Using Option-Implied Densities to Price Payoffs and Check Arbitrage
Summary
The document explains how a risk-neutral probability density for the underlying at expiration can be recovered from the second strike derivative of call prices, under a constant interest-rate assumption. Once recovered, the density can be integrated against a maturity payoff to value claims whose payoff depends on the terminal underlying price.
It also describes a diagnostic use: an implied density that turns negative may signal problems in the option volatility curve, especially when wing implied volatilities are poorly marked or interpolated. The discussion emphasizes that this is a pricing distribution consistent with traded, hedgeable derivatives, not necessarily the real-world frequency distribution of outcomes. The answer uses betting odds as an analogy for why a price-implied probability need not equal a person's subjective forecast. It gives no empirical example or detailed construction procedure, and the density's reliability depends on sound option prices and curve extrapolation.
Key ideas
- The second strike derivative of call prices can recover a risk-neutral terminal density under the stated rate assumption.
- A maturity payoff can be priced by taking its expectation under that density and discounting it.
- Negative density values can expose arbitrage inconsistencies in an option volatility curve.
- The recovered density reflects market pricing and hedging relationships rather than necessarily real-world probabilities.
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# Implied probability density (Question 2 - Applications and Interpretation)
# Implied probability density (Question 2 - Applications and Interpretation)
Using the second derivative of the Call-Option-Price one can try to recover the pricing density.
> Formally: Assuming a constant interst rate $r$ and also not making any assumptions on the model used to evolve $S_t$ $C(t,S_t,K,r,T)=e^{-r(T-t)}\int_0^{\infty}(S_T-K)^+f(S_T|S_t)dS_T$ The density is then recovered via $p(S_T|S_t)=e^{r(T-t)}\frac{\partial^2 C(t,S_t,K,r,T)}{\partial K^2}|_{K=S_T}$
As a follow-up to my last question:
- What are the applications of this recovered density ?
- How can we Interpret it ? (can it be considered the "real" probability density ? - seeing how it is used in a pricing context it should still be risk neutral)
## Answer by joelhoro (score 4, accepted)
https://quant.stackexchange.com/a/10575
1) A straigthforward application is to price any complex payoff at maturity using this.
By that I mean a payoff that is such that the price of the option is
$$P = e^{-r(T-t)}E[f(S_T)] $$
Which you can then calculate by integrating $f(S_T)$ w.r.t. to your density.
One of the challenges though is to have a proper marks and inter/extrapolation for the implied vols of the wings (i.e. far away from current forward) so that your density does not become negative.
Therefore another application for this density is that it is a good way to check for no arbitrage on a term option volatility curve.
2) Yes, it is just the probability which you can hedge against by using derivatives on the market. It's a bit like if a horse has 1 against 4 odds, that does not mean that there is a 20% chance that he will win, it just (sort of) means that if you have some product whose price needs to know that probability, the right number to put in the price is 20% because the hedging instrument you will use (i.e. better 1 against 4 on that horse, or against) will cost exactly that.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.