KPTI - 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	5.54%	5.24%	4.97%	8.63%	26.42%	25.79%	5.42	9.19	4.16	11.97
Weekly	12.38%	13.28%	12.69%	19.38%	58.15%	56.77%	3.95	12.63	2.21	22.59
Monthly	24.96%	27.14%	26.80%	40.85%	121.05%	118.18%	2.10	23.69	0.63	79.48
Quarterly	33.71%	46.09%	40.84%	67.27%	209.67%	204.69%	0.87	57.46	0.11	467.68
Annual	—	—	—	—	419.33%	409.38%	0.11	467.68	0.00	30980.47
Option Time Horizon	—	—	—	—	158.28%	154.52%	1.45	34.37	0.30	167.33

Calculation Notes: Computed at 2026-03-23T16:39 UTC | Reference Contract: Call option expiring 2026-05-15 (52 days) | Spot Price (S): $7.06 | Strike (K): $7.50 (first OTM call ≥ spot) | Observed Mid-Price: $3.9250 (Bid $1.9500 / Ask $5.9000) | Risk-Free Rate (r): 3.73% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 419.33% (CRR binomial tree, American, n=20 steps) | BS IV: 416.28% (European Black-Scholes) | Yahoo IV (Annual): 409.38% → 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 2.34. Further, an Excess Kurtosis of 39.88 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -16.59%	12	0.39%	0.39%
-16.59% to -12.43%	19	0.61%	1.00%
-12.43% to -8.28%	82	2.64%	3.63%
-8.28% to -4.12%	369	11.86%	15.50%
-4.12% to 0.03%	1144	36.78%	52.28%
0.03% to 4.19%	1019	32.77%	85.05%
4.19% to 8.34%	321	10.32%	95.37%
8.34% to 12.50%	100	3.22%	98.59%
12.50% to 16.65%	19	0.61%	99.20%
Greater than 16.65%	25	0.80%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-5.51%	5.57%	80.26%	68.27%
±2σ	-11.05%	11.11%	96.72%	95.45%
±3σ	-16.59%	16.65%	98.81%	99.73%

Descriptive Statistics

Mean	0.034%
Standard Error	0.099%
Median	-0.106%
Mode	0.000%
Standard Deviation	5.540%
Sample Variance	30.694%
Kurtosis	39.88
Skewness	2.34
Range	136.989%
Minimum	-43.478%
Maximum	93.511%
Count	3110

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.797%	1484	47.72%	1.812%
Negative	-3.509%	1576	50.67%	-1.778%
Zero	0.000%	50	1.61%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	11.67%
20	1 Month	9.46%
60	1 Quarter	9.58%
250	1 Year	9.67%
750	3 Years	9.33%
1,250	5 Years	8.63%
2,500	10 Years	7.48%
5,000	20 Years	7.41%
12,500	50 Years	7.41%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.86. Further, an Excess Kurtosis of 71.65 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.15%	0	0.00%	0.00%
-8.15% to -4.22%	0	0.00%	0.00%
-4.22% to -0.29%	0	0.00%	0.00%
-0.29% to 3.64%	369	11.86%	11.86%
3.64% to 7.57%	1578	50.72%	62.58%
7.57% to 11.50%	772	24.82%	87.40%
11.50% to 15.43%	242	7.78%	95.18%
15.43% to 19.36%	74	2.38%	97.56%
19.36% to 23.29%	34	1.09%	98.65%
Greater than 23.29%	42	1.35%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.33%	12.81%	89.78%	68.27%
±2σ	-2.91%	18.05%	96.95%	95.45%
±3σ	-8.15%	23.29%	98.65%	99.73%

Descriptive Statistics

Mean	7.568%
Standard Error	0.094%
Median	6.418%
Mode	5.000%
Standard Deviation	5.241%
Sample Variance	27.465%
Kurtosis	71.65
Skewness	5.86
Range	96.137%
Minimum	0.502%
Maximum	96.639%
Count	3111

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.568%	3111	100.00%	7.568%
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 0.89. Further, an Excess Kurtosis of 24.60 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -14.62%	18	0.58%	0.58%
-14.62% to -10.89%	26	0.84%	1.41%
-10.89% to -7.16%	117	3.76%	5.18%
-7.16% to -3.43%	377	12.12%	17.29%
-3.43% to 0.30%	1017	32.69%	49.98%
0.30% to 4.03%	1036	33.30%	83.29%
4.03% to 7.76%	372	11.96%	95.24%
7.76% to 11.49%	104	3.34%	98.59%
11.49% to 15.22%	24	0.77%	99.36%
Greater than 15.22%	20	0.64%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-4.67%	5.27%	76.92%	68.27%
±2σ	-9.64%	10.25%	95.72%	95.45%
±3σ	-14.62%	15.22%	98.78%	99.73%

Descriptive Statistics

Mean	0.302%
Standard Error	0.089%
Median	0.310%
Mode	0.000%
Standard Deviation	4.973%
Sample Variance	24.728%
Kurtosis	24.60
Skewness	0.89
Range	126.703%
Minimum	-48.005%
Maximum	78.698%
Count	3111

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.566%	1635	52.55%	1.874%
Negative	-3.413%	1433	46.06%	-1.572%
Zero	0.000%	43	1.38%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.06. Further, an Excess Kurtosis of 6.67 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -36.93%	4	0.62%	0.62%
-36.93% to -27.65%	5	0.78%	1.40%
-27.65% to -18.37%	17	2.64%	4.03%
-18.37% to -9.09%	87	13.49%	17.52%
-9.09% to 0.19%	229	35.50%	53.02%
0.19% to 9.47%	199	30.85%	83.88%
9.47% to 18.76%	72	11.16%	95.04%
18.76% to 28.04%	16	2.48%	97.52%
28.04% to 37.32%	7	1.09%	98.60%
Greater than 37.32%	9	1.40%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.18%	12.57%	78.60%	68.27%
±2σ	-24.56%	24.94%	95.50%	95.45%
±3σ	-36.93%	37.32%	97.98%	99.73%

Descriptive Statistics

Mean	0.193%
Standard Error	0.487%
Median	-0.235%
Mode	0.000%
Standard Deviation	12.375%
Sample Variance	153.134%
Kurtosis	6.67
Skewness	1.06
Range	132.484%
Minimum	-45.060%
Maximum	87.424%
Count	645

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	9.163%	310	48.06%	4.404%
Negative	-8.256%	329	51.01%	-4.211%
Zero	0.000%	6	0.93%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	20.20%
12	1 Quarter	22.62%
52	1 Year	22.11%
156	3 Years	20.54%
260	5 Years	19.38%
520	10 Years	17.13%
1,040	20 Years	17.05%
2,600	50 Years	17.05%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.34. Further, an Excess Kurtosis of 16.03 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -20.92%	0	0.00%	0.00%
-20.92% to -10.96%	0	0.00%	0.00%
-10.96% to -1.00%	0	0.00%	0.00%
-1.00% to 8.96%	73	11.30%	11.30%
8.96% to 18.93%	354	54.80%	66.10%
18.93% to 28.89%	138	21.36%	87.46%
28.89% to 38.85%	44	6.81%	94.27%
38.85% to 48.81%	16	2.48%	96.75%
48.81% to 58.77%	8	1.24%	97.99%
Greater than 58.77%	13	2.01%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	5.64%	32.21%	90.09%	68.27%
±2σ	-7.64%	45.49%	96.44%	95.45%
±3σ	-20.92%	58.77%	97.99%	99.73%

Descriptive Statistics

Mean	18.926%
Standard Error	0.523%
Median	15.578%
Mode	16.667%
Standard Deviation	13.281%
Sample Variance	176.376%
Kurtosis	16.03
Skewness	3.34
Range	116.787%
Minimum	4.348%
Maximum	121.135%
Count	646

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	18.926%	646	100.00%	18.926%
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 1.41. Further, an Excess Kurtosis of 8.12 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -36.69%	2	0.31%	0.31%
-36.69% to -27.18%	7	1.08%	1.39%
-27.18% to -17.66%	14	2.17%	3.56%
-17.66% to -8.15%	92	14.24%	17.80%
-8.15% to 1.37%	236	36.53%	54.33%
1.37% to 10.88%	195	30.19%	84.52%
10.88% to 20.39%	71	10.99%	95.51%
20.39% to 29.91%	17	2.63%	98.14%
29.91% to 39.42%	4	0.62%	98.76%
Greater than 39.42%	8	1.24%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-11.32%	14.05%	80.50%	68.27%
±2σ	-24.00%	26.74%	96.13%	95.45%
±3σ	-36.69%	39.42%	98.45%	99.73%

Descriptive Statistics

Mean	1.366%
Standard Error	0.499%
Median	0.619%
Mode	0.000%
Standard Deviation	12.685%
Sample Variance	160.901%
Kurtosis	8.12
Skewness	1.41
Range	131.392%
Minimum	-46.609%
Maximum	84.783%
Count	646

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	9.446%	344	53.25%	5.030%
Negative	-7.941%	298	46.13%	-3.663%
Zero	0.000%	4	0.62%	0.000%

Close-to-Close Returns

Gram-Charlier Analysis: Data reveals Right Skewed (Positive Asymmetry) with a Skewness factor of 0.77. Further, an Excess Kurtosis of 1.19 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -74.27%	0	0.00%	0.00%
-74.27% to -55.55%	0	0.00%	0.00%
-55.55% to -36.84%	6	4.05%	4.05%
-36.84% to -18.12%	26	17.57%	21.62%
-18.12% to 0.60%	51	34.46%	56.08%
0.60% to 19.32%	38	25.68%	81.76%
19.32% to 38.04%	13	8.78%	90.54%
38.04% to 56.76%	11	7.43%	97.97%
56.76% to 75.48%	0	0.00%	97.97%
Greater than 75.48%	3	2.03%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-24.36%	25.56%	71.62%	68.27%
±2σ	-49.31%	50.52%	93.92%	95.45%
±3σ	-74.27%	75.48%	97.97%	99.73%

Descriptive Statistics

Mean	0.603%
Standard Error	2.052%
Median	-2.186%
Mode	-53.057%
Standard Deviation	24.959%
Sample Variance	622.935%
Kurtosis	1.19
Skewness	0.77
Range	139.671%
Minimum	-53.057%
Maximum	86.614%
Count	148

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	21.080%	68	45.95%	9.685%
Negative	-16.803%	80	54.05%	-9.083%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	50.79%
12	1 Year	50.80%
36	3 Years	43.90%
60	5 Years	40.85%
120	10 Years	36.17%
240	20 Years	36.72%
600	50 Years	36.72%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.79. Further, an Excess Kurtosis of 3.43 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -35.64%	0	0.00%	0.00%
-35.64% to -15.28%	0	0.00%	0.00%
-15.28% to 5.08%	0	0.00%	0.00%
5.08% to 25.43%	28	18.79%	18.79%
25.43% to 45.79%	69	46.31%	65.10%
45.79% to 66.15%	30	20.13%	85.23%
66.15% to 86.50%	8	5.37%	90.60%
86.50% to 106.86%	7	4.70%	95.30%
106.86% to 127.22%	4	2.68%	97.99%
Greater than 127.22%	3	2.01%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	18.65%	72.93%	80.54%	68.27%
±2σ	-8.50%	100.08%	95.30%	95.45%
±3σ	-35.64%	127.22%	97.99%	99.73%

Descriptive Statistics

Mean	45.790%
Standard Error	2.224%
Median	39.032%
Mode	32.450%
Standard Deviation	27.143%
Sample Variance	736.755%
Kurtosis	3.43
Skewness	1.79
Range	140.646%
Minimum	12.601%
Maximum	153.247%
Count	149

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	45.790%	149	100.00%	45.790%
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 1.16. Further, an Excess Kurtosis of 2.84 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -74.25%	0	0.00%	0.00%
-74.25% to -54.15%	0	0.00%	0.00%
-54.15% to -34.05%	3	2.01%	2.01%
-34.05% to -13.95%	30	20.13%	22.15%
-13.95% to 6.15%	49	32.89%	55.03%
6.15% to 26.25%	40	26.85%	81.88%
26.25% to 46.35%	17	11.41%	93.29%
46.35% to 66.45%	7	4.70%	97.99%
66.45% to 86.55%	0	0.00%	97.99%
Greater than 86.55%	3	2.01%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-20.65%	32.95%	73.15%	68.27%
±2σ	-47.45%	59.75%	95.97%	95.45%
±3σ	-74.25%	86.55%	97.99%	99.73%

Descriptive Statistics

Mean	6.152%
Standard Error	2.196%
Median	3.203%
Mode	-47.820%
Standard Deviation	26.800%
Sample Variance	718.222%
Kurtosis	2.84
Skewness	1.16
Range	157.949%
Minimum	-47.820%
Maximum	110.129%
Count	149

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	23.543%	82	55.03%	12.957%
Negative	-15.133%	67	44.97%	-6.805%
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 0.55. Further, an Excess Kurtosis of -0.31 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -103.94%	0	0.00%	0.00%
-103.94% to -78.66%	0	0.00%	0.00%
-78.66% to -53.38%	1	2.04%	2.04%
-53.38% to -28.10%	13	26.53%	28.57%
-28.10% to -2.82%	11	22.45%	51.02%
-2.82% to 22.46%	13	26.53%	77.55%
22.46% to 47.75%	7	14.29%	91.84%
47.75% to 73.03%	3	6.12%	97.96%
73.03% to 98.31%	1	2.04%	100.00%
Greater than 98.31%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-36.52%	30.89%	63.27%	68.27%
±2σ	-70.23%	64.60%	97.96%	95.45%
±3σ	-103.94%	98.31%	100.00%	99.73%

Descriptive Statistics

Mean	-2.816%
Standard Error	4.815%
Median	-3.981%
Mode	-53.408%
Standard Deviation	33.708%
Sample Variance	1136.210%
Kurtosis	-0.31
Skewness	0.55
Range	142.059%
Minimum	-53.408%
Maximum	88.651%
Count	49

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	28.114%	22	44.90%	12.623%
Negative	-28.019%	27	55.10%	-15.439%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Quarters	Horizon	Average Quarterly True Range %
4	1 Year	82.38%
12	3 Years	71.56%
20	5 Years	67.27%
40	10 Years	61.64%
80	20 Years	62.61%
200	50 Years	62.61%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.05. Further, an Excess Kurtosis of 1.07 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -47.94%	0	0.00%	0.00%
-47.94% to -13.38%	0	0.00%	0.00%
-13.38% to 21.19%	0	0.00%	0.00%
21.19% to 55.76%	12	24.00%	24.00%
55.76% to 90.33%	17	34.00%	58.00%
90.33% to 124.90%	10	20.00%	78.00%
124.90% to 159.47%	8	16.00%	94.00%
159.47% to 194.03%	1	2.00%	96.00%
194.03% to 228.60%	1	2.00%	98.00%
Greater than 228.60%	1	2.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	44.24%	136.42%	68.00%	68.27%
±2σ	-1.85%	182.51%	96.00%	95.45%
±3σ	-47.94%	228.60%	98.00%	99.73%

Descriptive Statistics

Mean	90.329%
Standard Error	6.518%
Median	74.614%
Mode	24.638%
Standard Deviation	46.091%
Sample Variance	2124.397%
Kurtosis	1.07
Skewness	1.05
Range	216.434%
Minimum	24.638%
Maximum	241.072%
Count	50

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	90.329%	50	100.00%	90.329%
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 0.59. Further, an Excess Kurtosis of -0.40 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -107.76%	0	0.00%	0.00%
-107.76% to -77.13%	0	0.00%	0.00%
-77.13% to -46.50%	2	4.00%	4.00%
-46.50% to -15.87%	10	20.00%	24.00%
-15.87% to 14.77%	16	32.00%	56.00%
14.77% to 45.40%	9	18.00%	74.00%
45.40% to 76.03%	8	16.00%	90.00%
76.03% to 106.66%	4	8.00%	98.00%
106.66% to 137.29%	1	2.00%	100.00%
Greater than 137.29%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-26.08%	55.61%	68.00%	68.27%
±2σ	-66.92%	96.45%	94.00%	95.45%
±3σ	-107.76%	137.29%	100.00%	99.73%

Descriptive Statistics

Mean	14.766%
Standard Error	5.776%
Median	5.109%
Mode	-48.840%
Standard Deviation	40.841%
Sample Variance	1668.000%
Kurtosis	-0.40
Skewness	0.59
Range	165.721%
Minimum	-48.840%
Maximum	116.881%
Count	50

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	44.757%	27	54.00%	24.169%
Negative	-20.440%	23	46.00%	-9.402%
Zero	0.000%	0	0.00%	0.000%

KPTI Deep Analytical Report