Redundant Securities in a Two-State Market

Summary

The document raises a question about redundancy when three securities each have two possible future payoffs. It describes a common exercise setup: represent each asset by its payoff in the two states, then solve two equations to find holdings in two securities that replicate the third.

The central concept is that with only two states, payoff vectors lie in a two-dimensional space. If two securities span that space, any third payoff vector can be replicated and is redundant. Whether this conclusion holds depends on the payoffs and on whether the chosen two securities are independent; if they do not span the relevant payoff space, replication may fail. The source contains only the question and no answer, so it does not establish assumptions such as trading constraints or give a specific payoff example.

Key ideas

• A security is redundant when its state-contingent payoff can be replicated by other securities.
• With two future states, payoffs can be represented as vectors with two components.
• Two securities can replicate a third when their payoff vectors span the required payoff space.
• Replication depends on the securities’ payoffs and independence, not merely on the number of assets.

Tags

# Non-redundant asset?


# Non-redundant asset?












I've been solving many exercises with three assets that have two possible payoffs each, one payoff per possible future state. The question is always the same, i.e. is any asset redundant.

After setting a system of two equations with two unknowns, I always find some quantities of other two stocks, that make the third one redundant.

So why the question? Is it possible that we couldn't express one security with the other two?

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