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No-Arbitrage Call Bounds in a Three-State Binomial Model

Article Quant Q&A · Author: cmcw
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Summary

The document asks how allowing a stock to finish unchanged affects no-arbitrage pricing in a one-period model. The stock starts at 100 and may end at 50, 100, or 150; the European call has a strike of 150. Since the call pays zero in every listed state, its payoff can be replicated by holding no assets, so the model does not pin down a unique price through replication alone.

The prompt seeks the range of prices consistent with no arbitrage, but it provides no derivation or answer. In a frictionless setting with unrestricted trading, a strictly positive call price would permit an arbitrage by selling the call and investing nothing to cover its zero payoff; a negative price would permit an arbitrage by buying it. Thus the no-arbitrage price is zero under the stated payoff and assumptions. The document is a short question rather than a worked explanation, and it does not discuss market frictions or other contract features.

Key ideas

  • A call with strike 150 pays zero across all three stated terminal stock prices.
  • The zero payoff can be replicated by holding no position in the stock or cash.
  • The question highlights that a redundant payoff does not produce a unique price through replication alone.
  • Under frictionless trading, any price other than zero for this always-worthless claim allows an arbitrage.

Tags

OptionsArbitrageDerivatives pricing
Full text
# No unique no-arbitrage price when the stock price can remain unchanged


# No unique no-arbitrage price when the stock price can remain unchanged












In a 1-period binomial model, with initial stock price 100, if the stock price is either 50,100, or 150 after 1 period then how can I show there is no longer a unique no-arbitrage price for a European call option with strike price K=150. What would the range of values for this price be that ensure there is no arbitrage?

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