Incomplete Markets in a Three-Outcome Option Tree

Summary

The document extends a one-step binomial option tree with a third possible asset outcome: the price can rise, fall, or stay unchanged. Using a call with strike 100 and stock outcomes of 110, 100, and 90, it shows that one stock position cannot make the option-and-stock portfolio equal in value across all three states. The example therefore lacks the exact replication available in the binomial case.

The article also applies risk-neutral pricing. To keep the expected stock price unchanged, the up and down outcomes must have the same probability, but the middle outcome leaves that probability free to vary. The resulting expected call value spans a range rather than a unique price, consistent with the bounds from the hedging argument. This illustrates an incomplete market, where available securities cannot replicate every derivative payoff. The document says that pricing then depends on investors’ risk preferences; its brief, one-period example does not explore how those preferences determine a market price.

Key ideas

• Adding a third possible asset outcome can prevent exact replication of an option payoff with stock alone.
• In the example, one hedge ratio cannot equalize the portfolio values across all three next-step states.
• The risk-neutral probabilities are not uniquely determined when the middle outcome is possible.
• The example produces a range of possible call values rather than a unique price.
• Incomplete markets leave derivative pricing dependent on investor risk preferences.

Tags

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