# Market Crashes and Time-Translation Invariance

## Abstract

**:**

## 1. Introduction

## 2. Self-Similarity and Time-Translation Invariance

#### 2.1. Higher-Order Regression, Approximants and Multipliers

#### 2.2. Methods of Finding Origin

#### 2.3. Ad Hoc Approximants

## 3. Examples

- 1.
- How do we calculate and interpret the probabilistic pattern that we encounter in the day preceding a crash?
- 2.
- How do we actually calculate the typical market reactions expressed through the price movements, if we know that a shock has already struck?

- 1.
- The examples of the first type are supposed to exemplify the reaction to shock.
- 2.
- The second group brings up the so-called bubbles.
- 3.
- The third group includes some non-monotonous price configurations reminiscent of the typical patterns of technical analysis.

#### 3.1. 9/11

#### 3.2. America Goes to War

#### 3.3. Fukushima, or Godzilla Strikes Again

#### 3.4. Flash Crash

#### 3.5. Bubbles

#### 3.5.1. Hockey Stick Bubble: Second Order

#### 3.5.2. Hockey Stick Bubble: Third Order

#### 3.5.3. Hockey Stick Bubble: Fourth Order

#### 3.5.4. The Largest Bubble

#### 3.5.5. Death of the Hero

#### 3.6. Non-Monotonous Crashes

#### 3.6.1. When the Market was Young

#### 3.6.2. Flag-like Growth

#### 3.6.3. Head and Shoulders Growth

#### 3.6.4. Friday the 13th Bad Luck

## 4. Comments on Trading

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**When the market was young. All exponential approximants corresponding to the solutions to the equation ${E}_{1}({t}_{N},r)=s\left({t}_{N}\right)$ are shown. Together they form the discrete spectrum. The most extreme, stable downward and less stable upward solutions are drawn with solid lines. There are also four additional intermediate solutions. The real value of ${s}_{16}$ is shown as well as the lower straight line.

**Figure 2.**9/11. Pattern in DJ index preceding the terrorist attack/shock of 9/11, 2001. Fourth-order regression is plotted against true data points. The inverse multiplier of the first order and the first-order approximant are shown dependent on origin at $t={T}_{N}$, $N=17$. The typical value ${s}_{17}$ (level) is shown as well with a dot-dashed line.

**Figure 3.**War. Pattern in DJ index in the time preceding the entrance to war, on 1 February 1917. Fourth-order regression is plotted against true data points. The inverse first-order multiplier and the first-order approximant are shown dependent on origin at $t={T}_{N}$, $N=16$. The level ${s}_{16}$ is shown as well with a dot-dashed line.

**Figure 4.**Fukushima. Pattern in Nikkei index in the time preceding the flash crash, on 14 March 2011, with ${s}_{16}=$ 8605.15. Non-monotonous pattern corresponding to the 4th-order regression is plotted against true data points. The inverse multiplier of the first order and the first-order approximant are shown dependent on origin at $t={T}_{N}$, $N=15$. The level ${s}_{15}=$ 9620.49 is shown with a dot-dashed line.

**Figure 5.**Flash crash in DJ on 6 May 2010. Fourth-order regression is plotted against true data points. The dependencies on origin of the inverse first-order multiplier and the first-order approximant are shown at $t={T}_{N}$, $N=15$. The typical value of the level ${s}_{15}$ is shown with a dot-dashed line.

**Figure 7.**Hockey stick bubble. Second-order regression. The inverse absolute value of the multiplier is shown as a function of origin at $t=15$. The first-order approximant is shown for $N=15$. Typical value ${s}_{15}$ (level) is shown with a dot-dashed line.

**Figure 8.**Hockey stick bubble. Third-order regression. The inverse absolute value of the multiplier is shown at $N=15$. The first-order approximant is presented as well, dependent on origin. The typical value ${s}_{15}$ (level) is shown with a dot-dashed line.

**Figure 9.**Hockey stick bubble. Fourth-order regression. The inverse multiplier and the first-order approximant are shown as functions of origin at $t={T}_{N}$, $N=15$. The typical value ${s}_{15}$ (level) is shown with a dot-dashed line.

**Figure 11.**Largest bubble. Fourth-order regression. The inverse multiplier and the first-order approximant are shown dependent on origin at $t={T}_{N}$, $N=15$. The value ${s}_{15}=897.42$ (level) is shown with a dot-dashed line.

**Figure 12.**Second-largest bubble. Pattern in Shanghai Composite preceding the crash of 9 August 1994. Fourth-order regression is plotted against true data points. The inverse multiplier of the first order and the first-order approximant are shown dependent on origin at $t={T}_{N}$, $N=16$. The level ${s}_{16}$ is shown as well with a dot-dashed line.

**Figure 13.**Death of hero. Pattern in Shanghai Composite index preceding the crash of 19 February 1997. A monotonous growth pattern represented by the fourth-order regression is plotted against true data points. The inverse multiplier and the first-order approximant are shown as functions of the origin at $t={T}_{N}$, $N=15$. The typical value ${s}_{15}$ (level) is shown as well.

**Figure 14.**Hero in mortal danger. Pattern in DJ preceding the crash of 26 September 1955. Fourth-order regression is plotted against true data points. The inverse multiplier of the first order and the first-order approximant are drawn as functions of the origin at $t={T}_{N}$, $N=14$. The typical value ${s}_{14}=487.5$ (level) is shown as well.

**Figure 15.**When the market was young. Pattern in DJ index, which preceded the crash of 29 June 1896. Fourth-order regression is plotted against true data points. The inverse multiplier of the first order and first-order approximant are presented dependent on origin at $t={T}_{N}$, $N=15$. The level ${s}_{15}$ is shown as well with a dot-dashed line.

**Figure 16.**Flag-like pattern in Shanghai Composite. Fourth-order regression is plotted against true data points.

**Figure 17.**Flag-like pattern in Shanghai Composite. The results are found from the analysis based on 4th-order regression. The inverse multiplier and first-order approximant are presented as functions of the origin at $t={T}_{N}$, $N=15$. Level ${s}_{15}=3376.50$ is shown as well.

**Figure 19.**Head and shoulders. The inverse multiplier of the first order and first-order approximant are presented dependent on origin at $t={T}_{N}$, $N=15$. The typical value ${s}_{15}=717.36$ (level) is shown as well.

**Figure 20.**Price pattern in DJ, reminding us of the cup with a handle. Fourth-order regression is plotted against true data points. The inverse multiplier of the first order and first-order approximant are presented dependent on origin at $t={T}_{N}$, $N=15$. The value of ${s}_{15}=2759.8$ (level) is shown with a dot-dashed line.

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Gluzman, S. Market Crashes and Time-Translation Invariance. *FinTech* **2023**, *2*, 221-247.
https://doi.org/10.3390/fintech2020014

**AMA Style**

Gluzman S. Market Crashes and Time-Translation Invariance. *FinTech*. 2023; 2(2):221-247.
https://doi.org/10.3390/fintech2020014

**Chicago/Turabian Style**

Gluzman, Simon. 2023. "Market Crashes and Time-Translation Invariance" *FinTech* 2, no. 2: 221-247.
https://doi.org/10.3390/fintech2020014