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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)

Thanks!

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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.

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  • $\begingroup$ 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 Aug 5 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. $\endgroup$ – Dimitri Vulis Aug 5 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 Aug 5 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. $\endgroup$ – Dimitri Vulis Aug 5 at 17:10
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    $\begingroup$ 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 $\endgroup$ – noob2 Aug 5 at 18:56

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