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?


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.

  • $\begingroup$ 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. $\endgroup$ – Ric Dec 16 '14 at 12:05
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.