ZM - Detail Report

Quick Overview

Timeframe	C-C DoR σ (Historical)	H-L DoR σ (Historical)	O-C DoR σ (Historical)	5Y ATRP (Avg)	Binomial IV (Period)	Yahoo IV (Period)	1σ Lower (68.27%)	1σ Upper	2σ Lower (95.45%)	2σ Upper
Daily	3.44%	2.61%	2.82%	3.74%	3.20%	3.24%	87.44	93.23	84.69	96.26
Weekly	7.45%	7.13%	6.96%	8.71%	7.05%	7.13%	84.14	96.89	78.41	103.96
Monthly	14.18%	13.50%	12.48%	17.85%	14.68%	14.84%	77.96	104.57	67.32	121.10
Quarterly	32.87%	36.45%	28.33%	30.26%	25.42%	25.71%	70.02	116.43	54.30	150.13
Annual	—	—	—	—	50.85%	51.42%	54.30	150.13	32.66	249.63
Option Time Horizon	—	—	—	—	16.62%	16.81%	76.46	106.62	64.75	125.90

Calculation Notes: Computed at 2026-02-22T14:20 UTC | Reference Contract: Call option expiring 2026-04-02 (39 days) | Spot Price (S): $90.29 | Strike (K): $91.00 (first OTM call ≥ spot) | Observed Mid-Price: $5.8000 (Bid $5.5000 / Ask $6.1000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 50.85% (CRR binomial tree, American, n=20 steps) | BS IV: 50.69% (European Black-Scholes) | Yahoo IV (Annual): 51.42% → period-scaled as σ_period = σ_annual·√h | Price bands: Upper = S·e^+σ√h, Lower = S·e^−σ√h

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.13. Further, an Excess Kurtosis of 15.00 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.23%	8	0.47%	0.47%
-10.23% to -7.66%	17	0.99%	1.45%
-7.66% to -5.08%	59	3.43%	4.89%
-5.08% to -2.50%	185	10.76%	15.65%
-2.50% to 0.08%	626	36.42%	52.07%
0.08% to 2.66%	567	32.98%	85.05%
2.66% to 5.24%	159	9.25%	94.30%
5.24% to 7.82%	64	3.72%	98.02%
7.82% to 10.39%	22	1.28%	99.30%
Greater than 10.39%	12	0.70%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.36%	3.52%	78.48%	68.27%
±2σ	-6.80%	6.96%	94.88%	95.45%
±3σ	-10.23%	10.39%	98.84%	99.73%

Descriptive Statistics

Mean	0.080%
Standard Error	0.083%
Median	-0.035%
Mode	-0.035%
Standard Deviation	3.438%
Sample Variance	11.821%
Kurtosis	15.00
Skewness	1.13
Range	58.154%
Minimum	-17.370%
Maximum	40.784%
Count	1719

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.425%	843	49.04%	1.189%
Negative	-2.187%	872	50.73%	-1.109%
Zero	0.000%	4	0.23%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	3.78%
20	1 Month	4.78%
60	1 Quarter	3.51%
250	1 Year	2.83%
750	3 Years	2.85%
1,250	5 Years	3.74%
2,500	10 Years	4.36%
5,000	20 Years	4.36%
12,500	50 Years	4.36%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.95. Further, an Excess Kurtosis of 6.27 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -3.67%	0	0.00%	0.00%
-3.67% to -1.71%	0	0.00%	0.00%
-1.71% to 0.25%	0	0.00%	0.00%
0.25% to 2.21%	379	22.03%	22.03%
2.21% to 4.17%	693	40.29%	62.33%
4.17% to 6.12%	356	20.70%	83.02%
6.12% to 8.08%	154	8.95%	91.98%
8.08% to 10.04%	69	4.01%	95.99%
10.04% to 12.00%	43	2.50%	98.49%
Greater than 12.00%	26	1.51%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.56%	6.78%	80.17%	68.27%
±2σ	-1.06%	9.39%	95.12%	95.45%
±3σ	-3.67%	12.00%	98.49%	99.73%

Descriptive Statistics

Mean	4.166%
Standard Error	0.063%
Median	3.484%
Mode	1.533%
Standard Deviation	2.611%
Sample Variance	6.819%
Kurtosis	6.27
Skewness	1.95
Range	24.480%
Minimum	0.743%
Maximum	25.223%
Count	1720

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.166%	1720	100.00%	4.166%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.26. Further, an Excess Kurtosis of 3.33 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.45%	11	0.64%	0.64%
-8.45% to -6.34%	27	1.57%	2.21%
-6.34% to -4.22%	61	3.55%	5.76%
-4.22% to -2.11%	200	11.63%	17.38%
-2.11% to 0.00%	570	33.14%	50.52%
0.00% to 2.12%	562	32.67%	83.20%
2.12% to 4.23%	174	10.12%	93.31%
4.23% to 6.35%	83	4.83%	98.14%
6.35% to 8.46%	23	1.34%	99.48%
Greater than 8.46%	9	0.52%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.81%	2.82%	75.41%	68.27%
±2σ	-5.63%	5.64%	94.94%	95.45%
±3σ	-8.45%	8.46%	98.84%	99.73%

Descriptive Statistics

Mean	0.004%
Standard Error	0.068%
Median	-0.014%
Mode	0.000%
Standard Deviation	2.819%
Sample Variance	7.946%
Kurtosis	3.33
Skewness	0.26
Range	29.722%
Minimum	-11.800%
Maximum	17.922%
Count	1720

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.042%	851	49.48%	1.010%
Negative	-2.005%	863	50.17%	-1.006%
Zero	0.000%	6	0.35%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.17. Further, an Excess Kurtosis of 6.10 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -21.98%	1	0.28%	0.28%
-21.98% to -16.40%	3	0.84%	1.12%
-16.40% to -10.81%	11	3.08%	4.20%
-10.81% to -5.23%	50	14.01%	18.21%
-5.23% to 0.35%	121	33.89%	52.10%
0.35% to 5.94%	115	32.21%	84.31%
5.94% to 11.52%	35	9.80%	94.12%
11.52% to 17.11%	13	3.64%	97.76%
17.11% to 22.69%	3	0.84%	98.60%
Greater than 22.69%	5	1.40%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.09%	7.80%	78.15%	68.27%
±2σ	-14.54%	15.24%	94.68%	95.45%
±3σ	-21.98%	22.69%	98.32%	99.73%

Descriptive Statistics

Mean	0.354%
Standard Error	0.394%
Median	-0.043%
Mode	-7.771%
Standard Deviation	7.445%
Sample Variance	55.427%
Kurtosis	6.10
Skewness	1.17
Range	72.033%
Minimum	-24.100%
Maximum	47.933%
Count	357

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.582%	178	49.86%	2.783%
Negative	-4.845%	179	50.14%	-2.429%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	7.76%
12	1 Quarter	7.53%
52	1 Year	6.70%
156	3 Years	6.81%
260	5 Years	8.71%
520	10 Years	10.25%
1,040	20 Years	10.25%
2,600	50 Years	10.25%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.42. Further, an Excess Kurtosis of 9.62 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.71%	0	0.00%	0.00%
-10.71% to -5.36%	0	0.00%	0.00%
-5.36% to -0.01%	0	0.00%	0.00%
-0.01% to 5.34%	63	17.60%	17.60%
5.34% to 10.69%	169	47.21%	64.80%
10.69% to 16.04%	67	18.72%	83.52%
16.04% to 21.39%	34	9.50%	93.02%
21.39% to 26.74%	16	4.47%	97.49%
26.74% to 32.09%	3	0.84%	98.32%
Greater than 32.09%	6	1.68%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	3.56%	17.82%	82.96%	68.27%
±2σ	-3.58%	24.96%	94.41%	95.45%
±3σ	-10.71%	32.09%	98.32%	99.73%

Descriptive Statistics

Mean	10.690%
Standard Error	0.377%
Median	8.530%
Mode	5.826%
Standard Deviation	7.133%
Sample Variance	50.879%
Kurtosis	9.62
Skewness	2.42
Range	59.498%
Minimum	2.208%
Maximum	61.706%
Count	358

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.690%	358	100.00%	10.690%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.25. Further, an Excess Kurtosis of 4.93 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -20.77%	2	0.56%	0.56%
-20.77% to -15.55%	5	1.40%	1.96%
-15.55% to -10.33%	16	4.47%	6.42%
-10.33% to -5.11%	38	10.61%	17.04%
-5.11% to 0.11%	121	33.80%	50.84%
0.11% to 5.33%	113	31.56%	82.40%
5.33% to 10.55%	42	11.73%	94.13%
10.55% to 15.77%	13	3.63%	97.77%
15.77% to 20.99%	7	1.96%	99.72%
Greater than 20.99%	1	0.28%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.85%	7.07%	77.93%	68.27%
±2σ	-13.81%	14.03%	94.97%	95.45%
±3σ	-20.77%	20.99%	99.16%	99.73%

Descriptive Statistics

Mean	0.107%
Standard Error	0.368%
Median	0.031%
Mode	0.072%
Standard Deviation	6.960%
Sample Variance	48.444%
Kurtosis	4.93
Skewness	0.25
Range	74.116%
Minimum	-32.734%
Maximum	41.382%
Count	358

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.011%	179	50.00%	2.506%
Negative	-4.797%	179	50.00%	-2.398%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.86. Further, an Excess Kurtosis of 1.62 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -41.33%	0	0.00%	0.00%
-41.33% to -30.70%	0	0.00%	0.00%
-30.70% to -20.06%	4	4.88%	4.88%
-20.06% to -9.43%	14	17.07%	21.95%
-9.43% to 1.20%	28	34.15%	56.10%
1.20% to 11.84%	25	30.49%	86.59%
11.84% to 22.47%	5	6.10%	92.68%
22.47% to 33.11%	2	2.44%	95.12%
33.11% to 43.74%	3	3.66%	98.78%
Greater than 43.74%	1	1.22%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.97%	15.38%	75.61%	68.27%
±2σ	-27.15%	29.56%	92.68%	95.45%
±3σ	-41.33%	43.74%	98.78%	99.73%

Descriptive Statistics

Mean	1.204%
Standard Error	1.566%
Median	0.270%
Mode	-29.484%
Standard Deviation	14.179%
Sample Variance	201.051%
Kurtosis	1.62
Skewness	0.86
Range	74.089%
Minimum	-29.484%
Maximum	44.605%
Count	82

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.889%	43	52.44%	5.710%
Negative	-9.474%	39	47.56%	-4.506%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	13.23%
12	1 Year	12.76%
36	3 Years	13.94%
60	5 Years	17.85%
120	10 Years	21.51%
240	20 Years	21.51%
600	50 Years	21.51%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.15. Further, an Excess Kurtosis of 0.99 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -16.26%	0	0.00%	0.00%
-16.26% to -6.13%	0	0.00%	0.00%
-6.13% to 3.99%	0	0.00%	0.00%
3.99% to 14.12%	20	24.10%	24.10%
14.12% to 24.25%	32	38.55%	62.65%
24.25% to 34.37%	14	16.87%	79.52%
34.37% to 44.50%	10	12.05%	91.57%
44.50% to 54.62%	4	4.82%	96.39%
54.62% to 64.75%	2	2.41%	98.80%
Greater than 64.75%	1	1.20%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	10.74%	37.75%	67.47%	68.27%
±2σ	-2.76%	51.25%	95.18%	95.45%
±3σ	-16.26%	64.75%	98.80%	99.73%

Descriptive Statistics

Mean	24.245%
Standard Error	1.482%
Median	21.092%
Mode	6.892%
Standard Deviation	13.501%
Sample Variance	182.282%
Kurtosis	0.99
Skewness	1.15
Range	60.343%
Minimum	6.892%
Maximum	67.235%
Count	83

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	24.245%	83	100.00%	24.245%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Fairly Symmetrical with a Skewness factor of 0.25. Further, an Excess Kurtosis of 0.20 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -36.65%	0	0.00%	0.00%
-36.65% to -27.29%	1	1.20%	1.20%
-27.29% to -17.93%	4	4.82%	6.02%
-17.93% to -8.57%	11	13.25%	19.28%
-8.57% to 0.79%	28	33.73%	53.01%
0.79% to 10.15%	23	27.71%	80.72%
10.15% to 19.51%	9	10.84%	91.57%
19.51% to 28.87%	4	4.82%	96.39%
28.87% to 38.23%	3	3.61%	100.00%
Greater than 38.23%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-11.69%	13.27%	74.70%	68.27%
±2σ	-24.17%	25.75%	90.36%	95.45%
±3σ	-36.65%	38.23%	100.00%	99.73%

Descriptive Statistics

Mean	0.789%
Standard Error	1.370%
Median	-0.717%
Mode	-27.751%
Standard Deviation	12.481%
Sample Variance	155.773%
Kurtosis	0.20
Skewness	0.25
Range	58.321%
Minimum	-27.751%
Maximum	30.570%
Count	83

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.795%	40	48.19%	5.202%
Negative	-8.519%	43	51.81%	-4.413%
Zero	0.000%	0	0.00%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.23. Further, an Excess Kurtosis of 1.42 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -93.48%	0	0.00%	0.00%
-93.48% to -68.83%	0	0.00%	0.00%
-68.83% to -44.17%	0	0.00%	0.00%
-44.17% to -19.52%	5	17.86%	17.86%
-19.52% to 5.14%	11	39.29%	57.14%
5.14% to 29.79%	8	28.57%	85.71%
29.79% to 54.45%	1	3.57%	89.29%
54.45% to 79.10%	1	3.57%	92.86%
79.10% to 103.76%	2	7.14%	100.00%
Greater than 103.76%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-27.73%	38.01%	82.14%	68.27%
±2σ	-60.61%	70.88%	89.29%	95.45%
±3σ	-93.48%	103.76%	100.00%	99.73%

Descriptive Statistics

Mean	5.138%
Standard Error	6.212%
Median	-1.555%
Mode	-43.827%
Standard Deviation	32.873%
Sample Variance	1080.639%
Kurtosis	1.42
Skewness	1.23
Range	131.672%
Minimum	-43.827%
Maximum	87.845%
Count	28

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	30.822%	13	46.43%	14.310%
Negative	-17.121%	15	53.57%	-9.172%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	18.47%
12	3 Years	22.44%
20	5 Years	30.26%
40	10 Years	42.24%
80	20 Years	42.24%
200	50 Years	42.24%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.57. Further, an Excess Kurtosis of 2.01 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -58.66%	0	0.00%	0.00%
-58.66% to -31.33%	0	0.00%	0.00%
-31.33% to -4.00%	0	0.00%	0.00%
-4.00% to 23.34%	4	13.79%	13.79%
23.34% to 50.67%	14	48.28%	62.07%
50.67% to 78.00%	7	24.14%	86.21%
78.00% to 105.34%	0	0.00%	86.21%
105.34% to 132.67%	2	6.90%	93.10%
132.67% to 160.01%	2	6.90%	100.00%
Greater than 160.01%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	14.23%	87.12%	82.76%	68.27%
±2σ	-22.22%	123.56%	93.10%	95.45%
±3σ	-58.66%	160.01%	100.00%	99.73%

Descriptive Statistics

Mean	50.671%
Standard Error	6.768%
Median	40.827%
Mode	10.698%
Standard Deviation	36.445%
Sample Variance	1328.257%
Kurtosis	2.01
Skewness	1.57
Range	145.319%
Minimum	10.698%
Maximum	156.017%
Count	29

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	50.671%	29	100.00%	50.671%
Negative	0.000%	0	0.00%	0.000%
Zero	0.000%	0	0.00%	0.000%

Open-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.52. Further, an Excess Kurtosis of 0.65 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -82.21%	0	0.00%	0.00%
-82.21% to -60.97%	0	0.00%	0.00%
-60.97% to -39.72%	3	10.34%	10.34%
-39.72% to -18.48%	2	6.90%	17.24%
-18.48% to 2.77%	11	37.93%	55.17%
2.77% to 24.01%	8	27.59%	82.76%
24.01% to 45.26%	3	10.34%	93.10%
45.26% to 66.50%	1	3.45%	96.55%
66.50% to 87.75%	1	3.45%	100.00%
Greater than 87.75%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-25.56%	31.10%	75.86%	68.27%
±2σ	-53.89%	59.42%	96.55%	95.45%
±3σ	-82.21%	87.75%	100.00%	99.73%

Descriptive Statistics

Mean	2.769%
Standard Error	5.260%
Median	0.828%
Mode	-46.438%
Standard Deviation	28.327%
Sample Variance	802.415%
Kurtosis	0.65
Skewness	0.52
Range	123.519%
Minimum	-46.438%
Maximum	77.081%
Count	29

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	23.569%	15	51.72%	12.191%
Negative	-19.517%	14	48.28%	-9.422%
Zero	0.000%	0	0.00%	0.000%

ZM Deep Analytical Report