Skip to content
All library documents

Choosing Curves and Forwards for Cap Implied Volatility Pricing

Article Quant Q&A · Author: jimifiki
Read the original

Summary

The document discusses which forward rates and discount curves to use when converting a cap implied volatility into a price under Black 76. It explains that at forward at-the-money, the forward equals the strike, so their ratio drops out of the logarithmic term in the option formula and curve sensitivity is reduced. Discounting still requires an appropriate money market rate for the caplet’s expiry.

For the underlying, the response recommends using the corresponding traded market rate, such as a forward money market rate for a caplet. If that rate is not directly traded, it can be derived from bootstrapped zero rates on a money market or swap curve. The discussion draws an analogy with swaptions, where the relevant forward is the swap rate matching the option’s expiry and swap tenor. It gives general guidance rather than identifying Reuters’ exact conventions or calibration instruments, and does not specify the precise curve setup for a particular market.

Key ideas

  • At forward at-the-money, the forward equals the strike and the Black formula’s log-moneyness term is zero.
  • Discounting still uses a rate appropriate to the option’s expiry.
  • Use the traded underlying rate corresponding to the caplet or swaption being priced.
  • When a forward is not directly traded, derive it from bootstrapped zero rates on a suitable money market or swap curve.

Tags

OptionsFixed incomeDerivatives pricing
Full text
# How does Reuters quote caps?


# How does Reuters quote caps?












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?

## Answer by Richi Wa (score 2, accepted)

https://quant.stackexchange.com/a/15893

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.