Interest rate derivatives: models and methods

Last updated 10 November 2026

This page describes how GIAnalyzer and the Gamma Impact API value interest rate derivatives and measure their risk: the instruments covered, the model behind each, the market data each takes, and what the models do not do. It describes GIAnalyzer version 5.12.0; the Gamma Impact API covers the same instruments and models. It is provided for information and is not a contractual specification.

The function-level detail is in the function reference.

Curves and discounting

Valuation is multi-curve: each leg takes a discounting curve and, for floating legs, a separate forecasting curve. Both are given as zero-rate curves, with their own compounding and day-count conventions.

• Discount factors come from the discounting curve's zero rate at each payment date, interpolated between the curve's points, and run from the spot date to the payment date. Values are present values at the spot date.
• Forward rates for floating periods are simple forward rates over the accrual period, implied by the forecasting curve's discount factors at the start and end of the period, in the leg's day count.
• Interpolation: linear by default; log-linear, exponential, cubic spline and log-cubic spline are available.
• The engine values from curves the user supplies. In the Gamma Impact API, curves are built from market data before valuation.

Schedules and conventions

• Coupon schedules are generated backward from maturity at the leg's payment frequency (monthly, quarterly, semi-annual or annual). A short front stub is merged into the first period. If the maturity falls at a month end, payment dates follow the month end.
• Business-day conventions: Following, Modified Following, Preceding, Modified Preceding, or no adjustment. Non-business days are Saturdays and Sundays only; holiday calendars are not supported.
• Fixings are at the start of each period; there is no payment lag.
• The current period of a floating leg uses its last reset rate, given as an input.
• Day-count conventions and compounding are as for the rest of the engine: Actual/Actual, Actual/360, Actual/365 and the 30/360 family; annual to continuous compounding.

Instruments and models

Forward rate agreements

The settlement amount is the difference between the forward rate and the contract rate over the period, discounted at the forward rate itself to the start of the period, as FRAs settle in the market, and then to the spot date on the discounting curve.

Interest rate swaps

Fixed-for-floating swaps, paying or receiving fixed, are valued leg by leg as the discounted sum of their coupons, plus any notional exchanges:

• Fixed leg: the fixed rate on the outstanding notional over each accrual period.
• Floating leg: the forward rate plus spread on the outstanding notional; the current period uses its last reset rate.
• Amortization: an even repayment of principal at a set frequency, or a custom schedule of principal dates and amounts.
• Fair (par) rate: the fixed rate that sets the swap's value to zero, found by a numerical root search.
• Carry and roll-down: carry is the value of the cash flows up to a horizon date (optionally with the change in accrued interest); roll-down is the change in value from moving to the horizon date on an unchanged curve. Both are also given in rate terms, as differences between fair rates.

Overnight index swaps

The floating leg compounds the overnight rate daily over each period. Two methods are available:

• Exact: compounding day by day over business days, each day's rate implied from the forecasting curve, with past fixings taken from a fixing history.
• Approximate (default): the compounded factor for a period is taken from the forecasting curve's discount factors at its start and end, which reproduces daily compounding along the curve. Days already elapsed in the current period are compounded from the fixing history.

Amortizing OIS and OIS with a custom schedule are supported.

Basis swaps

Two floating legs, each with its own forecasting curve and spread, discounted on a common curve.

Cross-currency swaps

Fixed-fixed, fixed-floating and floating-floating cross-currency swaps, with IBOR or risk-free-rate floating legs, in bullet, amortizing and custom-schedule forms:

• Each leg is valued in its own currency on that currency's discounting and forecasting curves, with its notional exchanges.
• The legs are converted at the spot exchange rate on the valuation date.
• A cross-currency basis enters through the leg spread or the curves supplied.

Caps and floors

Each caplet or floorlet is valued with the Black (1976) model on its forward rate, with one flat volatility for the whole cap, and discounted from its payment date. The volatility can be given lognormal or normal; a normal volatility is converted to a lognormal one for each caplet. The caplet already fixed pays its intrinsic value on the last reset rate.

Swaptions

European swaptions on fixed-for-floating swaps are valued with the Black (1976) model on the forward swap rate, multiplied by the annuity of the underlying fixed leg (physical settlement):

• Volatility is flat, or read from a volatility cube by option expiry, swap tenor and strike (relative to the at-the-money rate), interpolated between its points.
• Swaptions on overnight index swaps use the OIS par rate as the forward.
• Implied volatility is found from a given premium by a bracketing root search.

Total return swaps

Equity-style total return swaps: the asset leg is the change in the reference price plus dividends since the last reset; the funding leg accrues an IBOR or risk-free rate plus spread on the reset value. Both legs are valued as the amounts accrued in the current period.

Risk measures

All sensitivities are computed by revaluing the instrument with a small change in one input (finite differences).

A fixed leg is shifted on its discounting curve; floating legs and options are shifted on both their curves together.

What the models do not do

The interest rate models:

• have no holiday calendars: only weekends are non-business days;
• have no payment lags, lookback, lockout or observation shifts for overnight index swaps;
• use one flat volatility per cap, and no smile model (such as SABR) for swaptions beyond interpolation of the cube;
• have no normal (Bachelier) pricer: normal volatilities are converted to lognormal;
• do not reset notionals on cross-currency swaps (mark-to-market cross-currency swaps);
• do not take a fixing for a FRA already in its period: the curve's forward is used.

References

• Andersen, L. and Piterbarg, V. (2010). Interest Rate Modeling. Atlantic Financial Press.
• Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics 3, 167–179.
• Brigo, D. and Mercurio, F. (2006). Interest Rate Models: Theory and Practice, 2nd edition. Springer.
• Hull, J. C. (2022). Options, Futures, and Other Derivatives, 11th edition. Pearson.

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