Liquidity-of-Time Premium (TLP)

• Liquidity-of-Time Premium (TLP) is defined as the extra return or discount reflecting the cost of accessing liquidity during market closures, mathematically modeled via an embedded put option.
• The pricing method uses a contingent claim structure with Black-Scholes-style approximations to link TLP to volatility and closure duration, highlighting the impact of overnight gap risks.
• Empirical proxies and dynamic collateral policies illustrate that TLP increases with higher volatility and longer closures, influencing stablecoin protocol design and risk management.

Liquidity-of-Time Premium (TLP) denotes the extra return or cost of providing liquidity when the primary market is closed. In the formulation introduced for time-bound stablecoins, a stablecoin is minted at the underlying asset’s official close price $S_c$ and redeemed at the next open, when the underlying price is $S_o$; TLP is then the percentage discount of fair stablecoin value below the stale closing price, interpreted as the market’s price for accessing liquidity during the closed period (Borjigin et al., 7 Oct 2025). More broadly, adjacent literatures study closely related objects—time-to-liquidate discounts, transaction-cost liquidity premia, execution-horizon tradeoffs, liquidity discount rates, and timing rents from just-in-time liquidity provision—even when they do not use the term TLP.

1. Formal definition and contingent-claim structure

The core time-bound stablecoin setup is defined by the payoff identity

$$\min(S_c,\;S_o) \;=\; S_c \;-\; \max(0,\;S_c - S_o),$$

so the stablecoin’s value at issuance is modeled as

$$\text{Stablecoin value at issuance} \;=\; S_c \;-\; P(S_c),$$

where $P(S_c)$ is the value of a put option with strike $S_c$ expiring at the next open. The fair issuance value is written as

$P_c = S_c - P(S_c),$

and TLP is defined as

$\text{TLP} = \frac{S_c - P_c}{S_c}.$

Under the idealized model, this is essentially the put value normalized by $S_c$, so TLP is the market’s price for accessing liquidity during the closed period (Borjigin et al., 7 Oct 2025).

This representation makes the economic content explicit. The stablecoin is interpreted as a fully collateralized claim minus an embedded put option. The embedded put is the borrower’s option to default if the asset falls between the close and the next open. A common misconception is that a time-bound stablecoin should trade at the stale close price itself. The contingent-claim formulation instead implies that fair value is below $S_c$ by the value of the close-to-open downside protection.

2. Economic mechanism, no-arbitrage band, and term structure

TLP arises because liquidity is time-dependent. When the primary market is closed, an investor cannot immediately trade the underlying asset even if news arrives, creating a liquidity gap over time. A time-bound stablecoin lets the holder monetize the close-price value immediately, but the holder bears overnight gap risk: by the next open, the asset may be worth less. The discount from $S_o$0 is therefore compensation for bearing that gap, analogous to paying a premium for immediacy (Borjigin et al., 7 Oct 2025).

The no-arbitrage model assumes that the underlying closes at $S_o$1 and reopens at $S_o$2, overnight return is lognormal with volatility $S_o$3, risk-neutral pricing is used for the fair-value derivation, there are no transaction costs in the cleanest version, the stablecoin is fully collateralized or bounded by collateral and liquidation rules, the borrower redeems if $S_o$4, and arbitrage is limited during the closed period because simultaneous cross-market trading is not possible. Within that framework, the paper derives the pricing corridor

$S_o$5

where $S_o$6 is the default probability and $S_o$7 is the conditional loss fraction given default. The upper bound reflects the fact that the stablecoin cannot rationally trade above $S_o$8 because users can mint against collateral and sell it; the lower bound reflects expected recovery value.

The term structure of TLP is driven by volatility and closure length. The paper gives a Black-Scholes-style short-maturity put expression with overnight duration $S_o$9, and for small $$\min(S_c,\;S_o) \;=\; S_c \;-\; \max(0,\;S_c - S_o),$$0 states the approximation

$$\min(S_c,\;S_o) \;=\; S_c \;-\; \max(0,\;S_c - S_o),$$1

The implication is direct: longer closures and higher volatility increase TLP. The figures are described as showing a steeper TLP term structure for high-volatility assets and for 3- to 4-day closures. Collateralization matters as well: higher LTV can increase supply and, if too aggressive, raise tail risk, whereas lower LTV makes the stablecoin safer and scarcer, which supports parity (Borjigin et al., 7 Oct 2025).

3. Empirical measurement and reported magnitudes

Because the proposed stablecoin market is not yet mature, existing market phenomena are used as empirical proxies for TLP. The paper outlines five main classes of proxy. First, for cross-listed stocks, ADR premiums compare an ADR price in the U.S. to the stale local close, capturing how much investors pay for overnight liquidity in the ADR market relative to the local market. Second, futures-cash basis measures the divergence between overseas index futures and the prior cash close when the cash market is closed. Third, pre-market versus official close gaps measure the effect of overnight news and the cost of transacting before the main market opens. Fourth, overnight returns and morning reversals are interpreted as the implicit cost of providing liquidity during off-hours. Fifth, wider spreads and lower depth after hours are described as directly measuring the cost of immediacy and therefore relating to TLP (Borjigin et al., 7 Oct 2025).

The empirical program is framed around event studies on cross-listed stocks and futures, especially around earnings and news shocks, measuring how these premiums widen overnight and converge at the open. Reported quantitative illustrations include nightly TLP often around a few tenths of a percent, and a histogram of nightly TLP over 2018–2023 with mean/median 0.23% / 0.18%, 95th/99th percentiles 0.9% / 1.8%, and one illustrated stressed night reaching about 3% TLP during an earnings crash. The backtests and figures include term-structure curves, capital-efficiency versus tail-risk trade-offs, and time-liquidity heatmaps. These results support the claim that TLP is usually small in normal conditions, rises with volatility and with longer closures, and becomes nonlinear for weekends, holidays, or stressed events (Borjigin et al., 7 Oct 2025).

4. Dynamic control through collateral policy and protocol design

A central contribution of the time-bound stablecoin framework is to use LTV as a policy instrument to control TLP. The policy intuition is directional and explicit. If the stablecoin trades below peg, TLP is positive and too high, so the protocol should reduce LTV to restrict supply and make the stablecoin scarcer. If the stablecoin trades at or above parity, the protocol can raise LTV to improve capital efficiency. The control problem is framed with a target range $$\min(S_c,\;S_o) \;=\; S_c \;-\; \max(0,\;S_c - S_o),$$2 and a target level $$\min(S_c,\;S_o) \;=\; S_c \;-\; \max(0,\;S_c - S_o),$$3, together with a linear feedback rule that lowers LTV when observed TLP exceeds target and raises it when observed TLP is below target (Borjigin et al., 7 Oct 2025).

The risk limit is also written in default-probability terms: $$\min(S_c,\;S_o) \;=\; S_c \;-\; \max(0,\;S_c - S_o),$$4 and, under lognormal assumptions, the paper gives a maximum-feasible LTV of the form

$$\min(S_c,\;S_o) \;=\; S_c \;-\; \max(0,\;S_c - S_o),$$5

This threshold is intended to ensure that default probability stays below a target $$\min(S_c,\;S_o) \;=\; S_c \;-\; \max(0,\;S_c - S_o),$$6. The authors compare the mechanism to a central bank adjusting interest rates to defend a peg; here the defended parity is the stablecoin’s relation to the underlying close price.

Protocol design implications follow from this control logic. The pricing corridor should contain market prices, persistent discounts signal too much risk or too little supply, and persistent premia signal that issuance can be expanded safely. Liquidations must be robust because the underlying market is closed during the stablecoin’s life; suggested mechanisms include on-chain auctions, possibly hybridized with off-chain liquidation through brokers. Oracles are critical because the protocol must anchor to official close prices and later official open prices. The paper also states that rare extreme overnight shocks can exceed collateral buffers, so a treasury or insurance fund is prudent, and that market adoption may reduce TLP over time by arbitraging away some of the premia the instrument exploits (Borjigin et al., 7 Oct 2025).

5. Related concepts in adjacent literatures

Several literatures provide close analogues to TLP without always pricing “time itself” in the same primitive way. They differ in whether the premium is attached to delayed liquidation, trading frictions, execution timing, contract discounting, or asset-supply and self-insurance mechanisms.

Literature	Core object	Relation to TLP
Random liquidation time in portfolio choice (Bordag et al., 2014)	Utility is weighted by the survival function $$\min(S_c,\;S_o) \;=\; S_c \;-\; \max(0,\;S_c - S_o),$$7	Delayed liquidation creates a time-based liquidity discount
Small transaction costs (Gerhold et al., 2011)	$$\min(S_c,\;S_o) \;=\; S_c \;-\; \max(0,\;S_c - S_o),$$8 and $$\min(S_c,\;S_o) \;=\; S_c \;-\; \max(0,\;S_c - S_o),$$9	Liquidity has an intertemporal welfare cost tied to rebalancing over time
Execution horizon model (Darby, 2021)	$$\text{Stablecoin value at issuance} \;=\; S_c \;-\; P(S_c),$$0, $$\text{Stablecoin value at issuance} \;=\; S_c \;-\; P(S_c),$$1	Liquidity premium is treated as a time premium in execution
CDS liquidity modeling (Brigo et al., 2010)	$$\text{Stablecoin value at issuance} \;=\; S_c \;-\; P(S_c),$$2	Illiquidity enters as an additional discount rate accumulating over time
Government debt liquidity (Cantore et al., 29 Jan 2025)	Return difference between illiquid capital and liquid public debt	Liquidity premium reflects both self-insurance demand and policy-driven supply

In the Merton-style illiquid-asset problem, the value of the illiquid asset depends not only on its paper price but also on when it can be turned into liquid wealth. The random-horizon objective

$$\text{Stablecoin value at issuance} \;=\; S_c \;-\; P(S_c),$$3

shows that liquidation uncertainty behaves like a time-dependent discount factor; the paper states that the illiquid asset contributes only through an effective extra liquid wealth term $$\text{Stablecoin value at issuance} \;=\; S_c \;-\; P(S_c),$$4, not its full paper value $$\text{Stablecoin value at issuance} \;=\; S_c \;-\; P(S_c),$$5 (Bordag et al., 2014). This is an implicit TLP or time-based liquidity discount.

In the transaction-cost literature, the liquidity premium is the amount of expected excess return the investor would give up to avoid transaction costs, and the first-order relation between liquidity premium and turnover is universal in the constant $$\text{Stablecoin value at issuance} \;=\; S_c \;-\; P(S_c),$$6 (Gerhold et al., 2011). This is not a closed-market TLP, but it formalizes the cost of preserving flexibility in time through ongoing rebalancing.

The execution literature makes the time interpretation fully explicit. The “Time Is Money” model equates liquidity premium to the value of a hypothetical at-the-money half-straddle under Bachelier dynamics, so the equilibrium trading horizon $$\text{Stablecoin value at issuance} \;=\; S_c \;-\; P(S_c),$$7 is the point at which variance risk equals the cost of liquidity (Darby, 2021). In the CDS literature, the most direct TLP-like mechanism is the liquidity discount factor $$\text{Stablecoin value at issuance} \;=\; S_c \;-\; P(S_c),$$8, which treats illiquidity as an extra discount rate on future cash flows (Brigo et al., 2010). In the HANK model of government debt, the liquidity premium is the return difference between illiquid capital and liquid public debt, with two competing forces—the self-insurance demand channel and the policy-driven supply channel—moving the premium in opposite directions (Cantore et al., 29 Jan 2025).

Two additional literatures clarify the boundaries of the concept. The equilibrium model for the cross-section of liquidity premia studies return adjustments induced by quadratic trading frictions and states explicitly that it is not about a TLP in the sense of pricing time itself; its liquidity premia are cross-sectional asset premia generated by costly portfolio adjustment and endogenous equilibrium price impact (Muhle-Karbe et al., 2020). The decentralized-exchange literature on just-in-time liquidity likewise does not introduce a formal variable called TLP, but it endogenously generates a time-based premium because a JIT LP can observe pending swaps in the public mempool, deposit liquidity only for the profitable block, collect a pro-rata share of the trading fee, and withdraw immediately after the swap (Capponi et al., 2023). This suggests that TLP can also arise as a timing rent from transient liquidity provision.

6. Conceptual boundaries, limitations, and open issues

The principal conceptual boundary is that not every liquidity premium is a Liquidity-of-Time Premium. In the time-bound stablecoin framework, time itself is the primitive friction: a closed market prevents contemporaneous access to the underlying, and the premium is attached to bridging that interval. In other settings, the premium may instead derive from quadratic transaction costs, proportional bid-ask spreads, cross-sectional differences in market depth, or equilibrium portfolio adjustment. The literature therefore supports a family resemblance rather than a single universal definition (Muhle-Karbe et al., 2020).

Several implementation and measurement limitations are explicit. For time-bound stablecoins, the pricing model is simplified and relies on idealized lognormal overnight dynamics; empirical TLP is indirect because the actual SSS market is not yet live; and real-world implementation depends on custody, legal/regulatory constraints, and oracle integrity. Extreme discontinuities, liquidity crises, and market closures longer than modeled could generate losses beyond the clean formulas, and static formulas may not fully capture feedback effects from behavior, market making, and protocol usage (Borjigin et al., 7 Oct 2025). In the execution setting, the simplest closed form assumes short horizons, ignores interest rates in most formulas, and assumes a risk-neutral trader absent bias or exogenous convexity (Darby, 2021). In CDS pricing, the survey concludes that there is no fully satisfactory contract-level measure yet, especially when credit-liquidity correlation is present (Brigo et al., 2010).

A second recurring issue is the distinction between private and social value. In just-in-time DEX liquidity, short-duration liquidity has an endogenous premium because timing and information allow fee extraction while avoiding toxic flow, yet the same paper shows that more providers can sometimes mean less liquidity when JIT provision crowds out passive LPs; proposed mitigations are a two-tiered fee structure and competition à la Cournot (Capponi et al., 2023). A plausible implication is that time-sensitive liquidity can be privately valuable while destabilizing the longer-horizon liquidity base.

Across these literatures, the durable common idea is that time-to-access is economically priced. Whether encoded as an embedded put between close and open, a survival-function discount from delayed liquidation, a turnover-linked welfare wedge, an execution-horizon indifference condition, an illiquidity discount rate, or a return spread between illiquid and liquid assets, TLP identifies the fact that liquidity is not only a quantity or a spread, but also a temporal state variable.