WYCENA ASYMETRYCZNYCH OPCJI LOGARYTMICZNYCH ZA POMOCĄ TRANSFORMATY FOURIERA

• Authors: Orzechowski Arkadiusz

Title variants

EN

PRICING ASYMMETRIC LOGARITHMIC OPTIONS VIA FOURIER TRANSFORM

• Languages of publication: PL EN

Abstracts

PL

Celem artykułu jest wycena europejskich opcji logarytmicznych o asymetrycznym profilu wypłaty. W ramach podejmo­wanej problematyki zaproponowane zostały trzy podejścia pozwalające określić wartość modelową analizowanych instrumentów pochodnych przy utrzymaniu założeń właściwych modelowi F. Blacka i M. Scholesa, tj. podejście martyngałowe, F. Blacka i M. Scholesa oraz bazujące na transformacie Fouriera. Ponadto, dokonano analizy szybkości i dokładności obliczeniowej uwzględnionych podejść do wyceny analizowanych derywatów.

EN

"The aim of the article is to price European logarithmic options via Fourier transform. As a part of the subject metter, three approaches were proposed to determine theoretical value of the analyzed derivatives in the F. Black and M. Scholes setting, i.e. the martingale approach, F. Black and M. Scholes approach and the approach based on the Fourier transform. In addition, an analysis of the computational speed and accuracy of the valuations was carried out."

Keywords

PL

model F. Blacka i M. Scholesa; transformata Fouriera; asymetryczne opcje logarytmiczne

EN

Black-Scholes model; Fourier transform; asymmetric logarithmic options

• Publisher: Katedra Ekonometrii i Statystyki, Szkoła Główna Gospodarstwa Wiejskiego w Warszawie
• Journal: Metody Ilościowe w Badaniach Ekonomicznych
• Year: 2018
• Volume: 19
• Issue: 3
• Pages: 238-250

Contributors

author

Orzechowski Arkadiusz

• Kolegium Ekonomiczno-Społeczne, Szkoła Główna Handlowa w Warszawie

References

• Attari M. (2004) Option Pricing Using Fourier Transform: A Numerically Efficient Simplification. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=520042 [dostęp: 1.07.2018].
• Bakshi G., Madan D. (2000) Spanning and Derivative-Security Valuation. Journal of Financial Economics, 2(55), 205-238.
• Bates D. S. (2006) Maximum Likelihood Estimation of Latent Affine Processes. Review of Financial Studies, 19(3), 909-965.
• Bates D. S. (1996) Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options. The Review of Financial Studies, 9(1), 69-107.
• Biagini F., Bregman Y., Meyer-Brandis T. (2008) Pricing of Catastrophe Insurance Options Written on a Loss Index with Reestimation. Insurance: Mathematics and Economics, 43 (2), 214-222.
• Black F., Scholes M. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.
• Borovkov K., Novikov A. (2002) On a New Approach to Calculating Expectations for Option Pricing. Journal of Applied Probability, 39(4), 889-895.
• Carr P., Geman H., Madan D. B., Yor M. (2002) The Fine Structure of Asset Returns: An Empirical Investigation. Journal of Business, 75(2), 305-332.
• Carr P., Madan D. B. (1999) Option Valuation Using the Fast Fourier Transform. Journal of Computational Finance, 2(4), 61-73.
• Cont R., Tankov P. (2004) Financial Modeling with Jump Processes. Boca Raton: Chapman and Hall.
• Derman E., Kani, T. (1998) Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility. International Journal of Theoretical and Applied Finance, 1, 61-110.
• Dupire B. (1994) Pricing with Smile. Risk, 7(1), 18-20.
• Hagan P. S., Kumar D., Lesniewski A., Woodward D. E (2003) Managing Smile Risk. Wilmott Magazine, 3, 84-108.
• Heston S. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, 6(2), 327-343.
• Hubalek F., Kallsen J. (2005) Variance-Optimal Hedging and Markowitz-Efficient Portfolios for Multivariate Processes with Stationary Independent Increments with and Without Constraints. Working paper, TU München.
• Hurd T. R., Zhou Z. (2010) A Fourier Transform Method for Spread Option Pricing. SIAM Journal on Financial Mathematics, 1(1), 142-157.
• Jackson K. R., Jaimungal S., Surkov V. (2008) Fourier Space Time-Stepping for Option Pricing with Levy Models. Journal of Computational Finance, 12(2), 1-29.
• Kou S. (2002) Jump-Diffusion Model for Option Pricing. Management Science, 48(8), 1086-1101.
• Lee R. W. (2004) Option Pricing by Transform Methods: Extensions, Unification, and Error Control. Journal of Computational Finance, 7(3), 50-86.
• Lewis A. (2001) A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes. SSRN Electronic Journal, 1-25.
• Lipton A. (2002) The Vol Smile Problem. http://www.math.ku.dk/~rolf/Lipton_VolSmileProblem.pdf, [dostęp: 8.12.2017].
• Madan D., Carr P., Chang E. (1998) The Variance Gamma Process and Option Pricing. European Finance Review, 1, 79-105.
• Merton R. C. (1976) Option Pricing when Underlying Stock Returns Are Discontinuous. Journal of Financial Economics, 3(1-2), 125-144.
• Naik V. (2000) Option Pricing with Stochastic Volatility Models. Decisions in Economics and Finance, 23(2), 75-99.
• Rydberg T. H. (1997) The Normal Inverse Gaussian Levy Process: Simulation and Approximation. Communication in Statistics Stochastic Models, 13(4), 887-910.
• Schmelzle M. (2010) Option Pricing Formulae Using Fourier Transform: Theory and Application. http://pfadintegral.com [dostęp: 1.07.2018]
• Scott L. (1997) Pricing Stock Options in a Jump-Diffusion Model with Stochastic Volatility and Interest Rates: Applications of Fourier inversion methods. Mathematical Finance, 7(4), 413-424.
• Stein E. M., Stein J. C. (1991) Stock Price Distribution with Stochastic Volatility: An Analytic Approach. The Review of Financial Studies, 4(4), 727-752.
• Wu L. (2008) Modeling Financial Security Returns using Lévy Processes. [in:] Handbooks in Operations Research and Management Science: Financial Engineering, 15, Elsevier, North-Holland, 117-162.
• Zhu J. (2000) Modular Pricing of Options: An Application of Fourier Analysis. Lecture Notes in Economics and Mathematical Systems, 493, Springer-Verlag, Berlin, Heidelberg.