# Estimation of volatility into Black-76 formula

I am trying to estimate the (annualized) volatility that should go into an European Swaption (such as 2y5y). Given we take the black76-formula, where the discounting is the term outside the expectation, and the pricing formula has the (a) the drift term i.e. the Forward, and (b) the distribution with the volatility term.

If I extract a time-series of the 2y5y forward rates, I would like to ask what should I use to estimate the volatility. Would it be

1. he (excel) stdev function which is taking (Summation(Xi - Xmean)^2) / n; or
2. the summation of the square of the realized daily moves... Summation(Xi^2)

Kind regards Kian

• Are you sure that the time series of 2y5y forward swap rates is the "correct" series for the volatility estimation? Every day from $t_0$ to $t_n=2$ the underlying of the option is going to be (2y-1d)5y, (2y-2d)5y, (2y-3d)5y and so on until the very last day when you settle on the spot 5y. Therefore you should collect a time series of all these values and compute the cross differences (because today (2y-1d)5y is yesterday (2y-2d)5y and so on). – LePiddu Dec 11 '19 at 14:15
• actually, i agree with this technically. so one takes the forward roll-down to spot or maybe even the spot from t0 to t-expiry – Kiann Dec 26 '19 at 13:36
• however, ** if ** one uses the time-series of the forward rate, how would one plug it into the black76 formula? ... i unfortunately just have to work with the time-series of forwards – Kiann Dec 26 '19 at 13:38

Why are you not using broker quoted implied volatilities? E.g. ICAP and TP quote all the standard expiries & tenors. If you want a reliable (i.e. executable) price you'd be better off using implied rather than historical vol. For example USD 2y5y OIS Black vol is around 40% right now (equivalent to 350bp spot premium). If that's not a concern or available, I would use the standard deviation of daily differences, scaled by $$\sqrt{T}$$ (typically $$T = 252$$). But note that implied can either trade at a discount or premium to realised volatility.
• would use $\sigma = \sqrt{\frac{1}{N-1} \sum_{i=1}^{N}{(x_i - \bar{x})^2}}$ which is the same as Excel's =STDEV.S() over the time of the expiry – oronimbus Dec 4 '19 at 16:45