Setting ATM Cap Strikes from the Forward Curve for Volatility Calibration

Summary

This short fixed-income question concerns calibrating the parameters of a one-factor Gaussian Heath-Jarrow-Morton volatility model to at-the-money cap prices. The proposed calibration uses weighted least squares, with Black vegas as weights, and the questioner is unsure how to determine the strike needed to calculate those vegas.

The answer is that an at-the-money cap is based on at-the-money swaps, so its strike is the relevant swap rate. That rate can be calculated from the supplied forward curve, which gives the strike needed for the Black vega weights. The exchange is narrowly focused: it resolves the strike input but does not provide a full calibration procedure, market conventions, or evidence about calibration quality.

Key ideas

• For an at-the-money cap, use the corresponding at-the-money swap rate as the strike.
• The swap rate can be computed from the forward curve.
• The resulting strike supplies the input needed to calculate Black vegas for weighted calibration.
• The answer addresses strike selection but does not detail the full model calibration.

Tags

# ATM i.r. Caps - Black vol calibration


# ATM i.r. Caps - Black vol calibration












I'm provided the forward curve and time 0 prices of ATM Caps.

Volatility is 1-factor Gaussian HJM model with specification:

$$ \sigma(t, T) = \nu \exp \{ \beta (T − t) \} $$

Now, I need to calibrate the volatility parameters $\beta$ and $\nu$ to the cap prices weighted by corresponding Black vegas (weighted least squares).

But the problem is I do not know the cap rate (strike rate) and since Black vegas are a function of cap rate (strike rate), I'm lost on how to proceed.

Will be very thankful if someone can point me in the correct direction. The formula for Caplets is:

## Answer by learningIR (score 4)

https://quant.stackexchange.com/a/33715

Because these are ATM Swaps, strike rates should be equal to the Swap rates which can be computed off the forward curves

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.