May 20, 2026

Recovering risk-neutral density from option prices

Breeden-Litzenberger gives you the market's implied distribution over future spot from a strip of vanilla calls. Here's how, and why it's the cleanest way to read sentiment from options.

TODO — write the post body here. Outline:

  • State the result: the risk-neutral density q(K) equals the second derivative of the call price with respect to strike, discounted by e^{rT}.
  • Walk through the derivation in one short paragraph (twin digital → butterfly limit).
  • Show what a real numerical estimate looks like — finite-difference the IV surface, convert to price space, take the second difference. Mention smoothing.
  • One concrete example: SPY 30d density on a calm day vs a CPI day, and what that tells you about tail pricing.
  • End with the caveat: this is the risk-neutral density, not the physical one. The difference is the variance risk premium — which is the topic of the next post.
All notes