IDEAS home Printed from https://ideas.repec.org/b/uts/finphd/29.html
   My bibliography  Save this book

Pricing American Options Using Fourier Analysis

Author

Listed:
  • Andrew Ziogas

Abstract

The analytic expression for an American option price under the Black-Scholes model requires the early exercise boundary as one of its inputs, and this is not known a priori. An implicit integral equation can be found for this free boundary, but it has no known closed-form solution, and its numerical solution is highly non-trivial. This has given rise to a number of analytical solution methods and numerical techniques designed to handle the early exercise feature. The aim ofthis thesis is to explore Fourier-type solution methods for pricing American options. The price is defined as a free boundary value problem, whose solution satisfies the Black-Scholes PDE with certain final and boundary conditions. This problem is solved using the incomplete Fourier transform method of McKean (1965). The method is generalised to American options with monotonic and convex payoffs in a systematic way, and is further extended by applying it to solve the PIDE for the American call option under Merton’s (1976) jump-diffusion model. In this case numerical integration solutions require an intense level of computation. The thesis considers the Fourier-Hermite series expansion method as an alternative approach. This is extended to allow for jump-diffusion with log-normally distributed jump sizes. The main contributions ofthe thesis are: • Evaluation ofAmerican Options under Geometric Brownian Motion - Chapters 2 and 3. The details of McKean’s (1965) incomplete Fourier transform are provided for a monotonic payoff function, and several forms for the price and free boundary are reproduced in the case of an American call. A numerical scheme for implementing the equations is given, along with a comparison ofseveral existing numerical solution methods. The applicability ofthe transform technique to more general payofftypes is demonstrated using an American strangle position with interdependent component options. A coupled integral equation system for the two free boundaries is found and solved using numerical integration. The resulting free boundaries are consistently deeper in-the-money than those for the corresponding independent American call and put. • Pricing American Options under Jump-Diffusion - Chapter 4. The incomplete Fourier transform method is applied to the jump-diffusion model of Merton (1976). The PIDE for an American call is solved, and the results are simplified to replicate the integral equations of Gukhal (2001) for the price and free Abstract boundary. An implicit expression for the limit of the free boundary at expiry is derived, and an iterative algorithm is presented for solving the integral expressions numerically. The results are demonstrated to be consistent with existing knowledge of American options under jump-diffusion, and display behaviour that is consist with market-observed volatility smiles. • Fourier-Hermite Series Expansionsfor Options under Jump-Diffusion - Chapter 5. The Fourier-Hermite series expansion method is extended to the jump-diffusion model of Merton (1976) in the case where the jump sizes are lognormally distributed. With the aid of a suitably calibrated scaling parameter, the method is used to evaluate American call options. The pricing accuracy of this approach is shown to be comparable to both the iterative numerical integration method, and the method of lines technique by Meyer (1998). The series expansion method displays a high computation speed in exchange for some loss of accuracy in the free boundary approximation.

Suggested Citation

  • Andrew Ziogas, 2005. "Pricing American Options Using Fourier Analysis," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 29.
  • Handle: RePEc:uts:finphd:29
    as

    Download full text from publisher

    File URL: https://opus.lib.uts.edu.au/bitstream/10453/37244/2/02Whole.pdf
    Download Restriction: no

    References listed on IDEAS

    as
    1. Jarrow, Robert A & Rosenfeld, Eric R, 1984. "Jump Risks and the Intertemporal Capital Asset Pricing Model," The Journal of Business, University of Chicago Press, vol. 57(3), pages 337-351, July.
    2. Roll, Richard, 1977. "An analytic valuation formula for unprotected American call options on stocks with known dividends," Journal of Financial Economics, Elsevier, vol. 5(2), pages 251-258, November.
    3. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    4. Parkinson, Michael, 1977. "Option Pricing: The American Put," The Journal of Business, University of Chicago Press, vol. 50(1), pages 21-36, January.
    5. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    6. Chiarella, Carl & El-Hassan, Nadima & Kucera, Adam, 1999. "Evaluation of American option prices in a path integral framework using Fourier-Hermite series expansions," Journal of Economic Dynamics and Control, Elsevier, vol. 23(9-10), pages 1387-1424, September.
    7. Siim Kallast & Andi Kivinukk, 2003. "Pricing and Hedging American Options Using Approximations by Kim Integral Equations," Review of Finance, Springer, vol. 7(3), pages 361-383.
    8. Siim Kallast & Andi Kivinukk, 2003. "Pricing and Hedging American Options Using Approximations by Kim Integral Equations," Review of Finance, European Finance Association, vol. 7(3), pages 361-383.
    9. Ju, Nengjiu, 1998. "Pricing an American Option by Approximating Its Early Exercise Boundary as a Multipiece Exponential Function," Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 627-646.
    10. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14.
    11. Peter Ritchken & Rob Trevor, 1999. "Pricing Options under Generalized GARCH and Stochastic Volatility Processes," Journal of Finance, American Finance Association, vol. 54(1), pages 377-402, February.
    12. Ball, Clifford A & Torous, Walter N, 1985. " On Jumps in Common Stock Prices and Their Impact on Call Option Pricing," Journal of Finance, American Finance Association, vol. 40(1), pages 155-173, March.
    13. Huang, Jing-zhi & Subrahmanyam, Marti G & Yu, G George, 1996. "Pricing and Hedging American Options: A Recursive Integration Method," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 277-300.
    14. Johnson, H. E., 1983. "An Analytic Approximation for the American Put Price," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 18(01), pages 141-148, March.
    15. Amin, Kaushik I, 1993. " Jump Diffusion Option Valuation in Discrete Time," Journal of Finance, American Finance Association, vol. 48(5), pages 1833-1863, December.
    16. Geske, Robert, 1981. "Comments on Whaley's note," Journal of Financial Economics, Elsevier, vol. 9(2), pages 213-215, June.
    17. Carl Chiarella & Nadima El-Hassan, 1999. "Pricing American Interest Rate Options in a Heath-Jarrow-Morton Framework Using Method of Lines," Research Paper Series 12, Quantitative Finance Research Centre, University of Technology, Sydney.
    18. Geske, Robert, 1979. "A note on an analytical valuation formula for unprotected American call options on stocks with known dividends," Journal of Financial Economics, Elsevier, vol. 7(4), pages 375-380, December.
    19. Blomeyer, Edward C., 1986. "An Analytic Approximation for the American Put Price for Options on Stocks with Dividends," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 21(02), pages 229-233, June.
    20. Barone-Adesi, Giovanni & Whaley, Robert E, 1987. " Efficient Analytic Approximation of American Option Values," Journal of Finance, American Finance Association, vol. 42(2), pages 301-320, June.
    21. Mark Schroder, 1989. "A Reduction Method Applicable to Compound Option Formulas," Management Science, INFORMS, vol. 35(7), pages 823-827, July.
    22. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    23. Broadie, Mark & Detemple, Jerome, 1996. "American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods," Review of Financial Studies, Society for Financial Studies, vol. 9(4), pages 1211-1250.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:uts:finphd:29. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Duncan Ford). General contact details of provider: http://edirc.repec.org/data/sfutsau.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.