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I'm wondering which curves should I use when passing from the Implied volatility to prices.

When I read an implied volatility (for instance 3Y Cap strike 0.5%) the discounts and forward rate entering in the Black formula have been taken from which curves? Calibrated on which instruments?

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I don't know exactly about Reuters but often implied volas in the Black 76 world are quoted (forward) ATM. Thus the forward equals the strike and they dissappear from the formula:

$$ C = E[(F-K)^+] = \exp(- r t) (F N(d_1) - K N(d_2)) $$ and $d_1 = (\log(F/K)+\sigma^2/2T)/(\sigma \sqrt{T})$ and $d_2 = d_1 - \sigma \sqrt{T}$, see e.g. here.

In the ATM case $F=K$ and in the term for $d_1$ we get $\log(F/K)=0$. Thus the formula depends as little as possible on curves. For $r$ I assume some appropriate money-market rate depending on the time-to-expiry of the caplet.

EDIT: I have worked using swaption data. There in the surface you have 2 dimensions: time to expiry of the option and then the term of the swap. Concerning the rate $F$ it is the traded swap rate that fits to the term (and the starting date) and thus is is a forward swap rate. The strike is then clear.

Summarizing: for the underlyings one should take the corresponding traded objects. In your case I would take a forward money market rate. If it is not traded then I would calculate it using the usual forward rate formula and take a money-market/swap based curve as basis (use zero-rates which you get by bootstrapping). Doing this you don't need a stochastic interest model for this but derive the underlying rather directly from traded objects. I am sure Reuters is doing something similar.

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  • $\begingroup$ Your answer completes my question (in this is great) but still, doesn't answer: how ATM F is determined, bootstrapping which instruments? which appropriate money-market rate is used? By the way I have a volatility surface so I cannot focus on ATMs only. $\endgroup$ – jimifiki Dec 16 '14 at 12:00
  • $\begingroup$ You are right, my answer focus too much on ATM. I will edit my answer a bit. $\endgroup$ – Richard Dec 16 '14 at 12:05
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    $\begingroup$ The atm levels are calculated using standard market conventions. In the US, that would be swaps versus 3 month libor, with discounting at Fed Funds. In Europe, it would be swaps versus 6 mo Euribor, with discounting at Eonia. $\endgroup$ – dm63 Feb 22 '18 at 14:30

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