Abstract
We study the problem of convergence of discrete-time option values to continuous-time option values. While previous papers typically concentrate on the approximation of geometric Brownian motion by a binomial tree, we consider here the case where the model is incomplete in both continuous and discrete time. Option values are defined with respect to the criterion of local risk-minimization and thus computed as expectations under the respective minimal martingale measures. We prove that for a jump-diffusion model with deterministic coefficients, these values converge; this shows that local risk-minimization possesses an inherent stability property under discretization.
Financial support by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303 at the University of Bonn, is gratefully acknowledged.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81 (1973), 637–659.
J. C. Cox, S. A. Ross, and M. Rubinstein, Option Pricing: A Simplified Approach, Journal of Financial Economics 7 (1979), 229–263.
H. Dangler, Poisson Approximations to Continuous Security Market Models, preprint, Cornell University, 1993.
D. Duffle and P. Protter, From Discrete-to Continuous-Time Finance: Weak Convergence of the Financial Gain Process, Mathematical Finance 2 (1992), 1–15.
E. Eberlein, On Modeling Questions in Security Valuation, Mathematical Finance 2 (1992), 17–32.
H. Föllmer and M. Schweizer, Hedging of Contingent Claims under Incomplete Information, in: Applied Stochastic Analysis, Stochastics Monographs (M. H. A. Davis and R. J. Elliott, eds.), vol. 5, Gordon and Breach, London/New York, 1991, pp. 389–414.
H. He, Convergence from Discrete-to Continuous-Time Contingent Claims Prices, Review of Financial Studies 3 (1990), 523–546.
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, 1981.
M. Jeanblanc-Picqué and M. Pontier, Optimal Portfolio for a Small Investor in a Market with Discontinuous Prices, Applied Mathematics and Optimization 22 (1990), 287–310.
F. Mercurio and W. J. Runggaldier, Option Pricing for Jump-Diffusions: Approximations and their Interpretation, Mathematical Finance 3 (1993), 191–200.
R. C. Merton, Theory of Rational Option Pricing, Bell Journal of Economics and Management Science 4 (1973), 141–183.
R. C. Merton, Option Pricing when Underlying Stock Returns are Discontinuous, Journal of Financial Economics 3 (1976), 125–144.
D. B. Nelson and K. Ramaswamy, Simple Binomial Processes as Diffusion Approximations in Financial Models, Review of Financial Studies 3 (1990), 393–430.
M. Schäl, On Quadratic Cost Criteria for Option Hedging, (to appear in Mathematics of Operations Research), preprint, University of Bonn, 1992.
M. Schweizer, Hedging of Options in a General Semimartingale Model, Diss. ETHZ No. 8615, 1988.
M. Schweizer, Option Hedging for Semimartingales, Stochastic Processes and their Applications 37 (1991), 339–363.
M. Schweizer, Approximating Random Variables by Stochastic Integrals, and Applications in Financial Mathematics, Habilitationsschrift, University of Göttingen, 1993.
H. Shirakawa, Security Market Model with Poisson and Diffusion Type Return Process, preprint IHSS 90–18, Tokyo Institute of Technology, 1990.
W. Willinger and M. S. Taqqu, Toward a Convergence Theory for Continuous Stochastic Securities Market Models, Mathematical Finance 1 (1991), 55–99.
X.-X. Xue, Martingale Representation for a Class of Processes with Independent Increments and its Applications, in: Applied Stochastic Analysis, Proceedings of a US—French Workshop (I. Karatzas and D. Ocone, eds.), Rutgers University, New Brunswick, N.J., Lecture Notes in Control and Information Sciences 177, Springer, 1992, pp. 279–311.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Basel AG
About this paper
Cite this paper
Runggaldier, W.J., Schweizer, M. (1995). Convergence of Option Values under Incompleteness. In: Bolthausen, E., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability, vol 36. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7026-9_26
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7026-9_26
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7028-3
Online ISBN: 978-3-0348-7026-9
eBook Packages: Springer Book Archive