# 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?

## 1 Answer

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.