▫️New Logarithmic Units for Financial Applications

Humanity has long been using various logarithmic units, such as decibels (dB) to measure sound intensity, semitones to measure the distance between musical notes, and bits to measure information. (see Measuring Levels)

• A dB (decibel) is a unit of measurement for relative sound level, signal power, or other physical quantities that can be expressed on a logarithmic scale. For example, a sound level of 0 dB corresponds to the threshold of human hearing, and every 10 dB increases the sound level by a factor of 10. So technically dB corresponds to $\sqrt[10]{10}$

• A bit (short for binary digit) is a unit of measurement for the amount of information that can be encoded in the binary system. At its core, the bit is a logarithmic unit that describes the amount of information needed to distinguish between two alternatives. More precisely, the amount of information measured in bits is determined by the logarithm to the base 2 of the number of possible alternatives.

• A semitone in music (especially after J.S.Bach “The Well-Tempered Clavier”) is a logarithmic unit because it represents a fixed ratio between two frequencies. Specifically, a semitone is equal to the $\sqrt[12]{2}$, which is approximately 1.05946. This means that if we start with a given frequency and go up one semitone, the resulting frequency will be approximately 1.05946 times higher. Similarly, going down one semitone will result in a frequency that is approximately 0.94387 times lower. This logarithmic relationship is why we use semitones to measure the distance between musical notes.

At the same time, despite the frequent use of logarithmic scales in the financial sector, we have not found corresponding widely accepted logarithmic units.

Traders often use logarithmic scales when analyzing price dynamics because logarithmic scales provide a better way to observe and interpret trends over a wider range of prices. In financial markets, prices can experience significant changes over short periods, and using a linear scale can make it difficult to see trends and changes in prices accurately. By using a logarithmic scale, traders can better observe and analyze the percentage changes in prices, which can be more meaningful than the absolute changes in prices. Additionally, logarithmic scales can help traders identify key support and resistance levels, which can be important for making trading decisions.

Why use logarithmic scales

One example of the importance of using a logarithmic scale in trading is as follows. In crypto-crypto pairs, the choice of which currency will be the base and which will be the quote is arbitrary and historical (the logical rule of taking the less volatile asset as the quote is not strict). Thus, we can have both the AB and BA pairs for currency pair A and B. However, when using a linear scale on the AB pair, a linearly declining trend (A becomes linearly cheaper in B) on the BA pair will not be linearly ascending, but hyperbolic (B becomes hyperbolically more expensive than A). This violates the symmetry of perception of dynamics, given that the AB or BA pair was chosen arbitrarily historically. When switching to logarithmic scales, any trend on the AB scale is transformed into its precise mirror reflection on the BA scale, preserving the perception of dynamics.

Meet Doubling, Docent, and Domille

Natural measure of scale difference or extance (exponential distance) between A and B can be calculated as $\ln(A/B)$ and is measured in Nepers ($\mathbf{Np}$)

$+1\,\mathbf{Np} = \times 2.718281828459045$

At the same time it is a measure for hyperbolic angles where 1 Np for hyperbolic trigonometric functions (sinh, cosh) corresponds to 1 radian for ordinary trigonometric functions (sin, cos). [Hyperbolic angles are logarithms of non-negative numbers by their nature].

However for the sake of usability it is better to use more practical and understandable unit - Doubling (D) that is tightly related to Neper:

1 Neper = 1.442695 Doubling

Here ratio is 1/ln(2) exactly the same as in nats vs. bits in information theory.

+1 Neper = ⨯2.7182818 +1 Doubling = ⨯2

and its derivative units: docents - doubling cents (⨝)

1 docent = 1⨝ = 2^(1/100)

+100⨝ = +100%

Yet docents are multiplicative:

+1⨝ = +0.695555% -1⨝ = -0.69075%

+100⨝ = 1D = ⨯2 = +100% -100⨝ = -1D = ⨯(1/2)= -50%

+200⨝ = 2D = ⨯4 = +300% -200⨝ = -2D = ⨯(1/4)= -75%

+300⨝ = 3D = ⨯8 = +700% -300⨝ = -3D = ⨯(1/8)= -93.75%

Probably for negative doublings we should use another word: Halving (H) and hacents (?)

1 Halving = -1 Doubling 100 hacents = 1 Halving

Rationale: phrase “Three Doublings” can be easily interpreted as ⨯8, and phrase “Three Halvings” easily interpreted as sequential halvings that give us 1/8