J UNIVERSITY OF CAPE TOWN A CONTINGENT CLAIMS ANALYSIS OF THE PRICING OF RIGHTS ISSUES WITH DISCONTINUOUS DIFFUSION PROCESSES A dissertation submitted in partial satisfaction of the requirements for the Degree of Master of Commerce (by coursework and dissertation). University byof Cape Town Russel John Botha Cape Town December 1998 The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgement of the source. The thesis is to be used for private study or non- commercial research purposes only. Published by the University of Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author. University of Cape Town ACKNOWLEDGEMENTS AND DECLARATION I, Russel John Botha, do hereby declare that this research report entitled : A Contingent Claims analysis of the Pricing of Rights Issues with Discontinuous Diffusion Processes is my own unaided work save to the extent stated in the acknowledgement. I certify that this work has not been submitted as a dissertation at any other university. I would like to express my sincere gratitude to the following perso.ns for their support in the compilation of this research report: • Professor Derek Botha of UCT for valuable guidance and support throughout; • Francois du Plessis of HSBC Simpson McKie for assistance in the initial stages of the thesis and data collection; • Michael Davidow of the Johannesburg Stock Exchange for current market statistics; • Professor Gewers of the Stellenbosch Graduate School of Business for the final data set obtained; • Praveen Kumar of Lehman Brothers (New York) for practical advice; • Professor Mike Page of the UCT Graduate School of Business for technical assistance. Russel John Botha ABSTRACT This research proposed to identify the most accurate method of pricing rights using option pricing models, including the Black Scholes model, the Cox constant elasticity of variance model and the Merton jump diffusion model, and to determine the set of input parameters that lead to the most optimal results. The empirical results indicated that on average all of the models are able to estimate the actual rights trading prices relatively well. Some models performed better than others and these findings were consistent with the original reasonings. The market was shown to not account for the effect of dilution. The best model prices were obtained when calculating volatility over a one year historical period that included the actual rights trading period. The hypothesis regarding trading volume showed that there is a significant impact of trading volume on the estimation of accurate option prices. The filter rule of rejecting rights prices below IO cents and 100 cents also improved the results thus showing a bias for lower priced rights to be incorrectly valued and possibly some inefficiency in this sector of the market. BRIEF CONTENTS PAGE 1. INTRODUCTION 2. THE illSTORY OF OPTION PRICING 11 3. THE DYNAMICS OF SHARE PRICE DIFFUSIONS IN CONTINUOUS TIME 19 4. THE BLACK-SCHOLES OPTION PRICING MODEL 30 5. SHARE PRICE VOLATILITY 50 6. CONSTANT ELASTICITY OF VARIANCE OPTION PRICING MODELS 67 7. SHARE PRICE DIFFUSION DISTRIBUTIONS 82 8. JUMP DIFFUSION OPTION PRICING MODELS 94 9. WARRANT PRICING AND RIGHTS ISSUES 112 1O.MARKET EFFICIENCY 123 I I .RESEARCH METHODOLOGY 135 12.RESULTS OF EMPIRICAL TESTING 162 13.CONCLUSION 184 14.BIBLIOGRAPHY 190 APPENDIX 1 - J.S.E. LISTING REQUIREMENTS FOR RIGHTS ISSUES 210 APPENDIX2 - CONVERGENCE OF THE MERTON MODEL INFINITE SERIES 217 APPENDIX3 - PROOF OF ITO'S LEMMA 219 APPENDIX4 - DATA SAMPLE 221 APPENDIX 5 - SAMPLE JUMP DIFFUSION PARAMETERS 226 APPENDIX6 - RESTRICTIONS ON RATIONAL OPTION PRICING 231 APPENDIX 7 - PROOF OF THE CONVERGENCE OF THE BINOMIAL MODEL TO THE BLACK SCHOLES MODEL 236 APPENDIX 8 - VISUAL BASIC ROUTINES WRIITEN 239 DETAILED CONTENTS PAGE 1. INTRODUCTION 1 1.1. INTRODUCTION TO CONTINGENT CLAIMS ANALYSIS 1.2. DEFINITION OF AN OPTION 1.3. RAISING CAPITAL IN THE MARKET AND THE LINK TO OPTION PRICING 1.4. ANALYSIS OF PRICING BIASES 1.5. MOTIVATION FOR THE RESEARCH 1.6. THE RESEARCH PROBLEM 1.7. RESEARCH METHODOLOGY 1.8. CONTRIBUTION TO KNOWLEDGE 1.9. STRUCTURE OF THE RESEARCH THESIS 2. THE IDSTORY OF OPTIONS AND OPTION PRICING 11 2.1. INTRODUCTION 2.2. THE ORIGINS OF OPTIONS TRADING 2.2.1. THE INTERNATIONAL ARENA 2.2.2. THE SOUTH AFRICAN BEGINNINGS 2.3. THE ORIGINS OF OPTION PRICING 3. DYNAMICS OF SHARE PRICE DIFFUSIONS IN CONTINUOUS TIME 19 3.1. INTRODUCTION 3.2. DEFINITIONS 3.3. TYPE I AND II OUTCOMES 3.4. TYPE III OUTCOMES II 4. THE BLACK-SCHOLES OPTION PRICING MODEL 30 4.1. INTRODUCTION 4.2. ARBITRAGE PRICING AND THE BLACK SCHOLES MODEL 4.3. BLACK SCHOLES MODEL ASSUMPTIONS 4.4. DERIVATION OF THE MODEL 4.5. SOLUTION TO THE BLACK SCHOLES PARTIAL DIFFERENTIAL EQUATION 4.6. THE BINOMIAL LATTICE MODEL OPTION PRICING MODEL 4.7. THE BINOMIAL OPTION PRICING FORMULA 4.8. A COMPARATIVE STATICS ANALYSIS OF THE BLACK SCHOLES PARAMETERS 5. SHARE PRICE VOLATILITY 50 5.1. INTRODUCTION TO VOLATILITY 5.2. RELATIONSHIP OF VOLATILITY TO SHARE PRICE 5.3. THE VOLATILITY SMILE 5.4. ESTIMATING STOCK VOLATILITY 5.5. IMPLICATIONS OF NON-CONSTANT VOLATILITY ON OPTION PRICING . 5.6. CONCLUSION 6. CONSTANT ELASTICITY OF VARIANCE OPTION PRICING MODEL 67 6.1 . INTRODUCTION 6.2. APPLICABILITY TO THE PRICING OF RIGHTS 6.3 . DEVELOPMENT OF THE MODEL 6.4. INTERPRETATION OF THE CEV FORMULA 6.5. TWO SPECIAL CASES OF THE CEV MODEL 6.5. l. SQUARE ROOT MODEL 6.5.2. ABSOLUTE MODEL 6.6. APPLICABILITY OF THE MODEL Ill 6.7. ESTIMATING THE CHARACTERISTIC PARAMETER AND RESULTS OF EMPIRICAL TESTING 6.8. CONCLUSION 7. SHARE PRICE DIFFUSION DISTRIBUTIONS 82 7.1. INTRODUCTION 7.2. THE SHARE PRICE DIFFUSION DEBATE AND EVIDENCE 7.3 . THE NECESSITY FOR A NEW THEORY 7.4. THE STABLE PARETIAN APPROACH 7.5. THE SUPPORTING EMPIRICAL EVIDENCE 7.6. THE LARGE PRICE CHANGE PHENOMENON EXPLAINED 7.7. IMPLICATIONS OF THE STABLE PARETIAN HYPOTHESIS ON THE BLACK SCHOLES MODEL 8. JUMP DIFFUSION OPTION PRICING MODELS 94 8.1. INTRODUCTION 8.2. THE JUMP DIFFUSION CONCEPT 8.3. DERIVATION OF THE MERTON JUMP DIFFUSION MODEL 8.3.1. THE STOCK PRICE DYNAMICS 8.3.2. THE OPTION PRICE DYNAMICS 8.4. DIVERSIFIABILITY OF THE JUMP RISK 8.5. MODIFICATIONS OF THE MERTON MODEL 8.6. ESTIMATING THE MODEL PARAMETERS 8.7. A COMPARISON TO THE BLACK SCHOLES MODEL 8.8. CONCLUSION 9. WARRANT PRICING AND RIGHTS ISSUES 112 9.1 INTRODUCTION 9.2 THE RELATIONSHIP BETWEEN WARRANTS AND OPTIONS 9.3 THE RELATIONSHIP BETWEEN RIGHTS AND WARRANTS 9.4 THE THEORY OF WARRANT PRICING IV 9.5 THE THEORY OF RIGHTS PRICING 9.6 EMPIRICAL EVIDENCE ON WARRANT PRICING 9.6.1 THE INTERNATIONAL EVIDENCE 9.6.2 THE SOUTH AFRICAN RIGHTS ISSUES MARKET 9.7 CONCLUSION 10. MARKET EFFICIENCY 123 10.1. INTRODUCTION 10.2. THE THEORY OF MARKET EFFICIENCY 10.3. THE THEORY OF OPTIONS MARKET EFFICIENCY TESTS 10.4. THE INTERNATIONAL EVIDENCE 10.5. THE SOUTH AFRICAN EVIDENCE 10.6. CONCLUSION 11. RESEARCH METHODOLOGY 135 11.1. OUTLINE TO METHODOLOGY 11.2. THE RESEARCH PROBLEM 11.3. THE RESEARCH SUB-PROBLEMS 11.3.1 . THE FIRST SUB-PROBLEM 11.3.2. THE SECOND SUB-PROBLEM 11.3.3. THE THIRD SUB-PROBLEM 11.3.4. THE FOURTH SUB-PROBLEM 11.4. THE HYPOTHESIS STATEMENTS 11.4.1. MAIN HYPOTHESIS STATEMENT 11.4.2. SUB-HYPOTHESIS STATEMENTS 11.5 . ASSUMPTIONS AND LIMITATIONS OF THE STUDY 11.6. THE NEED FOR THIS STUDY 11.7. BASIC RESEARCH METHODOLOGY 11.8. SAMPLE SELECTION 11.9. SAMPLE DESCRIPTIVE STATISTICS 11.9.1. VOLUME OF RIGHTS ISSUES 11 .9.2. STOCK PRICE ANALYSIS v 11.9.3. VOLATILITY ANALYSIS 11 .9.4. TRADING PERIOD ANALYSIS 11.10. INPUT PARAMETER ESTIMATION 11.10.1.UNDERLYING SHARE PRICE 11 .10.2.EXERCISE PRICE 11.10.3.RISK FREE RATE 11.10.4.TIME TO MATURITY 11.10.5.VOLATILITY 11.10.6.MERTON JUMP DIFFUSION PARAMETERS 11.11. VALUATION MODELS EMPLOYED 11.11.1.BLACK SCHOLES MODEL 11.11.2.CONSTANT ELASTICITY OF VARIANCE MODEL 11.11.3.MERTON JUMP DIFFUSION MODEL 11.12. STATISTICAL TESTING PROCEDURE 12. RESULTS OF TESTING AND ANALYSIS OF FINDINGS 162 12.1 . RESULT OF PRIMARY HYPOTHESIS TEST 12.2. TESTING OF UNADJUSTED MODELS 12.3. TESTING THE DIFFERENT VOLATILITY ESTIMATION INTERVALS 12.3.1. BLACK SCHOLES MODEL 12.3.2. SQUARE ROOT CEV MODEL 12.3.3. ABSOLUTE CEV MODEL 12.3.4. VOLATILITY SUMMARY 12.4. ADJUSTING FOR DILUTION 12.4.1 . TESTING OF DILUTION ADJUSTED MODELS 12.4.2. CONCLUSION ON DILUTION ADJUSTMENT 12.5. TRADING VOLUME ADJUSTMENTS 12.5.1. TESTING OF MODELS ADJUSTED FOR NIL VOLUME 12.5.2. CONCLUSION ON ADJUSTMENT FOR TRADING VOLUME 12.6. ADJUSTMENT FOR THE ABSOLUTE VALUE OF RIGHTS PRICES 12.6.1. FILTER RULE REJECTING RIGHTS BELOW 10 CENTS 12.6.2. FILTER RULE REJECTING RIGHTS BELOW 100 CENTS VI 12.6.3 . CONCLUSION ON ADJUSTMENT FOR ABSOLUTE VALUE OF RIGHTS PRICES 13. CONCLUSION 184 13 .1. THE RESEARCH PROBLEM 13 .2. THE RESEARCH FINDINGS 13.3. OVERALL SUMMARY OF FINDINGS 13.4. THE PROPOSED RIGHTS PRICING MODEL AND DIRECTION FOR FUTURE RESEARCH 14. BIBLIOGRAPHY 190 APPENDIX 1 - J.S.E. LISTING REQUIREMENTS FOR RIGHTS ISSUES APPENDIX2 - CONVERGENCE OF THE MERTON MODEL INFINITE SERIES APPENDIX3 - PROOF OF ITO'S LEMMA APPENDIX4 - DATA SAMPLE APPENDIXS - SAMPLE JUMP DIFFUSION PARAMETERS APPENDIX6 - RESTRICTIONS ON RATIONAL OPTION PRICING APPENDIX7 - PROOF OF THE CONVERGENCE OF THE BINOMIAL MODEL TO THE BLACK SCHOLES MODEL APPENDIX8 - VISUAL BASIC ROUTINES WRITTEN vii CHAPTER 1 - INTRODUCTION 1.1 INTRODUCTION TO CONTINGENT CLAIMS ANALYSIS Contingent claims analysis involves the valuation of claims whose payoffs are contingent on the value of one or more underlying assets.