International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 01, January 2019, pp. 1852-1863, Article ID: IJMET_10_01_191
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=01
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
Scopus Indexed
ITO-RENTABILITY FUNCTION FOR STOCK
OPTION DRIFT AND VOLATILITY USING DJI
INDICES
S. O. Edeki* and G. O. Akinlabi
Department of Mathematics, Covenant University, Ota, Nigeria
F. O. Egara and A. M. Okeke
Department of Science Education, University of Nigeria, Nsukka
*
Corresponding author
ABSTRACT
In this paper, the drift and volatility parameters of a stock option are modeled via ItoRentability function based on Dow Jones Industrial (DJI) Average indices. The stock
exchange data of DJI span between November 17, 2015 to April 6, 2018 involving daily
data with 600 sample size. The stock price basic moments viz: expected value and
volatility are derived in the sense of Ito stochastic dynamics. It is noted that the proposed
models will be of immense importance in financial and production engineering, likewise
other areas of applied, and management sciences.
Keywords: option pricing, stochastic model, Ito calculus, stock exchange market
Cite this Article: S. O. Edeki, G. O. Akinlabi, F. O. Egara and A. M. Okeke, Ito-Rentability
Function for Stock Option Drift and Volatility Using Dji Indices, International Journal of
Mechanical Engineering and Technology, 10(01), 2019, pp.1952–1963
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&Type=01
1. INTRODUCTION
In financial engineering, reliable methods of estimating the prices of underlying assets have
provoked critical researches. These have great effects on decision analysis as a vital tool for our
daily events. The state of decision making mostly occur under uncertainty which requires a set of
quantitative decision making approaches to assist the decision maker under the probabilistic
condition. Expected value (EV) has been noted as one of the most widely used decision-making
tools under uncertainty. In addition, the stock price parameter is one of the highly volatile
variables in a stock exchange market while the associated unstable property calls for concern on
the part of investors, since the sudden change in share prices occurs randomly and frequently [110]. As such, expected value, and volatility functions of stock price process are derived using the
stock exchange data of DJI Average indices spanning between November 17, 2015 to April 6,
2018. We make reference to [11-22] on researches related to volatility, stock markets, and
http://www.iaeme.com/IJMET/index.asp
1952
editor@iaeme.com
S. O. Edeki, G. O. Akinlabi, F. O. Egara and A. M. Okeke
solution methods for differential model equations including stochastic differential equations
(SDEs).
2. THE DJI AVERAGE STOCK DATA, MATERIAL AND METHOD
Table 2.1: DJI Average stock data between 17/11/2015-06/04/2018
Stock-Price
SN
Date
StockPrice ( St )
(R )
SN
Date
( St )
(R )
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
11/17/2015
11/18/2015
11/19/2015
11/20/2015
11/23/2015
11/24/2015
11/25/2015
11/27/2015
11/30/2015
12/1/2015
12/2/2015
12/3/2015
12/4/2015
12/7/2015
12/8/2015
12/9/2015
12/10/2015
12/11/2015
12/14/2015
12/15/2015
12/16/2015
12/17/2015
12/18/2015
12/21/2015
12/22/2015
17486.99
17485.49
17739.83
17732.75
17823.61
17770.9
17820.81
17806.04
17802.84
17719.72
17883.14
17741.57
17482.68
17845.49
17703.99
17558.18
17493.17
17574.75
17277.11
17374.78
17530.85
17756.54
17495.04
17154.94
17253.55
-----8.6E-05
0.014546
-0.0004
0.005124
-0.00296
0.002809
-0.00083
-0.00018
-0.00467
0.009222
-0.00792
-0.01459
0.020753
-0.00793
-0.00824
-0.0037
0.004664
-0.01694
0.005653
0.008983
0.012874
-0.01473
-0.01944
0.005748
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
12/23/2015
12/24/2015
12/28/2015
12/29/2015
12/30/2015
12/31/2015
1/4/2016
1/5/2016
1/6/2016
1/7/2016
1/8/2016
1/11/2016
1/12/2016
1/13/2016
1/14/2016
1/15/2016
1/19/2016
1/20/2016
1/21/2016
1/22/2016
1/25/2016
1/26/2016
1/27/2016
1/28/2016
1/29/2016
17427.63
17593.26
17535.66
17547.37
17711.94
17590.66
17405.48
17147.5
17154.83
16888.36
16519.17
16358.71
16419.11
16526.63
16159.01
16354.33
16009.45
15989.45
15768.87
15921.1
16086.46
15893.16
16168.74
15960.28
16090.26
0.01009
0.009504
-0.00327
0.000668
0.009379
-0.00685
-0.01053
-0.01482
0.000427
-0.01553
-0.02186
-0.00971
0.003692
0.006549
-0.02224
0.012087
-0.02109
-0.00125
-0.0138
0.009654
0.010386
-0.01202
0.01734
-0.01289
0.008144
SN
Date
( St )
(R )
SN
Date
StockPrice ( St )
(R )
51
52
53
54
55
56
57
2/1/2016
2/2/2016
2/3/2016
2/4/2016
2/5/2016
2/8/2016
2/9/2016
16453.63
16420.21
16186.2
16329.67
16417.95
16147.51
16005.41
0.022583
-0.00203
-0.01425
0.008864
0.005406
-0.01647
-0.0088
76
77
78
79
80
81
82
3/8/2016
3/9/2016
3/10/2016
3/11/2016
3/14/2016
3/15/2016
3/16/2016
17050.67
16969.17
17006.05
17014.99
17207.49
17217.15
17249.34
0.003495
-0.00478
0.002173
0.000526
0.011314
0.000561
0.00187
Stock-Price
St
St
http://www.iaeme.com/IJMET/index.asp
1953
St
St
editor@iaeme.com
Ito-Rentability Function for Stock Option Drift and Volatility Using Dji Indices
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75
101
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105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
2/10/2016
2/11/2016
2/12/2016
2/16/2016
2/17/2016
2/18/2016
2/19/2016
2/22/2016
2/23/2016
2/24/2016
2/25/2016
2/26/2016
2/29/2016
3/1/2016
3/2/2016
3/3/2016
3/4/2016
3/7/2016
4/13/2016
4/14/2016
4/15/2016
4/18/2016
4/19/2016
4/20/2016
4/21/2016
4/22/2016
4/25/2016
4/26/2016
4/27/2016
4/28/2016
4/29/2016
5/2/2016
5/3/2016
5/4/2016
5/5/2016
5/6/2016
5/9/2016
5/10/2016
5/11/2016
5/12/2016
5/13/2016
5/16/2016
16035.61
15897.82
15691.62
16012.39
16217.98
16483.76
16410.96
16417.13
16610.39
16418.84
16504.38
16712.7
16634.15
16545.67
16851.17
16896.17
16945
16991.29
17741.66
17912.25
17925.95
17890.2
18012.1
18059.49
18092.84
17985.05
17990.94
17987.38
17996.14
18023.88
17813.09
17783.78
17870.75
17735.02
17664.48
17650.3
17743.85
17726.66
17919.03
17711.12
17711.12
17531.76
0.001887
-0.00859
-0.01297
0.020442
0.012839
0.016388
-0.00442
0.000376
0.011772
-0.01153
0.00521
0.012622
-0.0047
-0.00532
0.018464
0.00267
0.00289
0.002732
0.009693
0.009615
0.000765
-0.00199
0.006814
0.002631
0.001847
-0.00596
0.000327
-0.0002
0.000487
0.001541
-0.0117
-0.00165
0.00489
-0.0076
-0.00398
-0.0008
0.0053
-0.00097
0.010852
-0.0116
0
-0.01013
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83
84
85
86
87
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89
90
91
92
93
94
95
96
97
98
99
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126
127
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129
130
131
132
133
134
135
136
137
138
139
140
141
142
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144
145
146
147
148
149
1954
3/17/2016
3/18/2016
3/21/2016
3/22/2016
3/23/2016
3/24/2016
3/28/2016
3/29/2016
3/30/2016
3/31/2016
4/1/2016
4/4/2016
4/5/2016
4/6/2016
4/7/2016
4/8/2016
4/11/2016
4/12/2016
5/18/2016
5/19/2016
5/20/2016
5/23/2016
5/24/2016
5/25/2016
5/26/2016
5/27/2016
5/31/2016
6/1/2016
6/2/2016
6/3/2016
6/6/2016
6/7/2016
6/8/2016
6/9/2016
6/10/2016
6/13/2016
6/14/2016
6/15/2016
6/16/2016
6/17/2016
6/20/2016
6/21/2016
17321.38
17481.49
17589.7
17602.71
17588.81
17485.33
17526.08
17512.58
17652.36
17716.05
17661.74
17799.39
17718.03
17605.45
17687.28
17555.39
17586.48
17571.34
17501.28
17514.16
17437.32
17507.04
17525.19
17735.09
17859.52
17826.85
17891.5
17754.55
17789.05
17799.8
17825.69
17936.22
17931.91
17969.98
17938.82
17830.5
17710.77
17703.65
17602.23
17733.44
17736.87
17827.33
0.004176
0.009243
0.00619
0.00074
-0.00079
-0.00588
0.002331
-0.00077
0.007982
0.003608
-0.00307
0.007794
-0.00457
-0.00635
0.004648
-0.00746
0.001771
-0.00086
-0.01131
0.000736
-0.00439
0.003998
0.001037
0.011977
0.007016
-0.00183
0.003627
-0.00765
0.001943
0.000604
0.001454
0.006201
-0.00024
0.002123
-0.00173
-0.00604
-0.00671
-0.0004
-0.00573
0.007454
0.000193
0.0051
editor@iaeme.com
S. O. Edeki, G. O. Akinlabi, F. O. Egara and A. M. Okeke
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151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
5/17/2016
6/23/2016
6/24/2016
6/27/2016
6/28/2016
6/29/2016
6/30/2016
7/1/2016
7/5/2016
7/6/2016
7/7/2016
7/8/2016
7/11/2016
7/12/2016
7/13/2016
7/14/2016
7/15/2016
7/18/2016
7/19/2016
7/20/2016
7/21/2016
7/22/2016
7/25/2016
7/26/2016
7/27/2016
7/28/2016
9/2/2016
9/6/2016
9/7/2016
9/8/2016
9/9/2016
9/12/2016
9/13/2016
9/14/2016
9/15/2016
9/16/2016
9/19/2016
9/20/2016
9/21/2016
9/22/2016
9/23/2016
9/26/2016
9/27/2016
17701.46
17844.11
17946.63
17355.21
17190.51
17456.02
17712.76
17924.24
17904.45
17807.47
17924.24
17971.22
18161.53
18259.12
18356.78
18414.3
18508.88
18521.55
18503.12
18582.7
18589.96
18524.15
18554.49
18497.37
18473.27
18461.01
18466.01
18493.4
18527.71
18486.69
18404.17
18028.95
18262.99
18073.39
18024.91
18217.21
18154.82
18175.36
18164.96
18343.76
18377.36
18217.76
18099.21
0.00968
0.000641
0.005745
-0.03295
-0.00949
0.015445
0.014708
0.011939
-0.0011
-0.00542
0.006557
0.002621
0.01059
0.005373
0.005349
0.003134
0.005136
0.000685
-0.001
0.004301
0.000391
-0.00354
0.001638
-0.00308
-0.0013
-0.00066
0.003775
0.001483
0.001855
-0.00221
-0.00446
-0.02039
0.012981
-0.01038
-0.00268
0.010669
-0.00342
0.001131
-0.00057
0.009843
0.001832
-0.00868
-0.00651
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150
176
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184
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199
200
226
227
228
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231
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233
234
235
236
237
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239
240
241
242
1955
6/22/2016
7/29/2016
8/1/2016
8/2/2016
8/3/2016
8/4/2016
8/5/2016
8/8/2016
8/9/2016
8/10/2016
8/11/2016
8/12/2016
8/15/2016
8/16/2016
8/17/2016
8/18/2016
8/19/2016
8/22/2016
8/23/2016
8/24/2016
8/25/2016
8/26/2016
8/29/2016
8/30/2016
8/31/2016
9/1/2016
10/10/2016
10/11/2016
10/12/2016
10/13/2016
10/14/2016
10/17/2016
10/18/2016
10/19/2016
10/20/2016
10/21/2016
10/24/2016
10/25/2016
10/26/2016
10/27/2016
10/28/2016
10/31/2016
11/1/2016
17832.67
18442.52
18434.5
18401.15
18313.08
18351.43
18402.8
18541.89
18538.05
18541.48
18519.08
18595.65
18588.59
18614.48
18537.09
18566.54
18585.17
18535.86
18568.94
18537.5
18471.21
18467.92
18421.29
18491.28
18436.7
18396.57
18282.95
18308.43
18132.63
18088.32
18177.35
18135.85
18145.06
18178.21
18161.87
18152.63
18197.14
18206.52
18103.8
18234.81
18193.79
18176.6
18158.24
0.0003
-0.001
-0.00043
-0.00181
-0.00479
0.002094
0.002799
0.007558
-0.00021
0.000185
-0.00121
0.004135
-0.00038
0.001393
-0.00416
0.001589
0.001003
-0.00265
0.001785
-0.00169
-0.00358
-0.00018
-0.00252
0.003799
-0.00295
-0.00218
-0.00068
0.001394
-0.0096
-0.00244
0.004922
-0.00228
0.000508
0.001827
-0.0009
-0.00051
0.002452
0.000515
-0.00564
0.007237
-0.00225
-0.00094
-0.00101
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9/28/2016
9/29/2016
9/30/2016
10/3/2016
10/4/2016
10/5/2016
10/6/2016
10/7/2016
11/14/2016
11/15/2016
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1/27/2017
1/30/2017
1/31/2017
2/1/2017
2/2/2017
2/3/2017
2/6/2017
2/7/2017
2/8/2017
2/9/2017
18240.22
18322.88
18181.8
18279.6
18267.68
18205.5
18280.42
18295.35
18876.77
18858.21
18909.85
18866.22
18905.33
18898.68
18970.39
19015.52
19093.72
19122.14
19064.07
19135.64
19149.2
19161.25
19244.35
19219.91
19241.99
19559.94
19631.35
19770.2
19852.21
19876.13
19811.5
19909.01
19836.66
20103.36
20028.62
19913.16
19923.81
19858.34
19964.21
20025.61
20107.62
20049.29
20061.73
0.007791
0.004532
-0.0077
0.005379
-0.00065
-0.0034
0.004115
0.000817
0.005064
-0.00098
0.002738
-0.00231
0.002073
-0.00035
0.003794
0.002379
0.004112
0.001488
-0.00304
0.003754
0.000709
0.000629
0.004337
-0.00127
0.001149
0.016524
0.003651
0.007073
0.004148
0.001205
-0.00325
0.004922
-0.00363
0.00135
-0.00372
-0.00576
0.000535
-0.00329
0.005331
0.003075
0.004095
-0.0029
0.000621
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1956
11/2/2016
11/3/2016
11/4/2016
11/7/2016
11/8/2016
11/9/2016
11/10/2016
11/11/2016
12/20/2016
12/21/2016
12/22/2016
12/23/2016
12/27/2016
12/28/2016
12/29/2016
12/30/2016
1/3/2017
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1/5/2017
1/6/2017
1/9/2017
1/10/2017
1/11/2017
1/12/2017
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1/17/2017
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1/23/2017
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3/6/2017
3/7/2017
3/8/2017
3/9/2017
3/10/2017
3/13/2017
3/14/2017
3/15/2017
3/16/2017
3/17/2017
18017.72
17978.75
17928.35
17994.64
18251.38
18317.26
18603.14
18781.65
19920.59
19968.97
19922.68
19908.61
19943.46
19964.31
19835.46
19833.17
19872.86
19890.94
19924.56
19906.96
19931.41
19876.35
19887.38
19926.21
19912.54
19848.82
19822.73
19813.55
19795.06
19794.79
19794.68
19994.48
20076.25
20955.71
20934.89
20940.44
20864.32
20919.01
20899.28
20848.6
20874.78
20969.27
20965.37
-0.00774
-0.00216
-0.0028
0.003698
0.014268
0.00361
0.015607
0.009596
0.004231
0.002429
-0.00232
-0.00071
0.001751
0.001045
-0.00645
-0.00012
0.002001
0.00091
0.00169
-0.00088
0.001228
-0.00276
0.000555
0.001952
-0.00069
-0.0032
-0.00131
-0.00046
-0.00093
-1.4E-05
-5.5E-06
0.010094
0.00409
-0.00252
-0.00099
0.000265
-0.00364
0.002621
-0.00094
-0.00242
0.001256
0.004527
-0.00019
editor@iaeme.com
S. O. Edeki, G. O. Akinlabi, F. O. Egara and A. M. Okeke
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2/10/2017
2/13/2017
2/14/2017
2/15/2017
2/16/2017
2/17/2017
2/21/2017
2/22/2017
2/23/2017
2/24/2017
2/27/2017
2/28/2017
3/1/2017
3/2/2017
3/3/2017
4/10/2017
4/11/2017
4/12/2017
4/13/2017
4/17/2017
4/18/2017
4/19/2017
4/20/2017
4/21/2017
4/24/2017
4/25/2017
4/26/2017
4/27/2017
4/28/2017
5/1/2017
5/2/2017
5/3/2017
5/4/2017
5/5/2017
5/8/2017
5/9/2017
5/10/2017
5/11/2017
5/12/2017
5/15/2017
6/21/2017
6/22/2017
6/23/2017
20211.23
20338.54
20374.22
20504.27
20627.31
20564.13
20663.43
20715.41
20817.21
20751.91
20808.71
20833.88
20957.29
21128.91
21008.75
20668.22
20644.32
20637.95
20561.69
20484.75
20561.39
20503.52
20406.68
20578.1
20723.59
20915.51
21009.95
20991.12
20987.39
20962.73
20941.19
20915
20987.83
20929.04
20991.26
21022.28
20958.49
20925.72
20893.19
20923.63
21466.39
21407.98
21380.92
0.007452
0.006299
0.001754
0.006383
0.006001
-0.00306
0.004829
0.002516
0.004914
-0.00314
0.002737
0.00121
0.005923
0.008189
-0.00569
0.000988
-0.00116
-0.00031
-0.0037
-0.00374
0.003741
-0.00281
-0.00472
0.0084
0.00707
0.009261
0.004515
-0.0009
-0.00018
-0.00117
-0.00103
-0.00125
0.003482
-0.0028
0.002973
0.001478
-0.00303
-0.00156
-0.00155
0.001457
-0.00255
-0.00272
-0.00126
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428
1957
3/20/2017
3/21/2017
3/22/2017
3/23/2017
3/24/2017
3/27/2017
3/28/2017
3/29/2017
3/30/2017
3/31/2017
4/3/2017
4/4/2017
4/5/2017
4/6/2017
4/7/2017
5/16/2017
5/17/2017
5/18/2017
5/19/2017
5/22/2017
5/23/2017
5/24/2017
5/25/2017
5/26/2017
5/30/2017
5/31/2017
6/1/2017
6/2/2017
6/5/2017
6/6/2017
6/7/2017
6/8/2017
6/9/2017
6/12/2017
6/13/2017
6/14/2017
6/15/2017
6/16/2017
6/19/2017
6/20/2017
7/27/2017
7/28/2017
7/31/2017
20916.27
20956.33
20640.42
20645.07
20674.45
20488.35
20542.14
20675.75
20662.79
20700.34
20665.17
20634.94
20745.06
20653.77
20647.81
20984.48
20846.17
20579.65
20698.28
20867.77
20908.67
20949.21
21062.96
21070.15
21045.49
21048.46
21030.55
21142.09
21195.03
21145.48
21171.57
21169.76
21208.96
21259.95
21256.83
21342.71
21291.69
21335.93
21444.75
21521.25
21717.42
21787.51
21863.39
-0.00234
0.001915
-0.01507
0.000225
0.001423
-0.009
0.002625
0.006504
-0.00063
0.001817
-0.0017
-0.00146
0.005337
-0.0044
-0.00029
0.002908
-0.00659
-0.01279
0.005764
0.008189
0.00196
0.001939
0.00543
0.000341
-0.00117
0.000141
-0.00085
0.005304
0.002504
-0.00234
0.001234
-8.6E-05
0.001852
0.002404
-0.00015
0.00404
-0.00239
0.002078
0.0051
0.003567
0.001247
0.003227
0.003483
editor@iaeme.com
Ito-Rentability Function for Stock Option Drift and Volatility Using Dji Indices
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451
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460
461
462
463
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466
467
468
469
470
471
6/26/2017
6/27/2017
6/28/2017
6/29/2017
6/30/2017
7/3/2017
7/5/2017
7/6/2017
7/7/2017
7/10/2017
7/11/2017
7/12/2017
7/13/2017
7/14/2017
7/17/2017
7/18/2017
7/19/2017
7/20/2017
7/21/2017
7/24/2017
7/25/2017
7/26/2017
8/31/2017
9/1/2017
9/5/2017
9/6/2017
9/7/2017
9/8/2017
9/11/2017
9/12/2017
9/13/2017
9/14/2017
9/15/2017
9/18/2017
9/19/2017
9/20/2017
9/21/2017
9/22/2017
9/25/2017
9/26/2017
9/27/2017
9/28/2017
9/29/2017
21434.68
21411.19
21372.36
21487.38
21348.6
21392.3
21492.83
21423.93
21354.66
21381.23
21410.17
21467.93
21537.19
21532.77
21633.97
21589.94
21569.25
21641.54
21591.72
21577.78
21638.56
21690.38
21936.01
21981.77
21912.37
21815.76
21820.38
21764.43
21927.79
22090.56
22103.47
22144.96
22252.44
22297.92
22349.7
22351.38
22414.02
22334.07
22320.47
22322.03
22330.93
22306.83
22358.47
0.002514
-0.0011
-0.00181
0.005382
-0.00646
0.002047
0.004699
-0.00321
-0.00323
0.001244
0.001353
0.002698
0.003226
-0.00021
0.0047
-0.00204
-0.00096
0.003351
-0.0023
-0.00065
0.002817
0.002395
0.003488
0.002086
-0.00316
-0.00441
0.000212
-0.00256
0.007506
0.007423
0.000584
0.001877
0.004853
0.002044
0.002322
7.52E-05
0.002802
-0.00357
-0.00061
6.98E-05
0.000399
-0.00108
0.002315
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491
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494
495
496
1958
8/1/2017
8/2/2017
8/3/2017
8/4/2017
8/7/2017
8/8/2017
8/9/2017
8/10/2017
8/11/2017
8/14/2017
8/15/2017
8/16/2017
8/17/2017
8/18/2017
8/21/2017
8/22/2017
8/23/2017
8/24/2017
8/25/2017
8/28/2017
8/29/2017
8/30/2017
10/6/2017
10/9/2017
10/10/2017
10/11/2017
10/12/2017
10/13/2017
10/16/2017
10/17/2017
10/18/2017
10/19/2017
10/20/2017
10/23/2017
10/24/2017
10/25/2017
10/26/2017
10/27/2017
10/30/2017
10/31/2017
11/1/2017
11/2/2017
11/3/2017
21961.42
22004.36
22007.58
22058.39
22100.2
22095.14
22022.34
21988.2
21883.32
21945.64
22029.91
22031.93
21984.74
21724.88
21671.36
21739.78
21850.27
21839.9
21819.08
21832.5
21718
21859.76
22762.03
22779.73
22784.76
22827.65
22854.85
22876.43
22892.92
22952.41
23087.13
23107.47
23205.18
23348.95
23346.78
23431.09
23380.89
23419.16
23405.75
23369.22
23442.9
23463.24
23549.59
0.004484
0.001955
0.000146
0.002309
0.001895
-0.00023
-0.00329
-0.00155
-0.00477
0.002848
0.00384
9.17E-05
-0.00214
-0.01182
-0.00246
0.003157
0.005082
-0.00047
-0.00095
0.000615
-0.00524
0.006527
0.0041
0.000778
0.000221
0.001882
0.001192
0.000944
0.000721
0.002599
0.00587
0.000881
0.004228
0.006196
-9.3E-05
0.003611
-0.00214
0.001637
-0.00057
-0.00156
0.003153
0.000868
0.00368
editor@iaeme.com
S. O. Edeki, G. O. Akinlabi, F. O. Egara and A. M. Okeke
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519
520
521
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523
524
525
551
552
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554
555
556
557
558
559
560
561
562
563
564
10/2/2017
10/3/2017
10/4/2017
10/5/2017
11/10/2017
11/13/2017
11/14/2017
11/15/2017
11/16/2017
11/17/2017
11/20/2017
11/21/2017
11/22/2017
11/24/2017
11/27/2017
11/28/2017
11/29/2017
11/30/2017
12/1/2017
12/4/2017
12/5/2017
12/6/2017
12/7/2017
12/8/2017
12/11/2017
12/12/2017
12/13/2017
12/14/2017
12/15/2017
1/25/2018
1/26/2018
1/29/2018
1/30/2018
1/31/2018
2/1/2018
2/2/2018
2/5/2018
2/6/2018
2/7/2018
2/8/2018
2/9/2018
2/12/2018
2/13/2018
22423.47
22564.45
22645.67
22669.08
23432.71
23367.47
23388.4
23334.59
23365.34
23433.77
23370.71
23500.15
23597.24
23552.75
23552.86
23625.19
23883.26
24013.8
24305.4
24424.11
24335.01
24171.9
24116.6
24263.26
24338.11
24452.96
24525.19
24631.01
24585.71
26313.06
26466.74
26584.28
26198.45
26268.17
26083.04
26061.79
25337.87
24085.17
24892.87
24902.3
23992.67
24337.76
24540.33
0.002907
0.006287
0.003599
0.001034
-0.00253
-0.00278
0.000896
-0.0023
0.001318
0.002929
-0.00269
0.005539
0.004131
-0.00189
4.64E-06
0.003071
0.010924
0.005466
0.012143
0.004884
-0.00365
-0.0067
-0.00229
0.006081
0.003085
0.004719
0.002954
0.004315
-0.00184
0.001179
0.00584
0.004441
-0.01451
0.002661
-0.00705
-0.00081
-0.02778
-0.04944
0.033535
0.000379
-0.03653
0.014383
0.008323
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576
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583
584
585
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587
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589
1959
11/6/2017
11/7/2017
11/8/2017
11/9/2017
12/18/2017
12/19/2017
12/20/2017
12/21/2017
12/22/2017
12/26/2017
12/27/2017
12/28/2017
12/29/2017
1/2/2018
1/3/2018
1/4/2018
1/5/2018
1/8/2018
1/9/2018
1/10/2018
1/11/2018
1/12/2018
1/16/2018
1/17/2018
1/18/2018
1/19/2018
1/22/2018
1/23/2018
1/24/2018
3/2/2018
3/5/2018
3/6/2018
3/7/2018
3/8/2018
3/9/2018
3/12/2018
3/13/2018
3/14/2018
3/15/2018
3/16/2018
3/19/2018
3/20/2018
3/21/2018
23533.96
23574.03
23542.6
23492.09
24739.56
24834.38
24838.09
24778.26
24764.04
24715.84
24766.52
24807.21
24849.63
24809.35
24850.45
24964.86
25114.92
25308.4
25312.05
25348.13
25398.6
25638.39
25987.62
25910.78
26149.55
25987.35
26025.32
26214.87
26282.07
24394.91
24471.31
24965.89
24758.15
24853.41
25004.89
25372.44
25257.75
25086.97
24837.29
24877.34
24893.69
24650.64
24723.49
-0.00066
0.001703
-0.00133
-0.00215
0.006258
0.003833
0.000149
-0.00241
-0.00057
-0.00195
0.00205
0.001643
0.00171
-0.00162
0.001657
0.004604
0.006011
0.007704
0.000144
0.001425
0.001991
0.009441
0.013621
-0.00296
0.009215
-0.0062
0.001461
0.007283
0.002563
-0.02514
0.003132
0.020211
-0.00832
0.003848
0.006095
0.014699
-0.00452
-0.00676
-0.00995
0.001613
0.000657
-0.00976
0.002955
editor@iaeme.com
Ito-Rentability Function for Stock Option Drift and Volatility Using Dji Indices
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2/16/2018
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2/21/2018
2/22/2018
2/23/2018
2/26/2018
2/27/2018
2/28/2018
3/1/2018
24535.82
25047.82
25165.94
25124.91
24988.06
24855.41
25050.51
25403.35
25735.78
25485.15
25024.04
-0.00018
0.020867
0.004716
-0.00163
-0.00545
-0.00531
0.007849
0.014085
0.013086
-0.00974
-0.01809
590
591
592
593
594
595
596
597
598
599
600
3/22/2018
3/23/2018
3/26/2018
3/27/2018
3/28/2018
3/29/2018
4/2/2018
4/3/2018
4/4/2018
4/5/2018
4/6/2018
24526.01
23995.18
23825.74
24276.62
23883.08
23949.18
24076.6
23698.33
23654.15
24313.91
24373.6
-0.00799
-0.02164
-0.00706
0.018924
-0.01621
0.002768
0.00532
-0.01571
-0.00186
0.027892
0.002455
3. THE STOCK PRICE STOCHASTIC DYNAMICS
Let S (t ) represent the firm stock price of an asset at a specified time t . Then the stochastic
dynamic of the stock price is a stochastic differential equation (SDE) of the form:
dS = µ Sdt + σ SdW , S = S ( t ) , W = W ( t )
(3.1)
Where µ , σ , and W ( t ) denote the drift parameter, volatility parameter, and Brownian
motion (Wiener process) respectively. In an integral form, (3.1) is expressed as:
t
t
S ( t ) = S ( t0 ) + ∫ µ S (τ )dτ + ∫ σ S (τ )dW (τ ) , ∀t ≥ 0
t0
t0
.
(3.2)
In (3.2), the core problem is the calculation of the stochastic integral (the second integral in
RHS). This can be resolved via Ito theorem.
3.1. Ito formula
Suppose X = { X i , t ≥ 0} is an adapted stochastic process, then an Ito SDE associated to X is
defined as:
dX = a ( X , t ) dt + b ( X , t ) dW ( t ) , X ( 0 ) = X 0 , t ∈
+
(3.3)
Where a ( X , t ) and b ( X , t ) are average drift term, and diffusion term respectively, while
dW ( t ) is a Brownian noise. Suppose that X solves (3.3), and h ( X , t ) is taken as a smooth
function such that h :
+
→
, and define:
Z (t ) = h ( X , t ) ,
Then, by Ito, Z solves the SDE:

 ∂h ( X , t )
∂h ( X , t ) 1 2
∂2h ( X , t )  
+ a ( X ,t )
+ b ( X ,t)
 dZ = 
 dt 
∂t
∂X
2
∂X 2  




∂h ( X , t )


+ b( X ,t)
dW ( t )
∂X

.
(3.4)
The proof of (3.4) can be found in [3]. So, the solution of (3.2) which is the stock price at
time t is obtained as follows upon the application of the Ito formula:
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1960
editor@iaeme.com
S. O. Edeki, G. O. Akinlabi, F. O. Egara and A. M. Okeke
S (t ) = S ( t0 ) e
 σ2 
 µ−
 t −t +σ W ( t − t0 )
0

2 


(
)
.
(3.5)
For t0 = 0 , (3.5) becomes:
S (t ) = S0 e
 σ2 
µ−
 t +σ W ( t )

2 


, S ( 0 ) = S0
.
(3.6)
3.2. The Expected Value of the Stock Price (Drift term)
Recall for a GBM W ( t ) , W
(MGF):
(
E e
µW ( t )
)=e
µ2
2
t
N ( o, t ) ; showing that W ( t ) has a moment generating function
, µ∈
(3.7)
Where the operator E ( ⋅) denotes mathematical expectation or expected value with respect to
( ⋅) .
Now, for a GBM in (3.6), we take the mathematical expectation of both sides as follows:
  µ − σ 2 t +σ W ( t ) 

2 


E ( S (t ) ) = E  S 0 e






  µ − σ 2 t 

2  
σW (t )

= E ( S0 ) E  e

E e




(
= S0e
 σ2 
 µ−
t

2 


(
E eσW (t )
).
)
(3.8)
Hence, applying (3.7) in (3.8) yields:
E ( S (t ) ) = S0 e
 σ2 
µ−
t σ 2
t

2 


2
e
= S0e µt = S0 exp ( µt )
.
(3.9)
3.3. The Variance and Volatility of the Stock Price
The variance of S ( t ) is defined and denoted as:
(
Var ( S ( t ) ) = E  S 2 (t )  − E [ S (t )]
)
2
(3.10)
But
 σ2 
t + 2σ W ( t )
2 µ −
2 
2 

0
S 2 (t ) = S e
 2 µ − σ 2 t + 2σ W ( t ) 

2 


E  S 2 (t )  = E  S02 e 




http://www.iaeme.com/IJMET/index.asp
(3.11)
= S02 e 2 µ t +σ
1961
2t
(3.12)
editor@iaeme.com
Ito-Rentability Function for Stock Option Drift and Volatility Using Dji Indices
Hence,
(
Var ( S ( t ) ) = E  S 2 (t )  − E [ S (t )]
)
2
(
= S02 e 2 µt +σ
2t
) − (S e
µt 2
0
)
(3.13)
Equations (3.9) and (3.13) thus represent the expected value and the variance of the stock
price process, S ( t ) respectively.
3.4. The Drift and Volatility Estimation
Here, the stock rentability is defined as [15]:

S ( t j ) − S ( t j −1 )
 R (t j ) =
, j ≥ 1, for discrete time,
S ( t j −1 )

RSi = 
dS ( t )

 R ( t ) = S t , for continuous time.
()

(3.14)
Hence, the drift term µ , and the volatility parameter σ of the rentability function are
estimated
using
unbiased
estimators
give: µ = E [ RSi ] = 5.816 × 10−4 ,
which
σ = Var ( RSi ) = 7.359 ×10−3 , with S0 = 17486.99 for data size of n = 756 , W ( t )
N ( 0, t ) . As
such:
E  S ( t )  = (17486.99 ) e5.816×10
(
−4
t
.
(3.15)
)
2
2
Var ( S ( t ) ) = S02 e 2 µ t eσ t − 1 = (17486.99 ) e
Volatility ( S ( t ) ) =
2
(17486.99 ) e
(
2
)  (5.416×10 ) t
(
2 5.816×10−4 t
−5
e


− 1 .

2
)  e(5.416×10 ) t − 1


2 5.816×10−4 t
−5

.
(3.16)
4. CONCLUDING REMARKS
The research shows an easy approach for estimating the drift and volatility parameters of a stock
option model via Ito-Rentability function based on Dow Jones Industrial Average indices. We
conclude that the result is a good approximation for local prediction which can serve as a
reference point for Stratonovich integral in stock option valuation with a wide application in other
aspects of financial and production engineering.
ACKNOWLEDGEMENTS
The authors1 are indeed grateful to Covenant University for the provision of resources, and
enabling working environment.
REFERENCES
[1]
[2]
[3]
[4]
L. Bachelier, Theorie de la Speculative, Paris: Gauthier- Villar. Translated in Cooler, (1964).
F. Black F, P. Karasinki, Bond and option pricing when short rates lognormal, Finan. Analysis
J., 47 (4), (1991), 52-59.
F. Black, M. Scholes, The pricing of options and corporate liability, J. Political Economy, 81,
(1973), 637-659.
J. Hull , A. White, The pricing of options on assets with stochastic volatility, J. Finance, 42,
(1987), 271-301.
http://www.iaeme.com/IJMET/index.asp
1962
editor@iaeme.com
S. O. Edeki, G. O. Akinlabi, F. O. Egara and A. M. Okeke
[5]
[6]
[7]
[8]
[9]
S.O. Edeki, O.O. Ugbebor, E.A.Owoloko, He's polynomials for analytical solutions of the
Black-Scholes pricing model for stock option valuation, Lecture Notes in Engineering and
Computer Science, Volume 2224, 2016, 632-634.
S. Susumu, H. Mijuki, An improvement in the stochastic stock price differences Model,
Journal of Applied Operational Research., 4 (3), (2012), 125-138.
S.O. Edeki, E.A. Owoloko, and O.O. Ugbebor, The Modified Black-Scholes Model via
Constant Elasticity of Variance for Stock Options Valuation, AIP Conference proceedings,
1705, 020041 (2016); doi: 10.1063/1.4940289.
J.P. McNicholas, J.L. Rzzo, Stochastic GBM Methods for Modelling Market Prices, Casuality
Actuarial Society EForum, Summer (2012). 7
S. O. Edeki, I. Adinya, O. O. Ugbebor, The Effect of Stochastic Capital Reserve on Actuarial
Risk Analysis via an Integro-differential Equation, IAENG International Journal of Applied
Mathematics, vol. 44 (2), (2014), 83-90.
[10]
S.O. Edeki, O.O. Ugbebor, E.A.Owoloko, A note on Black-Scholes pricing model for
theoretical values of stock options, AIP Conference Proceedings, Vol. 1705, (2016),
Article number 4940288.
[11]
R. S. Sorin, Stochastic modeling of stock prices, Mongometry investment technology Inc. 200
Federal Street Camden, NJ 08103, (1997).
S. O. Edeki, M.E. Adeosun, E.A. Owoloko, G.O. Akinlabi, I. Adinya, Parameter Estimation
of Local Volatility in Currency Option Valuation, International Review on Modelling and
Simulations (IREMOS), 9 (2), (2016), 130-133. https://doi.org/10.15866/iremos.v9i2.8161.
S. O. Edeki, G. O. Akinlabi, N. Hinov, Zhou Method for the Solutions of System of
Proportional Delay Differential Equations, MATEC Web of Conferences, Volume 125, 4
October 2017, Article number 02001.
S. O. Edeki, and O.O. Ugbebor, On a Generalized Squared Gaussian Diffusion Model for
Option Valuation, MATEC Web of Conferences, 125, 02015, (2017).
L. Panzar, E.C. Cipu, Using of stochastic Ito and Stratonovich integrals derived security
pricing, Proceedings of the 3rd International Colloquium. Mathematics in Engineering and
Numerical Physics, October 7-9, Bucharest, Romania, (2004), 222-229.Luigi T. De Luca,
Propulsion physics (EDP Sciences, Les Ulis, 2009).
S.O. Edeki, O.O. Ugbebor, E.A. Owoloko, On a Nonlinear Transaction-Cost Model for Stock
Prices in an Illiquid Market Driven by a Relaxed Black-Scholes Model Assumptions,
Malaysian Journal of Mathematical Sciences, 11 (1), (2017), 83 - 96.
G. Wang, X. Wang . K. Zhoui, Pricing vulnerable options with stochastic volatility, Physica
A 485, (2017), 91{103.
S. O. Edeki, O. O. Ugbebor, E.A. Owoloko, On a Dividend-Paying Stock Options Pricing
Model (SOPM) Using Constant Elasticity of Variance Stochastic Dynamics, International
Journal of Pure and Applied Mathematics, 106 (4), (2016), 1029-1036.
S. O. Edeki, G.O. Akinlabi, M.E. Adeosun, Analytical solutions of the Navier-Stokes model
by He's polynomials, Lecture Notes in Engineering and Computer Science, Vol. 2223, 2016,
16-19.
G. O. Akinlabi, S. O. Edeki, The Solution of Initial-value Wave-like Models via Perturbation
Iteration Transform Method, Lecture Notes in Engineering and Computer Science, Vol. 2228,
2017, 1015-1018.
O.S. Adesina, O.K. Famurewa, R.J. Dare, O.O. Agboola, O.A. Odetunmibi, The mackeyglass type delay differential equation with uniformly generated constants, International
Journal of Mechanical Engineering and Technology, 9 (9), (2018), 467-477.
O. A. Agbolade, T.A., Anake, Solutions of First-Order Volterra Type Linear
Integrodifferential Equations by Collocation Method, Journal of Applied Mathematics, 2017,
1510267, (2017).
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
http://www.iaeme.com/IJMET/index.asp
1963
editor@iaeme.com