TRADE AS PRICE MEASUREMENT
Measurement is a fundamental process of quantitatively determining attributes of an object, event, or phenomenon, playing a central role in various disciplines and providing the basis for objective and reproducible information. In the realm of finance, a crucial question arises: What exactly do we measure?
The primary focus of measurement in finance is on price. Price measurement involves ascribing a specific value to an asset or liability, and it takes place when buyers and sellers reach an agreement, effectively resolving the price through a transaction.
Formally, price measurement from a valuation perspective is addressed in IFRS-13, which offers general guidelines without specifying exact methodologies. Consequently, each institution is responsible for establishing its own pricing methodologies in line with its objectives and capabilities. So, how is price measured? What is the mechanics of price measurement?
Let’s consider a common example of performing price measurement through an order book on an exchange. The order book combines all outstanding buy and sell orders.
To perform price measurement, we can submit an order for 100 shares at a conservative price, such as $27.80. Since this price is not competitive, the order will not fill, requiring a price adjustment. For instance, we can move the price up to $27.83. When the price reaches the top of the order book, the order stands a reasonable chance of being filled. However, being at the top of the book does not guarantee immediate execution. By gradually increasing the price, the order will eventually be filled. At that point, price has been measured and the result is $27.83.
Alternatively, instead of purchasing, one could attempt to sell. In this scenario, we would have to gradually lower the price until the order fills, which might occur, for instance, at $27.87. This and the previous measurements, taken from opposing sides, yield different results ($27.83 and $27.87), separated by the amount of bid-ask spread. Price carries inherent uncertainty, and the amount of this uncertainty is measured by the spread. While localized within the spread, price remains uncertain until the next transaction.
Additionally, one could use market orders to get an immediate execution. In this case, the two measurement results would switch places, yet they would still reflect the same two numbers and the same underlying price uncertainty. This inherent uncertainty in price cannot be entirely eliminated; it is a characteristic feature of the market. Adding more liquidity can reduce the uncertainty, but complete elimination is not possible.
PRECISION OF PRICE MEASUREMENT
How do we improve the accuracy of price measurement? Normally, one would take more measurements, calculate the average, and the average would more closely represent the actual value of the measured quantity. How does this procedure work with price measurement?
If we performed repeated trades similar to the one described above, we would generally end up with a series of randomly chosen numbers between $27.83 and $27.87. No matter how many more trades we would perform they will still fall into the same range and be randomly scattered in it. Adjacent trades will not form a smoothly evolving line, nor will they fall closer to each other if performed with a smaller time step. Taking more measurements does not improve the accuracy of price. There is a level of price uncertainty, define by the spread, that cannot be overcome.
Now, what if we used 700 shares instead of 100 shares? A 700-share lot holds more weight in representing the company’s price compared to a 100-share lot. Let’s see what happens then. The 100 shares would hit the top levels of the order book and stop there. However, larger orders will penetrate more levels. For example, a market sell order for 700 shares will penetrate three bid levels and halt at the fourth, with a price of $27.79. As a result, the 700 shares would sell for an effective price of $27.814 per share, leading to a $0.016 discount compared to the price of 100 shares ($27.83). Surprisingly, our attempt to improve precision has the opposite effect.
Attempting to obtain more accurate price measurements actually results in a price shift, highlighting the complex mechanics of price formation. Achieving absolute precision in price measurement seems to be impossible altogether. Given these characteristics of price measurement, the question naturally arises: What kind of measurement is price measurement?
CHARACTERISTICS OF PRICE MEASUREMENT
As a process, price measurement has the following characteristics:
1. More measurements do not improve measurement accuracy
2. Adjacent measurements do not form a smooth line
3. Measurement act influences its own outcome
4. Attempts for more precise measurement, destroy the measurement precision
These are not features of classical measurements. These are features of a quantum measurement. To remind, here is the comparison chart.
Simple comparison shows that price measurement is a quantum measurement.
N.B. Whether the quantum nature of price measurement has any relevance to cognition or social aspects is neither within my expertise nor my interest. Any quantum effects arising from the interaction of elementary particles are undoubtedly washed away by decoherence. Additionally, the other mechanisms that elicit quantum features from the microstructural components of nature into the price formation level remain a puzzle, to which, again, I am not attempting to provide an answer. What is clear is that, based on its characteristics, price measurement qualifies as a quantum measurement, and that is the focus of the quantum theory of market microstructure.
WHY IS IT IMPORTANT?
So, price dynamics on microstructural level is quantum. What are the implications?
1. Assets are not merely described by their prices; they are described by the price operator. The spectrum of the price operator represents the range of prices an asset can have as a result of measurement.
2. Price dynamics cannot be adequately described by stochastic processes alone. Instead, it is described by quantum-chaotic equations, with the price operator acting as the Hamiltonian. While the stochastic framework is suitable for modeling trade price dynamics in highly liquid securities when trading small quantities (retail scale), it does not fit well with institutional operations. Institutional finance operates under a different paradigm.
3. Financial data should not be considered noisy; it is inherently quantum in nature. As an example, nobody talks about a muon’s trajectory in a bubble chamber being noisy. Noisy implies that if all aspects of the process were predictable, uncertainty would vanish. However, even with predictability, bid-ask uncertainty remains an integral part of finance and the market. Without that uncertainty, there is no market.
4. There is no such thing as price slippage. Instead, there are discounts for bulk sales and premiums for bulk purchases, reflecting price elasticity due to different pricing for larger quantities. The idea of slippage implies that somehow the security could have been available at a better price, but price slipped. This is fantasy.
5. There is no such thing as volatility at micro level. At micro level volatility loses its meaning and price uncertainty is measured by spread. Attempts to force the notion of volatility onto the microstructure can lead to artificially induced paradoxes and a flawed understanding of the market dynamics.
PRACTICAL IMPLICATIONS
The main practical implications for different stakeholders regarding the quantum theory of market microstructure are as follows:
Quantitative Traders:
The crucial implication for quantitative traders is that effective decision-making requires a quantum-level AI. Classical AI cannot interact with quantum processes, such as the price formation process. Quantum AI already encompasses the primary features of price formation in trading without needing extensive learning, unlike classical AI. This efficiency allows quantum AI to operate on financial data with far less training data and fewer parameters.
Pricing and Risk Management:
Classical processes cannot replicate quantum processes, particularly price dynamics. If pricing or risk valuation relies on classical algorithms, constant adjustments would be necessary to adjust for liquidity and enhance accuracy.
Quantum Computing Programmers:
The primary practical implication for quantum computing programmers is to focus on providing new functionality instead of merely transferring old models into the quantum computing environment. Developing new functionalities adds more value than making incremental improvements by implementing old models in a quantum setting.
Academia:
For academia, the main practical implication is that they may need to adapt to these developments, even if it might impact their ongoing projects. Opposing the adoption of the quantum theory of market microstructure might delay progress, but embracing and exploring this emerging field could lead to new opportunities and advancements.
RESOLVING THE FICTITIOUS PARADOXES OF CLASSICAL MODELS
In the quantum pricing model, the “paradoxes” often observed in academic literature are effectively resolved. For instance, one such paradox is associated with the volatility computed based on trades, which diverges to infinity as the time step shrinks. However, in quantum models of market microstructure, there is no paradox, as price uncertainty does not scale to zero; it remains finite even at the high-frequency trading (HFT) scale. The divergence occurs naturally when the finite uncertainty is divided by the diminishing time step. This seeming “paradox” is simply an artificial outcome resulting from the erroneous use of classical tools in a domain where they are inadequate.
Another “paradox” arises when the volatility computed from mid-prices appears to reach exact zero with a reducing time step. This contradicts the well-known scaling law of volatility. The explanation lies in the fact that at sufficiently small time scales, bid and ask prices remain fixed, leading to a constant mid-price, which, in turn, ensures zero volatility when computed from the mid-prices. Even if the bid and ask prices change by equal and opposite amounts, resulting in an unchanged mid-price, there is still observable volatility in bid and ask prices. Volatility, as a concept, is inapplicable at the microstructural level, and price uncertainty is better measured by the spread, as demonstrated in quantum models of market microstructure.
DIFFERENCES WITH ELEMENTARY PARTICLE MEASUREMENTS
Price measurement has substantial differences with elementary particle measurements.
1. It is inverted: measurement in financial markets occurs from inside out. Indeed, in elementary particle measurements the measuring apparatus has to be located outside of the particle. In finance, buy and sell orders interact by passing through the spread, so the measurement occurs as if it were taken from inside the spread in an outward direction. In this sense, measurement in finance is inverted.
2. It is non-local: deep book orders have just as quick impact on price formation as the orders at the top of the book, because both are visible/accessible at the same time. Order’s effect of price formation may vary with size and depth, but the immediacy is the same.
3. It has many formats: price can be weakly measured without a direct interaction. For example, it is possible to take continuous measurements of price simply by watching the available quotes or orders on Bloomberg screen. This is impossible in the world of elementary particles.
Due to these differences it is incorrect to simply borrow the Schrodinger equation and bring it into finance, as some researchers have tried in the past. The correct financial quantum equations reflect this specifics, and have numerous differences with the Schrodinger equation.
Measurement differences with the elementary particle world are discussed in another publication.
WHAT HAPPENS TO INTERFERENCE?
Interference is one of the main things setting quantum world apart from the classical. However, it is not observed in financial markets. Is that really so, and what happens to interference is discussed in another publication.