Scaling Volatility and Time Consistently in Hourly Binary Option Pricing
Summary
The discussion addresses volatility inputs for a cash-or-nothing call valued with a Black–Scholes formula when the option expires on an hourly schedule and the underlying is sampled every few seconds. The answer focuses on matching time units: Black–Scholes conventionally uses annualized volatility and time to expiry expressed as a fraction of a year. As an hourly option approaches expiry, its remaining time should be updated using the same annualization basis used for volatility.
The practical rule is consistency between the volatility scaling and the time variable, rather than a special adjustment simply because returns are sampled intraday. The response gives trading-day annualization conventions as examples but does not specify how to estimate volatility from the sampled returns or resolve details such as overnight periods, market hours, or microstructure noise. It therefore offers a units framework, not a complete realized volatility estimator. Those modeling choices matter when converting high-frequency observations into an annualized input for the option formula.
Key ideas
- Black–Scholes requires volatility and time to expiry to use consistent annual units.
- An hourly option's remaining maturity should be represented as a fraction of a year.
- The annualization convention should reflect the applicable trading calendar.
- The answer does not prescribe a method for estimating volatility from intraday sampled returns.
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# Volatility calculation for intra-day cash-or-nothing call binary option # Volatility calculation for intra-day cash-or-nothing call binary option Firstly, I do not have a quant finance background. This is new to me, and I imagine that this is a basic question for this group. I am calculating the price of a binary/digital option with closed-form equations derived from a Black-Scholes analysis. More specifically, I am using the Black-Scholes valuation for a Cash-or-nothing call. The option period that I have been asked to calculate ends every hour, on the hour. I am sampling the underlying every 5 seconds. How should I scale and/or calculate my volatility if I want to use the 'normal' approach (but assuming a 0 mean). These are all annualised to one year. Should I still do the same? More specifically, I am curious how I scale the standard deviation of the sum of the square log returns in this case? ## Answer by jtromans (score 1) https://quant.stackexchange.com/a/10259 Black–Scholes usually assumes your time and volatility are annualised. Accordingly, when you calculate the volatility term you would usually annualise it to 252 or 260 (or however many trading days a year are applicable to your situation). Accordingly, the time remaining term of the Binary Option must also be expressed as a fraction of a year (again, 252, or 260, days or..). By way of example, if you have a 1 hour option just starting, this T term would be expressed as a year (1/no-hours-tradeable-year). As the option period passes, you would decrease the T term so it is always expressed as part of a year. In summary, providing the way in which you scale volatility by time and the way you express your T term of the Black–Scholes are in the same, you'll be fine.
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