TheQuantBateman

01

INTUITION

Read the cash-flow timeline first.

An OIS floating leg compounds many overnight observations into one coupon. The observation schedule—not a single forecast rate—controls realised accrual, and collateralisation makes OIS instruments central to discount-curve construction.

01

Compounding uses a product of daily accumulation factors.

02

Known fixings and projected future observations coexist inside a live coupon.

03

Observation shifts and lockouts change which fixing is applied to which accrual day.

02

WHY MARKETS CARE

Start from cash flows and quotation.

OIS curves discount collateralised derivatives, express policy expectations and anchor the multi-curve framework.

INSTRUMENTS

overnight indexed swaps

OIS futures

collateralised swaps

compounded overnight coupons

QUOTE CONVENTION

State overnight index, payment frequency, day count, lookback or observation shift, lockout, payment lag and collateral currency.

03

MATHEMATICS

Value each dated cash flow under explicit conventions.

Formula · Short derivation

Compounded overnight coupon

R_{comp}=\frac{\prod_{i=1}^{n}(1+R_i\delta_i)-1}{A}

Daily simple accrual factors compound multiplicatively over the coupon period.

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Short derivation

From daily overnight loans to an OIS floating coupon

Roll one unit of notional through each overnight interval and retain every daily accumulation factor.

01
Accrue one overnight interval
One unit becomes 1+R_iδ_i over the i-th business-day interval.
02
Reinvest through the period
Each day starts from the accumulated prior balance, so factors multiply rather than rates add.
$A_{0,n}=\prod_i(1+R_i\delta_i)$
03
Convert to a coupon rate
Subtract principal and divide by the coupon accrual A to report an annualised compounded rate.
$R_{comp}=(A_{0,n}-1)/A$
04
Price the swap
Discount each fixed and floating payment on the collateral curve; at par, the difference between start and end discount factors balances the fixed annuity.
Numerical checkAt the computed par rate, fixed-leg PV minus floating-leg PV must be zero.

OIS pricing couples a precise overnight observation contract with collateral-consistent discounting.

Inputs

R_i: overnight fixing
δ_i: daily accrual fraction
A: coupon accrual fraction
K: fixed OIS rate

Assumptions and limits

The par formula shown assumes one curve and no payment irregularities.
Fallback and observation conventions differ across currencies.
Holiday calendars create non-uniform daily accruals.

Formula · Short derivation

Spot-starting OIS par rate

K_{OIS}=\frac{P(0,T_0)-P(0,T_n)}{\sum_{j=1}^{n}\alpha_jP(0,T_j)}

Under a single collateral curve, fixed and floating legs balance at inception.

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Formula · Definition

Known/projected split

\prod_i(1+R_i\delta_i)=\prod_{i\le k}(1+R_i^{fix}\delta_i)\prod_{i>k}(1+R_i^{fwd}\delta_i)

A live coupon combines realised observations and curve-projected future accrual.

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05

MODEL / PRICING

Build, calibrate, and reprice the contract.

METHOD

Build the observation schedule, splice known fixings with forwards, compound the coupon, then discount every payment on the collateral curve.

CALIBRATION

Bootstrap short maturities from cash or futures-like OIS instruments and longer maturities from OIS swaps, repricing every helper after each pillar.

Implementation with current QuantLib

Use the currency-specific OvernightIndex and OvernightIndexedSwap abstractions with explicit telescopic-value-date and observation settings. Curve helpers must share the same index conventions as the priced trade.

API authority: upstream QuantLib reference pinned in the source registry.

06PYTHON IMPLEMENTATIONOpen the implementation and checks.

ARCHITECTURE

Parse dated market inputs and conventions at the boundary.
Build deterministic curve objects in the framework-free quant layer.
Return PV, repricing residuals and sensitivities together.
Test inversion, par conditions, monotonic dates and invalid domains.

PYTHON 3 · NUMPY / SCIPY

Daily overnight compounding

Compound irregular daily accruals and compare with an additive approximation.

REUSABLE EXAMPLE

01from __future__ import annotations

02

03import math

04

05def compounded_rate(fixings: list[float], accruals: list[float]) -> float:

06 if len(fixings) != len(accruals) or not fixings:

07 raise ValueError("aligned non-empty fixing path required")

08 factor = math.prod(1.0 + r * d for r, d in zip(fixings, accruals))

09 total = sum(accruals)

10 if total <= 0 or factor <= 0:

11 raise ValueError("invalid accrual path")

12 return (factor - 1.0) / total

13

14rates = [0.0410, 0.0411, 0.0412, 0.0412]

15deltas = [1/360, 1/360, 3/360, 1/360]

16rate = compounded_rate(rates, deltas)

17assert rate > sum(r*d for r, d in zip(rates, deltas)) / sum(deltas)

18print(f"compounded overnight={rate:.6%}")

07

INTERACTIVE LAB

Move the state. Challenge the equation.

DAILY FIXINGS → COMPOUNDED COUPON

OIS policy-path laboratory

Apply hikes, cuts and lockout rules to a daily path; compare realised compound, forward projection and par fixed rate.

SYNTHETIC · CONTROLLED SCENARIOS

SCENARIO INTENSITY70%

Last fixing4.10%

Compounded coupon4.11%

Policy step0.0 bp

Annualised overnight rate by Observation day

Policy hold: Stable overnight path.

overnight fixing
running compound

Observation day: 1. overnight fixing: 4.100%. running compound: 4.100%.

Use Left/Right or Up/Down arrows to inspect values; Home and End jump to the bounds.

08

FRONT OFFICE

Follow the trade through risk and lifecycle events.

ON THE DESK

“The front end is a calendar-weighted policy path, not a row of equally spaced dots.”

VISIBLE INPUTS

overnight index fixings

observation convention

coupon schedule

collateral curve

payment lag

CALIBRATION

Bootstrap short maturities from cash or futures-like OIS instruments and longer maturities from OIS swaps, repricing every helper after each pillar.

RISK

meeting-date DV01

fixing exposure

front-end basis

calendar risk

DAILY WORKFLOW

load fixings
build observations
splice projections
compound coupon
reprice and bucket risk

Production failure modes

missing fixing
incorrect weekend weight
wrong observation shift
discount/projection curve confusion

09MACRO CONNECTIONOpen the transmission channel.

MACRO CONNECTION

Central-bank path into OIS

Expected meeting outcomes reprice dated overnight forwards; realised decisions subsequently enter compounded coupons through fixings.

01Meeting expectationtransmits

moves forward overnight segments

02OIS curvetransmits

prices the policy path

03Daily fixingtransmits

realises one path segment

04Coupon / P&Loutput

compounds the realised path

10COMMON PITFALLSOpen the failure checklist.

01

Averaging overnight rates instead of compounding.

02

Treating observation shift and lookback as synonyms.

03

Ignoring known-versus-projected fixing splits.

04

Discounting collateralised cash flows on a credit-sensitive term curve.

11SOURCES / FURTHER READINGOpen sources and continue the track.

research

Interest-rate products, term structures and short-rate lectures

Research map for the rates progression and numerical experiments; all platform prose and code are original.

Source: Financial Engineering: Interest Rates & xVA
Author: L. A. Grzelak
Ref: main

OPEN ORIGINAL SOURCE ↗

research

Stochastic processes, Monte Carlo and model-calibration lectures

Mathematical cross-reference for stochastic dynamics and implementation checks.

Source: Computational Finance Course
Author: L. A. Grzelak
Ref: main

OPEN ORIGINAL SOURCE ↗

implementation reference

Current term structures, indexes, rate helpers, instruments, engines and tests

Implementation authority for production abstractions; Academy derives the mathematics before introducing library objects.

Source: QuantLib upstream
Author: QuantLib contributors
Ref: v1.42.1

OPEN ORIGINAL SOURCE ↗

From daily overnight loans to an OIS floating coupon

Accrue one overnight interval

Reinvest through the period

Convert to a coupon rate

Price the swap

Central-bank path into OIS

Interest-rate products, term structures and short-rate lectures

Stochastic processes, Monte Carlo and model-calibration lectures

Current term structures, indexes, rate helpers, instruments, engines and tests