TTWO - 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	3.21%	3.11%	2.89%	2.69%	—	3.10%	194.27	205.59	188.84	211.50
Weekly	6.76%	7.45%	6.69%	6.34%	—	6.82%	187.76	212.71	176.41	226.41
Monthly	13.91%	16.39%	14.69%	13.50%	—	14.20%	175.51	227.56	154.14	259.12
Quarterly	21.85%	30.74%	22.63%	22.93%	—	24.60%	159.60	250.26	127.45	313.38
Annual	—	—	—	—	—	49.21%	127.45	313.38	81.28	491.39
Option Time Horizon	—	—	—	—	—	18.57%	168.64	236.83	142.31	280.66

Calculation Notes: Computed at 2026-03-23T16:42 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $199.85 | Strike (K): $200.00 (first OTM call ≥ spot) | Observed Mid-Price: $13.9500 (Bid $12.9000 / Ask $15.0000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 0.00% (CRR binomial tree, American, n=20 steps) | BS IV: 44.98% (European Black-Scholes) | Yahoo IV (Annual): 49.21% → period-scaled as σ_period = σ_annual·√h | Price bands: Upper = S·e^+σ√h, Lower = S·e^−σ√h

Historical Dividend Payouts

Ex-Dividend Date	Amount
2008-09-25	$0.0010

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.87. Further, an Excess Kurtosis of 22.10 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -9.53%	48	0.66%	0.66%
-9.53% to -7.12%	65	0.90%	1.56%
-7.12% to -4.71%	197	2.72%	4.28%
-4.71% to -2.30%	731	10.09%	14.37%
-2.30% to 0.10%	2753	38.01%	52.38%
0.10% to 2.51%	2406	33.22%	85.60%
2.51% to 4.92%	682	9.42%	95.02%
4.92% to 7.33%	214	2.95%	97.97%
7.33% to 9.74%	86	1.19%	99.16%
Greater than 9.74%	61	0.84%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-3.11%	3.32%	80.57%	68.27%
±2σ	-6.32%	6.53%	95.04%	95.45%
±3σ	-9.53%	9.74%	98.50%	99.73%

Descriptive Statistics

Mean	0.105%
Standard Error	0.038%
Median	0.000%
Mode	0.000%
Standard Deviation	3.211%
Sample Variance	10.313%
Kurtosis	22.10
Skewness	0.87
Range	86.258%
Minimum	-31.362%
Maximum	54.896%
Count	7243

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.203%	3579	49.41%	1.089%
Negative	-2.043%	3487	48.14%	-0.984%
Zero	0.000%	177	2.44%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	2.64%
20	1 Month	3.06%
60	1 Quarter	3.21%
250	1 Year	2.68%
750	3 Years	2.37%
1,250	5 Years	2.69%
2,500	10 Years	2.87%
5,000	20 Years	3.37%
12,500	50 Years	4.10%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 4.58. Further, an Excess Kurtosis of 49.17 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -5.31%	0	0.00%	0.00%
-5.31% to -2.98%	0	0.00%	0.00%
-2.98% to -0.65%	0	0.00%	0.00%
-0.65% to 1.68%	840	11.60%	11.60%
1.68% to 4.01%	3923	54.16%	65.75%
4.01% to 6.34%	1449	20.00%	85.75%
6.34% to 8.67%	549	7.58%	93.33%
8.67% to 11.00%	257	3.55%	96.88%
11.00% to 13.33%	119	1.64%	98.52%
Greater than 13.33%	107	1.48%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	0.91%	7.12%	88.29%	68.27%
±2σ	-2.20%	10.22%	95.86%	95.45%
±3σ	-5.31%	13.33%	98.52%	99.73%

Descriptive Statistics

Mean	4.012%
Standard Error	0.036%
Median	3.125%
Mode	5.000%
Standard Deviation	3.106%
Sample Variance	9.644%
Kurtosis	49.17
Skewness	4.58
Range	63.840%
Minimum	0.569%
Maximum	64.409%
Count	7244

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.012%	7244	100.00%	4.012%
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 0.77. Further, an Excess Kurtosis of 17.07 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.59%	39	0.54%	0.54%
-8.59% to -6.43%	87	1.20%	1.74%
-6.43% to -4.26%	243	3.35%	5.09%
-4.26% to -2.10%	735	10.15%	15.24%
-2.10% to 0.07%	2679	36.98%	52.22%
0.07% to 2.24%	2354	32.50%	84.72%
2.24% to 4.40%	732	10.10%	94.82%
4.40% to 6.57%	218	3.01%	97.83%
6.57% to 8.73%	93	1.28%	99.12%
Greater than 8.73%	64	0.88%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-2.82%	2.96%	79.64%	68.27%
±2σ	-5.71%	5.85%	94.75%	95.45%
±3σ	-8.59%	8.73%	98.58%	99.73%

Descriptive Statistics

Mean	0.070%
Standard Error	0.034%
Median	0.000%
Mode	0.000%
Standard Deviation	2.888%
Sample Variance	8.343%
Kurtosis	17.07
Skewness	0.77
Range	75.644%
Minimum	-28.016%
Maximum	47.628%
Count	7244

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	2.033%	3536	48.81%	0.992%
Negative	-1.895%	3527	48.69%	-0.923%
Zero	0.000%	181	2.50%	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 10.52 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -19.81%	8	0.53%	0.53%
-19.81% to -14.73%	18	1.19%	1.73%
-14.73% to -9.66%	47	3.12%	4.84%
-9.66% to -4.59%	167	11.08%	15.93%
-4.59% to 0.48%	526	34.90%	50.83%
0.48% to 5.56%	507	33.64%	84.47%
5.56% to 10.63%	156	10.35%	94.82%
10.63% to 15.70%	50	3.32%	98.14%
15.70% to 20.78%	15	1.00%	99.14%
Greater than 20.78%	13	0.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.28%	7.25%	79.30%	68.27%
±2σ	-13.04%	14.01%	95.02%	95.45%
±3σ	-19.81%	20.78%	98.61%	99.73%

Descriptive Statistics

Mean	0.484%
Standard Error	0.174%
Median	0.332%
Mode	0.000%
Standard Deviation	6.764%
Sample Variance	45.755%
Kurtosis	10.52
Skewness	0.86
Range	102.220%
Minimum	-32.353%
Maximum	69.867%
Count	1507

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.785%	802	53.22%	2.546%
Negative	-4.498%	691	45.85%	-2.062%
Zero	0.000%	14	0.93%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	7.75%
12	1 Quarter	8.16%
52	1 Year	6.29%
156	3 Years	5.55%
260	5 Years	6.34%
520	10 Years	6.56%
1,040	20 Years	7.80%
2,600	50 Years	9.39%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.96. Further, an Excess Kurtosis of 14.55 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -12.39%	0	0.00%	0.00%
-12.39% to -6.80%	0	0.00%	0.00%
-6.80% to -1.22%	0	0.00%	0.00%
-1.22% to 4.37%	217	14.39%	14.39%
4.37% to 9.95%	759	50.33%	64.72%
9.95% to 15.54%	303	20.09%	84.81%
15.54% to 21.12%	121	8.02%	92.84%
21.12% to 26.71%	68	4.51%	97.35%
26.71% to 32.29%	10	0.66%	98.01%
Greater than 32.29%	30	1.99%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.50%	17.40%	87.14%	68.27%
±2σ	-4.94%	24.84%	96.35%	95.45%
±3σ	-12.39%	32.29%	98.01%	99.73%

Descriptive Statistics

Mean	9.951%
Standard Error	0.192%
Median	7.688%
Mode	5.000%
Standard Deviation	7.447%
Sample Variance	55.460%
Kurtosis	14.55
Skewness	2.96
Range	71.951%
Minimum	1.571%
Maximum	73.522%
Count	1508

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	9.951%	1508	100.00%	9.951%
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 0.97. Further, an Excess Kurtosis of 7.40 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -20.04%	6	0.40%	0.40%
-20.04% to -15.03%	14	0.93%	1.33%
-15.03% to -10.01%	52	3.45%	4.77%
-10.01% to -5.00%	170	11.27%	16.05%
-5.00% to 0.02%	559	37.07%	53.12%
0.02% to 5.03%	481	31.90%	85.01%
5.03% to 10.05%	146	9.68%	94.69%
10.05% to 15.06%	43	2.85%	97.55%
15.06% to 20.08%	18	1.19%	98.74%
Greater than 20.08%	19	1.26%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-6.67%	6.71%	78.58%	68.27%
±2σ	-13.36%	13.39%	94.89%	95.45%
±3σ	-20.04%	20.08%	98.34%	99.73%

Descriptive Statistics

Mean	0.018%
Standard Error	0.172%
Median	-0.320%
Mode	0.000%
Standard Deviation	6.687%
Sample Variance	44.713%
Kurtosis	7.40
Skewness	0.97
Range	87.268%
Minimum	-40.721%
Maximum	46.547%
Count	1508

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	4.832%	707	46.88%	2.265%
Negative	-4.300%	788	52.26%	-2.247%
Zero	0.000%	13	0.86%	0.000%

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 8.28 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -39.74%	1	0.29%	0.29%
-39.74% to -29.31%	3	0.86%	1.15%
-29.31% to -18.88%	13	3.75%	4.90%
-18.88% to -8.44%	40	11.53%	16.43%
-8.44% to 1.99%	125	36.02%	52.45%
1.99% to 12.43%	108	31.12%	83.57%
12.43% to 22.86%	40	11.53%	95.10%
22.86% to 33.29%	10	2.88%	97.98%
33.29% to 43.73%	3	0.86%	98.85%
Greater than 43.73%	4	1.15%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-11.92%	15.90%	79.83%	68.27%
±2σ	-25.83%	29.82%	96.25%	95.45%
±3σ	-39.75%	43.73%	98.56%	99.73%

Descriptive Statistics

Mean	1.991%
Standard Error	0.747%
Median	1.298%
Mode	-7.662%
Standard Deviation	13.912%
Sample Variance	193.549%
Kurtosis	8.28
Skewness	1.11
Range	154.440%
Minimum	-57.410%
Maximum	97.030%
Count	347

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.466%	194	55.91%	5.851%
Negative	-8.812%	152	43.80%	-3.860%
Zero	0.000%	1	0.29%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	15.79%
12	1 Year	12.62%
36	3 Years	12.11%
60	5 Years	13.50%
120	10 Years	14.00%
240	20 Years	16.87%
600	50 Years	20.17%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.82. Further, an Excess Kurtosis of 12.88 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -26.03%	0	0.00%	0.00%
-26.03% to -13.73%	0	0.00%	0.00%
-13.73% to -1.44%	0	0.00%	0.00%
-1.44% to 10.86%	55	15.80%	15.80%
10.86% to 23.15%	169	48.56%	64.37%
23.15% to 35.44%	75	21.55%	85.92%
35.44% to 47.74%	24	6.90%	92.82%
47.74% to 60.03%	12	3.45%	96.26%
60.03% to 72.33%	7	2.01%	98.28%
Greater than 72.33%	6	1.72%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	6.76%	39.54%	88.22%	68.27%
±2σ	-9.63%	55.93%	95.69%	95.45%
±3σ	-26.03%	72.33%	98.28%	99.73%

Descriptive Statistics

Mean	23.149%
Standard Error	0.879%
Median	18.508%
Mode	32.000%
Standard Deviation	16.392%
Sample Variance	268.699%
Kurtosis	12.88
Skewness	2.82
Range	136.822%
Minimum	5.423%
Maximum	142.245%
Count	348

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	23.149%	348	100.00%	23.149%
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 Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.43. Further, an Excess Kurtosis of 18.64 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -44.34%	1	0.29%	0.29%
-44.34% to -33.33%	1	0.29%	0.57%
-33.33% to -22.31%	8	2.30%	2.87%
-22.31% to -11.29%	53	15.23%	18.10%
-11.29% to -0.27%	130	37.36%	55.46%
-0.27% to 10.75%	104	29.89%	85.34%
10.75% to 21.77%	32	9.20%	94.54%
21.77% to 32.79%	13	3.74%	98.28%
32.79% to 43.81%	2	0.57%	98.85%
Greater than 43.81%	4	1.15%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-14.96%	14.42%	81.90%	68.27%
±2σ	-29.65%	29.12%	96.26%	95.45%
±3σ	-44.34%	43.81%	98.56%	99.73%

Descriptive Statistics

Mean	-0.268%
Standard Error	0.788%
Median	-1.498%
Mode	-49.103%
Standard Deviation	14.692%
Sample Variance	215.861%
Kurtosis	18.64
Skewness	2.43
Range	179.089%
Minimum	-49.103%
Maximum	129.986%
Count	348

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	10.959%	152	43.68%	4.787%
Negative	-9.021%	195	56.03%	-5.055%
Zero	0.000%	1	0.29%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.55. Further, an Excess Kurtosis of 2.39 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -60.08%	0	0.00%	0.00%
-60.08% to -43.69%	1	0.86%	0.86%
-43.69% to -27.30%	5	4.31%	5.17%
-27.30% to -10.91%	21	18.10%	23.28%
-10.91% to 5.48%	29	25.00%	48.28%
5.48% to 21.87%	35	30.17%	78.45%
21.87% to 38.26%	20	17.24%	95.69%
38.26% to 54.65%	3	2.59%	98.28%
54.65% to 71.04%	1	0.86%	99.14%
Greater than 71.04%	1	0.86%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-16.38%	27.33%	71.55%	68.27%
±2σ	-38.23%	49.19%	96.55%	95.45%
±3σ	-60.08%	71.04%	99.14%	99.73%

Descriptive Statistics

Mean	5.478%
Standard Error	2.029%
Median	6.220%
Mode	-47.979%
Standard Deviation	21.854%
Sample Variance	477.617%
Kurtosis	2.39
Skewness	0.55
Range	148.941%
Minimum	-47.979%
Maximum	100.962%
Count	116

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	18.695%	71	61.21%	11.443%
Negative	-15.375%	45	38.79%	-5.964%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	18.30%
12	3 Years	20.95%
20	5 Years	22.93%
40	10 Years	24.57%
80	20 Years	29.60%
200	50 Years	34.86%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 2.99. Further, an Excess Kurtosis of 13.49 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -48.75%	0	0.00%	0.00%
-48.75% to -25.69%	0	0.00%	0.00%
-25.69% to -2.63%	0	0.00%	0.00%
-2.63% to 20.42%	15	12.82%	12.82%
20.42% to 43.48%	64	54.70%	67.52%
43.48% to 66.54%	20	17.09%	84.62%
66.54% to 89.60%	10	8.55%	93.16%
89.60% to 112.65%	4	3.42%	96.58%
112.65% to 135.71%	1	0.85%	97.44%
Greater than 135.71%	3	2.56%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	12.74%	74.22%	88.03%	68.27%
±2σ	-18.00%	104.97%	95.73%	95.45%
±3σ	-48.75%	135.71%	97.44%	99.73%

Descriptive Statistics

Mean	43.481%
Standard Error	2.842%
Median	33.112%
Mode	15.220%
Standard Deviation	30.743%
Sample Variance	945.142%
Kurtosis	13.49
Skewness	2.99
Range	220.494%
Minimum	15.220%
Maximum	235.714%
Count	117

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	43.481%	117	100.00%	43.481%
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 Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.23. Further, an Excess Kurtosis of 2.73 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -69.17%	0	0.00%	0.00%
-69.17% to -52.20%	0	0.00%	0.00%
-52.20% to -35.23%	3	2.56%	2.56%
-35.23% to -18.26%	23	19.66%	22.22%
-18.26% to -1.29%	46	39.32%	61.54%
-1.29% to 15.68%	23	19.66%	81.20%
15.68% to 32.65%	14	11.97%	93.16%
32.65% to 49.62%	4	3.42%	96.58%
49.62% to 66.59%	2	1.71%	98.29%
Greater than 66.59%	2	1.71%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-23.92%	21.34%	78.63%	68.27%
±2σ	-46.54%	43.96%	95.73%	95.45%
±3σ	-69.17%	66.59%	98.29%	99.73%

Descriptive Statistics

Mean	-1.289%
Standard Error	2.092%
Median	-5.520%
Mode	-50.957%
Standard Deviation	22.626%
Sample Variance	511.944%
Kurtosis	2.73
Skewness	1.23
Range	144.296%
Minimum	-50.957%
Maximum	93.339%
Count	117

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	22.209%	42	35.90%	7.972%
Negative	-14.449%	75	64.10%	-9.262%
Zero	0.000%	0	0.00%	0.000%

TTWO Deep Analytical Report