When the Market Loses its Randomness: How Price Quirks Create Infinite Profit Opportunities

Imagine a market where an asset's price sometimes gets «stuck» at certain points, as if pondering where to move next. Or a market where the price, upon reaching a certain level, bounces off it like a ball hitting a wall. Or even a market where, in certain zones, the price suddenly stops being random and starts moving predictably, like a wind-up toy.

Sounds strange? For mathematicians working with abstract models, these are strictly real scenarios. And surprisingly, in such «strange» markets, something arises that financiers call by the beautiful term «increasing profits», but which is essentially a nightmare for any theory of fair pricing – the ability to make money literally out of thin air, and to do so infinitely.

When Money Grows Out of Nowhere

Let's start with a simple question: what is arbitrage? In the classical sense, it is the opportunity to make a profit without risk and without investment. Buy cheaper in Lyon, sell dearer in Paris, put the difference in your pocket. Real markets usually «heal» such opportunities quickly – too many smart people are hunting for them.

But researchers from the world of financial mathematics have discovered something far stranger. They studied models where prices do not behave like ordinary Brownian motion (the classical random walk model), but as «general diffusions» – processes with trajectories of whimsical geometry. And in these models, they found increasing profits – trading strategies that work like a one-way ladder: your capital only grows, never falls, and with real probability, you end up with something positive.

Sounds like a financial «perpetual motion machine», doesn't it? And it really does look like a violation of the economic laws of nature. Think about it: if your portfolio monotonically increases, you are risking absolutely nothing. Yet, you retain the chance to earn. This isn't just arbitrage – it's arbitrage on steroids.

Anatomy of a Strange Market

To understand where these opportunities come from, we need to visualize how the model is built. Let's take a market with one risky asset (let's call it Y) and a bank account where capital grows at a constant interest rate $r$. So far, everything seems normal.

But asset Y is not an ordinary stock. It is a «general diffusion», a mathematical object described by two parameters: the scale function $s$ and the speed measure $m$. Don't let the terms scare you – essentially, these parameters determine exactly how the price moves. The scale function shows how the price space stretches or compresses, while the speed measure dictates how much time the process spends at different points.

Thanks to these parameters, the model can mimic amazing phenomena:

• Reflecting boundaries – the price cannot fall below a certain level and bounces off it
• Stickiness – the price gets «stuck» at certain points for a random amount of time
• Asymmetry – the process behaves differently when moving up versus down
• Slowdown on fractal sets – in certain zones, time seems to flow differently

And all this is not merely mathematical fantasy. Similar effects are observed in reality: central banks protect exchange rates by creating a «floor» or «ceiling»; some assets indeed demonstrate volatility asymmetry; portfolio protection mechanisms form points of price «attraction».

The Treasure Map: Measure $\nu$ and Strategy $\theta$

Now for the most interesting part. Researchers have created a complete «map» of where arbitrage opportunities hide in such models. This map is called measure $\nu$ (the Greek letter «Nu»), and it is fully determined by the model's parameters – the scale function, the speed measure, and the interest rate.

Measure $\nu$ is like an X-ray of the market, revealing all its «anomalies». It consists of several components corresponding to specific price behavior features:

• places where the derivative of the inverse scale function vanishes (where price loses randomness)
• «sticky» points where the process slows down
• contribution from reflecting or absorbing boundaries
• effects of asymmetry and other peculiarities

Why is this important? Because it is proven: arbitrage exists if and only if measure $\nu$ is not zero. If $\nu$ is trivial, the market is honest, and there are no increasing profits. If $\nu$ has mass at certain points – that is exactly where arbitrage is lurking.

But the researchers went further. They didn't just find the conditions for the existence of arbitrage – they constructed a canonical strategy $\theta$ (the Greek letter «Theta») that automatically «squeezes» all available arbitrage out of the model. This strategy works like this: it tracks measure $\nu$ and takes positions exactly at those points where $\nu$ indicates a profit opportunity.

The beauty is that if measure $\nu$ is non-zero, then the strategy $\theta$ itself is already an increasing profit. You don't even need to invent anything – $\theta$ will do all the work for you. It's like having a universal skeleton key to any door behind which money is hidden.

What Infinite Profit Looks Like

Let's break down the mechanics with concrete examples. Each of them shows how a specific feature of price behavior turns into an earning opportunity.

Example One: The Absorbing Boundary

Imagine that an asset's price can fall to zero and remains zero forever after that. This is called an «absorbing boundary». Now add an interest rate $r$ not equal to zero to the model. What happens?

When the price hits zero, the asset becomes absolutely predictable – it simply stays put. But the bank account continues to grow! This means the discounted price of the asset begins to monotonically fall or rise – depending on the sign of $r$. And here a trader can build a strategy: take a short or long position at the moment the boundary is reached and hold it. The portfolio value will monotonically increase, and there is no risk – because the price no longer changes! 🎯

Example Two: The Reflecting Boundary

Now imagine that the price cannot fall below a certain level and bounces off it. This is modeled by adding a special component to the process related to «local time» – a measure of how much time the process spends at the boundary.

If the coefficient before the local time is not zero (which depends on the interest rate and model parameters), then a deterministic drift arises at the boundary. A trader can construct a strategy that earns money only when the price is at the boundary, and does nothing the rest of the time. The beauty is that such a strategy takes no risks – it activates only at moments when the price behaves predictably.

Example Three: The Sticky Point

An even stranger case: the price sometimes gets «stuck» at a specific point for a random amount of time. Mathematically, this is described by a «speed measure» that has atoms at distinct points.

When the process hits such a point, time seems to slow down for it. The price stands still, while calendar time marches on. If an interest rate is active, the discounted price changes deterministically. A trader can exploit this: take a position when the price «sticks» and close it when it «unsticks». Profit is guaranteed, and risk is absent.

Example Four: Asymmetry (Skewness)

Skewed Brownian motion is a process that behaves differently when moving up and down from a certain point. For example, at zero, it may have different rebound speeds in the positive and negative directions.

This asymmetry creates an imbalance in local time, which a trader can exploit. The strategy will take different positions depending on which side of zero the price is on, earning from the very fact of the asymmetry.

Example Five: The Cantor Set

The most exotic case. Researchers constructed a model where the derivative of the scale function vanishes on a «fat» Cantor set – a fractal structure that has zero length but positive measure (yes, that happens!).

When the transformed price enters this set, it loses its martingale (random) component. Only the predictable part related to the interest rate remains. A trader can earn the entire time the price is in this zone. Profit accumulates not through local time at individual points, but through continuous «leaking» throughout the entire fractal region.

The Full Picture of Arbitrage

But the researchers didn't limit themselves to examples. They gave a complete description of all possible increasing profits in such models. It's as if you didn't just find a few secret passages in a game, but received a full map of all hidden possibilities.

It turned out that any increasing profit must satisfy three conditions:

1. Nullify the martingale part – the strategy must not rely on random price movements. It works only where randomness disappears.

2. Align with the sign of the canonical strategy $\theta$ – if measure $\nu$ is positive at some point, one must take a long position; if negative – a short one.

3. Yield positive profit with positive probability – otherwise, it's not arbitrage, but simply meaningless trading.

These conditions define the complete set of possible arbitrage strategies. Beyond this boundary, nothing works.

Connection to Deep Properties of Random Processes

Now – the most philosophical part. The study revealed a deep connection between the possibility of arbitrage and a fundamental property of stochastic processes called the Representation Property (RP).

RP means the following: any local martingale (a random process with specific properties) in a given model can be represented as an integral over the original price process. Sounds technical, but the meaning is simple: if RP holds, then all randomness in the model comes from one source – from the fluctuations of the asset price. There are no additional sources of uncertainty.

In ordinary models, where the price is described by a standard stochastic differential equation, RP always holds. But in general diffusions – not necessarily. And this is where the magic begins.

Researchers proved: the representation property is violated if and only if the set of points where the derivative of the inverse scale function is zero has positive measure.

Let me translate into human language: RP breaks down when zones appear in the model where the price loses its random component. And it is precisely in these zones that a special type of arbitrage arises – Quadratic Variation Increasing Profits (QVIP).

What does this mean? In ordinary models, all trading profit is related to local time – a measure of how often and how long the price visits certain levels. But in models with broken RP, profit can accumulate differently – through the measure of quadratic variation of the transformed process. That is, you earn not from visiting individual points, but from continuously «residing» in zones of deterministic movement.

And here is the main result: QVIP exist if and only if RP is violated and the interest rate is not zero.

The interest rate plays the role of a catalyst here. Without it, the discounted price remains a martingale even in zones where randomness is lost. But with a non-zero rate, a deterministic drift appears in these zones – and it is this drift that opens the door to arbitrage.

Why This Matters: Psychology and Reality

You might ask: why do we need all this? After all, real markets aren't described by such exotic models, right?

Not quite. Think about what hides behind these abstractions.

Reflecting boundaries arise when central banks protect exchange rates or when regulators impose limits on price movements. This is reality.

Sticky points can model psychological levels to which price is «attracted» due to a concentration of orders. Remember how price often «hangs» around round numbers – 100 euros, 1000 euros, 10,000 points. Traders know this and use it. 💭

Asymmetry reflects a well-known fact: markets fall faster than they rise. Fear is stronger than greed. This isn't just a saying – it is a statistically observable property of real prices.

Slowdown on fractal sets can model complex regimes where the market switches between different volatility states, creating self-similar patterns on different time scales.

Of course, real markets do not offer infinite opportunities for arbitrage – they are too efficient, liquid, and competitive. But the research shows an important point: price behavior quirks, even small ones, can create structural opportunities for extracting profit.

And here is the psychological moment: we are used to thinking that arbitrage is the exploitation of price gaps between markets or temporary inefficiencies. But the research shows otherwise: arbitrage can arise from the very geometry of randomness. From how price fluctuations are structured – where they slow down, where they accelerate, where they become predictable.

A Lesson for Practitioners and Dreamers

What can we take away from all this?

First, there are no perfect models. Every model is a compromise between realism and mathematical manageability. The classic Black-Scholes model is elegant but ignores many features of real prices. General diffusions allow for such features, but the price is the appearance of arbitrage opportunities that shouldn't exist in reality.

Second, details matter. It would seem, what's the big deal if the price sometimes «sticks» at a point or bounces off a boundary? But it is these trifles that create a structure that can be exploited. It's like in physics: friction seems insignificant, but without accounting for it, all calculations turn into fantasies about perpetual motion machines.

Third, mathematics outlines the boundaries of the possible. This research doesn't say: «Here is a way to earn a million». It says: «Here is why you can't earn a million out of nothing on an honest market.» Increasing profits are possible only in models with special, unrealistic properties. The real market does not allow this – too many participants hunt for any inefficiency.

But most importantly: this research reminds us that finance is not just about money, but about the structure of uncertainty. Prices fluctuate not just because – they reflect our collective expectations, fears, hopes, and limitations. And when we try to describe these fluctuations mathematically, we create models. A good model captures important properties of reality without spawning fantastic possibilities.

Researchers showed: if your model permits «perpetual motion machines» of trade, it means something in it is arranged incorrectly. Either you ascribed too much structure to the market (reflections, stickiness, asymmetry), or you forgot to consider that any structure will be quickly used up and vanish. Arbitrage self-destructs – this is one of the few «laws of nature» in finance.

In Search of a Fair Market

Measure $\nu$, canonical strategy $\theta$, representation property – all this may seem far from real life. But behind these symbols hides a fundamental question: how is a fair market structured?

A fair market is not one where everyone earns. It is a market where one cannot earn without risk. Where every profit is balanced by the possibility of loss. Where there are no «free lunches», as economists love to say.

And here is what's amazing: even in abstract mathematical models, even in the world of pure symbols and equations, fairness is a rare, fragile property. It is enough to add a small asymmetry, slightly change the way of reflecting from a boundary, create a zone of slowdown – and fairness collapses. Opportunities appear for exploitation, for extracting profit from the very structure of the model.

This, if you think about it, is very similar to real society. Fairness requires fine-tuning, a balance of many factors. The slightest skew – and the system starts working in favor of those who know where to look for opportunities. The difference is only that in mathematics, we can say exactly which skews lead to what consequences. In real life, everything is much more complicated.

Perhaps this is the main value of such research. They show how easily injustice is created and how difficult it is to maintain equilibrium. They teach us to see hidden structure where chaos seems to reign. And they remind us: behind every opportunity for quick profit stands either risk, or someone's loss, or an error in understanding reality.

Money exists only because we believe in it. But arbitrage exists only where our belief creates skews. And the task of a good model is to show these skews clearly enough so they can be fixed. Or at least understood.