Volatility calculation for intra-day cash-or-nothing call binary option

Question

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 1

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.