Using Digital Option Prices to Recover an Implied Distribution

Summary

This note proposes an alternative to numerically differentiating call prices twice to recover an option-implied probability density. The standard relationship mentioned connects the second strike derivative of a call price with the discounted density of the underlying at expiry. The author instead suggests using cash-or-nothing put prices adjusted for volatility skew to obtain the cumulative distribution function, then applying its inverse to uniform random draws to generate underlying-price scenarios consistent with that distribution.

The document presents this as a question for review rather than a validated method: it provides no derivation, numerical example, or evidence that the skew adjustment and discounting conventions are correct. In practice, the approach would depend on consistent option price and volatility inputs, a valid monotone cumulative distribution, and careful treatment of rates, dividends, and expiry conventions. Readers should regard the proposed formula as an idea to verify before using it for simulation or pricing.

Key ideas

• The standard Breeden-Litzenberger relationship recovers an expiry density from the second strike derivative of call prices.
• The proposed alternative uses skew-adjusted cash-or-nothing put prices to estimate the cumulative distribution.
• Inverse-transform sampling would map uniform random draws into scenarios from the estimated distribution.
• The note offers a proposal without derivation or validation, so its adjustment formula and assumptions require checking.

Tags

# Alternative Method for Determining Option-Implied pdf


# Alternative Method for Determining Option-Implied pdf












As I am refining a pricing model to incorporate skew, and not just ATM volatilities, I need to create random realizations of the underlying consistent with the skew-implied pdf. When searching, one ends up with the Breeden-Litzenberger formula, which states that:

$\frac{\partial^{2}C}{\partial K^{2}}=e^{-rT}g(S_{T})$

As I am slightly wary of using numerical second derivatives in my code, I looked for an alternative way of obtaining this. I came up with the idea of using correctly skew-adjusted cash-or-nothing binary put prices in order to derive the CDF of the underlying:

$P_{dig}=P_{dig,noskew}+\nu_{vanilla}*\frac{\partial \sigma}{\partial K}$

As $CDF=e^{rT}P_{dig}$, I run [0,1]-normally distributed random numbers through the inverse function $CDF^{-1}$ to get realizations of the underlying that are distributed consistent with the skew.

Can I please have your views on this / please comment if I am missing something here

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