#11 (Part 2 of the Main Series: Chapter 4): Mathematical Formulas for My Leveraged Nasdaq-100 TQQQ Simulation Model with Biased Probabilities, the ABC-K6 Model—An Explanation of "γ" and "δ"

In the previous articles (#9, #10), we introduced the following parameters in the K6 model:
• Determining the direction and magnitude of price movements within a certain period
α (Bias of luck/misfortune)
• Triggered when α continues in the same direction
β (Chain of coincidence)

In this article, we will introduce parameters that have slightly different characteristics within the K6 model, falling under the


**"Qualitative Hierarchy of Luck"**: • γ (Gamma) • δ (Delta)
.

◯ "Qualitative Hierarchy of Luck" with different characteristics
To summarize the basic structure of the K6 model once again, it takes the following form:

Future = [Now]
　　         × [Bias of luck/misfortune]
　　         × [Chain of coincidence]
　　         × [Qualitative hierarchy of luck]

Future = {Now × j1 × ( α × β × γ ) × j2 × δ }× ε

The α and β introduced so far were parameters directly linked to the "flow of price movements":
• α: Whether it went up or down during that period
• β: The fact that the same direction continued

On the other hand, γ and δ represent a hierarchy of "luck" that, while still luck, has different characteristics. In K6, these two types of "luck" are intentionally designed to be separate.

◯ Differences in the roles of γ and δ (Intuitive summary)

First, let's intuitively summarize the difference between γ and δ.
⭐︎ γ (Gamma) → Luck that occurs within the flow → Triggered in "conjunction" with the chain of α
⭐︎

δ (Delta) → Luck that intervenes from outside the flow → Triggered independently of the chain of αBoth act as bonuses or penalties, but "when and why they are triggered" is decisively different.

① γ : A "qualitative hierarchy of luck" that is "concomitant" to the chain of α

◯ What is γ? γ

γ is a parameter that is triggered in conjunction with the "moment the chain of α continuing in the same direction is broken". While

β exponentially amplifies the "chain itself," γ acts at the boundary where the chain ends.

In line with this role, I have given γ two names.
◯ Rescue γ
• Negative growth direction (▼α) chain is broken by the positive growth direction (△α) triggered
• Expresses the "sooner-than-expected rebound" and "sense of relief" that comes after a continued decline.

Rescue γ is a bonus-like parameter that mitigates the influence of β which has begun to expand exponentially in the negative direction.
◯ Resistance γ


• Positive growth direction (△α) chain is broken by the negative growth direction (▼α)• Expresses the "heavy pullback" and "sense of hitting a wall" that comes after a continued rise.
Resistance γ

is a penalty-like parameter that suppresses β
that has begun to overheat in the positive direction.

◯ γ’s Role

Rescue / Resistance γ both share the following roles:
• Suppressing exponential runaway caused by β • Preventing the entire model from being too biased toward upward or downward swings • Creating a "foundation" for trend reversals.
If

β
is "momentum," then γ is the brake and cushion that works at the moment that momentum is interrupted.

② δ: A “Qualitative Hierarchy of Luck” independent of the chain of α

◯ What is δ?
δ has properties completely different from γ.

δ is designed to:
• α be independent of the chain of
• occur stochastically each time
• provide a bonus or penalty when specific conditions are met.

I call δ the “Big Event” parameter.

◯ What δ represents

In the actual stock market, δ represents events that interrupt regardless of the preceding trend, such as:
• Earnings announcements
• Economic indicators
• Monetary policy
• Geopolitical risks
• Unexpected news
Whether the trend is good or bad,
“it comes when it comes.”
In K6, this discontinuity is isolated as
δ.

◯ The reason for separating γ and δ

In the K6 model, we treat these as separate parameters:
• β: Internal amplification due to the continuation of a chain
• γ: Accompanying luck that occurs at the moment a chain breaks
• δ: Exogenous luck that occurs independently of the chain

This incorporates the intuition into the mathematical formula that stock market uncertainty consists of “things that happen within the flow” and “things that interrupt from outside the flow.”

◯ Summary
• γ is “boundary luck” that triggers accompanying the chain of α •
δ is “external event luck” that triggers independently of the chain of α • By separating
β・γ・δ, K6 has a mathematical structure that can handle “momentum,” “reversal,” and “sudden events” simultaneously.

In the next installment, #12, I will introduce j1 and j2, parameters that shake the entire formula stochastically and independently, as well as ε, which accounts for expenses and costs essential in index investing.

(Postscript)
This article is not intended to recommend specific financial products or provide investment advice.
The formulas and models are merely thought models based on my own subjectivity, intuition, and metaphors.

“How to view the future” and “how to perceive risk”
I would be happy if you could read this as a full-length mirror for those purposes.