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Calendar Spread Options for Storable CALENDAR SPREAD OPTIONS FOR STORABLE COMMODITIES By JUHEON SEOK Bachelor of Arts in Economics SungKyunKwan University Seoul, Korea 2004 Master of Arts in Economics SungKyunKwan University Seoul, Korea 2008 Master of Science in Economics Oklahoma State University Stillwater, Oklahoma 2012 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 2013 CALENDAR SPREAD OPTIONS FOR STORABLE COMMODITIES Dissertation Approved: Dr. B. Wade Brorsen Dissertation Adviser Dr. Brian Adam Dr. Philip Kenkel Dr. Weiping Li Outside Committee Member ii ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my advisor, Dr. Brorsen, who gave me his limitless patience, support, encouragement, and guidance for the past three years and allowed me to experience the frontier research of calendar spread options. I would like to thank Dr. Adam for his sharp advice, support, and commenting and correcting my writing. I would also like to thank Dr. Kenkel and Dr. Li for guiding my research and providing me with valuable knowledge for my research. Special thanks to Dr. Stoecker who was willing to help me at all times. I would like to thank LJ, who is my best of best friend, was always willing to help me and cheer me up for everything. It would not have been possible to finish my dissertation without his support and encouragement. I thank my friends, Hyojin, Pilja, and Jungmin for their encouragement when I faced hardship in my research. Finally, I would like to thank my parents and my younger sister. Without their sacrifice, I would never have been able to complete my Ph.D. and dissertation. iii Acknowledgements reflect the views of the author and are not endorsed by committee members or Oklahoma State University. Name: Juheon Seok Date of Degree: DECEMBER, 2013 Title of Study: CALENDAR SPREAD OPTIONS FOR STORABLE COMMODITIES Major Field: Agricultural Economics Abstract: Previous studies provide pricing models of options on futures spreads. However, none fully reflect the economic reality that spreads can stay near full carry for long periods of time. A new option pricing model is derived that assumes convenience yield follows arithmetic Brownian motion that is truncated at zero. The new models as well as alternative models are tested by testing the truth of their distributional assumptions for calendar spreads and convenience yield with Chicago Board of Trade corn calendar spreads. Panel unit root tests fail to reject the null hypothesis of a unit root and thus support our assumption of arithmetic Brownian motion as opposed to a mean- reverting process as is assumed in much past research. The assumption that convenience yield follows a normal distribution truncated at zero is only approximate as the volatility of convenience yield never goes to zero. Estimated convenience yields can be negative, which is presumably due to measurement error. Option payoffs are estimated with the four different models and the relative performance of models is determined using bias and root mean squared error (RMSE). The new model outperforms three other models and that the other models overestimate actual payoffs. There is no significant difference in error variance for Hinz and Fehr, Poitras, and the new model, and the error variance of the new model is smaller than that of Gibson and Schwartz. Key words: calendar spreads, corn, futures, panel unit root tests, options, bias, RMSE, significance tests iv TABLE OF CONTENTS Chapter Page I. INTRODUCTION ......................................................................................................1 II. THEORETICAL MODEL ........................................................................................6 The Theory of Storage .............................................................................................6 Long Call Option Solution .......................................................................................9 Short Call Option Solution.....................................................................................10 A Solution to Final Calendar Spread Option .........................................................11 III. DATA ....................................................................................................................13 IV. METHODS ...........................................................................................................15 Testing the Truth of Assumptions..........................................................................15 A New Model for Calendar Spread Options ..........................................................18 Gibson and Schwartz model: Stochastic Convenience Yield ................................21 Poitras: Bachelier Model for Calendar Spread Options .........................................22 Hinz and Fehr Model: A Shifted Lognormal Distribution Model for Calendar Spread Options .......................................................................................................23 Prediction Tests ......................................................................................................24 V. RESULTS ...............................................................................................................27 VI. SUMMARY AND CONCLUSION ......................................................................31 REFERENCES ............................................................................................................34 v LIST OF TABLES Table Page Table 1. The volume of CBOT calendar spread options and futures ...........................38 Table 2. Summary statistics .........................................................................................39 Table 3. Distribution for corn futures spread and convenience yield ..........................41 Table 4. Panel unit root tests in corn futures spread and convenience yield (1975-2012) .......................................................................................................................42 Table 5. The results of the first order auto regression for nearby futures prices .........43 Table 6. Estimation of the parameters k and α of the Gibson and Schwartz model .......................................................................................................................44 Table 7. Panel cointegration tests for corn futures ......................................................45 Table 8. Payoffs of calendar spread options at the money ..........................................46 Table 9. Payoffs of calendar spread options in the money ..........................................47 Table 10. Payoffs of calendar spread options out of money ........................................48 Table 11. Bias and RMSE of calendar spread options according by exercise price ....49 Table 12. Overall bias and RMSE of calendar spread options according by exercise price .............................................................................................................50 Table 13. Significance tests for error means ................................................................51 Table 14. Significance tests for error variances ...........................................................52 vi LIST OF FIGURES Figure Page Figure 1. Plots of CBOT Dec-Mar corn spread ...........................................................53 Figure 2. Plots of CBOT Mar-May corn spread ..........................................................54 Figure 3. Plots of CBOT May-Jul corn spread ............................................................55 Figure 4. Plots of CBOT Jul-Dec corn spread .............................................................56 Figure 5. Plots of CBOT Dec-Jul corn spread .............................................................57 Figure 6. Nonparametric regression of corn spread versus days to expiration ............58 Figure 7. Histograms of Dec-Mar spread and convenience yield, (1976-2011) .........61 Figure 8. Histograms of Mar-May spread and convenience yield, (1975-2012) ........62 Figure 9. Histograms of May-Jul spread and convenience yield, (1975-2012) ..........63 Figure 10. Histograms of Jul-Dec spread and convenience yield, (1976-2012) .........64 Figure 11. Histograms of Dec-Jul spread and convenience yield, (1976-2011) .........65 Figure 12. Dec-Mar convenience yield plots by year, (1976-2011) ............................66 Figure 13. Mar-May convenience yield plots by year, (1975-2012) ..........................67 Figure 14. May-Jul convenience yield plots by year, (1975-2012) ............................68 Figure 15. Jul-Dec convenience yield plots by year, (1976-2012) .............................69 Figure 16. Dec-Jul convenience yield plots by year, (1976-2011) .............................70 Figure 17. Nonparametric regressions of Dec-Mar spread and convenience yield against trading days to expiration for the sample period ...........................71 vii LIST OF FIGURES Figure Page Figure 18. Nonparametric regressions of Mar-May spread and convenience yield against trading days to expiration for the sample period ...........................72 Figure 19. Nonparametric regressions of May-Jul spread and convenience yield against trading days to expiration for the sample period ...........................73 Figure 20. Nonparametric regressions of Jul-Dec spread and convenience yield against trading days to expiration for the sample period ...........................74 Figure 21. Nonparametric
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