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Hello, Sorry if it seems sillyUsing a Black formula to value a swaption at time 0, if I want to have the value of the swaption at time t, I just need to capitalize the value of the swaption (i.e. dividing by DF(0,t)) to get the Df(0,ti)*(ti ? ti-1) at the right date and changing the time parameter in d1 and d2, right? I've been thinking about this too much and start getting confused Many thanks to anyone who could help with that, G

Hello,Value of a swaption at time t is . It can only be calculated at time t.A fixed payment paid at time t has the same value as V(t) at time 0. However these are different payoffs and in general they will be different at t>0 (unless volatility is equal to 0). Hope this helps.

QuoteOriginally posted by: gibran05Hello, Sorry if it seems sillyUsing a Black formula to value a swaption at time 0, if I want to have the value of the swaption at time t, I just need to capitalize the value of the swaption (i.e. dividing by DF(0,t)) to get the Df(0,ti)*(ti ? ti-1) at the right date and changing the time parameter in d1 and d2, right? I've been thinking about this too much and start getting confused Many thanks to anyone who could help with that, GThe forward price of an european swaption, i.e. the fair price agreed upon today to be paid at a future time t, is todays price divided by df(0,t). No change in d1, d2.

Hi,I have a question related to Black 76 pricing equation for Swaptions. In Wilmotts book the formula for a Swaption with 6 month payment exchange for underlying swap is given as:exp(-r(T-t))((1-(1+F/2)^-2(Ts-T))/F)(FN(d1)-EN(d2)).Can someone please explain where one gets the term ((1-(1+F/2)^-2(T(s)-T))/F) from?I can derive the Swaption formula asexp(-r(T-t))(FN(d1)-EN(d2))(1/2)(D(T,T+1/2)+D(T,T+1)+D(T,T+3/2)+....+D(T,Ts)) but can't reduce it further (D(.,.) are the discount functions).Many Thanks.

QuoteOriginally posted by: MicroHedgeHi,I have a question related to Black 76 pricing equation for Swaptions. In Wilmotts book the formula for a Swaption with 6 month payment exchange for underlying swap is given as:exp(-r(T-t))((1-(1+F/2)^-2(Ts-T))/F)(FN(d1)-EN(d2)).Can someone please explain where one gets the term ((1-(1+F/2)^-2(T(s)-T))/F) from?I can derive the Swaption formula asexp(-r(T-t))(FN(d1)-EN(d2))(1/2)(D(T,T+1/2)+D(T,T+1)+D(T,T+3/2)+....+D(T,Ts)) but can't reduce it further (D(.,.) are the discount functions).Many Thanks.Isn't Dr Wilmott assuming that you discount at the forward swap rate ? Which is the way swaptions are cash settled no ?

Hi Dave,Yes, Dr. Wilmott is indeed assuming Cash Settlement whereas my derivation is for Physical Settlement. If I use F (Forward Swap Rate) as the discount rate in my derivation then I do get the Cash Settlement formula. The confusion I have is that the two values for the swaption are clearly not the same (at least to me) and the Physical Settlement formula seems to me to be the correct one so is the Cash Settlement formula just a definition thing, i.e. this is how a Cash Settlement is defined so use F as discount rate?Thanks

QuoteOriginally posted by: MicroHedgeHi Dave,Yes, Dr. Wilmott is indeed assuming Cash Settlement whereas my derivation is for Physical Settlement. If I use F (Forward Swap Rate) as the discount rate in my derivation then I do get the Cash Settlement formula. The confusion I have is that the two values for the swaption are clearly not the same (at least to me) and the Physical Settlement formula seems to me to be the correct one so is the Cash Settlement formula just a definition thing, i.e. this is how a Cash Settlement is defined so use F as discount rate?ThanksI dont think one is more correct than the other. the thing is you pay your money and you take your choice.

QuoteOriginally posted by: MicroHedgeThe confusion I have is that the two values for the swaption are clearly not the same (at least to me) and the Physical Settlement formula seems to me to be the correct one so is the Cash Settlement formula just a definition thing, i.e. this is how a Cash Settlement is defined so use F as discount rate?It is up to the market convention. For example EUR, GBP are trading cash-settled swaptions while in USD they are physically settled.

Thanks, this is what I wanted to know. Dave, by correct I meant that I would never have derived the price of a Swaption as that of a EU Cash Settled Swaption with F as the discount rate had I not been told that that is the convention. I can think of no mathematical argument for using F as the single discount rate for Swap payment stream.

QuoteOriginally posted by: MicroHedge I can think of no mathematical argument for using F as the single discount rate for Swap payment stream.The argument is not mathematical. The intuition behind this convention: Cash settled annuity process as a function of a forward swap rate does not represnt a value process of a traded asset, unlike the physically settled annuity. However the market convention is to treat it as if it does. Using the risk neutral measure with 'CS annuity numeraire' leads to this market convention formula. You can view it as a definition of the CS swaption implied vol.