Statistics and Probability Letters 111 (2016) 86–92 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Bernstein’s inequalities and their extensions for getting the Black–Scholes option pricing formula Anna Glazyrina, Alexander Melnikov ∗ Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada article info Article history: Received 28 August 2015 Received in revised form 28 December 2015 Accepted 2 January 2016 Available online 8 January 2016 MSC: 62P05 41A25 Keywords: Bernstein’s inequalities Option pricing Binomial model Cox–Ross–Rubinstein formula Black–Scholes formula Rate of convergence abstract In this paper we show how the results of Bernstein (1943) and recent results of Zubkov and Serov (2012) on the normal approximation to the binomial distribution lead to an alternative derivation of the Black–Scholes formula from a binomial option pricing model. © 2016 Elsevier B.V. All rights reserved. 1. Introduction Many models and facts in modern probability theory are based upon a Bernoulli scheme and the corresponding Binomial distribution developed under classical stochastic experiments. For this reason, any results in this direction have the potential for further research and extension. One of the key facts usually referred to in this regard is the classical De Moivre–Laplace theorem which gives normal approximation to the binomial distribution when the number of trials, n, grows without bound. This theorem has numerous applications. In particular, it is used for verification of the convergence of option prices in the financial market model with discrete time (binomial market, Cox–Ross–Rubinstein formula) to option prices in the Black–Scholes model (see Cox et al., 1979). Application of the De Moivre–Laplace theorem in this case provides both the convergence of corresponding option prices and the rate of convergence of 1/ √ n. The deeper studies of these subjects apply various modifications of the De Moivre–Laplace theorem. The most prominent and to some extent finishing touch is the use of Uspensky’s theorem (Uspensky, 1937), which provides for convergence of order 1/n (see Chang and Palmer, 2007; Chung et al., 2007; Leisen and Reimer, 2000). In a recent article, Zubkov and Serov (2012) brought into view one more classical modification of the De Moivre–Laplace theorem, given by Bernstein in 1943. In this paper we demonstrate how these results can be used to show the convergence of binomial (Cox–Ross–Rubinstein) option prices to the Black–Scholes prices. ∗ Corresponding author. E-mail addresses: glazyrin@ualberta.ca (A. Glazyrina), melnikov@ualberta.ca (A. Melnikov). http://dx.doi.org/10.1016/j.spl.2016.01.002 0167-7152/© 2016 Elsevier B.V. All rights reserved.