Statistical SOFR Term Structure Model

• Statistical SOFR term structure model is a dynamic framework that transforms backward-looking overnight SOFR rates into forward-looking term fixings using futures-based and arbitrage-free techniques.
• It integrates multiple architectures—such as affine, multi-curve, and nonlinear short-rate models—to address convexity, discontinuities, and basis spreads in market data.
• The approach enables coherent derivative valuation and risk management by calibrating to SOFR futures while accounting for smile, skew, and zero-bound challenges.

Searching arXiv for papers on SOFR term structure modeling to ground the article in the cited literature. A statistical SOFR term structure model is a dynamic framework for representing the term structure associated with the Secured Overnight Financing Rate by combining overnight-rate mechanics, term-rate construction, futures pricing, and statistical or risk-neutral state dynamics. In the literature, the label covers several distinct but related families: arbitrage-free futures-based affine models, multi-curve systems with endogenous basis spreads, generalized HJM and short-rate models with stochastic discontinuities, nonlinear short-rate models with smile and skew, tenor-based term-rate models with explicit variance damping, and real-world-measure simulation models for incomplete markets. Across these formulations, the common problem is to transform a backward-looking overnight benchmark into coherent forward-looking term rates, discount curves, and derivative valuations while controlling convexity, predictable jumps, basis effects, and long-horizon tail behavior (Skov et al., 2021, Fontana et al., 2022, Hasenbichler et al., 2023, Pennanen et al., 23 Jul 2025).

1. Conceptual foundations and rate definitions

SOFR is an overnight secured rate, so it does not directly provide the forward-looking term fixings that historically characterized LIBOR-style markets. A central modeling task is therefore to connect three objects: the overnight short rate, the backward-looking compounded rate over an accrual interval, and the forward-looking term rate implied by discount factors or futures prices. In a standard arbitrage-free setup with money-market numeraire $B_t=e^{\int_0^t r_s\,ds}$ and zero-coupon bonds $p(t,T)=B_t\mathbb{E}^Q[B_T^{-1}\mid\mathcal{F}_t]$, the forward-looking SOFR term rate is written as

$$R^F(S,T)=\frac{1}{T-S}\left(\frac{1}{p(S,T)}-1\right),$$

while a backward-looking compounded rate is constructed from daily overnight fixings over $[S,T]$. A key identity used in futures-based term-rate extraction is

$$B_t\mathbb{E}^Q\!\left[B_T^{-1}R^F(S,T)\mid\mathcal{F}_t\right] = B_t\mathbb{E}^Q\!\left[B_T^{-1}R^B(S,T)\mid\mathcal{F}_t\right],$$

so the forward-looking term rate is the market-consistent prediction of the realized compounded rate (Skov et al., 2021).

The same distinction appears in tenor-indexed market models. For tenor dates

$0=T_0<T_1<\dots<T_N<\infty,$

a family of term rates $\{F_i\}_{i=1}^N$ can be modeled either as ordinary forward-looking rates $F_i(t)$ associated with $[T_{i-1},T_i]$, or as backward-looking rates $F_i^B(t)$ derived from overnight benchmarks such as SOFR or ESTR. The backward-looking version evolves on $p(t,T)=B_t\mathbb{E}^Q[B_T^{-1}\mid\mathcal{F}_t]$0,

$p(t,T)=B_t\mathbb{E}^Q[B_T^{-1}\mid\mathcal{F}_t]$1

and the cited framework states explicitly that for $p(t,T)=B_t\mathbb{E}^Q[B_T^{-1}\mid\mathcal{F}_t]$2,

$p(t,T)=B_t\mathbb{E}^Q[B_T^{-1}\mid\mathcal{F}_t]$3

This places backward-looking SOFR term rates and forward-looking lognormal term-rate models within a common modeling machinery, with the main difference lying in measurability and accrual convention rather than in the stochastic architecture itself (Hasenbichler et al., 2023).

2. Principal model architectures

The literature contains several non-equivalent architectures for SOFR term structure modeling. They differ mainly in whether the primitive object is a short rate, a forward curve, a tenor-indexed term rate, or a latent macro-financial state vector.

Framework	State representation	Characteristic feature
AFNS / shadow-rate futures model	Gaussian affine $p(t,T)=B_t\mathbb{E}^Q[B_T^{-1}\mid\mathcal{F}_t]$4 or shadow rate $p(t,T)=B_t\mathbb{E}^Q[B_T^{-1}\mid\mathcal{F}_t]$5	Extracts term rates from CME SOFR futures
Multi-curve affine transition model	$p(t,T)=B_t\mathbb{E}^Q[B_T^{-1}\mid\mathcal{F}_t]$6	Endogenous SOFR, EFFR, LIBOR, and repo spreads
Generalized HJM with discontinuities	Forward curve $p(t,T)=B_t\mathbb{E}^Q[B_T^{-1}\mid\mathcal{F}_t]$7 under $p(t,T)=B_t\mathbb{E}^Q[B_T^{-1}\mid\mathcal{F}_t]$8	Known-time jump dates and roll-over structure
Nonlinear short-rate smile/skew model	OU factor $p(t,T)=B_t\mathbb{E}^Q[B_T^{-1}\mid\mathcal{F}_t]$9 and accumulated state $$R^F(S,T)=\frac{1}{T-S}\left(\frac{1}{p(S,T)}-1\right),$$0	Analytic SOFR futures and caplet pricing with smile/skew
Mean-field tenor market model	Tenor family $$R^F(S,T)=\frac{1}{T-S}\left(\frac{1}{p(S,T)}-1\right),$$1 with moment-dependent volatility	Explicit variance damping for long horizons
Real-world statistical forward-curve model	Forward-curve coefficients $$R^F(S,T)=\frac{1}{T-S}\left(\frac{1}{p(S,T)}-1\right),$$2 and macro factors $$R^F(S,T)=\frac{1}{T-S}\left(\frac{1}{p(S,T)}-1\right),$$3	Incomplete-market simulation and indifference pricing

In the AFNS line, the state vector follows a Gaussian affine diffusion under $$R^F(S,T)=\frac{1}{T-S}\left(\frac{1}{p(S,T)}-1\right),$$4,

$$R^F(S,T)=\frac{1}{T-S}\left(\frac{1}{p(S,T)}-1\right),$$5

with short rate $$R^F(S,T)=\frac{1}{T-S}\left(\frac{1}{p(S,T)}-1\right),$$6, or, in the zero-bound extension, $$R^F(S,T)=\frac{1}{T-S}\left(\frac{1}{p(S,T)}-1\right),$$7. The three-factor Gaussian arbitrage-free Nelson–Siegel specification supplies level, slope, and curvature factors, and closed-form futures pricing formulas for one-month and three-month SOFR futures (Skov et al., 2021).

In multi-curve transition models, SOFR is not treated as a pure risk-free object. Instead, the model is built directly on a SOFR short rate $$R^F(S,T)=\frac{1}{T-S}\left(\frac{1}{p(S,T)}-1\right),$$8 and an EFFR–SOFR spread process,

$$R^F(S,T)=\frac{1}{T-S}\left(\frac{1}{p(S,T)}-1\right),$$9

so SOFR-specific repo-market premia and unsecured overnight credit premia are both endogenous. LIBOR and term repo are then generated from the same latent factors through credit-downgrade and funding-liquidity roll-over components (Skovmand et al., 2022).

A distinct class models the overnight market in HJM or short-rate form with stochastic discontinuities. Here the numeraire itself is built from a roll-over strategy,

$[S,T]$0

and bond prices are written as

$[S,T]$1

This setup is designed for anticipated jump dates such as FOMC meetings, tax-payment dates, and roll-over dates of the overnight compounding numeraire (Fontana et al., 2022).

Nonlinear short-rate models start from an OU factor

$[S,T]$2

and map it into the short rate by

$[S,T]$3

An additional state

$[S,T]$4

is introduced so that backward-looking SOFR futures or compounded-rate option payoffs become Markovian in $[S,T]$5. This architecture is explicitly intended to incorporate convexity, skew, and smile (Romero-Bermúdez et al., 2024, Turfus et al., 2023).

Finally, the real-world statistical framework parameterizes the forward curve itself: $[S,T]$6 with either piecewise constant or piecewise linear basis functions. In the empirical examples, the selected tenors include $[S,T]$7 or $[S,T]$8, and the short end is tied structurally to the policy-rate lower bound through transformed state variables (Pennanen et al., 23 Jul 2025).

3. Arbitrage-free structure, observables, and calibration

The arbitrage-free strand of the literature fixes the initial discount curve and derivative prices by construction. In short-rate models with smile/skew, the no-arbitrage condition is

$[S,T]$9

so the curve-fitting term $$B_t\mathbb{E}^Q\!\left[B_T^{-1}R^F(S,T)\mid\mathcal{F}_t\right] = B_t\mathbb{E}^Q\!\left[B_T^{-1}R^B(S,T)\mid\mathcal{F}_t\right],$$0 is determined from the observed forward curve. For SOFR futures, the pricing function satisfies a backward Kolmogorov PDE without a discount term because futures are marked to market: $$B_t\mathbb{E}^Q\!\left[B_T^{-1}R^F(S,T)\mid\mathcal{F}_t\right] = B_t\mathbb{E}^Q\!\left[B_T^{-1}R^B(S,T)\mid\mathcal{F}_t\right],$$1 with solution represented through a Gaussian kernel plus perturbative smile/skew corrections (Romero-Bermúdez et al., 2024).

In futures-based AFNS estimation, the model is fitted directly to end-of-day CME futures prices by a Kalman filter or extended Kalman filter maximum-likelihood procedure. The observation set uses the seven nearest one-month contracts and five nearest quarterly contracts. The empirical findings reported are that Vasicek / one-factor Gaussian fits poorly, two-factor AFNS fits much better, and three-factor AFNS fits best across the curve; the shadow-rate AFNS is very similar to the three-factor Gaussian model except near the zero lower bound (Skov et al., 2021).

The multi-curve affine transition model uses a quasi-maximum-likelihood Kalman filter on Refinitiv data from June 2018 through October 2021. The sample contains daily observations of spot three- and six-month USD LIBOR, three- and six-month term repo rates for Treasuries, Eurodollar futures, one- and three-month SOFR futures, and Fed Funds futures. To preserve affine measurement equations, several spot and futures rates are transformed to yields. Reported RMSEs are about 2.2 bps for SOFR futures, 2.0 bps for Fed Funds futures, 2.9 bps for Eurodollar futures, 2.5 bps for spot LIBOR, and 2.4 bps for term repo (Skovmand et al., 2022).

The mean-field tenor model is calibrated through a three-step procedure: fit the initial curve $$B_t\mathbb{E}^Q\!\left[B_T^{-1}R^F(S,T)\mid\mathcal{F}_t\right] = B_t\mathbb{E}^Q\!\left[B_T^{-1}R^B(S,T)\mid\mathcal{F}_t\right],$$2 to a Nelson–Siegel type family by least squares; fit the principal factor $$B_t\mathbb{E}^Q\!\left[B_T^{-1}R^F(S,T)\mid\mathcal{F}_t\right] = B_t\mathbb{E}^Q\!\left[B_T^{-1}R^B(S,T)\mid\mathcal{F}_t\right],$$3 to caplet volatilities; and fit correlation parameters to swaption volatilities using a classical approximation formula rather than nested Monte Carlo. In the empirical study,

$$B_t\mathbb{E}^Q\!\left[B_T^{-1}R^F(S,T)\mid\mathcal{F}_t\right] = B_t\mathbb{E}^Q\!\left[B_T^{-1}R^B(S,T)\mid\mathcal{F}_t\right],$$4

which is described as a standard Rebonato-type specification. This calibration design is central to the model’s claim of being practical without nested simulations (Hasenbichler et al., 2023).

The incomplete-market statistical model uses a different calibration logic. Current curves are fitted to SOFR futures quotes and, when available, bid and ask prices by convex least squares or quadratic optimization: $$B_t\mathbb{E}^Q\!\left[B_T^{-1}R^F(S,T)\mid\mathcal{F}_t\right] = B_t\mathbb{E}^Q\!\left[B_T^{-1}R^B(S,T)\mid\mathcal{F}_t\right],$$5 The implementation uses Mosek as the interior-point solver and Python + CVXPY as the modeling layer, with reported solve times of less than 0.4 seconds for one calibration instance and under 8 minutes for about 1300 instances. Historical data then determine the VAR coefficients and innovation covariance, while deterministic shifts encode user views (Pennanen et al., 23 Jul 2025).

4. Convexity, smile, skew, and zero-bound behavior

Convexity correction is a structural issue because SOFR futures are marked to market, whereas forward rates are not. In the AFNS framework, one-month futures are approximately driven by

$$B_t\mathbb{E}^Q\!\left[B_T^{-1}R^F(S,T)\mid\mathcal{F}_t\right] = B_t\mathbb{E}^Q\!\left[B_T^{-1}R^B(S,T)\mid\mathcal{F}_t\right],$$6

while three-month futures satisfy

$$B_t\mathbb{E}^Q\!\left[B_T^{-1}R^F(S,T)\mid\mathcal{F}_t\right] = B_t\mathbb{E}^Q\!\left[B_T^{-1}R^B(S,T)\mid\mathcal{F}_t\right],$$7

The resulting convexity corrections are small for near-dated contracts but become material beyond about 2 years. The same study reports that the three-factor model aligns closely with the Federal Reserve’s indicative SOFR term rates, and that jumps and seasonality in spot SOFR do not need to be explicitly modeled in order to fit the futures curve (Skov et al., 2021).

Smile and skew are handled analytically in the nonlinear short-rate literature. The short-rate map

$$B_t\mathbb{E}^Q\!\left[B_T^{-1}R^F(S,T)\mid\mathcal{F}_t\right] = B_t\mathbb{E}^Q\!\left[B_T^{-1}R^B(S,T)\mid\mathcal{F}_t\right],$$8

is constructed so that $$B_t\mathbb{E}^Q\!\left[B_T^{-1}R^F(S,T)\mid\mathcal{F}_t\right] = B_t\mathbb{E}^Q\!\left[B_T^{-1}R^B(S,T)\mid\mathcal{F}_t\right],$$9 controls smile/nonlinearity and $0=T_0<T_1<\dots<T_N<\infty,$0 controls skewness/drift deformation. The pricing kernel is expanded perturbatively around a Gaussian Hull–White kernel, with small parameter

$0=T_0<T_1<\dots<T_N<\infty,$1

For 3M SOFR futures and related options, the literature states that omitting smile/skew and using only ATM calibration can overestimate convexity by 10–25%, especially for short maturities, and that relative differences between 3M SOFR and 3M Eurodollar convexity can be around 10–15% for short maturities (Romero-Bermúdez et al., 2024).

In the caplet literature, the same nonlinear OU structure is translated into effective variances. For compounded-rate caplets, the baseline variance is

$0=T_0<T_1<\dots<T_N<\infty,$2

and the asymptotic price is re-expressed through an effective variance

$0=T_0<T_1<\dots<T_N<\infty,$3

where the correction is a quadratic function of the moneyness parameter. The paper explicitly emphasizes that the linear term in moneyness produces skew and the quadratic term produces smile, so the model can be interpreted as imposing on a Hull–White model an effective variance that is a quadratic function of moneyness rather than a constant for any given maturity (Turfus et al., 2023).

The zero lower bound creates a separate modeling issue. In the shadow-rate AFNS extension, $0=T_0<T_1<\dots<T_N<\infty,$4 preserves affine state dynamics while truncating the short rate at zero. The reported empirical conclusion is that before the March 2020 rate collapse, Gaussian and shadow-rate models are similar, whereas after rates compress near zero the shadow model captures lower volatility better and the Gaussian model struggles with volatility compression at the lower bound (Skov et al., 2021).

5. Discontinuities, basis spreads, and long-horizon stability

A recurring theme in SOFR modeling is that overnight rates exhibit jumps or spikes at predetermined dates. One strand formalizes this through stochastic discontinuities in a generalized HJM framework. The forward rate dynamics are written as

$0=T_0<T_1<\dots<T_N<\infty,$5

where $0=T_0<T_1<\dots<T_N<\infty,$6 has jumps only at expected jump dates. Absence of arbitrage requires both the continuous-time HJM drift restriction and extra jump-date conditions at roll-over dates and expected jump dates (Fontana et al., 2022).

A more explicit policy-and-spike decomposition writes the SOFR short rate as

$0=T_0<T_1<\dots<T_N<\infty,$7

where $0=T_0<T_1<\dots<T_N<\infty,$8 is the policy target rate, $0=T_0<T_1<\dots<T_N<\infty,$9 month-end spikes, $\{F_i\}_{i=1}^N$0 non-month-end spikes such as the September 2019 repo spike, and $\{F_i\}_{i=1}^N$1 a mean-reverting residual. The model is designed so that the realized short rate is piecewise constant with jumps on known dates, while the forward curve continues to evolve diffusively as market expectations of those future jumps change. This suggests that discontinuity-aware spot models and smooth futures-based AFNS models address different empirical layers of the SOFR market rather than mutually exclusive descriptions (Gellert et al., 2021, Skov et al., 2021).

Another common simplification is to treat SOFR as synonymous with the risk-free rate. The transition-era multi-curve literature explicitly rejects this. In that framework,

$\{F_i\}_{i=1}^N$2

with $\{F_i\}_{i=1}^N$3 representing SOFR-specific repo-market premia such as collateral scarcity, haircuts, gap risk, and other systemic repo frictions. The filtered spread $\{F_i\}_{i=1}^N$4 stays around zero but turns negative during the SOFR surge in September 2019, indicating that the systemic repo risk premium exceeded the unsecured overnight credit spread at that time. The model therefore argues against treating SOFR as automatically “closer to risk-free” than EFFR in all states of the world (Skovmand et al., 2022).

Long-horizon statistical stability introduces yet another issue: unrealistically large simulated rates. The mean-field market model addresses this through moment-dependent volatility, explicit variance control, damping factors, and decorrelation. If

$\{F_i\}_{i=1}^N$5

then the moments

$\{F_i\}_{i=1}^N$6

solve a deterministic ODE system, and in the practically important $\{F_i\}_{i=1}^N$7 case the total variance proxy

$\{F_i\}_{i=1}^N$8

can be expressed explicitly from the principal volatility factor and damping function. A representative damping factor is

$\{F_i\}_{i=1}^N$9

and correlation vectors may switch to effective decorrelation beyond a variance threshold. In simulations of $F_i(t)$0, the proportion of cases with $F_i(t)$1 drops from 0.0830 undamped to 0.0083 under volatility freeze, and to 0.0073 with volatility freeze plus decorrelation. The appendix also provides an explicit tail bound for choosing thresholds before simulation (Hasenbichler et al., 2023).

6. Statistical, incomplete-market, and practical perspectives

The expression “statistical SOFR term structure model” is used in a narrower sense in the incomplete-market literature, where the primary measure is the subjective or real-world measure $F_i(t)$2 rather than a unique risk-neutral measure. The motivation is that the SOFR derivatives market remains incomplete and illiquid, with zero liquidity in many listed options, meaningful bid-ask spreads, sparse quotes across maturities, and observed prices not perfectly consistent with a single arbitrage-free curve. The model therefore combines a forward-curve state vector with a macroeconomic state vector that drives the central bank policy rate and, through it, jumps in SOFR (Pennanen et al., 23 Jul 2025).

In that framework, the macro vector is

$F_i(t)$3

where $F_i(t)$4 is the lower limit of the federal funds target range, $F_i(t)$5 inflation, and $F_i(t)$6 real GDP growth. The transformed curve factors are

$F_i(t)$7

and both $F_i(t)$8 and $F_i(t)$9 are modeled by linear VARs with deterministic shift terms used to encode user views. The reported simulation design uses 20,000 scenarios over a 10-year horizon, monthly for macro factors and daily for SOFR-term-structure factors. The same scenarios are then used for futures, options, swaptions, portfolio optimization, and utility-indifference pricing (Pennanen et al., 23 Jul 2025).

Utility-based pricing is formulated explicitly. For exponential utility

$[T_{i-1},T_i]$0

the indifference selling price for a payoff $[T_{i-1},T_i]$1 can be written as

$[T_{i-1},T_i]$2

This is a different pricing philosophy from replication under a unique martingale measure, and it is meant for risk management, scenario generation, and indifference pricing rather than exact cross-sectional fit (Pennanen et al., 23 Jul 2025).

A related but more classical one-factor approach prices and hedges SOFR derivatives with SOFR futures in a Vasicek environment. Under

$[T_{i-1},T_i]$3

the model writes SOFR, EFFR, hedge funding, collateral, and unsecured funding rates as the common factor plus deterministic bases. It derives closed-form expressions for SOFR futures, swaps, caplets, and swaptions, and embeds collateralization through an effective discount rate $[T_{i-1},T_i]$4. The paper’s limitations are explicit: only one factor, deterministic bases, simplified multi-curve reality, and dependence on liquid futures for dynamic replication (Bickersteth et al., 2021).

Taken together, these strands indicate that no single canonical specification exhausts the subject. Risk-neutral affine and HJM models dominate when the objective is arbitrage-free pricing from liquid futures and options; discontinuity-aware and multi-curve models are essential when the focus is overnight-rate mechanics, basis decomposition, and policy-date spikes; and real-world statistical models become natural when the objective is large-scale simulation, risk management, or pricing in an incomplete market. This suggests that “statistical SOFR term structure model” is best understood as a family of technically distinct models organized around a common empirical problem: extracting stable, economically interpretable SOFR term structures from a backward-looking overnight benchmark under limited liquidity and nontrivial market microstructure (Skov et al., 2021, Skovmand et al., 2022, Fontana et al., 2022, Pennanen et al., 23 Jul 2025).