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%	5.25%	4.97%	8.57%	25.24%	24.82%	7.51	12.45	5.84	16.02
Weekly	12.38%	13.31%	12.69%	19.29%	55.57%	54.63%	5.55	16.86	3.18	29.38
Monthly	25.00%	27.09%	26.84%	40.55%	115.67%	113.72%	3.04	30.74	0.96	97.75
Quarterly	34.46%	46.19%	41.27%	66.46%	200.34%	196.97%	1.30	71.70	0.18	531.62
Annual	—	—	—	—	400.69%	393.95%	0.18	531.62	0.00	29226.39
Option Time Horizon	—	—	—	—	154.12%	151.53%	2.07	45.16	0.44	210.91

Calculation Notes: Computed at 2026-02-22T14:22 UTC | Reference Contract: Call option expiring 2026-04-17 (54 days) | Spot Price (S): $9.67 | Strike (K): $10.00 (first OTM call ≥ spot) | Observed Mid-Price: $5.3000 (Bid $3.9000 / Ask $6.7000) | Risk-Free Rate (r): 3.72% (1-Month CMT, US Treasury) | Dividend Yield (q): 0.00% | Binomial IV (Annual): 400.69% (CRR binomial tree, American, n=20 steps) | BS IV: 396.57% (European Black-Scholes) | Yahoo IV (Annual): 393.95% → 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 2.35. Further, an Excess Kurtosis of 39.98 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	0.62%	1.00%
-12.43% to -8.27%	81	2.62%	3.63%
-8.27% to -4.12%	366	11.85%	15.47%
-4.12% to 0.04%	1136	36.78%	52.25%
0.04% to 4.20%	1012	32.76%	85.01%
4.20% to 8.36%	321	10.39%	95.40%
8.36% to 12.52%	98	3.17%	98.58%
12.52% to 16.68%	19	0.62%	99.19%
Greater than 16.68%	25	0.81%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-5.50%	5.59%	80.32%	68.27%
±2σ	-11.05%	11.13%	96.70%	95.45%
±3σ	-16.59%	16.68%	98.80%	99.73%

Descriptive Statistics

Mean	0.043%
Standard Error	0.100%
Median	-0.099%
Mode	0.000%
Standard Deviation	5.545%
Sample Variance	30.751%
Kurtosis	39.98
Skewness	2.35
Range	136.989%
Minimum	-43.478%
Maximum	93.511%
Count	3089

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.805%	1475	47.75%	1.817%
Negative	-3.503%	1564	50.63%	-1.774%
Zero	0.000%	50	1.62%	0.000%

Average True Range Percentage (ATRP)

Trading Days	Horizon	Average Daily True Range %
5	1 Week	10.51%
20	1 Month	11.48%
60	1 Quarter	9.13%
250	1 Year	9.88%
750	3 Years	9.25%
1,250	5 Years	8.57%
2,500	10 Years	7.49%
5,000	20 Years	7.40%
12,500	50 Years	7.40%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 5.88. Further, an Excess Kurtosis of 71.98 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -8.18%	0	0.00%	0.00%
-8.18% to -4.25%	0	0.00%	0.00%
-4.25% to -0.32%	0	0.00%	0.00%
-0.32% to 3.62%	363	11.75%	11.75%
3.62% to 7.55%	1573	50.91%	62.65%
7.55% to 11.49%	764	24.72%	87.38%
11.49% to 15.42%	242	7.83%	95.21%
15.42% to 19.35%	73	2.36%	97.57%
19.35% to 23.29%	33	1.07%	98.64%
Greater than 23.29%	42	1.36%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	2.31%	12.80%	89.81%	68.27%
±2σ	-2.94%	18.04%	96.96%	95.45%
±3σ	-8.18%	23.29%	98.64%	99.73%

Descriptive Statistics

Mean	7.552%
Standard Error	0.094%
Median	6.400%
Mode	5.000%
Standard Deviation	5.245%
Sample Variance	27.508%
Kurtosis	71.98
Skewness	5.88
Range	96.137%
Minimum	0.502%
Maximum	96.639%
Count	3090

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	7.552%	3090	100.00%	7.552%
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 0.90. Further, an Excess Kurtosis of 24.77 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%	1.42%
-10.89% to -7.17%	116	3.75%	5.18%
-7.17% to -3.44%	373	12.07%	17.25%
-3.44% to 0.29%	1010	32.69%	49.94%
0.29% to 4.02%	1032	33.40%	83.33%
4.02% to 7.75%	371	12.01%	95.34%
7.75% to 11.48%	99	3.20%	98.54%
11.48% to 15.21%	25	0.81%	99.35%
Greater than 15.21%	20	0.65%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-4.68%	5.27%	77.02%	68.27%
±2σ	-9.65%	10.24%	95.70%	95.45%
±3σ	-14.62%	15.21%	98.77%	99.73%

Descriptive Statistics

Mean	0.294%
Standard Error	0.089%
Median	0.300%
Mode	0.000%
Standard Deviation	4.973%
Sample Variance	24.732%
Kurtosis	24.77
Skewness	0.90
Range	126.703%
Minimum	-48.005%
Maximum	78.698%
Count	3090

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	3.558%	1623	52.52%	1.869%
Negative	-3.417%	1424	46.08%	-1.575%
Zero	0.000%	43	1.39%	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 6.70 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -36.89%	4	0.62%	0.62%
-36.89% to -27.61%	5	0.78%	1.40%
-27.61% to -18.33%	17	2.65%	4.06%
-18.33% to -9.04%	86	13.42%	17.47%
-9.04% to 0.24%	229	35.73%	53.20%
0.24% to 9.52%	197	30.73%	83.93%
9.52% to 18.80%	71	11.08%	95.01%
18.80% to 28.08%	16	2.50%	97.50%
28.08% to 37.37%	7	1.09%	98.60%
Greater than 37.37%	9	1.40%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-12.14%	12.61%	78.63%	68.27%
±2σ	-24.51%	24.99%	95.48%	95.45%
±3σ	-36.89%	37.37%	97.97%	99.73%

Descriptive Statistics

Mean	0.238%
Standard Error	0.489%
Median	-0.221%
Mode	0.000%
Standard Deviation	12.376%
Sample Variance	153.173%
Kurtosis	6.70
Skewness	1.06
Range	132.484%
Minimum	-45.060%
Maximum	87.424%
Count	641

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	9.177%	309	48.21%	4.424%
Negative	-8.230%	326	50.86%	-4.186%
Zero	0.000%	6	0.94%	0.000%

Average True Range Percentage (ATRP)

Weeks	Horizon	Average Weekly True Range %
4	1 Month	27.18%
12	1 Quarter	21.43%
52	1 Year	22.74%
156	3 Years	20.33%
260	5 Years	19.29%
520	10 Years	17.15%
1,040	20 Years	17.03%
2,600	50 Years	17.03%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 3.35. Further, an Excess Kurtosis of 16.05 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -21.03%	0	0.00%	0.00%
-21.03% to -11.05%	0	0.00%	0.00%
-11.05% to -1.07%	0	0.00%	0.00%
-1.07% to 8.90%	70	10.90%	10.90%
8.90% to 18.88%	356	55.45%	66.36%
18.88% to 28.86%	136	21.18%	87.54%
28.86% to 38.84%	43	6.70%	94.24%
38.84% to 48.82%	16	2.49%	96.73%
48.82% to 58.80%	8	1.25%	97.98%
Greater than 58.80%	13	2.02%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	5.58%	32.19%	90.19%	68.27%
±2σ	-7.73%	45.50%	96.42%	95.45%
±3σ	-21.03%	58.80%	97.98%	99.73%

Descriptive Statistics

Mean	18.884%
Standard Error	0.525%
Median	15.483%
Mode	16.667%
Standard Deviation	13.306%
Sample Variance	177.046%
Kurtosis	16.05
Skewness	3.35
Range	116.787%
Minimum	4.348%
Maximum	121.135%
Count	642

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	18.884%	642	100.00%	18.884%
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 1.43. Further, an Excess Kurtosis of 8.20 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -36.74%	2	0.31%	0.31%
-36.74% to -27.23%	7	1.09%	1.40%
-27.23% to -17.71%	14	2.18%	3.58%
-17.71% to -8.20%	91	14.17%	17.76%
-8.20% to 1.31%	236	36.76%	54.52%
1.31% to 10.83%	194	30.22%	84.74%
10.83% to 20.34%	69	10.75%	95.48%
20.34% to 29.85%	17	2.65%	98.13%
29.85% to 39.37%	4	0.62%	98.75%
Greater than 39.37%	8	1.25%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-11.37%	14.00%	81.15%	68.27%
±2σ	-24.06%	26.68%	96.11%	95.45%
±3σ	-36.74%	39.37%	98.44%	99.73%

Descriptive Statistics

Mean	1.313%
Standard Error	0.501%
Median	0.549%
Mode	0.000%
Standard Deviation	12.685%
Sample Variance	160.908%
Kurtosis	8.20
Skewness	1.43
Range	131.392%
Minimum	-46.609%
Maximum	84.783%
Count	642

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	9.430%	340	52.96%	4.994%
Negative	-7.931%	298	46.42%	-3.681%
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 0.76. 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 -74.21%	0	0.00%	0.00%
-74.21% to -55.46%	0	0.00%	0.00%
-55.46% to -36.70%	6	4.08%	4.08%
-36.70% to -17.95%	25	17.01%	21.09%
-17.95% to 0.80%	51	34.69%	55.78%
0.80% to 19.56%	38	25.85%	81.63%
19.56% to 38.31%	13	8.84%	90.48%
38.31% to 57.06%	11	7.48%	97.96%
57.06% to 75.81%	0	0.00%	97.96%
Greater than 75.81%	3	2.04%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-24.20%	25.81%	72.11%	68.27%
±2σ	-49.21%	50.81%	93.88%	95.45%
±3σ	-74.21%	75.81%	97.96%	99.73%

Descriptive Statistics

Mean	0.803%
Standard Error	2.062%
Median	-2.162%
Mode	-53.057%
Standard Deviation	25.004%
Sample Variance	625.215%
Kurtosis	1.19
Skewness	0.76
Range	139.671%
Minimum	-53.057%
Maximum	86.614%
Count	147

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	21.139%	68	46.26%	9.779%
Negative	-16.701%	79	53.74%	-8.975%
Zero	0.000%	0	0.00%	0.000%

Average True Range Percentage (ATRP)

Months	Horizon	Average Monthly True Range %
3	1 Quarter	48.72%
12	1 Year	51.73%
36	3 Years	43.68%
60	5 Years	40.55%
120	10 Years	36.33%
240	20 Years	36.63%
600	50 Years	36.63%

High-to-Low Returns

Gram-Charlier Analysis: Data reveals Highly Right Skewed (Extreme Positive Tail Risk) with a Skewness factor of 1.80. Further, an Excess Kurtosis of 3.52 proves the distribution possesses Extreme Fat Tails (Frequent Outliers, Leptokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -35.60%	0	0.00%	0.00%
-35.60% to -15.28%	0	0.00%	0.00%
-15.28% to 5.04%	0	0.00%	0.00%
5.04% to 25.36%	28	18.92%	18.92%
25.36% to 45.68%	68	45.95%	64.86%
45.68% to 66.00%	30	20.27%	85.14%
66.00% to 86.32%	8	5.41%	90.54%
86.32% to 106.64%	7	4.73%	95.27%
106.64% to 126.96%	4	2.70%	97.97%
Greater than 126.96%	3	2.03%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	18.59%	72.77%	81.08%	68.27%
±2σ	-8.51%	99.87%	95.27%	95.45%
±3σ	-35.60%	126.96%	97.97%	99.73%

Descriptive Statistics

Mean	45.680%
Standard Error	2.227%
Median	38.874%
Mode	32.450%
Standard Deviation	27.094%
Sample Variance	734.094%
Kurtosis	3.52
Skewness	1.80
Range	140.646%
Minimum	12.601%
Maximum	153.247%
Count	148

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	45.680%	148	100.00%	45.680%
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 1.17. Further, an Excess Kurtosis of 2.88 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -74.54%	0	0.00%	0.00%
-74.54% to -54.41%	0	0.00%	0.00%
-54.41% to -34.28%	3	2.03%	2.03%
-34.28% to -14.15%	30	20.27%	22.30%
-14.15% to 5.98%	48	32.43%	54.73%
5.98% to 26.11%	40	27.03%	81.76%
26.11% to 46.24%	17	11.49%	93.24%
46.24% to 66.37%	7	4.73%	97.97%
66.37% to 86.50%	0	0.00%	97.97%
Greater than 86.50%	3	2.03%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-20.86%	32.82%	72.97%	68.27%
±2σ	-47.70%	59.66%	95.95%	95.45%
±3σ	-74.54%	86.50%	97.97%	99.73%

Descriptive Statistics

Mean	5.981%
Standard Error	2.206%
Median	2.701%
Mode	-47.820%
Standard Deviation	26.839%
Sample Variance	720.308%
Kurtosis	2.88
Skewness	1.17
Range	157.949%
Minimum	-47.820%
Maximum	110.129%
Count	148

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	23.469%	81	54.73%	12.845%
Negative	-15.160%	67	45.27%	-6.863%
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 0.51. Further, an Excess Kurtosis of -0.50 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -105.38%	0	0.00%	0.00%
-105.38% to -79.53%	0	0.00%	0.00%
-79.53% to -53.69%	0	0.00%	0.00%
-53.69% to -27.84%	14	28.57%	28.57%
-27.84% to -2.00%	11	22.45%	51.02%
-2.00% to 23.84%	12	24.49%	75.51%
23.84% to 49.69%	9	18.37%	93.88%
49.69% to 75.53%	2	4.08%	97.96%
75.53% to 101.37%	1	2.04%	100.00%
Greater than 101.37%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-36.46%	32.46%	61.22%	68.27%
±2σ	-70.92%	66.91%	97.96%	95.45%
±3σ	-105.38%	101.37%	100.00%	99.73%

Descriptive Statistics

Mean	-2.002%
Standard Error	4.923%
Median	-3.981%
Mode	-53.408%
Standard Deviation	34.458%
Sample Variance	1187.359%
Kurtosis	-0.50
Skewness	0.51
Range	142.059%
Minimum	-53.408%
Maximum	88.651%
Count	49

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	29.927%	22	44.90%	13.437%
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	78.35%
12	3 Years	70.22%
20	5 Years	66.46%
40	10 Years	61.23%
80	20 Years	62.28%
200	50 Years	62.28%

High-to-Low 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 1.06 proves the distribution possesses Fat Tails (Higher Outlier Risk).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -48.60%	0	0.00%	0.00%
-48.60% to -13.96%	0	0.00%	0.00%
-13.96% to 20.68%	0	0.00%	0.00%
20.68% to 55.33%	12	24.00%	24.00%
55.33% to 89.97%	17	34.00%	58.00%
89.97% to 124.62%	10	20.00%	78.00%
124.62% to 159.26%	8	16.00%	94.00%
159.26% to 193.91%	1	2.00%	96.00%
193.91% to 228.55%	1	2.00%	98.00%
Greater than 228.55%	1	2.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	43.78%	136.16%	68.00%	68.27%
±2σ	-2.41%	182.36%	96.00%	95.45%
±3σ	-48.60%	228.55%	98.00%	99.73%

Descriptive Statistics

Mean	89.973%
Standard Error	6.533%
Median	73.650%
Mode	24.638%
Standard Deviation	46.192%
Sample Variance	2133.678%
Kurtosis	1.06
Skewness	1.06
Range	216.434%
Minimum	24.638%
Maximum	241.072%
Count	50

Frequency Breakdown

Category	Avg Return	Count	Probability	Freq Adj Return
Positive	89.973%	50	100.00%	89.973%
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 0.58. Further, an Excess Kurtosis of -0.44 proves the distribution possesses Normal Tails (Mesokurtic).

Bin Distribution Over Raw Range

Bin / Range	Frequency	Prob	Cum. Prob
Less than -109.54%	0	0.00%	0.00%
-109.54% to -78.58%	0	0.00%	0.00%
-78.58% to -47.63%	1	2.00%	2.00%
-47.63% to -16.68%	12	24.00%	26.00%
-16.68% to 14.27%	15	30.00%	56.00%
14.27% to 45.22%	9	18.00%	74.00%
45.22% to 76.17%	8	16.00%	90.00%
76.17% to 107.12%	4	8.00%	98.00%
107.12% to 138.08%	1	2.00%	100.00%
Greater than 138.08%	0	0.00%	100.00%

Standard Deviation Interval Analysis

SD	Lower Bnd	Upper Bnd	Actual %	Normal %
±1σ	-27.00%	55.54%	70.00%	68.27%
±2σ	-68.27%	96.81%	94.00%	95.45%
±3σ	-109.54%	138.08%	100.00%	99.73%

Descriptive Statistics

Mean	14.270%
Standard Error	5.836%
Median	5.109%
Mode	-48.840%
Standard Deviation	41.269%
Sample Variance	1703.115%
Kurtosis	-0.44
Skewness	0.58
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	-21.518%	23	46.00%	-9.898%
Zero	0.000%	0	0.00%	0.000%

KPTI Deep Analytical Report