Linear Rates, Products, and Models: SOFR, Swaps, Yield Curves, and Hedging

Interest-rate markets connect today’s money to future cash flows. Their basic products—deposits, bonds, futures, and swaps—provide the prices used to construct yield curves and hedge more complex contracts. The central skill is to move consistently between cash flows, discount factors, forward rates, market quotes, and risk sensitivities.

Learning goals and notation

Explain what rates desks mean by linear products.
Distinguish overnight SOFR, compounded SOFR, and forward-looking Term SOFR.
Price deposits and swaps, interpret futures quotes, and separate projection from discounting.
Bootstrap a small curve and compare interpolation choices.
Understand meeting-date steps, calendar turns, Z-spreads, and fast pricing approximations.

Symbol	Meaning
N	Notional principal used to calculate payments
δ, α	Year fractions under the specified day-count convention
D(t,T)	Discount factor at valuation time t for a payment at T
fⱼ	Model-implied forward rate for floating accrual period j
K	Contractual fixed swap rate
S	Current par swap rate for a specified schedule
1 bp	One basis point, or 0.0001 in decimal rate units

1. What makes a rates product “linear”?

On a rates desk, linear products generally means instruments without an explicit option payoff: deposits, vanilla non-callable bonds, swaps, and futures. A floating coupon is proportional to its applicable rate, and swap value is linear in a fixed contractual coupon when the curves are held fixed.

This is a product classification, not a claim that every price is a straight line in every interest rate. A fixed-rate bond has fixed payments but a nonlinear price–yield relationship. Discounting and compounded floating coupons introduce nonlinearities too. Caps, floors, and swaptions add explicit optionality and require additional volatility and distribution assumptions.

Market scale: measure like with like

The supplied outline quotes approximately 128 trillion in bonds, 573 trillion in swaps, more than 10 trillion in SOFR futures open interest, and daily Treasury/swap turnover around 900/800 billion. Without dates, definitions, and original sources, these are undated lecture-scale illustrations, not current measurements or directly comparable totals.

Bond principal outstanding, derivative notional, futures open interest, turnover, and market value measure different things. Notional is a reference amount, not generally the amount at risk or an asset’s current value. For a dated comparison, the BIS end-June 2025 OTC statistics report interest-rate derivatives as 79% of total OTC derivative notional. That is not a measure of every bond or exchange-traded futures market.

Rates products are central to funding and hedging, but liquidity varies by currency, maturity, instrument, and market conditions. “Rates are liquid” does not guarantee a tight executable price in every tenor during stress.

2. From LIBOR to SOFR

LIBOR represented unsecured wholesale bank borrowing across currencies and maturities. As underlying term borrowing became less representative, reliance on submissions and judgment created robustness concerns; manipulation also undermined trust. USD markets transitioned toward SOFR-based contracts, while legacy contracts required fallbacks and spread adjustments.

The FCA records the end of all remaining LIBOR settings: the last synthetic USD settings ceased after 30 September 2024. LIBOR should therefore be discussed as a legacy benchmark, not a live alternative for new benchmark selection.

Overnight SOFR

SOFR measures the cost of overnight cash borrowing secured by US Treasury collateral. The New York Fed calculates it using a volume-weighted median of eligible repo transactions, rather than a volume-weighted arithmetic average. Its underlying volume changes daily; the supplied 800 billion figure is not a fixed input to the methodology. See the New York Fed’s SOFR definition and data.

SOFR is secured overnight funding, whereas LIBOR embedded unsecured term bank credit and liquidity features. Replacing one with the other is not just renaming a rate. SOFR is also distinct from the federal funds rate and the Federal Reserve’s policy target range.

Compounded in arrears versus Term SOFR

For daily overnight fixings rₖ covering dₖ calendar days and ACT/360 accrual fractions, a simplified compounded period growth factor is:

G=∏ₖ[1+rₖdₖ/360].

If the full accrual fraction is δ, the equivalent annualized period rate is R_comp=(G−1)/δ. Weekend and holiday coverage matters. Observation shifts, lookbacks, lockouts, payment lags, and rounding depend on the actual contract. A period’s compounded rate is only fully known after the relevant observations have occurred.

Example: two one-day fixings of 4% and 5% give growth (1+0.04/360)(1+0.05/360). Over δ=2/360, the annualized compounded rate is approximately 4.50028%, slightly above the arithmetic average 4.5%.

CME Term SOFR instead provides forward-looking estimates for specified tenors using SOFR futures inputs. A rate known near the start of an interest period can be operationally useful for business loans. It is not the same contractual quantity as the later realized compounded overnight rate.

Scope of use and basis

The ARRC’s 2023 Term SOFR recommendations support limited derivatives use tied to hedging eligible cash-product exposures. These are scope-of-use best-practice recommendations, not a blanket Federal Reserve rule prohibiting every dealer trade. Applicable licensing and contractual requirements also need to be checked for an actual use.

A Term SOFR versus compounded-SOFR basis can reflect demand, hedgeability, conventions, and liquidity. The lecture’s 4–5 bp figure is not a constant, guaranteed spread or an arbitrage identity. Its sign and size need an observation date and a precisely defined pair of contracts.

3. Discounting and funding: understand the replication assumptions

Suppose a certain cash flow F is due after year fraction δ. In an idealized market where everyone can borrow and lend freely at the same rate r, with matching currency and timing:

P=F/(1+rδ).

Borrow P, buy the claim, and repay P(1+rδ)=F. Replication pins down its price. A different price would create an arbitrage if the required borrowing, lending, buying, and selling were all feasible on those same terms.

Why a private borrowing rate is not a universal discount rate

The lecture compares funding interest P r_b δ with assumed valuation carry P r_d δ. Their difference is P(r_d−r_b)δ. Setting it to zero works in a single-rate replication model, but assuming a valuation growth rate does not establish a tradable arbitrage by itself.

Different institutions can have different unsecured credit spreads, borrowing/lending rates, collateral agreements, and balance-sheet costs. An expensive private loan does not force every counterparty to value the same collateralized cash flow at that private rate. Personal financing profitability and a common market replication price are different questions.

For a perfectly cash-collateralized derivative, idealized valuation generally ties discounting to the collateral remuneration rate under the collateral agreement; SOFR-collateralized USD swaps commonly use the corresponding overnight discount framework. Uncollateralized or imperfectly collateralized trades can require credit, funding, and margin adjustments. A bond also has issuer credit and liquidity risk. These cannot all be collapsed into one universal institution-specific discount curve.

For the underlying modeling issues, see the research paper Funding, Collateral and Hedging. Its treatment explicitly includes collateral, funding, and counterparty-risk terms rather than assuming they are interchangeable.

4. Deposits, CDs, and forward rates

A simple deposit of principal 1 paying rate r over δ pays 1+rδ at maturity. A unit maturity claim under that same risk and funding convention has discount factor:

D(0,T)=1/(1+rδ).

At 5% for 90 actual days on ACT/360, δ=0.25 and D=1/1.0125≈0.987654. Investing 1,000,000 earns 12,500. A real certificate of deposit carries bank credit, liquidity, and contractual features; its quote is not automatically a risk-free discount-curve observation.

Deriving a forward rate

In a single-curve framework, a simple forward rate for [T₁,T₂] satisfies:

1+fδ=D(0,T₁)/D(0,T₂),

so f=[D(0,T₁)/D(0,T₂)−1]/δ.

If D(0,0.5)=0.98 and D(0,1)=0.955 with δ=0.5, the six-month forward rate is 5.23560%. It is the rate consistent with those discount factors and conventions, not an unbiased forecast of the future overnight rate.

In a multi-curve framework, one curve discounts payments while another projects a particular floating index. Distinct index, tenor, or collateral exposures may require basis curves. Do not use the single-curve forward identity to erase an observed basis.

5. SOFR futures: quotes, settlement, and convexity

A SOFR futures price index is 100−R, where R is expressed in percentage points. A quote of 95.25 corresponds to an implied futures rate of 4.75%, not 0.0475 index points. A higher rate corresponds to a lower futures price.

Contract	Simplified final settlement rate
One-month SOFR futures	Calendar-day-weighted arithmetic average of daily SOFR over the delivery month
Three-month SOFR futures	Annualized compounded SOFR over the reference quarter between quarterly IMM dates

The quarterly IMM reference dates are conventionally the third Wednesdays of March, June, September, and December. Monthly contracts do not simply use that same quarterly accrual schedule. Exact calendars and settlement rules must follow the contract. CME’s trading explanation distinguishes the averaging and compounding methods.

Futures rates need not equal forward rates

Daily variation margin changes when gains and losses are paid and how they interact with financing rates. Forward or swap cash flows have a different settlement pattern. The conversion can therefore require a convexity adjustment, depending on the contract, dynamics, volatility, and correlation assumptions.

Define the convention explicitly: CA=R_futures−R_forward for matching underlying periods and rate definitions. Then R_forward=R_futures−CA, and the quote can be written 100−(R_forward+CA) when both rates are in percentage points. The adjustment is not an extra exchange quote component; it is a model-based conversion.

For illustration, a 4.75% futures-implied rate and an assumed +2 bp adjustment give a forward-equivalent rate of 4.73%. The adjustment is not universally +2 bp, always positive in every model, or explained by one reinvestment story alone. Even products classified as linear can need volatility-dependent pricing corrections.

6. Vanilla fixed–floating swaps

A vanilla interest-rate swap exchanges fixed and floating interest on a notional N. In a standard single-currency swap the principal is generally not exchanged. Fix the perspective: receive floating, pay fixed.

For fixed-leg accrual fractions αᵢ and payment dates Tᵢ:

PV_fixed=NKΣᵢαᵢD(0,Tᵢ).

For floating-leg effective forward coupons fⱼ, accrual fractions δⱼ, and payment dates Uⱼ:

PV_float=NΣⱼfⱼδⱼD(0,Uⱼ).

Here fⱼ must be the coupon projection appropriate to the pricing framework; a raw expectation under an unrelated measure is not enough. Fixings already observed, payment lags, spreads, and overnight compounding require appropriate treatment.

The swap’s present value is PV_float−PV_fixed. Set it to zero to obtain the par fixed rate:

S=[ΣⱼfⱼδⱼD(0,Uⱼ)]/[ΣᵢαᵢD(0,Tᵢ)].

The denominator A=ΣᵢαᵢD(0,Tᵢ) is the fixed-leg annuity per unit notional. For the same curves and schedule, an existing pay-fixed swap has V=NA(S−K). A zero-PV par swap still has interest-rate risk and may require initial margin or fees.

Single-curve simplification and a worked swap

With the same projection and discount curve, aligned payment/accrual dates, no spread, and no complicating lags, the floating sum telescopes:

PV_float=N[D(0,T_start)−D(0,T_end)].

For a spot-starting swap in the simplified convention D(0,T_start)=1. Consider two annual payments, D₁=0.96 and D₂=0.92:

A=0.96+0.92=1.88; S=(1−0.92)/1.88≈4.25532%.

With N=1,000,000 and contractual K=4%, V=1,000,000×1.88×(0.0425532−0.04)=4,800 for the floating receiver. Fixed-leg PV is 75,200 and floating-leg PV is 80,000.

This telescoping shortcut does not automatically survive multi-curve projection, different conventions, or payment timing. The full discounted cash-flow formula is the safer starting point.

7. Curve calibration: recover discount factors from quotes

A curve is a model that maps dates to consistent discount factors and projected rates. A static calibrated curve describes today’s cross-section of prices. A dynamic short-rate or term-structure model adds how the curve might evolve; fitting today’s curve alone does not determine tomorrow’s distribution or option volatility.

A small bootstrap

Suppose a one-year deposit with annual accrual quotes 4%, so D₁=1/1.04≈0.961538. A spot-starting two-year annual-pay par swap quotes 4.5%. In the same simplified single-curve setup:

0.045(D₁+D₂)=1−D₂,

so D₂=(1−0.045D₁)/1.045≈0.915532.

The implied one-year forward for year two is D₁/D₂−1≈5.02513%. This is bootstrapping: earlier instruments determine earlier curve values, which let later instruments determine additional values.

For real instruments, coupon schedules can involve dates between quoted maturities, so interpolation and calibration are interdependent. Joint nonlinear solvers may be needed rather than a purely sequential bootstrap.

A calibration workflow

Specify the index, currency, collateral terms, calendars, settlement dates, and day counts.
Select liquid calibration quotes and validate their timestamps and bid–ask ranges.
Convert futures inputs with a consistent convexity treatment where needed.
Choose curve variables, such as log discount factors at knots, and interpolation rules.
Solve for variables that reprice calibration instruments, or fit within an explicit tolerance or weighted objective.
Check residuals, positive discount factors, forward behavior, and sensitivity to quote changes.

Negative rates can imply discount factors above one or locally increasing discount factors. Monotonic decrease should not be imposed as a universal no-arbitrage rule. Equally, positivity alone does not certify every aspect of an arbitrage-consistent model.

8. Local interpolation versus smooth curves

For a continuously compounded instantaneous forward curve f(u):

D(0,T)=exp[−∫₀ᵀf(u)du] and f(T)=−d log D(0,T)/dT where differentiable.

Piecewise constant f means log D is piecewise linear. This is one interpretation of “constant daily forward” or step-curve interpolation; the precise implementation must specify its compounding and day-count conventions.

Method	Benefit	Trade-off
Piecewise constant forwards / log-linear discounts	Simple, fast, with local interpolation dependence	Forward jumps at knots
Cubic or tension splines	Smoother forward and derivative behavior	Possible overshoot, more coupled sensitivities, dependence on boundaries and constraints
Hybrid curve	Separate short-end meeting/calendar structure from long-end smoothing	More conventions and calibration choices to validate

A date between 7- and 10-year knots may depend locally on those two curve parameters. It does not follow that an 8-year trade can always be hedged using only 7- and 10-year market instruments. Calibration maps quotes to parameters, and that mapping can spread quote sensitivities. Smooth methods are not all equally global; locality depends on the basis, constraints, and solver.

The Jacobian explains hedge spillover

Let c be curve parameters and x market quotes. Calibration satisfies g(c)=x. Where the system is differentiable and its square Jacobian J=∂g/∂c is nonsingular:

dc/dx=J⁻¹, so dV/dx=(∂V/∂c)J⁻¹.

Local parameter dependence can become nonlocal quote risk through J⁻¹. The quant–trader trade-off is therefore about smoothness, stability, repricing accuracy, and hedge behavior together, rather than smooth appearance alone.

9. FOMC structure and calendar turns

Policy decisions often create expected jumps near meeting dates, so a short-end curve can place forward-rate knots around expected policy effective dates. But the policy target range and realized SOFR are different: SOFR can move between meetings with repo supply, collateral demand, liquidity, and implementation spreads. Unscheduled policy changes are possible as well.

A hybrid model might use meeting-date steps near the front end and smoother long-end interpolation. This is a modeling choice, not a universal description of every modern curve or a guarantee that daily SOFR itself is flat between meetings.

Turns are short calendar intervals—often around reporting dates—where financing conditions differ from surrounding days. If a modeled one-day turn adds 50 bp to an overnight rate, its approximate extra accrual is 0.005/360≈0.00001389, or 13.89 on 1,000,000. That is 50 bp for one day, not 50 bp of annual portfolio return. The exact impact depends on duration, day count, direction, and compounding. Turns need not always be upward spikes.

10. Z-spreads: a bond’s spread to a reference curve

For fixed promised cash flows CFᵢ, a continuously compounded Z-spread z can be defined by:

P_dirty=ΣᵢCFᵢD_base(0,Tᵢ)exp(−zTᵢ).

Solve for the constant z that matches the full price, including accrued interest. The reference curve, settlement treatment, and spread-compounding convention must be named. A Z-spread can be measured against a swap or government reference curve; it is not restricted to one universal benchmark.

For a one-year zero paying 100, a base continuous zero rate of 4%, and a price 100exp(−0.045), the Z-spread is 0.5%=50 bp. Real bonds usually require numerical solving over multiple cash flows.

A Z-spread mixes credit, liquidity, and other pricing effects; it is not a pure default probability. It is also not the same as YTM minus a benchmark yield. For embedded options, an option-adjusted spread requires an option model rather than treating uncertain cash flows as fixed.

11. Fast pricing, DV01, and execution

Let xᵢ be calibration market quotes and V a trade’s value. A first-order update around a fully calibrated point is:

V(x+Δx)≈V(x)+Σᵢ(∂V/∂xᵢ)Δxᵢ.

The same formula can update a derived rate r instead of V; keep their units distinct. These derivatives must include the calibration mapping if they are meant to represent market-quote risk.

Define a signed bucket DV01 here as the value change for a +1 bp move in one quote, with recalibration and other quotes fixed. If two buckets are +1,200 and −300 currency units per bp, moves of +2 bp and −1 bp imply approximately 1,200×2+(−300)×(−1)=2,700 in value change. Other desks report unsigned DV01 magnitudes, so the sign convention must be checked.

Hedges choose instrument notionals to offset the relevant bucket sensitivities, ideally accounting for transaction costs, liquidity, and basis exposure. A single parallel-rate hedge does not eliminate curve-shape risk. Near-dependent hedge instruments can also make the hedge system unstable.

What a real-time approximation needs

Full recalibration and risk calculations can be too expensive for every quote event, depending on the implementation. A desk can cache a calibrated state and its sensitivities, use local updates for small changes, and refresh when market moves, elapsed time, fixings, or conventions make the approximation stale. Large moves, second-order effects, and ill-conditioned calibration can invalidate a linear update.

An executable bid or offer also reflects inventory, uncertainty, hedging cost, volatility, and expected flow. For example, a dealer with excess duration may shade quotes to encourage offsetting client trades. Model mid-value, inventory adjustment, and bid–ask compensation serve different purposes and should be monitored separately. A predictive flow model does not remove adverse selection or guarantee a profitable quote.

12. Practice questions

Does calling a bond a linear rates product imply that its price is linear in yield?
How is overnight SOFR aggregated, and is it the same as forward-looking Term SOFR?
At a 4% simple annual rate for 90 days on ACT/360, what is the discount factor for a unit maturity claim?
A futures index is 96.10 and an assumed convexity adjustment is +3 bp under CA=futures rate−forward rate. What is the forward-equivalent rate?
For annual fixed-leg payments with D₁=0.97 and D₂=0.93 in a spot-starting single-curve swap, find the par rate.
Can a bank’s higher unsecured borrowing spread force all collateralized swap counterparties to use that bank’s funding rate for discounting?
Does local interpolation in curve parameters guarantee local market-instrument hedges?
Why can a Z-spread not be read directly as a default probability?

Answers

No. The classification generally distinguishes products without explicit optionality; bond price–yield convexity remains.
It is a volume-weighted median of eligible overnight Treasury repo transactions. Term SOFR is a different forward-looking benchmark derived from futures inputs.
D=1/(1+0.04×0.25)=1/1.01≈0.990099.
Futures-implied rate 3.90%; forward-equivalent rate 3.87% under the stated adjustment convention.
S=(1−0.93)/(0.97+0.93)=0.07/1.90≈3.68421%.
No. Replication, collateral terms, and credit/funding adjustments must be modeled consistently; the private borrowing rate is not a universal market discount rate.
No. The calibration Jacobian can spread sensitivity from local parameters across market quotes.
It depends on the benchmark curve and reflects credit, recovery assumptions, liquidity, and other pricing effects rather than a probability alone.