \infty } (\mathop {\sup }\limits_{n \in {\mathbb N}} {{\bf{1}}_{\{ t - 1/k \lt {\tau _n} \lt t\} }}|{X_{{\tau _n}}} - {X_{t - 1/k}}| + \mathop {\sup }\limits_{n \in {\mathbb N}} {{\bf{1}}_{\{ t - 1/k \lt {\sigma _n} \lt t\} }}|{X_{{\sigma _n}}} - {X_{t - 1/k}}|)} \hfill \cr {\quad \quad \le 2 \bar \xi \mathop {\lim }\limits_{k \to + \infty } \mathop {\sup }\limits_{u \in (t - 1/k,t)} |{X_u} - {X_{t - }}| = 0.} \hfill \cr } $$

Hence, by the Moore–Osgood theorem, we can conclude that, for every t ≥ 0,

(2.2) $$\zeta {{\bf{1}}_{\{ t = \tau \} }} = \mathop {\lim }\limits_{n \to + \infty } {({h^n} \cdot X)_t} - \mathop {\lim }\limits_{k \to + \infty } \mathop {\lim }\limits_{n \to + \infty } {({h^n} \cdot X)_{t - 1/k}} = \Delta {X_t} {\bar h_t}\quad {\rm{a}}{\rm{.s}}.,$$

with ${\bar h_t} : = \mathop {\lim }\nolimits_{n \to + \infty } h_t^n = \mathop {\lim }\nolimits_{n \to + \infty } {\xi ^n}{{\bf{1}}_{\{ {\sigma _n} \lt t \le {\tau _n}\} }}$, for all t ≥ 0. Letting ξ = lim_n→+∞ ξⁿ (see Definition 2.2), a first implication of (2.2) is that {τ < + ∞} ⊆ {ξ ≠ 0} and [[τ ]] ⊆{ΔX ≠ 0},up to an evanescent set, so that τ is a jump time of X. Furthermore, always by (2.2), on {τ < + ∞} it holds that $\{ \Delta {X_\tau } > 0\} = \{ {\bar h_\tau } > 0\} $. Noting that the random variables ${\xi ^n}{{\bf{1}}_{\{ {\sigma _n} \lt \tau \} }}$ and ${{\bf{1}}_{\{ \tau \le {\tau _n}\} }}$ are ℱ _τ−-measurable for every n ∈ ℕ (see e.g. [Reference Jacod and Shiryaev20, § I.1.17]), this implies that ${{\bf{1}}_{\{ \tau \lt + \infty , \Delta {X_\tau } > 0\} }}$ is ℱ _τ−-measurable as well. To complete the proof, it remains to show that τ is a predictable time. For each n ∈ ℕ, let A_n ≔ {σ_n < τ ≤ τ_n} ∩ {ξⁿ ≠ 0} and note that A_n ⊆ {τ < + ∞}, since each stopping time τ_n is bounded. Moreover, it holds that

$$\mathop {\lim }\limits_{n \to + \infty } {{\bf{1}}_{{A_n}}} = \mathop {\lim }\limits_{n \to + \infty } {{\bf{1}}_{\{ {\sigma _n} \lt \tau \le {\tau _n}\} }}{\xi ^n}{{{{\bf{1}}_{\{ {\xi ^n} \ne 0\} }}} \over {{\xi ^n}}} = {\zeta \over {\xi \Delta {X_\tau }}}{{\bf{1}}_{\{ \tau \lt + \infty \} }}\quad {\rm{a}}{\rm{.s}}{\rm{.}}$$

This identity shows that the sequence (A_n)_n∈ℕ is convergent, with lim_n→+∞ A_n = {τ < + ∞} and ξ ΔX _τ = ζ on {τ < + ∞} (up to a ℙ-nullset). Since the stopping times (σ_n)_n∈ℕ and (τ_n)_n∈ℕ converge a.s. to τ for n → +∞, this implies that

$$[[\tau ]] \subseteq \mathop {\liminf}\limits_{n \to + \infty } ]]{\sigma _n},{\tau _n}]] \subseteq \mathop {\limsup }\limits_{n \to + \infty } ]]{\sigma _n},{\tau _n}]] \subseteq [[\tau ]],,$$

so that [[τ ]] = lim_n→+∞ ]]σ_n, τ_n]]. Since each stochastic interval ]]σ_n, τ_n]] is a predictable set (see e.g. [Reference Jacod and Shiryaev20, Proposition I.2.5]), it follows that [[τ]] is also a predictable set, i.e. τ is a predictable time.

Let us now prove that X exhibits fully predictable jumps if and only if there exists a flash strategy generating a constant profit, following a line of reasoning similar to that in the first part of the proof. Let T be a predictable time with [[T]] ⊆ {ΔX ≠ 0} such that the random variable ΔX_T 1_{T<+∞} is ℱ _T−-measurable. Fix some constant k ≥ 1 such that ℙ(T < +∞, |ΔX_T| ∈ [1/k, k]) > 0 and define the stopping time $\tau : = T{{\bf{1}}_{\{ |\Delta {X_T}| \in [1/k,k]\} }} + \infty {{\bf{1}}_{\{ |\Delta {X_T}| \notin [1/k,k]\} }}$. By [Reference Jacod and Shiryaev20, Proposition I.2.10], τ is a predictable time and, therefore, there exists an announcing sequence (ρ_n)_n∈ℕ of stopping times satisfying ρ _n < τ and such that ρ_n increases to τ for n → +∞. As in the first part of the proof, let σ_n ≔ ρ_n ∧ n and τ_n ≔ τ ∧ n, for each n ∈ ℕ, and define the sequence (hⁿ)_n∈ℕ by

(2.3) $${h^n} = {\xi ^n}{{\bf{1}}_{]]{\sigma _n},{\tau _n}]]}},\quad {\rm{where}}{\xi ^n} : = k{{|{\mathbb E}[\Delta {X_\tau }|{{\cal F}_{{\sigma _n}}}]| \wedge 1/k} \over {{\mathbb E}[\Delta {X_\tau }|{{\cal F}_{{\sigma _n}}}]}}\quad {\rm{for every}} n \in ,$$

with the conventions ΔX_τ = 0 on {τ = +∞} and ${\textstyle{0 \over 0}} = 0$. By construction, it holds that |ξⁿ| ≤ k, for every n ∈ ℕ, so that (hⁿ)_n∈ℕ is well-defined as a sequence of buy-and-hold strategies. Moreover, the sequence (ξⁿ)_n∈ℕ converges a.s. to the random variable

$$\xi : = k{{|{\mathbb E}[\Delta {X_\tau }|{{\cal F}_{\tau - }}]| \wedge 1/k} \over {{\mathbb E}[\Delta {X_\tau }|{{\cal F}_{\tau - }}]}} = k{{|\Delta {X_\tau }| \wedge 1/k} \over {\Delta {X_\tau }}} = {{{{\bf{1}}_{\{ \Delta {X_\tau } \ne 0\} }}} \over {\Delta {X_\tau }}},$$

where the second equality makes use of the fact that ΔX_τ is ℱ _τ−-measurable, as follows from the identity $\Delta {X_\tau } = \Delta {X_T}{{\bf{1}}_{\{ |\Delta {X_T}| \in [1/k,k]\} }}$ together with the ℱ _T−-measurability of ΔX_T 1_{T<+∞}. This shows that (hⁿ)_n∈ℕ is a flash strategy in the sense of Definition 2.2. Moreover, it holds that lim

$$\mathop {\lim }\limits_{n \to + \infty } {({h^n} \cdot X)_t} = \mathop {\lim }\limits_{n \to + \infty } ({\xi ^n}({X_{{\tau _n} \wedge t}} - {X_{{\sigma _n} \wedge t}})) = \xi \Delta {X_\tau }{{\bf{1}}_{\{ \tau \le t\} }} = {{\bf{1}}_{\{ \tau \le t\} }}\quad {\rm{a}}{\rm{.s}}.,$$

thus showing that (hⁿ)_n∈ℕ generates a constant profit with respect to c = 1.

Conversely, let (hⁿ)_n∈ℕ be a flash strategy, with ${h^n} = {\xi ^n}{{\bf{1}}_{]]{\sigma _n},{\tau _n}]]}}$, n ∈ N, generating a constant profit with respect to c > 0 and a stopping time τ. As in the case of a sure profit, it holds that $c = \Delta {X_\tau }\mathop {\lim }\nolimits_{n \to + \infty } h_\tau ^n$ a.s. on {τ < +∞}. This implies that [[τ ]] ⊆ {X ≠ 0} up to an evanescent set, so that τ is a jump time of X. Moreover, it holds that $\Delta {X_\tau } = c/(\mathop {\lim }\nolimits_{n \to + \infty } {\xi ^n}{{\bf{1}}_{\{ {\sigma _n} \lt \tau \le {\tau _n}\} }})$ a.s. on {τ < +∞}, from which the ℱ _τ−-measurability of ΔX _τ1_{τ<+∞} follows. Finally, the predictability of τ can be shown exactly as above in the case of a sure profit.

Remark 2.2. Examples of models allowing for constant profits via flash strategies are given by the escrowed dividend models introduced in [Reference Geske14], [Reference Roll35], and [Reference Whaley37] (see also the analysis in [Reference Heath and Jarrow17]). Indeed, such models consider an asset paying a deterministic dividend at a known date and assume that the ex-dividend price drops by a fixed fraction δ ∈ (0, 1) of the dividend at the dividend payment date. This corresponds to a fully predictable jump of the process X and, hence, in view of Theorem 2.1, can be exploited to generate a constant profit via a flash strategy.

It is important to remark that, although a flash strategy (hⁿ)_n∈ℕ generating a sure profit does not involve any risk in the limit, each individual buy-and-hold strategy hⁿ carries the risk of potential losses. However, a flash strategy can be constructed in such a way that losses are uniformly bounded, as we are going to show in the remaining part of this section. This is an important property of flash strategies, especially in view of their practical applicability.

By Theorem 2.1, there are sure profits via flash strategies if and only if X exhibits predictable jumps. Thus, let T be a predictable time with [[T]] ⊆ {ΔX ≠ 0} such that ${{\bf{1}}_{\{ T \lt + \infty , \Delta {X_T} > 0\} }}$ is ℱ _T−-measurable. Consider the event A(N, C) ≔ {T ≤ N, |X _T−| ≤ C, ΔX_T > 0} ∈ ℱ _T−,for some constants N > 0 and C ≥ 1 such that ℙ(A(N, C)) > 0, and define the predictable time $\tau : = T{{\bf{1}}_{A(N,C)}} + \infty {{\bf{1}}_{A{{(N,C)}^c}}}$. Then define the sequences of stopping times (σ_n)_n∈ℕ and (τ_n)_n∈ℕ by σ_n ≔ ρ_n ∧ n and τ_n ≔ τ ∧ n, for each n ∈ ℕ, where (ρ_n)_n∈ℕ is an announcing sequence for τ. As in (2.1), we construct the buy-and-hold strategy ${h^n} = {\xi ^n}{{\bf{1}}_{]]{\sigma _n},{\tau _n}]]}}$, with

(2.4) $${\xi ^n} : = {{{\mathbb P}(\tau \lt + \infty |{{\cal F}_{{\sigma _n}}})} \over {1 + {{(|{X_{{\sigma _n}}}| - C)}^ + }}}\quad {\rm{for every}} n \in {\mathbb N}.$$

Since ${X_{{\sigma _n}}} \to {X_{\tau - }}$ a.s. for n → + ∞ and |X _τ−| ≤ C on {τ < +∞}, the sequence (ξⁿ)_n∈ℙ converges a.s. to ξ = 1_{τ<+∞}. The same arguments given in the first part of the proof of Theorem 2.1 then allow us to show that (hⁿ)_n∈ℕ generates a sure profit at τ. Furthermore, on {τ < +∞}, for every n ∈ ℕ such that n ≥ N, we have

$${({h^n} \cdot X)_\tau } = {\xi ^n}({X_\tau } - {X_{{\rho _n}}}) = {\xi ^n}(\Delta {X_\tau } + {X_{\tau - }} - {X_{{\rho _n}}}) \ge - C - {{|{X_{{\rho _n}}}|} \over {1 + {{(|{X_{{\rho _n}}}| - C)}^ + }}} \ge - 2C\quad {\rm{a}}{\rm{.s}}{\rm{.}}$$

We have thus shown that, even if each individual buy-and-hold strategy hⁿ does involve some risk, the potential losses from trading are uniformly bounded on {τ < +∞} for all sufficiently large n. An analogous result can be shown to hold true in the case of flash strategies generating constant profits, modifying the strategy (2.3) in analogy to (2.4).

3.1. Right-continuity and sure profits

As explained in Section 2.1, the process X is allowed to be fully general, up to the mild requirement of path regularity, in the sense of right-continuity and existence of limits from the left. One might wonder whether right-continuity can be relaxed, assuming only that X has làdlàg paths (i.e. with finite limits from the left and from the right). As shown below, this is unfeasible, because right-continuity represents an indispensable requirement for any arbitrage-free price process. For a làdlàg process X = (X_t)_t≥0, we denote by X_{t +} the right-hand limit at t and Δ⁺X_t ≔ X_{t +} − X_t, for t ≥ 0.

Proposition 3.1. Assume that the process X is làdlàg. If X fails to be right-continuous, then there exists a flash strategy (hⁿ)_n∈ℕ such that (hⁿ · X)_t converges a.s. to 1_{τ<t}, for all t ≥ 0, for some stopping time τ with ℙ(τ < +∞) > 0. Conversely, if there exists such a flash strategy, then X cannot be right-continuous.

Proof. The argument is similar to the second part of the proof of Theorem 2.1. Suppose that there exists a stopping time T such that [[T]] ⊆ {Δ⁺X_T ≠ 0}. Fix a constant k such that ℙ(T < +∞, |Δ⁺X_T| ∈ [1/k, k]) > 0 and define $\tau : = T{{\bf{1}}_{\{ |{\Delta ^ + }{X_T}| \in [1/k,k]\} }} + \infty {{\bf{1}}_{\{ |{\Delta ^ + }{X_T}| \notin [1/k,k]\} }},$, setting Δ⁺X_T = 0 on {T = +∞}. Since the filtration 𝔽 is right-continuous, τ is a stopping time. Let us define the sequences of bounded stopping times (σ_n)_n∈ℕ and (τ_n)_n∈ℕ by

$${\sigma _n} : = \tau \wedge n\quad {\rm{and}}\quad {\tau _n} : = (\tau + {n^{ - 1}}) \wedge n\quad {\rm{for each}} n \in .$$

It holds that ∨_n∈ℕℱ_σn = ℱ_τ. Indeed, for any A ∈ ℱ_τ, define the sets

$${A_1} : = A \cap \{ \tau = {\sigma _1}\} ,\quad {A_n} : = \bigcap\limits_{j \lt n} (A \cap \{ \tau = {\sigma _n}\} \cap \{ \tau > j\} )\quad {\rm{and}}\quad {A_\infty } : = A \cap \{ \tau = + \infty \} .$$

It can be checked that ${A_\infty } \in {{\cal F}_{\tau - }} = \mathop \vee \nolimits_{n \in {\mathbb N}} {{\cal F}_{{\sigma _n} - }} \subseteq \mathop \vee \nolimits_{n \in {\mathbb N}} {{\cal F}_{{\sigma _n}}}$ and ${A_n} \in {{\cal F}_{{\sigma _n}}}$, for each n ∈ ℕ. Since $A = {A_\infty }\bigcup\nolimits_ ( \cup _{n = 1}^{ + \infty }{A_n})$, this shows that ${{\cal F}_\tau } \subseteq \mathop \vee \nolimits_{n \in {\mathbb N}} {{\cal F}_{{\sigma _n}}}$. On the contrary, since σ_n ≤ τ, for every n ∈ ℕ, the inclusion $\mathop \vee \nolimits_{n \in {\mathbb N}} {{\cal F}_{{\sigma _n}}} \subseteq {{\cal F}_\tau }$ is obvious. Now define the sequence of buy- and-hold strategies ${h^n} : = {\xi ^n}{{\bf{1}}_{]]{\sigma _n}{\tau _n}]]}}$, where

$${\xi ^n} : = k{{|{\mathbb E}[{\Delta ^ + }{X_\tau }|{{\cal F}_{{\sigma _n}}}]| \wedge 1/k} \over {{\mathbb E}[{\Delta ^ + }{X_\tau }|{{\cal F}_{{\sigma _n}}}]}}\quad {\rm{for every}} n \in {\mathbb N}.$$

By the martingale convergence theorem, the random variables (ξ_n)_n∈ℕ converge a.s. to

$$\xi : = k{{|{\mathbb E}[{\Delta ^ + }{X_\tau }|\mathop \vee \limits_{n \in {\mathbb N}} {{\cal F}_{{\sigma _n}}}]| \wedge 1/k} \over {{\mathbb E}[{\Delta ^ + }{X_\tau }|\mathop \vee \limits_{n \in {\mathbb N}} {{\cal F}_{{\sigma _n}}}]}} = k{{|{\mathbb E}[{\Delta ^ + }{X_\tau }|{{\cal F}_\tau }]| \wedge 1/k} \over {{\mathbb E}[{\Delta ^ + }{X_\tau }|{{\cal F}_\tau }]}} = {{{{\bf{1}}_{\{ {\Delta ^ + }{X_\tau } \ne 0\} }}} \over {{\Delta ^ + }{X_\tau }}},$$

where we have used the right-continuity of the filtration 𝔽. Observe that (hⁿ)_n∈ℕ is a flash strategy in the sense of Definition 2.2. Moreover, for every t ≥ 0, it holds that $\mathop {\lim }\nolimits_{n \to + \infty } {X_{{\sigma _n} \wedge t}} = {X_{\tau \wedge t}}$ and

$$\mathop {\lim }\limits_{n \to + \infty } {X_{{\tau _n} \wedge t}} = {X_{\tau + }}{{\bf{1}}_{\{ \tau \lt t\} }} + {X_t}{{\bf{1}}_{\{ \tau \ge t\} }},$$

so that

$$\mathop {\lim }\limits_{n \to + \infty } {({h^n} \cdot X)_t} = \mathop {\lim }\limits_{n \to + \infty } ({\xi ^n}({X_{{\tau _n} \wedge t}} - {X_{{\sigma _n} \wedge t}})) = \xi {\Delta ^ + }{X_\tau }{{\bf{1}}_{\{ \tau \lt t\} }} = {{\bf{1}}_{\{ \tau \lt t\} }}\quad {\rm{a}}{\rm{.s}}.,$$

thus proving the first part of the proposition.

To prove the converse implication, let (hⁿ)_n∈ℕ be a flash strategy such that (hⁿ · X)_t → 1_{τ<t} a.s. as n → +∞, for all t ≥ 0, for some stopping time τ with ℙ(τ < +∞) > 0. Then, a straightforward adaptation of the arguments given in the last part of the proof of Theorem 2.1 allows us to show that Δ⁺X_τ ≠ 0 a.s. on {τ < +∞}, thus proving the claim.

Proposition 3.1 shows that the failure of right-continuity leads to a constant profit from a flash strategy that can be realized at any time at which the price process jumps from the right. This result depends crucially on the right-continuity of the filtration ℱ, which implies that Δ⁺X_t is known at time t, immediately before the occurrence of the jump. Therefore, by trading sufficiently fast and liquidating the position immediately after the jump from the right, a trader can take advantage of this information and realize a constant profit. In this sense, right-continuity is an essential requirement for any arbitrage-free price process.

3.2. Behaviour of profits from flash strategies under transaction costs

In practice, transaction costs and market frictions can significantly affect the feasibility of trading strategies, thus limiting the profitability of arbitrage strategies. In this section, we study the behaviour of sure and constant profits via flash strategies with respect to small transaction costs. To this effect, let us formulate the following definition (see [Reference Guasoni and RáSONY15]).

Definition 3.1. For ɛ > 0, two strictly positive processes $\tilde X = {({\tilde X_t})_{t \ge 0}}$ are said to be ɛ-close if

$${1 \over {1 + \varepsilon }} \le {{{\tilde X_t}} \over {{X_t}}} \le 1 + \varepsilon \quad {\rm{a}}{\rm{.s}}{\rm{.}}\;{\rm{for all}}t \ge 0.$$

This definition corresponds to considering proportional transaction costs, with a bid (selling) price equal to X_t/(1 + ɛ) and an ask (buying) price equal to X_t(1 + ɛ). The definition also embeds the possibility of model mis-specifications, in the sense that the model price process X corresponds to some true price process $\tilde X$ up to a model error of magnitude ɛ.

In this context, assuming a strictly positive price process X, we shall say that sure/constant profits via flash strategies are robust if they persist for every process $\tilde X$ which is ɛ-close to X,for sufficiently small ɛ> 0. This robustness property is made precise by the following proposition.

Proposition 3.2. Assume that the process X is strictly positive and admits constant profits via flash strategies. Then there exists a flash strategy (hⁿ)_n∈ℕ and a predictable time τ such that, for every strictly positive process $\tilde X$ which is ɛ-close to X, it holds that

$$\mathop {\lim }\limits_{n \to + \infty } {({h^n} \cdot \tilde X)_t} \ge \bar c {{\bf{1}}_{\{ \tau \le t\} }}\quad {{a}}{{.s}}{{.}}\;{{for all}} t \ge 0,$$

with $\bar c > 0$, for sufficiently small ɛ > 0.

Proof. Suppose that X admits a flash strategy ${({\hat h^n})_{n \in {\mathbb N}}}$ which generates a constant profit with respect to a stopping time $\hat \tau $. By Theorem 2.1, $\hat \tau $ is a predictable time and X exhibits a fully predictable jump at $\hat \tau $. Let N > 0 be a constant such that ${\mathbb P}(\hat \tau < + \infty ,|{X_{\hat \tau - }}| \le N) > 0$ and define the predictable time $T : = \hat \tau {{\bf{1}}_{\{ \hat \tau \lt + \infty ,|{X_{\hat \tau - }}| \le N\} }} + \infty {{\bf{1}}_{\{ \hat \tau = + \infty \} \cup \{ \hat \tau \lt + \infty ,|{X_{\hat \tau - }}| > N\} }}$. Clearly, X still exhibits a fully predictable jump at T. Hence, in view of Theorem 2.1, there exists a flash strategy (hⁿ)_n∈ℕ, composed of elements ${h^n} = {\xi ^n}{{\bf{1}}_{]]{\sigma _n},{\tau _n}]]}}$, n ∈ ℕ, with |ξⁿ| ≤ k a.s. for all n ∈ ℕ, for some constant k > 0, which generates a constant profit c > 0 with respect to a predictable time τ with [[τ ]] ⊆ [[T]] and ℙ(τ < +∞) > 0. Let ɛ > 0 and consider a strictly positive process $\tilde X$ which is ɛ-close to X. As in [Reference Chau and Tankov5, Section 2], we can compute, for all n ∈ ℕ and t ≥ 0:

(3.1) $$\matrix{ {{{({h^n} \cdot \tilde X)}_t}} \hfill & = \hfill & {{\xi ^n}({{\tilde X}_{{\tau _n} \wedge t}} - {{\tilde X}_{{\sigma _n} \wedge t}})} \hfill \cr {} \hfill & \ge \hfill & {{\xi ^n}{{\bf{1}}_{\{ {\xi ^n} \ge 0\} }}({{{X_{{\tau _n} \wedge t}}} \over {1 + \varepsilon }} - (1 + \varepsilon ){X_{{\sigma _n} \wedge t}}) + {\xi ^n}{{\bf{1}}_{\{ {\xi ^n} \lt 0\} }}((1 + \varepsilon ){X_{{\tau _n} \wedge t}} - {{{X_{{\sigma _n} \wedge t}}} \over {1 + \varepsilon }})} \hfill \cr {} \hfill & = \hfill & {{{\bf{1}}_{\{ {\xi ^n} \ge 0\} }}{{{{({h^n} \cdot X)}_t}} \over {1 + \varepsilon }} + {{\bf{1}}_{\{ {\xi ^n} \lt 0\} }}(1 + \varepsilon )({h^n} \cdot X{)_t} - |{\xi ^n}|\varepsilon {{2 + \varepsilon } \over {1 + \varepsilon }}{X_{{\sigma _n} \wedge t}}} \hfill \cr {} \hfill & \ge \hfill & {{{\bf{1}}_{\{ {\xi ^n} \ge 0\} }}{{{{({h^n} \cdot X)}_t}} \over {1 + \varepsilon }} + {{\bf{1}}_{\{ {\xi ^n} \lt 0\} }}(1 + \varepsilon )({h^n} \cdot X{)_t} - 2 \varepsilon k{X_{{\sigma _n} \wedge t}}.} \hfill \cr } $$

As shown in the first part of the proof of Theorem 2.1, it holds that ${X_{{\sigma _n}}} \to {X_{\tau - }}$ a.s. on {τ < +∞} for n → +∞. Hence, using Definition 2.2 and taking the limit for n → +∞ in (3.1) yields

$$\matrix{ {\mathop {\lim }\limits_{n \to + \infty } {{({h^n} \cdot \tilde X)}_t}} \hfill & \ge \hfill & {{{\bf{1}}_{\{ \xi \ge 0\} }}{c \over {1 + \varepsilon }} + {{\bf{1}}_{\{ \xi \lt 0\} }}c(1 + \varepsilon ) - 2 \varepsilon k{X_{\tau - }}} \hfill \cr {} \hfill & \ge \hfill & {{c \over {1 + \varepsilon }} - 2 \varepsilon kN} \hfill \cr {} \hfill & { = :} \hfill & {\bar c\quad {\rm{a}}{\rm{.s}}{\rm{.}}\;{\rm{on}}\{ \tau \le t\} .} \hfill \cr } $$

For sufficiently small ɛ, it holds that $\bar c > 0$. Furthermore, it can be easily verified that ${({h^n} \cdot \tilde X)_t} \to 0$ a.s. on {τ > t} for n → +∞, thus proving the claim.

The last proposition shows that the presence of fully predictable jumps represents a violation to the absence of arbitrage principle which persists under small transaction costs. This result is in line with the empirical evidence reported in [Reference Tedeschini36] (compare with Remark 2.1). We also want to point out that the same reasoning applies to Proposition 3.1, thus implying that the necessity of right-continuity is robust with respect to small transaction costs. Furthermore, an argument similar to that given in the proof of Proposition 3.2 allows to show that constant profits via flash strategies are robust with respect to small fixed (instead of proportional) transaction costs.

3.3. The semimartingale case

Theorem 2.1 holds true for any càdlàg adapted process X. If in addition X is assumed to be a semimartingale, then the absence of (fully) predictable jumps admits a further simple characterization. For a semimartingale X, we say that a bounded predictable process h is an instantaneous strategy if it is of the form h = ξ 1_[[τ]], for some bounded random variable ξ and a stopping time τ. In the spirit of Definition 2.2, we say that an instantaneous strategy h generates sure profits if (h · X)_t = ζ 1_{τ≤t} a.s. for every t ≥ 0, for some random variable ζ such that {τ < +∞} ⊆ {ζ > 0}. If ℙ(ζ = c) = 1, for some constant c > 0, then the instantaneous strategy h is said to generate a constant profit.

In the proof of Corollary 3.1, the semimartingale property is used to ensure that the gains from trading generated by a sequence of buy-and-hold strategies forming a flash strategy converge to the gains from trading generated by an instantaneous strategy.

In the semimartingale case, under the additional assumption of quasi-left-continuity of the filtration 𝔽 (i.e. ℱ_T = ℱ _T− for every predictable time T; see [Reference Protter32, Section IV.3]), it has been shown in [Reference Huang19] that predictable jumps cannot occur if the financial market is viable, in the sense that there exists an optimal consumption plan for some agent with sufficiently regular preferences. In our context, this result is a direct consequence of Corollary 3.1, as the existence of sure profits is clearly incompatible with any form of market viability, regardless of the quasi-left-continuity of ℱ.

3.4. Comparison with other no-arbitrage conditions

The absence of sure profits from instantaneous strategies must be regarded as a minimal no-arbitrage condition. In particular, in the semimartingale case, it is implied by the requirement of no increasing profit (NIP), itself an extremely weak no-arbitrage condition for a financial model (see [Reference Fontana11] and [Reference Karatzas and Kardaras22, Section 3.4]).^{Footnote 4}

The absence of predictable jumps can be directly proved by martingale methods under the classical no free lunch with vanishing risk (NFLVR) condition. Note, however, that NFLVR is much stronger than the absence of sure profits as considered above. We recall that NFLVR is equivalent to the existence of a probability measure ℚ ∼ ℙ such that X is a sigma-martingale under ℚ (see [Reference Delbaen and Schachermayer6]). For completeness, we present the following proposition with its simple proof.

Proposition 3.3. Assume that the process X is a semimartingale satisfying NFLVR. Then X cannot exhibit predictable jumps.

Proof. If X satisfies NFLVR, then there exists ℚ ∼ ℙ and an increasing sequence of predictable sets (Σ_k)_k∈ℕ with ∪_k∈ℕ Σ_k = Ω × ℝ₊ such that ${{\bf{1}}_{{\Sigma _k}}} \cdot X$ is a uniformly integrable martingale under ℚ, for every k ∈ ℕ (see [Reference Jacod and Shiryaev20, Definition III.6.33]). Let T be a predictable time such that [[T]] ⊆{ X = 0} and ${{\bf{1}}_{\{ T \lt + \infty , \Delta {X_T} > 0\} }}$ is ℱ _T−-measurable. With the convention ΔX_T = 0 on {T = +∞}, we define the predictable times ${\tau ^1} : = T{{\bf{1}}_{\{ \Delta {X_T} > 0\} }} + \infty {{\bf{1}}_{\{ \Delta {X_T} \le 0\} }}$ and ${\tau ^2} : = T{{\bf{1}}_{\{ \Delta {X_T} \lt 0\} }} + \infty {{\bf{1}}_{\{ \Delta {X_T} \ge 0\} }}$. Hence,

$$\matrix{ {{Y^{(k)}}} \hfill & { : = } \hfill & {{{\bf{1}}_{{\Sigma _k}}} \cdot (|\Delta {X_T}|{{\bf{1}}_{[[T, + \infty [[}})} \hfill \cr {} \hfill & = \hfill & {{{\bf{1}}_{{\Sigma _k}}} \cdot (\Delta {X_{{\tau ^1}}}{{\bf{1}}_{[[{\tau ^1}, + \infty [[}} - \Delta {X_{{\tau ^2}}}{{\bf{1}}_{[[{\tau ^2}, + \infty [[}})} \hfill \cr {} \hfill & = \hfill & {{{\bf{1}}_{{\Sigma _k}}} \cdot (({{\bf{1}}_{[[{\tau ^1}]]}} - {{\bf{1}}_{[[{\tau ^2}]]}}) \cdot X)} \hfill \cr {} \hfill & = \hfill & {({{\bf{1}}_{[[{\tau ^1}]]}} - {{\bf{1}}_{[[{\tau ^2}]]}}) \cdot ({{\bf{1}}_{{\Sigma _k}}} \cdot X),} \hfill \cr } $$

for every k ∈ ℕ, where we have used [Reference Jacod and Shiryaev20, § I.4.38] and the associativity of the stochastic integral. Therefore, the process Y ^(k) is a nondecreasing local martingale. Since $Y_0^{(k)} = 0$, this implies that Y ^(k) ≡ 0 (up to an evanescent set), for all k ∈ N. In turn, this implies that |ΔX_T|= 0 a.s., contradicting the assumption that T is a jump time of X.