A Study of Option Pricing Models - Lognormal or Hyperbolic Levy ? Público
Chen, Chen (2012)
Published
Abstract
Abstract
A Study of Option Pricing Models - Lognormal or Hyperbolic
Levy?
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.
Table of Contents
Table of Contents
1.Introduction...1
1.1 Preliminaries...2
1.2 Classical Modeling of Stock Movement...7
1.2.1 The Markov Property and
Martingale...8
1.2.2 Wiener Process...10
1.2.3 The Classical Model for Stock Price...12
1.2.4 Ito's Lemma...13
2. The Black-Scholes Model...15
2.1 Lognormal Property of Stock Price...15
2.2 Black-Scholes model's ideas and assumptions...16
2.3 Black-Scholes formula's derivation...17
3. Testing Black-Scholes Assumptions...21
4. Modeling Stock Price by Discontinuous Lévy Process...28
4.1 The Hyperbolic Density and Empirical
Data...28
4.2 Lévy Process...33
4.3 Hyperbolic Lévy Motion model...36
4.3.1 Esscher Transform...37
4.3.2 Applying Esscher Transform to the Hyperbolic Model...39
4.4 Numerical Approximation...40
5. Conclusion...48
List of Figures
1. One-period Binomial Model...5
2. Simulation of Wiener Process...11
3. BAC Normal Q-Q Plot...22
4. GOOG Normal Q-Q Plot...22
5. IBM Normal Q-Q Plot...23
6. BAC implied volatility...26
7. GOOG implied volatility...26
8. IBM implied volatility...27
9. Normal PDF V.S. Hyperbolic PDF...30
10. BAC Hyperbolic Q-Q Plot...31
11. GOOG Hyperbolic Q-Q Plot...31
12. IBM Hyperbolic Q-Q Plot...32
13.Integrand of (4.4.7)...44
14. The graph in the special case, where ς = σ =
1, shows what the density function
ft(x)
should look like...45
15. The graph in the case, where ς = 0.3258 and σ = 0.003, shows the density function ft(x) behaves abnormally...46
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