Replication and Completeness in a Multiperiod Binomial Model

Summary

The document explains replication as matching a derivative’s payoff across every possible model state using positions in the underlying stock and a money market account. In a one-period example, stock and bond holdings are chosen so their combined terminal value equals the derivative payoff in each state; their initial cost then gives the derivative’s no-arbitrage price.

For multiple periods, replication is framed as a self-financing strategy: rebalancing between stock and bond uses only the portfolio’s existing funds, with no outside cash added or withdrawn. The document connects completeness to the ability to replicate every derivative and obtain a unique price. It offers a conceptual explanation rather than a worked numerical example or a detailed proof, and it does not discuss assumptions needed for completeness in particular models.

Key ideas

• A derivative is replicated when a portfolio of traded assets matches its payoff in every possible state.
• In a one-period model, the replicating portfolio’s initial cost determines the derivative’s fair price.
• A self-financing strategy funds each rebalance from the assets already held.
• Completeness means every derivative payoff can be replicated, yielding a unique no-arbitrage price.

Tags

# Complete Multiperiod Binomial model


# 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 ?

## Answer by Richi Wa (score 0, accepted)

https://quant.stackexchange.com/a/17459

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.