Temporal Function Market Making (TFMM)

• Temporal Function Market Making (TFMM) is a framework where market mechanisms use time-indexed functions to dynamically adjust pricing rules and liquidity incentives.
• TFMM encompasses implementations like time-dependent CFMM invariants, dynamic-weight AMMs, propagator-based dealer models, and prediction market constructs to tailor price responses.
• TFMM enhances market efficiency by mitigating arbitrage leakage, replicating option payoffs, synchronizing external data, and stabilizing evolving liquidity conditions.

Searching arXiv for the cited TFMM-related papers and terminology to ground the article in current arXiv records. Temporal Function Market Making (TFMM) denotes market-making mechanisms in which the pricing rule, invariant, or cost function is explicitly indexed by time or by information arriving over time. In the cited literature, TFMM includes time-dependent CFMM invariants, dynamic-weight AMM rebalancing, propagator-based dealer models under transient impact, and cost-function prediction markets with decreasing utility for information. The shared structural feature is temporal adaptation of prices, incentives, and loss geometry, rather than a single implementation template (Jepsen et al., 2023, Dudík et al., 2014, Willetts, 5 Mar 2026, Barzykin, 19 Jan 2026).

1. Formal definitions and scope

In the CFMM formulation, a market maker is specified by a trading function $\phi:\mathbb{R}_+^n\to\mathbb{R}$, with pool state given by a reserves vector $R\in\mathbb{R}_+^n$ and invariant level set $\phi(R)=k$. Valid trades with fee $\gamma\in(0,1]$ satisfy

$\phi(R+\gamma \Delta-\Lambda)=\phi(R),$

where $\Delta,\Lambda\in\mathbb{R}_+^n$ are the tendered and received baskets. The instantaneous price is obtained from the gradient: if $P=\nabla \phi(R)$ and token $n$ is the numeraire, then

$$p_i=\frac{P_i}{P_n}=\frac{\partial_i\phi}{\partial_n\phi}.$$

For a 2-asset pool with reserves $(x,y)$ and numeraire asset $R\in\mathbb{R}_+^n$0, a general time-dependent invariant can be written as $R\in\mathbb{R}_+^n$1, with instantaneous exchange rate

$R\in\mathbb{R}_+^n$2

and invariant slope

$R\in\mathbb{R}_+^n$3

This is the basic invariant-based definition of TFMM in the AMM setting (Jepsen et al., 2023).

A second formalization treats TFMM as a time-indexed family of convex cost functions. In cost-function-based markets, the market maker maintains a state $R\in\mathbb{R}_+^n$4 and a closed, proper, convex cost function $R\in\mathbb{R}_+^n$5. A trade $R\in\mathbb{R}_+^n$6 costs $R\in\mathbb{R}_+^n$7, and prices are $R\in\mathbb{R}_+^n$8 when $R\in\mathbb{R}_+^n$9 is differentiable, or more generally $\phi(R)=k$0. The TFMM abstraction is the tuple $\phi(R)=k$1, where $\phi(R)=k$2 is time-indexed and $\phi(R)=k$3 is a state-transition operator satisfying price preservation. Utility for information is represented by the mixed Bregman divergence

$\phi(R)=k$4

where $\phi(R)=k$5 is the convex conjugate of $\phi(R)=k$6 (Dudík et al., 2014).

A common misconception is that TFMM refers only to clock-time-dependent AMM curves. In the cited literature, the temporal function may instead be an impact kernel $\phi(R)=k$7, a family of convex costs $\phi(R)=k$8, or an oracle-synchronized effective curve derived from external liquidity and inventory state (Barzykin, 19 Jan 2026, Abgaryan et al., 2023).

2. Time-dependent CFMM invariants and option replication

The paper "Analysis of the RMM-01 Market Maker" gives a concrete TFMM instantiation in a 2-asset CFMM whose invariant depends on time-to-maturity $\phi(R)=k$9 (Jepsen et al., 2023). With reserves per LP token $\gamma\in(0,1]$0, strike price $\gamma\in(0,1]$1, implied volatility $\gamma\in(0,1]$2, and $\gamma\in(0,1]$3 in the paper’s analysis, the trading function is

$\gamma\in(0,1]$4

where $\gamma\in(0,1]$5 is the standard normal CDF and $\gamma\in(0,1]$6 its quantile. The time dependence enters through $\gamma\in(0,1]$7, so $\gamma\in(0,1]$8. As $\gamma\in(0,1]$9, $\phi(R+\gamma \Delta-\Lambda)=\phi(R),$0 collapses to a constant-sum market. The paper identifies this as the hallmark of TFMM: a CFMM whose invariant explicitly depends on time, thereby shaping path-dependent price dynamics and payoffs (Jepsen et al., 2023).

RMM-01 is designed so that LP tokens replicate a Black–Scholes priced covered call. With

$\phi(R+\gamma \Delta-\Lambda)=\phi(R),$1

the Black–Scholes call price is

$\phi(R+\gamma \Delta-\Lambda)=\phi(R),$2

and the covered call value is

$\phi(R+\gamma \Delta-\Lambda)=\phi(R),$3

The paper shows that when the CFMM’s reported price equals the external market price $\phi(R+\gamma \Delta-\Lambda)=\phi(R),$4,

$\phi(R+\gamma \Delta-\Lambda)=\phi(R),$5

Perfect replication occurs at $\phi(R+\gamma \Delta-\Lambda)=\phi(R),$6, where

$\phi(R+\gamma \Delta-\Lambda)=\phi(R),$7

The RMM-01 marginal price is

$\phi(R+\gamma \Delta-\Lambda)=\phi(R),$8

At fixed reserves, the time derivative is

$\phi(R+\gamma \Delta-\Lambda)=\phi(R),$9

Thus, even with no swaps, the internal price drifts deterministically with $\Delta,\Lambda\in\mathbb{R}_+^n$0. Arbitrageurs continuously adjust reserves so that $\Delta,\Lambda\in\mathbb{R}_+^n$1, coupling the TFMM’s time drift to external market prices. Profitable arbitrage exists when the CFMM-reported price lies outside

$\Delta,\Lambda\in\mathbb{R}_+^n$2

A central analytical result concerns price stability relative to Uniswap’s constant product market. For Uniswap V2, $\Delta,\Lambda\in\mathbb{R}_+^n$3 and $\Delta,\Lambda\in\mathbb{R}_+^n$4, whereas RMM-01 has infinitesimal price impact

$\Delta,\Lambda\in\mathbb{R}_+^n$5

At equal prices, RMM-01 has strictly smaller infinitesimal price impact when

$\Delta,\Lambda\in\mathbb{R}_+^n$6

The bound is easiest to satisfy near at-the-money and as $\Delta,\Lambda\in\mathbb{R}_+^n$7 decreases. Near expiration,

$\Delta,\Lambda\in\mathbb{R}_+^n$8

so price manipulation is impossible because $\Delta,\Lambda\in\mathbb{R}_+^n$9 does not depend on reserves when $P=\nabla \phi(R)$0. The same construction is composable with lending: shorting the LP token and holding one unit of the underlying gives a synthetic call,

$P=\nabla \phi(R)$1

and put–call parity yields a synthetic put via

$P=\nabla \phi(R)$2

(Jepsen et al., 2023).

3. Dynamic-weight TFMM and the geometry of rebalancing

A second TFMM line studies dynamic-weight AMMs, especially geometric mean market makers with invariant

$P=\nabla \phi(R)$3

where $P=\nabla \phi(R)$4 lies on the simplex, $P=\nabla \phi(R)$5, $P=\nabla \phi(R)$6. At equilibrium, with exogenous market prices $P=\nabla \phi(R)$7 and pool value $P=\nabla \phi(R)$8, the value fraction in asset $P=\nabla \phi(R)$9 equals the weight:

$n$0

In TFMM, a pool manager prescribes a weight trajectory over time. At each block boundary, weights are updated from $n$1 to $n$2, arbitrageurs trade, and the pool returns to equilibrium at the new weights and current prices (Willetts, 5 Mar 2026).

When weights jump from $n$3 to $n$4 at fixed prices, the pool’s value shrinks by the retention ratio

$n$5

so the log-loss is exactly

$n$6

Thus the per-step arbitrage loss is the KL divergence between the new and old weight vectors. For a small change $n$7 with $n$8,

$n$9

which identifies the Fisher–Rao metric

$$p_i=\frac{P_i}{P_n}=\frac{\partial_i\phi}{\partial_n\phi}.$$0

as the natural Riemannian metric on the weight simplex (Willetts, 5 Mar 2026).

Under the Hellinger embedding

$$p_i=\frac{P_i}{P_n}=\frac{\partial_i\phi}{\partial_n\phi}.$$1

the simplex maps to the positive orthant of the unit sphere, $$p_i=\frac{P_i}{P_n}=\frac{\partial_i\phi}{\partial_n\phi}.$$2, and the metric becomes

$$p_i=\frac{P_i}{P_n}=\frac{\partial_i\phi}{\partial_n\phi}.$$3

The leading-order loss-minimizing interpolation is therefore SLERP, the great-circle geodesic on the sphere traversed at constant speed. If

$$p_i=\frac{P_i}{P_n}=\frac{\partial_i\phi}{\partial_n\phi}.$$4

then

$$p_i=\frac{P_i}{P_n}=\frac{\partial_i\phi}{\partial_n\phi}.$$5

For $$p_i=\frac{P_i}{P_n}=\frac{\partial_i\phi}{\partial_n\phi}.$$6 steps, each quadratic-loss increment is

$$p_i=\frac{P_i}{P_n}=\frac{\partial_i\phi}{\partial_n\phi}.$$7

and total quadratic loss is $$p_i=\frac{P_i}{P_n}=\frac{\partial_i\phi}{\partial_n\phi}.$$8.

The midpoint of SLERP is exactly the "(AM+GM)/normalize" heuristic:

$$p_i=\frac{P_i}{P_n}=\frac{\partial_i\phi}{\partial_n\phi}.$$9

where

$(x,y)$0

This identity holds for any number of tokens and any magnitude of weight change. It implies that all dyadic SLERP points can be generated by recursive AM–GM bisection without trigonometric functions. On the full KL cost, if weights are bounded away from zero and $(x,y)$1, the paper proves

$(x,y)$2

so relative sub-optimality is $(x,y)$3 (Willetts, 5 Mar 2026).

4. Propagator-based TFMM and endogenous price impact

In dealer markets, TFMM can be formulated through a temporal impact kernel rather than an AMM invariant. In "Market Making and Transient Impact in Spot FX", the midprice is

$(x,y)$4

where $(x,y)$5 is cumulative interbank execution and $(x,y)$6 is a causal, nonincreasing, integrable propagator. The paper uses the exponential kernel

$(x,y)$7

with resilience rate $(x,y)$8 and half-life

$(x,y)$9

In the equivalent Markovian embedding,

$R\in\mathbb{R}_+^n$00

where $R\in\mathbb{R}_+^n$01. Dealer inventory evolves as

$R\in\mathbb{R}_+^n$02

and interbank hedging cost is

$R\in\mathbb{R}_+^n$03

The HJB with state $R\in\mathbb{R}_+^n$04 yields approximate optimal quotes

$R\in\mathbb{R}_+^n$05

with

$R\in\mathbb{R}_+^n$06

and optimal hedging speed

$R\in\mathbb{R}_+^n$07

where

$R\in\mathbb{R}_+^n$08

The boundary $R\in\mathbb{R}_+^n$09 defines the pure internalization zone. In this formulation, the TFMM “temporal function” is the impact kernel $R\in\mathbb{R}_+^n$10 and its resilience parameter $R\in\mathbb{R}_+^n$11 (Barzykin, 19 Jan 2026).

A related perspective is given by the dynamic-auction model in "A Model of Market Making and Price Impact", where the market-making function arises endogenously from a stochastic differential game among symmetric market makers (Singh, 2021). In linear-symmetric Nash equilibrium,

$R\in\mathbb{R}_+^n$12

and individual trading rates satisfy

$R\in\mathbb{R}_+^n$13

Writing $R\in\mathbb{R}_+^n$14 and identifying $R\in\mathbb{R}_+^n$15, the equilibrium price takes Almgren–Chriss form

$R\in\mathbb{R}_+^n$16

Here the temporal pricing rule is inventory-dependent and rate-dependent: a persistent component enters through the state $R\in\mathbb{R}_+^n$17, while temporary impact enters through $R\in\mathbb{R}_+^n$18. This literature suggests that a temporal market-making function need not be explicitly imposed; it can emerge from equilibrium quoting under inventory risk and competition (Singh, 2021).

5. Information-sensitive TFMM in prediction markets

In prediction markets, TFMM is built from a time-indexed family of convex cost functions whose utility for information decreases over time. In the cost-function framework, with outcome space $R\in\mathbb{R}_+^n$19, payoff vectors $R\in\mathbb{R}_+^n$20, state $R\in\mathbb{R}_+^n$21, cost $R\in\mathbb{R}_+^n$22, and conjugate

$R\in\mathbb{R}_+^n$23

trade payoff is

$R\in\mathbb{R}_+^n$24

The mixed Bregman divergence

$R\in\mathbb{R}_+^n$25

has the identities

$R\in\mathbb{R}_+^n$26

where $R\in\mathbb{R}_+^n$27 (Dudík et al., 2014).

For sudden revelation, a random variable $R\in\mathbb{R}_+^n$28 becomes publicly revealed at a known time $R\in\mathbb{R}_+^n$29. The market maker leaves the state unchanged, $R\in\mathbb{R}_+^n$30, and replaces $R\in\mathbb{R}_+^n$31 by a new cost function $R\in\mathbb{R}_+^n$32 constructed from offsets

$R\in\mathbb{R}_+^n$33

the piecewise-shifted dual

$R\in\mathbb{R}_+^n$34

and the convex roof $R\in\mathbb{R}_+^n$35. In primal form,

$R\in\mathbb{R}_+^n$36

Under the consistency condition in the paper’s "Implicit submarket closing" theorem, the post-switch market simultaneously satisfies CondPrice, Ex, and Ze: conditional prices are preserved, excess utility is preserved within each conditional submarket, and the guaranteed profit from knowing only the revealed information is zero. The worst-case loss is preserved (Dudík et al., 2014).

For gradual decrease, the structured market is represented as an LCMM with submarkets $R\in\mathbb{R}_+^n$37 and non-increasing schedules $R\in\mathbb{R}_+^n$38, $R\in\mathbb{R}_+^n$39. The direct-sum cost at time $R\in\mathbb{R}_+^n$40 is

$R\in\mathbb{R}_+^n$41

and the constrained market is

$R\in\mathbb{R}_+^n$42

If $R\in\mathbb{R}_+^n$43 minimizes $R\in\mathbb{R}_+^n$44, the price-preserving state update to time $R\in\mathbb{R}_+^n$45 is

$R\in\mathbb{R}_+^n$46

The divergence evolution is

$R\in\mathbb{R}_+^n$47

This construction preserves prices while making utility for targeted information nonincreasing in time, and the worst-case loss over time never exceeds the initial one (Dudík et al., 2014).

6. Oracle-synchronized architectures, composability, and design tensions

The paper "Dynamic Function Market Maker" presents a protocol architecture that implements a TFMM-like mechanism through a virtual order book, an external liquidity density function (ELDF), rebalancing premiums, non-linear buffers, and an algorithmic accounting-asset $R\in\mathbb{R}_+^n$48 (Abgaryan et al., 2023). In the notation of the paper, $R\in\mathbb{R}_+^n$49 are reserves, $R\in\mathbb{R}_+^n$50 is the consolidated external price, $R\in\mathbb{R}_+^n$51 is the external liquidity density function, $R\in\mathbb{R}_+^n$52 is the internal liquidity density function, $R\in\mathbb{R}_+^n$53 are nominal open inventories expressed in $R\in\mathbb{R}_+^n$54, $R\in\mathbb{R}_+^n$55 is the rebalancing premium function, $R\in\mathbb{R}_+^n$56 is its aggressiveness parameter, $R\in\mathbb{R}_+^n$57 is the sLP cover coefficient, $R\in\mathbb{R}_+^n$58 is utilization, $R\in\mathbb{R}_+^n$59 is Treasury Reserve, and $R\in\mathbb{R}_+^n$60 is the rebalancing fee fraction of the AMM fee. The protocol’s solvency invariant is

$R\in\mathbb{R}_+^n$61

with

$R\in\mathbb{R}_+^n$62

and

$R\in\mathbb{R}_+^n$63

The one-price principle is expressed through the synchronization error

$R\in\mathbb{R}_+^n$64

with enforcement band

$R\in\mathbb{R}_+^n$65

The internal effective curve is synchronized to the external curve by

$R\in\mathbb{R}_+^n$66

where $R\in\mathbb{R}_+^n$67 is the fitted external curve. Rebalancing premiums and sLP buffers modify the effective price schedule through

$R\in\mathbb{R}_+^n$68

and

$R\in\mathbb{R}_+^n$69

The deterministic settlement flow includes data aggregation, pricing update, order routing, buffer adjustment, fee allocation, and auction updates of $R\in\mathbb{R}_+^n$70 subject to Treasury bounds (Abgaryan et al., 2023).

Across these constructions, TFMM is not a single invariant family but a design principle for embedding time, information, or resilience directly into the market-making rule. In RMM-01, the temporal parameter is $R\in\mathbb{R}_+^n$71 and the invariant flattens toward constant-sum near expiry (Jepsen et al., 2023). In dynamic-weight AMMs, the temporal object is the weight trajectory, and arbitrage leakage is measured by KL divergence on the simplex (Willetts, 5 Mar 2026). In propagator models, the temporal object is the impact kernel $R\in\mathbb{R}_+^n$72 and its resilience scale $R\in\mathbb{R}_+^n$73 (Barzykin, 19 Jan 2026). In prediction markets, it is the schedule of convex costs $R\in\mathbb{R}_+^n$74 and the state update operator $R\in\mathbb{R}_+^n$75 that preserve prices while attenuating utility for selected information (Dudík et al., 2014).

The main design tensions are also explicit in the cited papers. RMM-01 assumes continuous trading and frictionless markets for replication, with real-world fees and lending interest introducing small deviations (Jepsen et al., 2023). The dynamic-weight geometry assumes zero fees and perfect, immediate arbitrage within each block step, exogenous prices constant within each step, and a positive weight floor (Willetts, 5 Mar 2026). DFMM requires sufficient external liquidity and bounded latency for its contraction argument, and it specifies fallback modes for oracle failures and collateral shortfall (Abgaryan et al., 2023). The transient-impact HJB requires progressively measurable controls together with integrability and boundedness sufficient for existence of the solution (Barzykin, 19 Jan 2026). The prediction-market constructions require consistency or tightness conditions to achieve the full combination of CondPrice, Ex, Ze, and DecUtil guarantees (Dudík et al., 2014). These conditions do not negate the TFMM framework; they delineate the regimes in which its temporal adaptations are analytically tractable and economically interpretable.