- Level: AS and A Level
- Subject: Maths
- Word count: 1756
One basic assumption of Black-Scholes model is that the stock price is log-normally distributed with constant volatility. However, in option market, does this assumption hold?
Extracts from this document...
Introduction
Introduction
Method used to exam mispricing problem of Black-Scholes model
Interpretation of the results
Conclusion
Reference:
Appendix 1: The raw data of lognormal distribution for Six Continent options on 18th, Feb 2003.
Appendix 2: The raw data of lognormal distribution for Six Continent options on 20th, Feb, 2003.
Appendix 3: The raw data of mixlognormal distribution for Six Continent options on 18th, Feb, 2003.
Appendix 4: The raw data of mixlognormal distribution for Six Continent options on 20th, Feb 2003
Introduction
One basic assumption of Black-Scholes model is that the stock price is log-normally distributed with constant volatility. However, in option market, does this assumption hold? In our paper, we try to show how wrong Black-Scholes is by challenging this assumption and illustrate the difference between Black-Scholes and real world.
Method used to exam mispricing problem of Black-Scholes model
About Mixlognormal: The probability distribution of the stock price might be made up of a mixture of two lognormal distributions, one for the possibility of an increase in share price and the other one of a decrease. In this way, we can capture the empirical distribution of stock price; its shape must be
Middle
650
8.34
1.8
95.5000
0.0188
2.5
700
3.36
0.75
145.5000
0.0051
1
750
1.25
0.06
195.5000
0.0003
0.5
800
0.43
0
245.5000
0
18.53
0.1862
Appendix 2: The raw data of lognormal distribution for Six Continent options on 20th, Feb, 2003. | |||||||
Trade | 37672 | ||||||
Maturity | 37727 | ||||||
r | 0 | ||||||
T | 0.15068493 | ||||||
F | 615.5 | ||||||
sigma | 0.36813434 | ||||||
mkt call | strikes | BS theory | Sq Error | moneyness | weigted sq error | ||
256.5 | 360 | 255.5013903 | 0.997221354 | 255.5 | 0.003903019 | ||
227 | 390 | 225.5132704 | 2.210364876 | 225.5 | 0.009802062 | ||
197 | 420 | 195.5835263 | 2.006397745 | 195.5 | 0.010262904 | ||
157.5 | 460 | 156.0819138 | 2.010968563 | 155.5 | 0.012932274 | ||
119.5 | 500 | 118.0715084 | 2.040588233 | 115.5 | 0.017667431 | ||
76.5 | 550 | 75.70383997 | 0.633870793 | 65.5 | 0.009677417 | ||
42.5 | 600 | 42.91727918 | 0.174121912 | 15.5 | 0.011233672 | ||
21 | 650 | 21.38061239 | 0.144865789 | 34.5 | 0.004199008 | ||
9 | 700 | 9.401761762 | 0.161412513 | 84.5 | 0.001910207 | ||
3.5 | 750 | 3.688784898 | 0.035639738 | 134.5 | 0.000264979 | ||
1 | 800 | 1.308782808 | 0.095346823 | 184.5 | 0.000516785 | ||
0.5 | 850 | 0.425705755 | 0.005519635 | 234.5 | 2.35379E-05 | ||
10.51631797 | 0.082393295 | ||||||
Appendix 3: The raw data of mixlognormal distribution for Six Continent options on 18th, Feb, 2003. | |||||
Trade |
Conclusion
Cmarket
X
Cimplied
(Cmarket-Cimplied)^2
Implied sigma by BS
196
360
194.6650299
1.782145105
0.605651935
166.5
390
165.1475618
1.82908915
0.53766516
138
420
136.4577704
2.378472184
0.502173454
101.5
460
100.7421585
0.574323782
0.454991983
69
500
69.60810609
0.369793012
0.424942163
38
550
38.86792249
0.753289445
0.410555737
17.5
600
17.00376787
0.246246324
0.393289282
7
650
6.928636732
0.005092716
0.38547187
2.5
700
2.914632833
0.171920386
0.382378979
1
750
1.14301875
0.020454363
0.394114927
0.5
800
0.422795028
0.005960608
0.417111769
8.136787075
Appendix 4:The raw data of mixlognormal distribution for Six Continent options on 20th, Feb 2003
Trade
20-Feb-03
Maturity
16-Apr-03
r
0
T
0.1506849
F
615.5
F1
631.64945
sigm1
0.005
F2
613.47651
sigma2
0.3942533
p
0.1113462
Cmarket
X
Cimplied
(Cmarket-Cimplied)^2
Implied sigma by BS
256.5
360
255.5039835
0.992048772
0.684121728
227
390
255.5291768
2.163320768
0.634591317
197
420
195.6487533
1.825867624
0.544725925
157.5
460
156.3364319
1.353890697
0.457835271
119.5
500
118.6654525
0.696469529
0.411053745
76.5
550
76.50974266
9.49E-05
0.380098628
42.5
600
42.7445391
0.059799374
0.363618501
21
650
20.42471433
0.330953606
0.363939585
9
700
9.572591543
0.327861075
0.362150688
3.5
750
4.077689599
0.333725272
0.363395708
1
800
1.597153827
0.356592693
0.351554917
0.5
850
0.581926491
0.00671195
0.376206304
8.447336279
This student written piece of work is one of many that can be found in our AS and A Level Probability & Statistics section.
Found what you're looking for?
- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month