# IR Cap Forward Premium

A well known broker quotes cap/floors as spot premium for ATM straddles but forward premium for the skew, given that the difference between spot premium and forward premium is that the option is not discounted at maturity for the latter, how do I account for this in a cap that consists of multiple caplets?

For example I have a caplet with spot premium defined as

$$$$V_{caplet}^{spot}(t, T_j)={\tau_jP\left(t,\ T_j\right)\mathbb{E}_t^{T_j}\ \left[F_j\left(T_{j-1}\right)-K\right]}^+,$$$$ then the equivalent forward premium would be $$$$V_{caplet}^{forward}(t, T_j)={\tau_j\ \mathbb{E}_t^{T_j} \left[F_j\left(T_{j-1}\right)-K\right]}^+.$$$$ However, if I have a 1Y cap consisting of caplets such as $$$$V_{cap}^{spot}(t)=\sum_{j=1}^{N}{\tau_jP\left(t,\ T_j\right)\mathbb{E}_t^{T_j}\ \left[F_j\left(T_{j-1}\right)-K\right]}^+,$$$$ where $$N=4, T_N=1Y$$ for simplicity, then how do you define the forward premium of this cap? Do I simply take the discount factors as $$1$$ everywhere? i.e.

$$$$V_{cap}^{forward}(t)=\sum_{j=1}^{N}{\tau_j\mathbb{E}_t^{T_j} \left[F_j\left(T_{j-1}\right)-K\right]}^+.$$$$

• what instruments are you trying to price using the atm spot straddle prices and otm strike forward premium caplets? Commented May 5, 2021 at 13:55
• I'm trying to strip caplets to determine caplet volatilities, these are to be used to eventually calibrate stochastic models such as Hull-White etc. Commented May 5, 2021 at 15:04
• they're giving you forward premium because a discounted premium is much more complex for a long expiration swaption at that strike. extract the vol from the model or just use the vol given by the dealer to price that particular strike caplet. Commented May 5, 2021 at 22:11