Preprint Review Version 1 Preserved in Portico This version is not peer-reviewed

# On the Option Pricing by the Binomial Model

Version 1 : Received: 18 July 2021 / Approved: 19 July 2021 / Online: 19 July 2021 (11:39:43 CEST)

How to cite: Halidias, N. On the Option Pricing by the Binomial Model. Preprints 2021, 2021070407. https://doi.org/10.20944/preprints202107.0407.v1 Halidias, N. On the Option Pricing by the Binomial Model. Preprints 2021, 2021070407. https://doi.org/10.20944/preprints202107.0407.v1

## Abstract

In this note we study the binomial model applied to European, American and Bermudan type of derivatives. Our aim is to give the necessary and sufficient conditions under which we can define a fair value via replicating portfolios for any derivative using simple mathematical arguments and without using no arbitrage techniques. Giving suitable definitions we are able to define rigorously the fair value of any derivative without using concepts from probability theory or stochastic analysis therefore is suitable for students or young researchers. It will be clear in our analysis that if $e^{r \delta} \notin [d,u]$ then we can not define a fair value by any means for any derivative while if $d \leq e^{r \delta} \leq u$ we can. Therefore the definition of the fair value of a derivative is not so closely related with the absence of arbitrage. In the usual probabilistic point of view we assume that $d < e^{r \delta} < u$ in order to define the fair value but it is not clear what we can (or we can not) do in the cases where $e^{r \delta} \leq d$ or $e^{r \delta} \geq u$.

## Keywords

Option Pricing; Fair Value; Binomial Model; Bermudan Options

## Subject

Computer Science and Mathematics, Algebra and Number Theory