Black76 Model Discounting

[Anonymous]

Hi, I have a question that has been bugging me for a while and I hope somebody can shed some light. In the handbook (Instruments, Ch7 Caps, Flors and Swaptions, page 294) the Black76 formula is shown and each caplet is discounted by B(t,Tj) which is given by 1/(1+d*F(t,Tj-1,Tj), which is a local discount factor (ie discounting from Tj to T-j not from Tj to t (which is confusingly also stated on this page). This is also confirmed in the example on page 206 where we can see that the discounting is just for 3 months based off of the forward rate for that period, the caplet cashflow being in 9 months time.
My question is why is only local discounting used and not full discounting back to the price calculation date, ie t?

Thanks!

[David Newing]

Black76 Model Discounting

[Anonymous]

[Anonymous]

I was having a look at an old copy of Hull (1993) to see if I could reconcile the formula on p204 to Hull. From what I can see, the “local discounting” is just to discount the caplet payout at the end of a caplet period (at time Tj) to an equivalent amount at time Tj-1. But the formula in Hull also includes a term for exp(-r*(Tj-1 – t)) where r is the risk free rate. This seems like what you were expecting to see. Same with me. So I reckon there’s definitely a question mark over the formula on p204.
Hull explains the rationale for the “local discounting” more fully than I would attempt here.

[Anonymous]

Thanks for the reply Jim,

I’ll get hold of a copy of Hull and read through.

[Author]

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