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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

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  • $\begingroup$ 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). $\endgroup$ – LePiddu Dec 11 '19 at 14:15
  • $\begingroup$ 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 $\endgroup$ – Kiann Dec 26 '19 at 13:36
  • $\begingroup$ 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 $\endgroup$ – Kiann Dec 26 '19 at 13:38
  • $\begingroup$ In electricity markets we use the ATM IV for the appropriate maturity. In the absence of liquidity, which is sadly common, we resort to GARCH models or the IV of a strongly correlated liquid market whose historical volatility resembles that of the market of interest. $\endgroup$ – CasusBelli Dec 28 '20 at 3:45
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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.

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  • $\begingroup$ it is for a illiquid currency where there is no market-quotes. $\endgroup$ – Kiann Dec 4 '19 at 13:22
  • $\begingroup$ I would use the standard deviation of daily differences : by this, you mean you will take summation(Xi - Xmean)^2 / n; rather than summation(Xi)^2/n of the forward-rate daily change? $\endgroup$ – Kiann Dec 4 '19 at 13:23
  • $\begingroup$ 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 $\endgroup$ – oronimbus Dec 4 '19 at 16:45
  • $\begingroup$ the key query I have, is why? .... consider the original black76 formula, which has the forward-rate, and the brownian motion volatility term; under the risk-neutral measure; with the risk-free rate outside of the expectation..... would we estimate the volatility (using the forward rate time-series) using daily realized movements, or the standard deviation (which removes the mean). $\endgroup$ – Kiann Dec 4 '19 at 22:40
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Whenever you look at pricing something illiquid, it is important to try pricing it multiple different ways. You do this to give yourself confidence that the litany of assumptions you're making are at least somewhat valid. For the example you give above, i would consider the following:

  1. Do you have a well known rate curve? (i.e. are all your swap rates well known?)

  2. Do you have any other tenors where the swaptions are traded?

     If the answer to this is yes, then you can use this liquid tenor to test your estimations.
    
  3. Are there other currencies with very similar looking rate curves?

     If there are, then perhaps these are candidates for proxy currencies for the volatility.
    
  4. Are there any other currencies which are strongly correlated with the currency you're looking at?

     Same as above.
    
  5. If you look at the historical volatility of other currencyies swap curves vs. the swaption markets, how well do they price? Is this accuracy a function of swaption maturity or the swap tenor?

     If the answer to this is badly, then it would errode my confidence in the methodology. Yo're looking at 2y maturity swaptions, many people make the arbitrary assumption that a good measure of the 2y expected future volatility is 2y of historical volatility. Do you think that the 2y we have just experienced will be a good estimator of the next 2y? Personally i do not.
    
  6. Is there anything going on at the moment which would cause a disconnect between historical volatilities and what we expect to happen going forwards?

     See my point above - I'm sure you've noticed that we're in the middle of a rather turbulent time. The only time when historical volatility is (IMO) a reasonable estimator of the future is when we expect the future to be exactly like the period over which we've measured the historical volatility.
    

Additionally, you mention black76, but you're dealing with swaptions - you may want to consider normal volatilities.

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