Error estimates for multinomial approximations of American options in a class of jump diffusion models

Abstract

We derive error estimates for multinomial approximations of American options in a class of multidimensional jump diffusion models. We assume that the pay-offs are Markovian and satisfy Lipschitz-type conditions. Error estimates for such type of approximations were not obtained before. Our main tool is the strong approximation theorems for i.i.d. random vectors which were obtained in Sakhanenko (A new way to obtain estimates in the invariance principle, High Dimen. Probab. II (2000), pp. 221–243). For the multidimensional Black–Scholes model, our results can also be extended to a general path-dependent pay-offs which satisfy Lipschitz-type conditions. For the case of multinomial approximations of American options for the Black–Scholes model, our estimates are a significant improvement of those which were obtained in Kifer (Optimal stopping and strong approximation theorems, Stochastics 79 (2007), pp. 253–273; for game options in a more general set-up).

Keywords:

• optimal stopping
• American options
• jump diffusion model
• strong approximation theorems

AMS Subject Classification::

• 60F15
• 60G40
• 91B24
• 91B28

Acknowledgements

I would like to express my deepest gratitude to my PhD adviser, Yuri Kifer, for guiding me and helping me to present this work. I am also very grateful to A. Zaitsev for his valuable discussions. I would like to thank the anonymous referees for reading carefully the manuscript and helping me to improve the paper. This research was partially supported by ISF grant no. 130/06.