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Defining the structure of maps using local features is among the popular fields of study in mathematics. Therefore determining the structure of maps on rings which behave like derivations at idempotent-product elements has been getting attention recently. This subject is useful for examining the structure of rings and algebraic operators in both algebra and analysis as well. Suppose that R is a ring, d : R ! R is an additive map, z 2 R and d meets the condition below: 8a; b 2 R : d(ab) = ad(b) + d(a)b Therefore d is called a derivation on R. If for every a; b 2 R where ab = z, d(ab) = ad(b) + d(a)b then d behaves like a derivation at idempotent-product elements of ab = z. The main challenge... 

In the past decades, option pricing has become one of the major areas in modern financial theory and practice. The Black-Scholes-Merton method is a type of option pricing, which is an appropriate and very important model in financial markets due to the pricing process under the assumption of no arbitrage and the recognition of the appropriate discount rate.Inspite of its advantages, this model is not appropriate for pricing the options which need to be investigated before the maturity.To overcome this limitation, some discrete extension of Black Scholes model were introduced such as binomial and trinomial trees.In all of these models during the contract period, volatility is considered...