Spot and Forward Premium Conventions for Caps and Caplets

Summary

The document asks how a forward premium quote for an interest-rate cap relates to the usual spot premium. It describes the distinction as whether the caplet payoff is discounted to its payment date: the spot-premium expression includes a discount factor, while the proposed forward-premium expression omits it. The question then extends this comparison from a single caplet to a cap made up of several caplets, each with its own accrual period and payment date.

The author proposes defining the cap’s forward premium by summing the caplet values without their discount factors, and asks whether this is the correct treatment. No answer or supporting derivation is included, so the proposed sum is unresolved rather than an established pricing result. The document is useful as a statement of the quote-convention issue, but it does not specify a complete market convention or explain how to compare premium quotes across maturities. A practitioner would need the relevant quoting definitions and valuation framework to validate the proposed aggregation.

Key ideas

• A spot premium expression for a caplet includes discounting to the payment date.
• The question defines forward premium as omitting that discount factor.
• A cap consists of caplets with separate accrual periods and payment dates.
• The proposed forward premium for a cap sums the undiscounted caplet values, but the document does not confirm this rule.

Tags

# IR Cap Forward Premium


# 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

\begin{equation} 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]}^+, \end{equation} then the equivalent forward premium would be \begin{equation} V_{caplet}^{forward}(t, T_j)={\tau_j\ \mathbb{E}_t^{T_j} \left[F_j\left(T_{j-1}\right)-K\right]}^+. \end{equation} However, if I have a 1Y cap consisting of caplets such as \begin{equation} 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]}^+, \end{equation} 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.

\begin{equation} 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]}^+. \end{equation}

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