Recovering Implied Probability Density from Put Prices

Summary

The note explains that the Breeden–Litzenberger relationship can be applied to put prices as well as call prices. Taking the second derivative of a European option price with respect to strike yields the risk-neutral implied probability density at that strike, subject to the usual conditions behind the identity.

Its reasoning is put–call parity: for European options, the parity relation differs by terms that are linear in strike, so taking a second strike derivative removes those terms. The put-based result therefore matches the call-based result. The document gives no numerical example or discussion of estimating derivatives from noisy market quotes, so it establishes the theoretical equivalence rather than a practical calibration procedure.

Key ideas

• The second strike derivative of a European call price gives the risk-neutral implied density under the Breeden–Litzenberger identity.
• The same result follows from European put prices.
• Put–call parity explains the equivalence because its other terms vanish after two strike derivatives.
• The note does not address practical smoothing or numerical differentiation of market prices.

Tags

# Implied Probability Density with Puts


# Implied Probability Density with Puts












The second derivative of the call price at K gives the probability of that strike (implied probability density).

In practice, what adjustments or acknowledgements (if any) need to be made to produce a IPD with puts?

## Answer by Quantuple (score 2)

https://quant.stackexchange.com/a/30679

By call-put parity, the second derivative of a European call option price with respect to strike is strictly equivalent to that of a European put.

So, yes: the result, known as the Breeden-Litzenberger identity, stays unchanged.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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