With a payer swaptions delta and gamma is there a method for approximating pnl for a given move in underlying swap rate? (An equivalent to the Taylor expansion for a vanilla call)



1 Answer 1


Yes, the Taylor expansion usually works well as the first approximation to explain the P&L when the curve doesn't move a lot. The "delta" (first derivative) is the sensitivity to the parallel shift of the swap curve. The "gamma" (second derivative) is the convexity.

You can get even better P&L explanation by

  • including separate sensitivities to swap rates at different tenors (most of your sensitivity is to the rate for the maturity of your underlying), rather than assuming parallel shift.

  • using more sophisticated IR gamma. Depending on your needs, you can use one number risk-weighted to ficus on yourt underlying's maturity. Or you can have a matrix with an entry for each tenor pair.

  • include the cross-gammas between rates, implied volatility, and time.

  • $\begingroup$ Thank you. Do you have a simple numeric example? For instance for a payer with price 3.2%, delta 27.4%, gamma 4.8% what would the price be for a +100bp shift? $\endgroup$
    – Laralander
    Commented Aug 5, 2020 at 16:17
  • $\begingroup$ For Taylor, you want the delta to be the P&L (change in premium) when swap curve moves (in parallel) by a small shift, like 1 bp. Is this what you mean by 27.4%? But Taylor-based risk-theoretical P&L explain won't work well for a 100bps shift.. it's good up to 20 bps maybe. $\endgroup$ Commented Aug 5, 2020 at 16:27
  • $\begingroup$ Ok so if the price is 364,744.57 and the DV01 is 9,050 and gamma (per basis point) is 153.12. What would the pnl be for a 20bp increase? Thanks again $\endgroup$
    – Laralander
    Commented Aug 5, 2020 at 16:36
  • $\begingroup$ The P&L explained by the IR is something like dv01 $\times$ 20bps ± gamma $\times$ 20bps$^2$ - sorry, figure out the signs and the scaling. $\endgroup$ Commented Aug 5, 2020 at 17:10
  • 1
    $\begingroup$ About the signs: if you receive fixed (aka short the swap) then the linear term has a negative sign (you make money when int rates fall) and the quadratic term has a positive sign (profit is better when yields fall than th loss when they rise by an equal amount). And vice versa if your position is pay fixed, receive floating $\endgroup$
    – nbbo2
    Commented Aug 5, 2020 at 18:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.