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%	2.60%	2.81%	3.70%	2.48%	2.74%	75.39	79.22	73.55	81.21
Weekly	7.47%	7.15%	7.01%	8.64%	5.45%	6.03%	73.18	81.62	69.30	86.19
Monthly	14.28%	13.43%	12.69%	17.73%	11.35%	12.55%	68.99	86.57	61.59	96.98
Quarterly	33.09%	35.74%	28.48%	31.16%	19.66%	21.73%	63.49	94.07	52.16	114.51
Annual	—	—	—	—	39.31%	43.47%	52.16	114.51	35.21	169.66
Option Time Horizon	—	—	—	—	14.84%	16.41%	66.63	89.65	57.44	103.99

Calculation Notes: Computed at 2026-03-23T16:35 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $77.29 | Strike (K): $77.50 (first OTM call ≥ spot) | Observed Mid-Price: $4.6250 (Bid $4.2000 / Ask $5.0500) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 39.31% (CRR binomial tree, American, n=20 steps) | BS IV: 38.98% (European Black-Scholes) | Yahoo IV (Annual): 43.47% → 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 1.11. Further, an Excess Kurtosis of 14.93 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -10.24%	9	0.52%	0.52%
-10.24% to -7.66%	17	0.98%	1.49%
-7.66% to -5.08%	59	3.39%	4.89%
-5.08% to -2.51%	186	10.69%	15.57%
-2.51% to 0.07%	635	36.49%	52.07%
0.07% to 2.65%	574	32.99%	85.06%
2.65% to 5.22%	161	9.25%	94.31%
5.22% to 7.80%	65	3.74%	98.05%
7.80% to 10.38%	22	1.26%	99.31%
Greater than 10.38%	12	0.69%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.36%	3.51%	78.51%	68.27%
±2σ	-6.80%	6.94%	94.89%	95.45%
±3σ	-10.24%	10.38%	98.79%	99.73%

Descriptive Statistics

Mean	0.071%
Standard Error	0.082%
Median	-0.035%
Mode	-0.035%
Standard Deviation	3.436%
Sample Variance	11.806%
Kurtosis	14.93
Skewness	1.11
Range	58.154%
Minimum	-17.370%
Maximum	40.784%
Count	1740

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.415%	852	48.97%	1.183%
Negative	-2.188%	884	50.80%	-1.112%
Zero	0.000%	4	0.23%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	2.95%
20	1 Month	4.19%
60	1 Quarter	3.96%
250	1 Year	2.92%
750	3 Years	2.86%
1,250	5 Years	3.70%
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 -3.65%	0	0.00%	0.00%
-3.65% to -1.70%	0	0.00%	0.00%
-1.70% to 0.26%	0	0.00%	0.00%
0.26% to 2.21%	382	21.94%	21.94%
2.21% to 4.17%	705	40.49%	62.44%
4.17% to 6.12%	359	20.62%	83.06%
6.12% to 8.07%	155	8.90%	91.96%
8.07% to 10.03%	70	4.02%	95.98%
10.03% to 11.98%	43	2.47%	98.45%
Greater than 11.98%	27	1.55%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	1.56%	6.77%	80.13%	68.27%
±2σ	-1.04%	9.38%	95.12%	95.45%
±3σ	-3.65%	11.98%	98.45%	99.73%

Descriptive Statistics

Mean	4.165%
Standard Error	0.062%
Median	3.476%
Mode	1.533%
Standard Deviation	2.605%
Sample Variance	6.785%
Kurtosis	6.27
Skewness	1.95
Range	24.480%
Minimum	0.743%
Maximum	25.223%
Count	1741

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.165%	1741	100.00%	4.165%
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 -8.44%	11	0.63%	0.63%
-8.44% to -6.33%	27	1.55%	2.18%
-6.33% to -4.22%	62	3.56%	5.74%
-4.22% to -2.11%	201	11.55%	17.29%
-2.11% to 0.00%	580	33.31%	50.60%
0.00% to 2.11%	566	32.51%	83.11%
2.11% to 4.22%	178	10.22%	93.34%
4.22% to 6.33%	84	4.82%	98.16%
6.33% to 8.44%	23	1.32%	99.48%
Greater than 8.44%	9	0.52%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.81%	2.81%	75.36%	68.27%
±2σ	-5.62%	5.63%	94.95%	95.45%
±3σ	-8.44%	8.44%	98.85%	99.73%

Descriptive Statistics

Mean	0.002%
Standard Error	0.067%
Median	-0.015%
Mode	0.000%
Standard Deviation	2.813%
Sample Variance	7.911%
Kurtosis	3.33
Skewness	0.26
Range	29.722%
Minimum	-11.800%
Maximum	17.922%
Count	1741

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.039%	860	49.40%	1.007%
Negative	-1.999%	875	50.26%	-1.005%
Zero	0.000%	6	0.34%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.14. Further, an Excess Kurtosis of 5.97 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -22.09%	1	0.28%	0.28%
-22.09% to -16.49%	3	0.83%	1.11%
-16.49% to -10.89%	12	3.32%	4.43%
-10.89% to -5.29%	51	14.13%	18.56%
-5.29% to 0.31%	121	33.52%	52.08%
0.31% to 5.91%	116	32.13%	84.21%
5.91% to 11.51%	36	9.97%	94.18%
11.51% to 17.11%	13	3.60%	97.78%
17.11% to 22.72%	3	0.83%	98.61%
Greater than 22.72%	5	1.39%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-7.16%	7.78%	78.12%	68.27%
±2σ	-14.62%	15.25%	94.46%	95.45%
±3σ	-22.09%	22.72%	98.34%	99.73%

Descriptive Statistics

Mean	0.312%
Standard Error	0.393%
Median	-0.043%
Mode	-7.771%
Standard Deviation	7.468%
Sample Variance	55.775%
Kurtosis	5.97
Skewness	1.14
Range	72.033%
Minimum	-24.100%
Maximum	47.933%
Count	361

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.571%	180	49.86%	2.778%
Negative	-4.918%	181	50.14%	-2.466%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	11.27%
12	1 Quarter	9.93%
52	1 Year	7.12%
156	3 Years	6.90%
260	5 Years	8.64%
520	10 Years	10.28%
1,040	20 Years	10.28%
2,600	50 Years	10.28%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.39. Further, an Excess Kurtosis of 9.39 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 -5.35%	0	0.00%	0.00%
-5.35% to 0.01%	0	0.00%	0.00%
0.01% to 5.37%	64	17.68%	17.68%
5.37% to 10.73%	169	46.69%	64.36%
10.73% to 16.09%	69	19.06%	83.43%
16.09% to 21.45%	35	9.67%	93.09%
21.45% to 26.81%	16	4.42%	97.51%
26.81% to 32.17%	3	0.83%	98.34%
Greater than 32.17%	6	1.66%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	3.58%	17.88%	82.87%	68.27%
±2σ	-3.57%	25.02%	94.48%	95.45%
±3σ	-10.71%	32.17%	98.34%	99.73%

Descriptive Statistics

Mean	10.728%
Standard Error	0.376%
Median	8.561%
Mode	5.826%
Standard Deviation	7.147%
Sample Variance	51.074%
Kurtosis	9.39
Skewness	2.39
Range	59.498%
Minimum	2.208%
Maximum	61.706%
Count	362

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.728%	362	100.00%	10.728%
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 0.27. Further, an Excess Kurtosis of 4.74 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -20.89%	2	0.55%	0.55%
-20.89% to -15.63%	5	1.38%	1.93%
-15.63% to -10.37%	16	4.42%	6.35%
-10.37% to -5.11%	39	10.77%	17.13%
-5.11% to 0.15%	122	33.70%	50.83%
0.15% to 5.41%	114	31.49%	82.32%
5.41% to 10.67%	44	12.15%	94.48%
10.67% to 15.93%	11	3.04%	97.51%
15.93% to 21.19%	8	2.21%	99.72%
Greater than 21.19%	1	0.28%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.86%	7.17%	78.18%	68.27%
±2σ	-13.87%	14.18%	94.75%	95.45%
±3σ	-20.89%	21.19%	99.17%	99.73%

Descriptive Statistics

Mean	0.152%
Standard Error	0.369%
Median	0.031%
Mode	-8.007%
Standard Deviation	7.013%
Sample Variance	49.185%
Kurtosis	4.74
Skewness	0.27
Range	74.116%
Minimum	-32.734%
Maximum	41.382%
Count	362

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	5.108%	181	50.00%	2.554%
Negative	-4.804%	181	50.00%	-2.402%
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 0.83. Further, an Excess Kurtosis of 1.53 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -41.81%	0	0.00%	0.00%
-41.81% to -31.10%	0	0.00%	0.00%
-31.10% to -20.39%	4	4.82%	4.82%
-20.39% to -9.68%	13	15.66%	20.48%
-9.68% to 1.03%	28	33.73%	54.22%
1.03% to 11.74%	27	32.53%	86.75%
11.74% to 22.45%	5	6.02%	92.77%
22.45% to 33.16%	2	2.41%	95.18%
33.16% to 43.87%	3	3.61%	98.80%
Greater than 43.87%	1	1.20%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-13.25%	15.31%	75.90%	68.27%
±2σ	-27.53%	29.59%	92.77%	95.45%
±3σ	-41.81%	43.87%	98.80%	99.73%

Descriptive Statistics

Mean	1.032%
Standard Error	1.567%
Median	0.393%
Mode	-29.484%
Standard Deviation	14.280%
Sample Variance	203.932%
Kurtosis	1.53
Skewness	0.83
Range	74.089%
Minimum	-29.484%
Maximum	44.605%
Count	83

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.748%	44	53.01%	5.698%
Negative	-9.929%	39	46.99%	-4.665%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	19.53%
12	1 Year	14.20%
36	3 Years	14.29%
60	5 Years	17.73%
120	10 Years	21.58%
240	20 Years	21.58%
600	50 Years	21.58%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.14. Further, an Excess Kurtosis of 0.98 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -15.94%	0	0.00%	0.00%
-15.94% to -5.86%	0	0.00%	0.00%
-5.86% to 4.21%	0	0.00%	0.00%
4.21% to 14.29%	20	23.81%	23.81%
14.29% to 24.36%	33	39.29%	63.10%
24.36% to 34.44%	14	16.67%	79.76%
34.44% to 44.52%	10	11.90%	91.67%
44.52% to 54.59%	4	4.76%	96.43%
54.59% to 64.67%	2	2.38%	98.81%
Greater than 64.67%	1	1.19%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	10.93%	37.80%	67.86%	68.27%
±2σ	-2.50%	51.23%	95.24%	95.45%
±3σ	-15.94%	64.67%	98.81%	99.73%

Descriptive Statistics

Mean	24.365%
Standard Error	1.466%
Median	21.127%
Mode	6.892%
Standard Deviation	13.434%
Sample Variance	180.464%
Kurtosis	0.98
Skewness	1.14
Range	60.343%
Minimum	6.892%
Maximum	67.235%
Count	84

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	24.365%	84	100.00%	24.365%
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 0.27. Further, an Excess Kurtosis of 0.08 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -37.09%	0	0.00%	0.00%
-37.09% to -27.57%	1	1.19%	1.19%
-27.57% to -18.06%	4	4.76%	5.95%
-18.06% to -8.54%	11	13.10%	19.05%
-8.54% to 0.97%	29	34.52%	53.57%
0.97% to 10.49%	22	26.19%	79.76%
10.49% to 20.00%	10	11.90%	91.67%
20.00% to 29.52%	6	7.14%	98.81%
29.52% to 39.03%	1	1.19%	100.00%
Greater than 39.03%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-11.71%	13.66%	75.00%	68.27%
±2σ	-24.40%	26.35%	91.67%	95.45%
±3σ	-37.09%	39.03%	100.00%	99.73%

Descriptive Statistics

Mean	0.972%
Standard Error	1.384%
Median	-0.831%
Mode	-27.751%
Standard Deviation	12.687%
Sample Variance	160.956%
Kurtosis	0.08
Skewness	0.27
Range	58.321%
Minimum	-27.751%
Maximum	30.570%
Count	84

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	11.356%	40	47.62%	5.408%
Negative	-8.469%	44	52.38%	-4.436%
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 1.24. Further, an Excess Kurtosis of 1.39 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -94.64%	0	0.00%	0.00%
-94.64% to -69.82%	0	0.00%	0.00%
-69.82% to -45.00%	0	0.00%	0.00%
-45.00% to -20.18%	4	14.29%	14.29%
-20.18% to 4.64%	12	42.86%	57.14%
4.64% to 29.46%	8	28.57%	85.71%
29.46% to 54.27%	1	3.57%	89.29%
54.27% to 79.09%	1	3.57%	92.86%
79.09% to 103.91%	2	7.14%	100.00%
Greater than 103.91%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-28.45%	37.73%	82.14%	68.27%
±2σ	-61.54%	70.82%	89.29%	95.45%
±3σ	-94.63%	103.91%	100.00%	99.73%

Descriptive Statistics

Mean	4.638%
Standard Error	6.254%
Median	-2.824%
Mode	-43.827%
Standard Deviation	33.091%
Sample Variance	1095.016%
Kurtosis	1.39
Skewness	1.24
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	-18.055%	15	53.57%	-9.672%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	22.97%
12	3 Years	23.94%
20	5 Years	31.16%
40	10 Years	42.89%
80	20 Years	42.89%
200	50 Years	42.89%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.63. Further, an Excess Kurtosis of 2.18 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -55.66%	0	0.00%	0.00%
-55.66% to -28.86%	0	0.00%	0.00%
-28.86% to -2.05%	0	0.00%	0.00%
-2.05% to 24.75%	6	20.69%	20.69%
24.75% to 51.56%	12	41.38%	62.07%
51.56% to 78.37%	7	24.14%	86.21%
78.37% to 105.17%	0	0.00%	86.21%
105.17% to 131.98%	2	6.90%	93.10%
131.98% to 158.79%	2	6.90%	100.00%
Greater than 158.79%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	15.82%	87.30%	86.21%	68.27%
±2σ	-19.92%	123.05%	93.10%	95.45%
±3σ	-55.66%	158.79%	100.00%	99.73%

Descriptive Statistics

Mean	51.562%
Standard Error	6.637%
Median	40.827%
Mode	16.443%
Standard Deviation	35.742%
Sample Variance	1277.521%
Kurtosis	2.18
Skewness	1.63
Range	139.574%
Minimum	16.443%
Maximum	156.017%
Count	29

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	51.562%	29	100.00%	51.562%
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 0.45. Further, an Excess Kurtosis of 0.53 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -82.07%	0	0.00%	0.00%
-82.07% to -60.72%	0	0.00%	0.00%
-60.72% to -39.36%	3	10.34%	10.34%
-39.36% to -18.00%	2	6.90%	17.24%
-18.00% to 3.35%	10	34.48%	51.72%
3.35% to 24.71%	10	34.48%	86.21%
24.71% to 46.06%	2	6.90%	93.10%
46.06% to 67.42%	1	3.45%	96.55%
67.42% to 88.78%	1	3.45%	100.00%
Greater than 88.78%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-25.12%	31.83%	75.86%	68.27%
±2σ	-53.60%	60.30%	96.55%	95.45%
±3σ	-82.07%	88.78%	100.00%	99.73%

Descriptive Statistics

Mean	3.352%
Standard Error	5.288%
Median	0.828%
Mode	-46.438%
Standard Deviation	28.475%
Sample Variance	810.822%
Kurtosis	0.53
Skewness	0.45
Range	123.519%
Minimum	-46.438%
Maximum	77.081%
Count	29

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	24.697%	15	51.72%	12.774%
Negative	-19.517%	14	48.28%	-9.422%
Zero	0.000%	0	0.00%	0.000%

ZM Deep Analytical Report