Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T02:25:16.776Z Has data issue: false hasContentIssue false
  • English
  • Français

On a new approach to calculating expectations for option pricing

Part of: Probability theory and stochastic processes

Published online by Cambridge University Press:  14 July 2016

K. Borovkov  and
A. Novikov
K. Borovkov*
Affiliation:
University of Melbourne
A. Novikov*
Affiliation:
University of Technology, Sydney
*
∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia. Email address: kostya@ms.unimelb.edu.au
∗∗ Postal address: School of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Sydney, NSW 2007, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss a simple new approach to calculating expectations of a specific form used for the pricing of derivative assets in financial mathematics. We show that in the ‘vanilla case’, the expectations can be found by simply integrating the respective moment generating function with a certain weight. In situations corresponding to barrier-type options, we just need to carry out one more integration. The suggested approach appears to be the first (and, apart from Monte Carlo simulation, the only) one to allow the pricing of discretely monitored exotic options when the underlying asset is modelled by a general Lévy process. We illustrate the method numerically by calculating the price of a discretely monitored lookback call option in the cases when the underlying follows the geometric Brownian and variance-gamma processes.

Keywords

Option pricingbarrier optionmoment generating functionschange of measure

MSC classification

Primary: 60-08: Computational methods (not classified at a more specific level)
Type
Short Communications
Information
Journal of Applied Probability , Volume 39 , Issue 4 , December 2002 , pp. 889 - 895
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]. AitSahlia, F., and Lai, T. L. (1998). Random walk duality and the valuation of discrete lookback options. Appl. Math. Finance 5, 227–240.CrossRefGoogle Scholar
[2]. Bingham, N. H., and Kiesel, R. (1998). Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives. Springer, New York.CrossRefGoogle Scholar
[3]. Carr, P., and Madan, D. (1999). Option valuation using the fast Fourier transform. J. Comput. Finance 2, 61–73.CrossRefGoogle Scholar
[4]. Heinrich, L. (1985). An elementary proof of Spitzer's identity. Statistics 16, 249–252.CrossRefGoogle Scholar
[5]. Madan, D. B., Carr, P. P., and Chang, E. C. (1998). The variance gamma process and option pricing. Europ. Finance Rev. 2, 79–105.CrossRefGoogle Scholar
[6]. Öhgren, A. (2001). A remark on the pricing of discrete lookback options. J. Comput. Finance 4, 141–146.CrossRefGoogle Scholar
[7]. Raible, S. (2000). Lévy processes in finance: theory, numerics, and empirical facts. Doctoral Thesis, University of Freiburg. Available at http://www.freidok.uni-freiburg.de/volltexte/51/.Google Scholar
[8]. Shiryaev, A. N. (1999). Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific, Singapore.CrossRefGoogle Scholar