Solution: Why Equivalence Matters for Arbitrage

Exercise: Why Equivalence Matters for Arbitrage

Solution

If an event has positive physical probability but zero pricing probability, a payoff that is positive only on that event can be assigned zero value even though it can actually occur. Conversely, if the pricing measure assigns positive weight to a physically impossible event, prices reflect scenarios outside the model.

Equivalence prevents this mismatch by ensuring both measures agree on null events. They may disagree on weights, but not on which events are impossible.

Takeaways

• Equivalent measures preserve almost-sure statements.
• No-arbitrage pricing needs consistent state support.
• Change of measure reweights scenarios; it does not invent or delete possible states.