Non-classical arithmetics replace ordinary addition and/or multiplication in standard arithmetic with other binary operations called pseudo-addition and multiplication. Dynamic non-classical arithmetic (DNA) allows a different non-classical arithmetic to be used at each time. We apply DNA to enhance our understanding of the logistic distribution in probability theory. We also apply DNA to develop a novel arbitrage-free option pricing model in finance. Under our approach, European option valuation reduces to pseudo-addition of spot and strike, while Bermudan option valuation reduces to repeated pseudo-addition. Simple and realistic closed-form formulas are easily generated in both cases.