#12 (Main Part 2: Chapter 5): My Leveraged Nasdaq-100 TQQQ Simulation Model with Biased Probabilities, the ABC-K6 Model Mathematical Formula... Explanation of "ε" and "j1, j2"

In the previous articles (#9, #10, #11), I explained the following parameters in the K6 model:
•
α (bias of luck/misfortune) which determines the direction and magnitude of price movements within a certain period
•
β (chain of coincidence) which is triggered when α continues in the same direction

• Parameters corresponding to the "qualitative hierarchy of luck":
γ
δ (delta).

This time, I will introduce the remaining parameters: ε, j1, and j2.

◯ Cost that causes impairment of TQQQ value while held - ε -

Bull-type leveraged ETFs like TQQQ generally have an expense ratio (annual rate) that is more than twice as high as non-leveraged ETFs.
Example: QQQ:TQQQ = 0.18 : 0.82 (%)

However, leveraged ETFs have hidden costs that are not explicitly stated in the product.
This is what is called "volatility drag (deterioration)" associated with daily rebalancing.

⭐︎ Volatility drag unique to leveraged ETFs (hereinafter, volatility deterioration)

Generally, when explaining why leveraged ETFs are not suitable for long-term holding, volatility deterioration is cited.
Roughly speaking, volatility deterioration is explained as follows: when a range-bound market where the underlying index repeatedly moves up and down continues, the negative compounding effect in leveraged ETFs is amplified, and the value (valuation) naturally impairs during the range-bound market.

Below, I will write it out with a simple mathematical formula using a 2x leverage example.

If a non-leveraged ETF linked to a benchmark index has a principal of 100 and daily price movements of ▼5.0% → △5.27%

Non-leveraged ETF valuation trend
100.00 → 95.00 → approx. 100.01

However, in a 2x leveraged ETF, the daily price movement rate is amplified by two, so it becomes ▼10% → △10.54%.

In that case,
2x leveraged ETF valuation trend
100.00 → 90.00 → approx. 99.49

While the non-leveraged ETF returns to its original valuation as a result of the price moving up and down, the leveraged ETF does not reach its original valuation as a result of the up and down movement.
When such a range-bound market where the original index repeatedly moves up and down within a certain range is prolonged, the impairment amount of the leveraged ETF's valuation naturally increases, so in my mathematical formula, I have incorporated volatility deterioration as a cost into ε.

In fact, in α of my mathematical formula, when 1, which means no weekly price movement, is selected by the dice draw, the valuation due to volatility deterioration is naturally impaired by ε

In my K6 model, I treat this ε as a temporal wear and tear unique to leveraged ETFs, where "the market being flat itself becomes a cost."

◯ "Irrationally" fluctuating luck, j1 and j2
In my mathematical formula, somewhat unusual parameters are j1 and j2. The
α, β, γ, δ, ε I have introduced so far are parameters that I set by breaking down how TQQQ valuation moves into structural elements based on my subjectivity; I can explain their roles, and they are parameters that contain artificial design intentions.

But, to put j1 and j2 in a word... in my latest version "ABC-K6αA," I incorporated these into the mathematical formula as unusual parameters, as elements where luck fluctuates irrationally outside of my design intentions.

Honestly, even without j1 and j2, I can generate a TQQQ-like time-series replica.
However, that would only be "somewhat plausible" in line with my design intentions, and it cannot reproduce the "unexplainable fluctuations" or "inexplicable discomfort" felt in the actual stock market.

I also feel that stock market price movements are sometimes irrational and accompanied by complex fluctuations due to luck-based elements that are difficult to explain.

Therefore, I incorporated parameters that fluctuate independently into the mathematical formula, j1 and j2.

j1 is the first half of the mathematical formula, and the part directly involving α is assigned a coefficient of 1 ± several percent through an independent dice roll.

j2 is assigned a coefficient that fluctuates within a range of 1 ± several percent through an independent dice roll when a Big Event occurs, affecting δ, but I have made its frequency distribution and fluctuation range slightly different from j1.

j1 and j2 are difficult to observe or define. Even so, if I ignore them, the generated time-series data shows a "sense of artifice" where it tries too hard to act like the TQQQ I intended, which I can see right through. Therefore, I have intentionally detached them from the design intent as independent fluctuations, giving them the role of naturally reproducing the countless scenarios born within the turbulence of the real stock market.

Next time, I plan to summarize the points covered so far and review the design philosophy, intent, and features of my model once again.

Postscript: My articles are not intended to recommend specific financial products or provide investment advice. Furthermore, my model is a thought experiment constructed based on my own subjectivity and has not undergone academic verification.