Fundamental Assumption

We assume that the market has no arbitrage. As a result, given two derivatives $(O_{t})_{t\in[0,T]},(\tilde{O}_{t})_{t\in[0,T]}$ with identical expiration $T$ and payoff (i.e. $O_{T}=\tilde{O}_{T}$) then $$O_{t}=\tilde{O}_{t}\quad\forall t\in[0,T]$$