Most of a bank's IRRBB calculation treats cashflows as if they are fixed and knowable. They usually are not. A floating-rate loan with a cap, a deposit the customer can break, a bond the issuer can call — each of these contains an option, and the value of that option changes as rates move. If the option is ignored, EVE is wrong in exactly the scenarios that matter most.
An Automatic Interest-Rate Option (AIRO) is any such embedded option whose exercise is driven purely by the level of interest rates. The purpose of this work is to isolate those options, value them properly, and add their value back into EVE so the risk picture is complete.
The method has four moves. Strip the option out of the instrument. Break a multi-period option into single-period pieces. Value each piece against the relevant forward rate. Add the values back to EVE, with the right sign depending on whether the bank holds or has sold the option.
A cap on a three-year quarterly-resetting loan is not one option — it is a strip of twelve single-period options, one per reset. Each single-period cap is a caplet; each single-period floor is a floorlet. The instrument is exploded into these pieces based on its reset frequency (monthly, quarterly, semi-annual or annual) and time to expiry, with the notional split across the pieces. A collar is simply a long floor and a short cap handled together.
Each caplet is an option on the forward rate for its own future period. That forward rate is read off the (shocked) yield curve by interpolating between the two nearest curve tenors to the caplet's expiry, with flat extrapolation beyond the last node. Every caplet is valued under every stress scenario, so the option value is known not just today but under each rate shock.
Each caplet and floorlet is priced with the Black-76 model — the market standard for interest-rate options, where the forward rate itself is the underlying:
| Term | Meaning |
|---|---|
| F | Forward rate for the caplet's period, from the shocked curve |
| K | Strike — the cap or floor rate written into the instrument |
| σ | Rate volatility for that tenor |
| T | Time to the caplet's expiry; τ is its accrual period |
| N(·) | Standard normal CDF (Abramowitz–Stegun approximation in the SAS implementation) |
| DF | Discount factor to the caplet's expiry |
Summing the caplet values gives the value of the cap; summing floorlets gives the floor; a collar is the net of the two. Each instrument's option delta — its sensitivity to a small rate move — is carried through as N(d₁) for caps and −N(−d₁) for floors, so the portfolio's overall option delta is available for hedging discussion.
Direction matters. If the bank holds the option (long), its value is a positive contribution. If the bank has sold it (short) — for example, a cap given to a borrower — the value is a liability when it moves into the money, so it enters with a negative sign.
The worst option-adjusted ΔEVE across scenarios is what should sit next to the standard ΔEVE in the regulatory pack — because for an option-heavy book the two can differ materially, and the option-adjusted number is the honest one.
Black-76 with caplet/floorlet decomposition is used because it is transparent, fast across many scenarios, and the accepted market standard for caps, floors and collars — the option types that dominate a banking book. For products where early exercise genuinely matters — Bermudan-style callable or puttable structures, or American-exercise features — a closed-form model is no longer adequate, and a lattice / binomial-tree or a short-rate model (for example Hull–White, solved on a trinomial tree with backward induction) is the right tool. The method is chosen to fit the product and the materiality of the early-exercise feature, not applied blindly.
Prepayment option — Bermudan receiver swaption on a short-rate tree →
Cap/floor loan — Black-76 valuation dashboard →
Interactive calculator — Black-76 caplet & rate tree →
Illustrative methodology description on synthetic assumptions. No client data, production code or calibrated parameters are reproduced here.