Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-06T09:57:53.192Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Pricing of American Options in Exponential Lévy Markets

Part of: Markov processes Mathematical economics Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Roman V. Ivanov
Roman V. Ivanov*
Affiliation:
Institute of Control Sciences of Russian Academy of Sciences
*
∗ Postal address: Laboratory 38, Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya 65, 117997 Moscow, Russia. Email address: roivanov@yahoo.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we discuss the problem of the pricing of American-style options in the exponential Lévy security market model. This model is typically incomplete, and we derive the explicit bounds of the interval of no arbitrage prices and the related optimal stopping moments for American put options and American call options in both finite and infinite horizon time. We consider a large class of Lévy processes.

Keywords

Incomplete market; Lévy processAmerican optionupper pricelower priceoptimal stopping

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60J75: Jump processes 91B70: Stochastic models
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 2 , June 2007 , pp. 409 - 419
Copyright
Copyright © Applied Probability Trust 2007 

References

Barndorff-Nielsen, O. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68.Google Scholar
Bellamy, N. and Jeanblanc-Picqué, M. (2000). Incompleteness of markets driven by a mixed diffusion. Finance Stoch. 4, 209–222.Google Scholar
Bergenthum, J. and Rüschendorf, L. (2006). Comparison of option prices in semimartingale models. Finance Stoch. 10, 222–249.Google Scholar
Carr, P., Geman, H., Madan, D. and Yor, M. (2002). The fine structure of asset returns: an empirical investigation. J. Business 75, 305–332.Google Scholar
Cherny, A. S. and Shiryaev, A. N. (2002). Change of time and measure for Levy processes. (Lectures from the Summer School ‘From Lévy processes to semimartingales – recent theoretical developments and applications to finance’ (Aarhus, August 2002)).Google Scholar
Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250.Google Scholar
Eberlein, E. and Jacod, J. (1997). On the range of options prices. Finance Stoch. 1, 131–140.Google Scholar
Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli 1, 281–299.Google Scholar
Eberlein, E. and Prause, K. (2002). The generalized hyperbolic model: financial derivatives and risk measures. In Mathematical Finance—Bachelier Congress, 2000 (Paris), Springer, Berlin, pp. 245–267.Google Scholar
Ekström, E. (2006). Bounds for perpetual American option prices in a Jump diffusion model. J. Appl. Prob. 43, 867–873.Google Scholar
Frey, R. and Sin, C. A. (1999). Bounds on European option prices under stochastic volatility. Math. Finance 9, 97–116.Google Scholar
Gushchin, A. A. and Mordecki, E. (2002). Bounds on option prices for semimartingale market models. Proc. Steklov Inst. Math. 237, 73–113.Google Scholar
Harrison, M. and Kreps, D. (1979). Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 20, 381–408.Google Scholar
Henderson, V. and Hobson, D. G. (2003). Coupling and option price comparisons in a Jump-diffusion model. Stoch. Stoch. Reports 75, 79–101.Google Scholar
Ivanov, R. V. (2005). Discrete approximation of finite-horizon American-style options. Lithuanian Math. J. 45, 525–536.Google Scholar
Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.Google Scholar
Jakubénas, P. (2002). On option pricing in certain incomplete markets. Proc. Steklov Math. Inst. 237, 123–142.Google Scholar
Kramkov, D. (1996). Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Prob. Theory Relat. Fields 105, 459–479.Google Scholar
Rüschendorf, L. (2002). On upper and lower prices in discrete-time models. Proc. Steklov Inst. Math. 237, 134–139.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Selivanov, A. (2004). On martingale measures in exponential Lévy models. Theory Prob. Appl. 49, 317–334.Google Scholar
Shiryaev, A. N. (1999). Essentials of Stochastic Finance. Facts, Models, Theory. World Scientific, River Edge, NJ.Google Scholar
Sin, C. A. (1996). Strictly local martingales and hedge ratios in stochastic volatility models. , Cornell University.Google Scholar
Yan, J. A. (1998). A new look at the fundamental theorem of asset pricing. J. Korean Math. Soc. 35, 659–673.Google Scholar