On the rate of convergence of prices of barrier options with discrete and continuous time

Authors:
O. M. Soloveyko and G. M. Shevchenko

Translated by:
O. Klesov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **79** (2008).

Journal:
Theor. Probability and Math. Statist. **79** (2009), 171-178

MSC (2000):
Primary 91B28; Secondary 60G50, 60F05

Published electronically:
December 30, 2009

MathSciNet review:
2494546

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A barrier option is a derivative realized or cancelled if the price of the underlying asset crosses a certain barrier. Most of the models in financial mathematics are considered for markets with continuous time. However the trading days for a particular stock take place at separate moments, i.e. discretely. The Black-Scholes model is extended in the paper in the sense that we consider barrier options with varying drifts. We find the rate of convergence of prices of such options with discrete time to the prices of options with continuous time.

**1.**Mark Broadie, Paul Glasserman, and Steven Kou,*A continuity correction for discrete barrier options*, Math. Finance**7**(1997), no. 4, 325–349. MR**1482707**, 10.1111/1467-9965.00035**2.**Mark Broadie, Paul Glasserman, and S. G. Kou,*Connecting discrete and continuous path-dependent options*, Finance Stoch.**3**(1999), no. 1, 55–82. MR**1805321**, 10.1007/s007800050052**3.**P. Carmona, F. Petit, J. Pitman, and M. Yor,*On the laws of homogeneous functionals of the Brownian bridge*, Studia Sci. Math. Hungar.**35**(1999), no. 3-4, 445–455. MR**1762255****4.**Endre Csáki, Antónia Földes, and Paavo Salminen,*On the joint distribution of the maximum and its location for a linear diffusion*, Ann. Inst. H. Poincaré Probab. Statist.**23**(1987), no. 2, 179–194 (English, with French summary). MR**891709****5.**Per Hörfelt,*Extension of the corrected barrier approximation by Broadie, Glasserman, and Kou*, Finance Stoch.**7**(2003), no. 2, 231–243. MR**1968947**, 10.1007/s007800200077**6.**Peter E. Kloeden and Eckhard Platen,*Numerical solution of stochastic differential equations*, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. MR**1214374****7.**S. G. Kou,*On pricing of discrete barrier options*, Statist. Sinica**13**(2003), no. 4, 955–964. Statistical applications in financial econometrics. MR**2026057****8.**Robert C. Merton,*Theory of rational option pricing*, Bell J. Econom. and Management Sci.**4**(1973), 141–183. MR**0496534**

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2000):
91B28,
60G50,
60F05

Retrieve articles in all journals with MSC (2000): 91B28, 60G50, 60F05

Additional Information

**O. M. Soloveyko**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov avenue, 6, Kyiv 03127, Ukraine

Email:
osoloveyko@univ.kiev.ua

**G. M. Shevchenko**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov avenue, 6, Kyiv 03127, Ukraine

Email:
zhora@univ.kiev.ua

DOI:
http://dx.doi.org/10.1090/S0094-9000-09-00789-3

Received by editor(s):
March 7, 2008

Published electronically:
December 30, 2009

Article copyright:
© Copyright 2009
American Mathematical Society