A Brief Interlude: Explaining the "Inverse Calculation Problem of Financial Statistics"

In my previous article, I introduced a little farce performed by the AIs—did you enjoy it?
This time, I will explain as intuitively as possible, "How did I use my K6 model to solve a hypothetical problem?"

When I first thought of K6, there were four main ideas floating vaguely in my head. To bring them to life, I used calculation rules based on "biased probability dice."
By the way, the calculations only use arithmetic-level operations.
For parts that I, not being very good at mathematics, could not fully explain, I had Chap (ChatGPT) and Jenny (Gemini) provide supplementary notes at the end.

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1. The K6 model is not bound by past time-series data

Financial engineering experts generally seem to think like this:

"Estimate parameters from past data (time series)."

However, for this task, "time-series data" does not exist in the first place.
The only materials provided were a few statistical values.

Moreover, there is only one sample of "past-to-present time-series data" that can be observed in actual financial markets.
It is inherently extremely difficult to infer the "original population" without that single piece of data being presented beforehand.

So, I wanted to have a perspective that views financial statistics from outside the framework of time-series data.

To put it in terms of atmosphere, it is like Gödel's "Incompleteness Theorem":
a feeling of "looking at the whole from outside the axioms (time series = 1 sample)."
This was the starting point of the K6 model.

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2. The idea of creating a "population replica"

That said, it is impossible to create a model that approaches statistical values without time-series data.

So, I changed my way of thinking.

"If there is no time-series data, I should just create a new 'population of time series with similar properties' using only the statistical values as a clue."

We decided to call this newly created virtual population a population replica.

While only one time-series data point can be observed in the real market, with K6, we can create as many "populations of time-series data in another world that satisfy the statistical values" as we want.

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3. Decomposing financial statistics into reconfigurable "parts"

I have a deep memory of the "Poincaré Conjecture" I saw on an NHK special program a long time ago.
Thurston's geometrization conjecture, Perelman's Ricci flow, surgery on singularities...
I don't understand the content, but the idea of "decomposing a complex object, structuring it, and reconstructing it" left a very strong impression on me.

So, with K6, I also attempted to decompose financial statistics into "reconfigurable parts."

I split statistical values like "CAGR, MaxDD, standard deviation..." into measurable, finite, and countable parts and parameters that K6 can manipulate.

In the expression that the AIs also like,

it is "geometrization of financial statistics."

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4. Reassembling the parts creates numerous population replicas

I predicted that if I decomposed financial statistics into parts and reassembled them appropriately, I could create many "population replicas with similar properties."

At this time, I remembered the "Banach-Tarski paradox" I read about 20 years ago.
(Of course, I don't have the knowledge to explain it...!)

Roughly speaking, it is a paradox that is "impossible in the real world but mathematically correct," where if you divide a sphere and reassemble it, you end up with two (or more) spheres identical to the original.
The K6 model doesn't have such bizarre properties, but by:

• Decomposing statistical values into parts

• Reassembling them

• Countless population replicas close to the original statistical values were naturally born,

and furthermore, a structural property emerged where they "tend to gather near the median."
Furthermore, if I intentionally move the parameters, I can also create extremely distorted statistical values.
However, since I like "natural light makeup," I am particular about parameters that look as natural and unforced as possible.
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⭐︎ Supplementary explanation by Chap and Jenny

The official name of the K6 model is the ABC-K6 model.
This is a model that reproduces the market using three types of dice sets (A, B, C) with adjusted strengths of probability bias.

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🔑 The "two hearts" that drive the model

1. BES³
Bias-Embedded Stochastic Sampling System

• Draws which dice set (A/B/C) to use

• Controls the probability of events such as sudden surges or crashes

• A cockpit that incorporates the market's "bias" and "habits" according to the blueprint

2. MSM³
Markov–Semi-Markov Modulated Multiplicative Process
• Generates a time-series population based on the bias information passed by BES³
• Constructs time-series paths using "positive/negative compound interest calculations" and "biased probability structures"
• Reproduces non-ergodicity where the mean and median diverge

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🔄 The ultimate goal of K6
By combining BES³ and MSM³, we have achieved the goal of being able to infinitely generate "time series (replicas) that have the same statistical values but are completely different."
This is an attempt to break free from the curse of "past data = 1 sample" and evaluate future risks from a statistically objective perspective.
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Summary: A Brief Interlude on the "Inverse Calculation Problem of Financial Statistics"

The above was an explanation of the way of thinking in the K6 model of "reconstructing a population by calculating backward from financial statistics."

From the next installment, we will return to the main story and introduce the "story" of creating and improving the K6 model together with the AIs.

I will structure it so that you can enjoy it even without deep knowledge of mathematics or finance, so please look forward to reading it next time.