TheQuantBateman

01

INTUITION

Read the cash-flow timeline first.

A forward rate agreement exchanges the difference between a contracted rate and a future fixing. Futures trade a related exposure but settle variation margin daily, creating a covariance effect between rates and reinvestment.

01

FRA value needs both projection and discounting.

02

Settlement timing changes the payoff denominator.

03

Futures equals forward only when daily-margin covariance is negligible.

02

WHY MARKETS CARE

Start from cash flows and quotation.

FRAs and rate futures are the liquid building blocks for forward-curve construction, central-bank path trading and front-end hedging.

INSTRUMENTS

forward rate agreements

three-month rate futures

OIS futures

forward-start swaps

QUOTE CONVENTION

State index, fixing period, accrual basis, settlement timing and whether the market price is quoted as 100 minus rate.

03

MATHEMATICS

Value each dated cash flow under explicit conventions.

Formula · Short derivation

End-settled FRA PV

V=N\,D(0,T_2)\,\delta\,[F(0;T_1,T_2)-K]

Projected coupon difference discounted from the payment date.

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Formula · Short derivation

Start-settled FRA payoff

\Pi_{T_1}=N\frac{\delta(L_{T_1}-K)}{1+\delta L_{T_1}}

The end-period interest difference is discounted back to the fixing/start date.

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Short derivation

From a future deposit to an FRA payoff

Compare the interest on a deposit struck at K with one struck at the future observed rate.

01
Define the accrual-period interest
On notional N, the interest difference paid at T₂ is Nδ(L−K).
02
Settle in advance if required
Discount that difference from T₂ to T₁ using the realised deposit rate L.
$\Pi_{T_1}=N\delta(L-K)/(1+\delta L)$
03
Value before fixing
Replace the unknown fixing with its projection under the appropriate curve and discount the resulting cash flow.
04
Separate futures margining
Daily variation margin is reinvested at stochastic rates; estimate the covariance correction with a stated rate model.
Numerical checkAt K=F, a newly struck idealised FRA has zero PV.

FRA and futures quotes target similar forward exposure but differ through settlement mechanics and convexity.

Inputs

F: projected simple fixing
K: contractual rate
δ: accrual fraction
D: settlement discount factor

Assumptions and limits

Convexity adjustment is model and volatility dependent.
Exchange contract dates may not align with OTC tenors.
Fallback index conventions can alter legacy FRA economics.

Formula · Full derivation

Futures-forward adjustment

F_{fut}\approx F_{fwd}+\operatorname{ConvAdj}(\sigma,\rho,T_1,T_2)

Daily margining adds a model-dependent covariance correction.

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05

MODEL / PRICING

Build, calibrate, and reprice the contract.

METHOD

Project the index fixing, apply exact payoff timing and discounting, then add a separately governed futures-convexity adjustment when mapping exchange quotes.

CALIBRATION

Use liquid futures with price, expiry, delivery period and convexity assumptions; repricing must be reported before and after the adjustment.

Implementation with current QuantLib

Use the relevant IborIndex or overnight-index futures helper, contract-specific dates and a documented convexity adjustment. Keep futures price conversion separate from forward-rate projection.

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

FRA zero-PV and settlement check

Verify the par condition and settlement-in-advance payoff.

REUSABLE EXAMPLE

01from __future__ import annotations

02

03def fra_start_payoff(notional: float, fixing: float, strike: float, accrual: float) -> float:

04 denominator = 1.0 + accrual * fixing

05 if notional < 0 or accrual <= 0 or denominator <= 0:

06 raise ValueError("invalid FRA domain")

07 return notional * accrual * (fixing - strike) / denominator

08

09notional, forward, accrual = 10_000_000.0, 0.0435, 0.25

10assert fra_start_payoff(notional, forward, forward, accrual) == 0.0

11up = fra_start_payoff(notional, forward + 0.0001, forward, accrual)

12assert 240.0 < up < 250.0

13print(f"+1bp fixing payoff={up:.2f}")

07

INTERACTIVE LAB

Move the state. Challenge the equation.

FORWARD FIXING AND MARGINING CONVEXITY

FRA/futures convexity laboratory

Move rate volatility, correlation and settlement timing; compare forward, futures and FRA PV.

SYNTHETIC · CONTROLLED SCENARIOS

SCENARIO INTENSITY70%

10Y adjustment3.4 bp

Rate volatility0.80%

Correlation0.15

Annualised rate by Contract start (years)

Low volatility: Forward and futures nearly coincide.

forward rate
futures-equivalent

Contract start (years): 0.3Y. forward rate: 3.273%. futures-equivalent: 3.273%.

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

“A futures strip is a policy view plus a margining convention plus a convexity assumption.”

VISIBLE INPUTS

contract dates

index curve

discount curve

accrual basis

volatility/correlation adjustment

CALIBRATION

Use liquid futures with price, expiry, delivery period and convexity assumptions; repricing must be reported before and after the adjustment.

RISK

meeting-date risk

convexity

roll and delivery risk

basis to OTC

DAILY WORKFLOW

map exchange dates
convert price to rate
apply convexity
bootstrap forward
hedge residual basis

Production failure modes

wrong IMM date
price/rate sign error
silent zero convexity
misaligned settlement timing

09MACRO CONNECTIONOpen the transmission channel.

MACRO CONNECTION

Policy surprises in futures strips

A policy surprise reprices meeting-dated forward periods first; volatility and rate-level changes also alter the futures-forward convexity adjustment.

01Policy surprisetransmits

shifts expected fixings

02Futures striptransmits

reprices meeting periods

03Convexity mappingtransmits

translates futures to forwards

04FRA / swap bookoutput

realises basis and hedge P&L

10COMMON PITFALLSOpen the failure checklist.

01

Reading 100-price with the wrong sign.

02

Treating futures and forwards as identical.

03

Omitting the settlement-in-advance denominator.

04

Hedging OTC dates with exchange contracts without basis attribution.

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 a future deposit to an FRA payoff

Define the accrual-period interest

Settle in advance if required

Value before fixing

Separate futures margining

Policy surprises in futures strips

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