A Study of Option Pricing Models – Lognormal or Hyperbolic Levy ? by

A Study of Option Pricing Models – Lognormal or Hyperbolic Levy ? By Chen Chen This paper is an investigation into two option pricing models: widely-used Black-Scholes model and one of its augmented extensions – hyperbolic Levy model. Firstly, we have a detailed discussion about the celebrated Black-Scholes model. However, clearly there are many deficiencies in Black-Scholes assumptions. In order to refine Black-Scholes model, Eberlein and Keller(1995) introduce the hyperbolic Levy motion and claim that the new pricing model can provide a better valuation of derivative securities. Following suggestions in that paper, we want to replicate their claims. We perform several statistical tests and show that the hyperbolic distributions can be well fitted to the financial data. This observation suggests us to replace the geometric Brownian motion by the hyperbolic Levy process and build the hyperbolic Levy pricing model. After an introduction into the Levy process theory, we attempt to numerically calculate the value of options according to the hyperbolic Levy model. But it turns out that the price implied by the hyperbolic model cannot be approximated as usual by regular method. This upsetting outcome leads us to look for explanations for the failure of computation, and according to Eberlein, Keller, and Prause (1998) Fast Fourier Transform should be able to efficiently compute the integral. Due to the limited time constraint, we leave it to interested readers. In conclusion, the hyperbolic Levy motion is a better process to fit into empirical data, but it demands an advanced numerical method to compute. Though the observed data is a poorly fit for Black-Scholes, its tractability (elegant solution forms, numerical calculation and other implications) trumps the realism of the hyperbolic Levy model. A Study of Option Pricing Models – Lognormal or Hyperbolic Levy ? By Chen Chen Skip Garibaldi Adviser A thesis submitted to the Faculty of Emory College of Arts and Sciences of Emory University in partial fulfillment of the requirements of the degree of Bachelor of Arts with Honors Department of Mathematics and Computer Science 2012 Acknowledgements I would like to thank Prof. Skip Garibaldi, who served as my advisor for this project and mentored me throughout the year. Without his encouragement, support and guidance, it would be unlikely that I was able to finish the project. I would be also grateful to the other members of my committee, Prof. Emily Hamilton and Prof. Tao Zha, for their comments on the work.