I am trying to figure out the delta for a vanilla swap using a leg by leg analysis. Definition: Delta(i) = Change in swap value if the swap rate is bumped by 1bp at the ith point ONLY. A. when SPREAD = 0FLOATING LEG:This is a floating rate bond whose present value is always 100 (at least on payment dates).Hence Delta(Floating leg) = 0FIXED LEG:What happens here when the discounting curve is bumped up 1bp at ONLY the nth point? The other forward rates are not affected.There are cash payments on N payment dates with SEMI-ANNUAL intervals. Let C(i) be the cashflow at ith point.PV(ith cash flow) = C(i)*Discountfactor(i)Thus Delta(i) = Delta(only if ith point is bumped by 1bp) = C(i)* [Discountfactor_bumped_by_1bp(i)-Discountfactor(i)]*10000So the Delta is the same as a fixed rate bond @ fixed rate.Could someone please elaborate on the math of Discountfactor_bumped(i) - Discountfactor_bumped_by_1bp(i) using the actual BASIS and other conventions for a plain swap, not using exp(), etc. as in Hull? SPREAD!=0Can we neglect the BASIS convention to assume the swap with fixed-SPREAD? And then calculate the Delta as above?Thanks

i think you are making this too complicated.... if you have an underlying yield curve from which you derive discount factors (zero rates), just model the swap by present valuing the fixed and floating sides (floating rate forwards will be derived from the zero curve)....once that is done all you need to do is perturb each element of the underlying curve and then you will have each curve element's DV01..... it doesnt matter what the payment freq or daycount convention is becuase you are modelling the swap with these details and seeing how its value changes with curve perturbation....

johnself,I am just trying to understand why a 10y swap is going to have a huge change if the 10y rate is bumped but not when the 5y rate is bumped. Could you explain that please?thks

ok i think you need to look at the yield curve differently... swap traders and risk managers view swap risk in several differnet ways: total DV01 and "bucket DV01".... total DV01 is when you perturb the curve in paralell... this is only useful for general, total interest rate risk.... however since swap books have thousands of offsetting swaps of all maturities, it is necessary to look at the breakdown of this total DV01 w/r/t the all the maturities (i.e. curve risk)... thus, it is important to know what the portfolio's sensistivity is to each instrument which comprises the yield curve, for these are the instruments which are used to hedge.... so each element ofthe curve (5y,6y,7y, etc swap) is bumped individually to see where the total DV01's risk resides.... so imagine a portfolio which only contains one 10y swap.... why should the 5y swap effect its price?..... sure they have a high correlation (i think this is your confusion) but you are not consideringing this.... you just want to know what your risk is to the 10y point so you know where on the curve you risk is located....which has nothing to do (again, forget about curve correlation for a moment) with the 5y swap rate

Think about the cash flow structure of the fixed side. You've got a bunch of small payments (the fixed coupons), then a huge payment at t=10 equal to the notional. Assume you're bumping the zero rate (bumping the par curve makes little difference to the explanation that follows). When you change the 5y rate, you change the 5y discount factor only, and so you change the PV of the cash flow at t=5. That change is going to be tiny. Repeat for the 10y rate. That change is going to be huge, not because the change in the discount factor is necessarily bigger, but because the cash flow at t=10 is huge. Ergo, the value of your swap has a lot of sensitivity to the 10y point. That make more sense?

well said geist.... my attempt wasnt as articulate

Geist, your argument is assuming there is an exchange of notionals on the fixed leg. Applying the same reasoning to the notional payment on the floating leg will cancel the rate risk. In most plain vanilla swaps notional is not exchanged. Consider the two legs of the swap separately. The fixed leg has insignificant delta risk as outlined by Geist. The floating leg only has risk at the swap maturity. Assuming standard swap convention (euribor set in advance) then if you increase the zero rate at time i you decrease the payment made at time i. However against this you increase the payment to be made at time i+1 as the euribor fixing is higher. These two effects cancel out. If you increase the zero rate at swap maturity there is no future payments to offset the change in DF, hence a swap only has risk at maturity. By considering a floating rate bond its straightforward to see that a stream of floating rate payments has the same risk as an exchange of notionals.

QuoteOriginally posted by: anfieldredGeist, your argument is assuming there is an exchange of notionals on the fixed leg. Applying the same reasoning to the notional payment on the floating leg will cancel the rate risk. In most plain vanilla swaps notional is not exchanged. Sure, but that's irrelevant.1) Can you add both front end and back end notionals on both sides without affecting the valuation? -> Yes, because, as you point out, the notional cash flows offset. Doesn't matter whether they're exchanged or not - this is for pricing purposes only.2) Can you discount the floating leg separately (including back end notional) and will you get par whatever you do to your curve (assuming no payments have fixed and that you're using the same projection and discount curve) -> Yes, and resultingly there is no risk on the float side on a reset date.3) Can you now apply the procedure I outlined above on the fixed leg, INCLUDING the back end notional and ignoring the float leg? -> Yes, because we already established that the DV01 of the float leg is zero, and the only remaining cash flows have the exact same profile as a fixed rate bond.

first, the fixed side does NOT have an insignificant DV01 - esp if it is an off-market higher-than-par rate..... the swap risk distills down to two aspects:1) fixed side (which has the risk of an annuity) 2) floating side which (assuming no libor spread) has the risk of the principal being exchanged at settlement and maturity......if there is a libor spread this can be lumped in with the fixed side annuity