Risk-Neutral Measure
A risk-neutral (equivalent martingale) measure is a probability reweighting that makes discounted asset prices martingales — prices are then discounted expected payoffs under that measure, not under the real-world P.
Definition
Existence of an equivalent martingale measure ↔ no arbitrage (first FTAP). Uniqueness ↔ completeness (second FTAP). Under Q, all assets grow at r; risk premia are inside the change of measure.
Why it matters
Option prices are Q-expectations. Historical frequencies are P. Calibrating a vol surface is Q; forecasting realized vol is P. They are allowed to differ.
Case
A 1% physical crash probability can be a 8% Q-probability if agents hate that state. That is why puts look ‘expensive’ versus historical frequency.
How to read it
If a slide mixes historical default rates with CDS-implied (Q) hazards without a premium, the identity is wrong.
Key papers
Landmark references with real publisher or arXiv links — not token-overlap noise.
- The Pricing of Options and Corporate LiabilitiesBlack, Scholes · 1973 · Journal of Political EconomyHedging argument that leads to risk-neutral pricing.