The rate of convergence of option prices when general martingale discrete-time scheme approximates the Black-Scholes model

• Volume: 104, Issue: 1, page 151-165
• ISSN: 0137-6934

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Abstract

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We take the martingale central limit theorem that was established, together with the rate of convergence, by Liptser and Shiryaev, and adapt it to the multiplicative scheme of financial markets with discrete time that converge to the standard Black-Scholes model. The rate of convergence of put and call option prices is shown to be bounded by ${n}^{-1/8}$. To improve the rate of convergence, we suppose that the increments are independent and identically distributed (but without binomial or similar restrictions on the distribution). Under additional assumptions, in particular under the assumption that absolutely continuous component of the distribution is nonzero, we apply asymptotical expansions of distribution function and establish that the rate of convergence is $O\left({n}^{-1/2}\right)$.

How to cite

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Yuliya Mishura. "The rate of convergence of option prices when general martingale discrete-time scheme approximates the Black-Scholes model." Banach Center Publications 104.1 (2015): 151-165. <http://eudml.org/doc/281800>.

@article{YuliyaMishura2015,
abstract = {We take the martingale central limit theorem that was established, together with the rate of convergence, by Liptser and Shiryaev, and adapt it to the multiplicative scheme of financial markets with discrete time that converge to the standard Black-Scholes model. The rate of convergence of put and call option prices is shown to be bounded by $n^\{-1/8\}$. To improve the rate of convergence, we suppose that the increments are independent and identically distributed (but without binomial or similar restrictions on the distribution). Under additional assumptions, in particular under the assumption that absolutely continuous component of the distribution is nonzero, we apply asymptotical expansions of distribution function and establish that the rate of convergence is $O(n^\{-1/2\})$.},
author = {Yuliya Mishura},
journal = {Banach Center Publications},
keywords = {Black-Scholes model; martingale discrete-time scheme; martingale central limit theorem; convergence rates},
language = {eng},
number = {1},
pages = {151-165},
title = {The rate of convergence of option prices when general martingale discrete-time scheme approximates the Black-Scholes model},
url = {http://eudml.org/doc/281800},
volume = {104},
year = {2015},
}

TY - JOUR
AU - Yuliya Mishura
TI - The rate of convergence of option prices when general martingale discrete-time scheme approximates the Black-Scholes model
JO - Banach Center Publications
PY - 2015
VL - 104
IS - 1
SP - 151
EP - 165
AB - We take the martingale central limit theorem that was established, together with the rate of convergence, by Liptser and Shiryaev, and adapt it to the multiplicative scheme of financial markets with discrete time that converge to the standard Black-Scholes model. The rate of convergence of put and call option prices is shown to be bounded by $n^{-1/8}$. To improve the rate of convergence, we suppose that the increments are independent and identically distributed (but without binomial or similar restrictions on the distribution). Under additional assumptions, in particular under the assumption that absolutely continuous component of the distribution is nonzero, we apply asymptotical expansions of distribution function and establish that the rate of convergence is $O(n^{-1/2})$.
LA - eng
KW - Black-Scholes model; martingale discrete-time scheme; martingale central limit theorem; convergence rates
UR - http://eudml.org/doc/281800
ER -

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