Bonds: models and methods

Last updated 10 November 2026

This page describes how GIAnalyzer and the Gamma Impact API value bonds and measure their risk: the bond types covered, how cash flows are built, how bonds are valued from a yield or a curve, the risk measures, and the models for convertible bonds and bond options. 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.

Cash flows

• Schedules are built backward from maturity at the coupon frequency (monthly, quarterly, semi-annual or annual). If maturity falls at a month end, coupon dates follow the month end.
• First coupon: when the first period is shorter than a full period, a stub of at least half a period becomes a short first coupon, and a shorter one is merged into a long first coupon. A bond's prospectus decides this in practice; the engine applies this rule.
• 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.
• Accrual day counts: Actual/Actual (ISDA and ICMA), Actual/360, Actual/365, NL/365 and the 30/360 family.
• Ex-coupon periods: when settlement falls within a given number of days before a coupon date, that coupon is excluded from the value and the accrued interest becomes negative.
• Settlement: every bond is valued at the settlement date given; cash flows on or before it are excluded.

Price, accrued interest and yield

• Accrued interest is the coupon accrued from the previous coupon date to settlement on the outstanding principal. The clean price is the dirty price less accrued interest.
• Values are in the units of the face value given: a face value of 100 gives prices per 100.
• From a yield: each cash flow is discounted at the yield, with the compounding and day count the user chooses, over the exact time from settlement to the cash flow.
• Yield to maturity is the yield that reproduces a given price, found by bisection. For zero-coupon bonds it is computed directly.
• From a curve: each cash flow is discounted at the zero rate of a discounting curve at its date, optionally plus a constant spread.
• Z-spread is the constant spread over the zero curve that reproduces a given price, found by a secant search with a bisection fallback.
• Risk from a price: the yield to maturity is found first, and the risk measures are computed from it.

Bond types

Fixed-coupon bonds

• Bullet: principal repaid at maturity.
• Even principal payments: equal instalments of principal at a set frequency.
• Swap-like (specified percentage): a set percentage of the original principal on each principal date, the remainder at maturity.
• Level payments (annuity): equal total payments of principal and interest.
• Custom schedule: principal dates and percentages given by the user.

Coupons accrue on the principal outstanding in each period.

Floating-rate bonds

The current coupon uses the last reset rate. Later coupons are the simple forward rate from a forecasting curve over each coupon period, plus the margin, and are discounted on a separate discounting curve. Bullet, even, swap-like and custom amortization are supported.

Zero-coupon bonds

Valued from a yield or a curve at maturity.

Bonds given as cash flows

Any bond can be valued from a list of its cash-flow dates and amounts, from a yield or a curve, with the same yield, spread and risk measures.

Convertible bonds

Convertible bonds are valued on a binomial tree for the share price (Cox, Ross and Rubinstein, 1979) with the Tsiveriotis-Fernandes split (1998): the value is divided into a cash part, discounted at the interest rate plus a credit spread, and an equity part, discounted at the risk-free rate. Conversion is possible at every node. The face value can be linked to a consumer price index.

Bond options

European options on fixed-coupon and zero-coupon bonds are valued with the Black (1976) model on the forward dirty price of the bond, with a lognormal price volatility. The forward price is the bond's value less the coupons paid before expiry, carried to expiry.

Bond futures and options on them

A bond future is valued as the bond's value divided by a conversion factor, computed as the bond's price at a 6% flat yield. Options on bond futures use the same Black model on the futures price. There is no selection of the cheapest-to-deliver bond.

Carry and roll-down

Carry is the value of the cash flows paid between settlement and 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.

Curve fitting: Nelson-Siegel and Svensson

Zero curves can be fitted with the Nelson-Siegel (1987) and Nelson-Siegel-Svensson (1994) forms, by weighted least squares on yields to maturity of government bills and fixed-coupon bonds (weights inversely proportional to term or duration), with the Levenberg-Marquardt optimizer.

Risk measures

For floating-rate bonds, the forecasting and discounting curves are shifted together. Measures are positive for a long position whose value falls when rates rise.

What the models do not do

The bond models:

• have no holiday calendars and compute no settlement lag: settlement is given;
• value CPI-linked bonds only as nominal bonds: there is no indexation of coupons and principal in the bond library;
• have no call or put features, on plain bonds or convertibles, and no option-adjusted spread;
• value bond options as European only, with a price volatility;
• treat a bond future as a single deliverable bond, with no cheapest-to-deliver selection;
• give floating-rate bonds no discount margin.

References

• Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics 3, 167–179.
• Cox, J. C., Ross, S. A. and Rubinstein, M. (1979). Option pricing: a simplified approach. Journal of Financial Economics 7(3), 229–263.
• Fabozzi, F. J. (ed.) (2021). The Handbook of Fixed Income Securities, 9th edition. McGraw-Hill.
• Nelson, C. R. and Siegel, A. F. (1987). Parsimonious modeling of yield curves. Journal of Business 60(4), 473–489.
• Svensson, L. E. O. (1994). Estimating and interpreting forward interest rates: Sweden 1992–1994. NBER Working Paper 4871.
• Tsiveriotis, K. and Fernandes, C. (1998). Valuing convertible bonds with credit risk. Journal of Fixed Income 8(2), 95–102.

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