Complete Multiperiod Binomial model

I have the following deifnition of a Complete multiperiod binomial model:

A multi period binomial model can be called complete if every derivative security can be replicated by trading in the underlying stock and the money market. In the complete market every derivative has a unique price that precludes arbitrage.

What does it mean that the derivative security can be replicated ?

1 Answer

In a one period model replication means that no matter which state of the model it holds that $$a S_1+ b B_1 = D_1$$ where $a,b$ are the quanties of stock and bond held and $S_1,B_1$ and $F_1$ are the prices of the stock, the bond and the derivative at $t=1$. In such a case the fair price of the derivative at time $0$ is $a S_0 + b B_0$ (not thinking about correct discounting). Fair here means that it can be replicated by other assets in the market (in the given model).

For multi-period settings usually one defines the notion of a strategy to be self-financing. Thus you shift between the stock and the bond but all money is either borrowed from/put on the money market account or financed by sales of the stock. Self-financing means that there is no money coming from outside.