Pricing Digital Options from a Single Realized Volatility Estimate

Summary

The document asks whether a continuum of European digital option prices can be estimated when no implied-volatility surface is available and the only volatility input is a single realized-volatility figure for a fixed tenor. The estimate is assumed to match risk-neutral variance near at-the-money and perhaps at a few deep-wing deltas. The desired output is a defensible strike-dependent price for a payoff that pays when the terminal underlying exceeds the strike.

The author has considered moment expansions, parametric volatility models, and mixtures of distributions, but reports that these approaches have not produced stable results. The question is therefore as much about identifiability and model assumptions as about calculation: one variance estimate cannot uniquely determine the full risk-neutral distribution or its tail shape. The document offers no proposed solution or empirical evidence, and explicitly recognizes that the input may be inadequate. Any resulting prices would depend materially on additional density-shape assumptions and require validation.

Key ideas

• A single realized-volatility estimate does not uniquely determine digital prices across strikes.
• Digital option values depend on the risk-neutral distribution, including its tails.
• Moment expansions, parametric models, and distribution mixtures are identified as possible frameworks.
• The author reports instability in attempted approaches but provides no evaluated solution.
• Additional assumptions about distribution shape are necessary to produce strike-dependent prices.

Tags

# How can one price European digital options across strikes using only a single realised‐volatility estimate?


# How can one price European digital options across strikes using only a single realised‐volatility estimate?












I’m in a setting where no implied‐volatility surface exists (e.g. illiquid or bespoke contracts). All I have is a single realised‐volatility (RV) number for a fixed tenor, which correctly reproduces risk‐neutral variance at ATM (Δ≈0.5) and perhaps at two deep‐wing deltas (Δ≈0.07,0.93). I need to quote defensible digital option prices, i.e., discontinuous European payoffs of the form $1_{S_T > K}$ across a continuum of strikes. What theoretical or heuristic frameworks allow one to go from:

- A single scalar RV input (accurate at only a few deltas) to

- A full digital pricing function D(K), without any observed implied volatilities?

I’m aware of several approaches, moment-based expansions, parametric volatility models, and distribution mixtures among them, and have tried variations of these, but none have yielded stable or reliable results in practice. It’s possible I’m applying them incorrectly, or that the assumptions required don’t hold under my setup.

I’m not looking for a full IV‐surface calibration, just tractable methods to produce defensible option prices D(k) given only a single RV input and basic density‐shape assumptions.

Note: I recognize that starting from a single RV estimate might be fundamentally flawed or too simplistic. Any insight on whether this methodology is conceptually sound, or guidance on pitfalls of this approach, would be appreciated.

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