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# 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

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.

Conclusion

pan="1" rowspan="1">

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

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

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