Two$-$period Binomial Option Valuation $($both European & American$)$

Calculate a European Call Option$$s value using both the No$-$arbitrage & Expectation approaches. The key information on the option is as follows:

Current underlying price: $$50$

Exercise price: $$50$

Up factor $($price$)$: $1\mathrm{.}12$

Down factor $($price$)$: $0\mathrm{.}92$

Risk free rate for one period: $5\%$