One share at a time

1Introduction

Fractional shares changed who could own a stock. They did not change the stock; they changed the unit, and when the unit became small enough the question of who could participate answered itself. Options are the next instrument waiting for the same change, and the evidence in this paper is that the wait is nearly over.

The paper proceeds in nine parts. Section 2 shows that options are the instrument retail traders have chosen, and in what form. Section 3 shows where the premium in options flows and why retail is on the wrong side of it. Section 4 measures the listed contract against the balance of a real retail account and the rules that sit on top of the arithmetic. Section 5 reviews the three US experiments with small contracts and distinguishes the one that worked. Section 6 describes the three rails that now exist: on-chain perpetual futures on equities, dollar stablecoins, and markets that do not close. Section 7 turns to the investors outside the United States for whom the listed options market is unavailable. Section 8 shows that the option is the natural hedge for the perpetual future, with the liquidation record of October 2025 as evidence of demand. Section 9 states what a one-share option on a 24/7 underlying changes and what it does not. Section 10 describes the design that follows and what remains undone.

Three claims are made that we believe are new in combination. First, that the economic case for the small option contract is a case about selling, not buying: the income strategies that carry the volatility risk premium are the ones the 100-share unit closes. Second, that the history of small contracts is a history of distribution, not demand: the Micro E-mini succeeded because it arrived with a hedge, 23-hour trading and every futures broker, and the mini options and Nanos failed because they arrived with the unit alone. Third, that the perpetual future on a stock is the first underlying that makes a 24/7 option hedgeable at every hour, which is the condition every prior 24/7 options proposal has skipped.

2Options are the instrument retail chose

US listed options volume sat between 3.9 and 5.2 billion contracts a year from 2010 to 2019. It has more than tripled since: 7.5 billion in 2020, 9.9 billion in 2021, 10.4 billion in 2022, 11.1 billion in 2023, 12.2 billion in 2024, and 15.2 billion in 2025, up 24% in a year and the sixth consecutive record [2, 3]. Figure 1 shows the series. Single-stock options were 54% of 2025 volume, ETF options 37% and index options 8%.

The composition of the growth matters more than its size. Within S&P 500 index options, same-day-expiry (0DTE) contracts rose from about 5% of volume in 2016 to 62% in August 2025 and 63% in February 2026 [4, 5] (Figure 2), and Cboe estimates that retail traders account for 50 to 60% of that flow.

Cboe's analysis of the positioning behind it answers the question of why retail prefers options to the leveraged linear products available at the same brokers: over 95% of 0DTE trades are in a defined-risk format, long options outright or spreads, where the maximum loss is known at entry; only about 4% are naked short options [6]. Retail traders have had margin, leveraged ETFs and futures alongside options for decades. The growth went to options, and within options to the shortest-dated, most defined-risk form. The retail trader is choosing an instrument whose downside is capped by construction, which no linear product offers.

The brokers' accounts say the same. Robinhood earned $399 million from options in the third quarter of 2026, up 31% year on year and its largest transaction revenue line, with contracts per trader up 66% [7]. Table 1 collects the figures.

3The seller's side

Every option has two sides and the premium flows one way. Implied volatility runs above realised volatility most of the time: the buyer pays for insurance against a move, and the seller collects the difference between what was paid for and what occurred. Cboe measures the gap directly. Since 2013 the difference between the VIX and the subsequent 30-day realised volatility of the S&P 500 has averaged 3.33 volatility points [9].

Over nearly four decades the simplest selling strategy has done what Figure 3 and Table 2 show. The Cboe S&P 500 PutWrite Index (PUT), which sells one at-the-money one-month put each month fully collateralised in Treasury bills, returned 9.40% a year from July 1986 to August 2023 against 9.91% for the S&P 500, with an annualised standard deviation of 10.26% against 15.38%, a maximum drawdown of 32.7% against 51.0%, and a longest drawdown of 30 months against 66 [9]. The buy-write index (BXM, covered calls) tells the same story over the same decades.

This is not a secret. It is the reason institutions, pension funds and the market makers on the other side of retail flow sell options for a living. Retail does the opposite. In the Journal of Finance study of the 2019 to 2021 retail boom, 69% of retail option volume was in calls, half of wholesaler trades were in contracts with under a week to expiry, and retail option trades lost more than $4 billion after costs between November 2019 and June 2021 [10]. The losses went to the bid-ask spread and to the sellers on the other side.

The same pattern appears wherever individual traders meet short-dated options. The Securities and Exchange Board of India studied all individual traders in equity futures and options over the three financial years to March 2024, about ten million people. Ninety-three percent lost money, an aggregate net loss above Rs 1.8 lakh crore (about $21 billion) and an average of about Rs 2 lakh per losing trader, costs included; in the final year proprietary desks and foreign institutions made gross profits of Rs 33,000 crore and Rs 28,000 crore, 96% and 97% of it through algorithmic trading [11]. The counterparties to the retail buyer were, overwhelmingly, professional sellers.

Why does retail buy when selling carries the premium? Partly behaviour: the lottery-like payoff of a cheap call is attractive in a way that a steady premium is not. Mostly, however, it is the contract. A trader cannot sell what they cannot cover. Table 3 sets the requirement to sell the two income strategies on one listed contract against the same requirement for a one-share contract.

At 100 shares, covering means holding $20,400 of one stock; on Eli Lilly at $1,419 it means $141,900. The two income strategies that carry the volatility risk premium are unavailable to any account that cannot post that, which, as the next section shows, is most of them. The 100-share unit does not merely make options expensive for small accounts. It sorts small accounts onto the buying side, where the expected return is negative, and keeps them off the selling side, where it is positive.

4The account and the rules

Robinhood reported 28.5 million funded customers and $355 billion of platform assets in July 2026, a mean of about $12,500 per funded account [8]. The mean is pulled upward by a small number of large accounts; the company does not disclose a median, and third-party estimates place it well below the mean. We use $12,500 as a generous benchmark. For the United States as a whole, the Federal Reserve's 2022 Survey of Consumer Finances found that 21% of families held stocks directly, with a median direct holding of $15,000 [12].

Against that account, Table 4 gives the capital one position requires under the listed contract and under a one-share contract. One NVDA covered call needs 163% of the mean account in a single stock. A diversified account that caps any name at 10% needs $204,000 before it can write one covered call on NVDA. One at-the-money NVDA call expiring in January 2028, fifteen months out, was offered on Iridium's chain at about 20% of spot on 31 October: about $4,100 per contract, a third of the account, so the position is undiversifiable by construction.

The constraint can be stated formally. A trader with capital $W$ and a risk budget $\rho W$ for a view, facing an option priced at $p$ per share with multiplier $m$, can hold at most

contracts. The trader is priced out entirely when $\rho W < m p$, and otherwise bears a rounding error in realised risk of up to $mp/W$ of capital. With $m = 100$, $p = \$6$ and $\rho = 1\%$, the trader is priced out below $W = \$60{,}000$. With $m = 1$ the threshold is $600, and the sizing error is bounded by $p/W$, under 0.05% at the benchmark account. Figure 4 draws the same arithmetic; its sliders try other accounts.

The rules that sit on top of the arithmetic, in the United States, are listed in Table 5. Most of them are loosening. The pattern day trader rule, which since 2001 required $25,000 of equity to day trade on margin, is being removed: the SEC approved FINRA's replacement on 14 April 2026, substituting intraday margin standards for the dollar floor, with an eighteen-month transition [13]. Brokers' approval tiers remain, and so does the one rule no regulator can change: a covered call needs 100 shares and a cash-secured put needs the cash for 100 shares, per contract, because the contract says so.

5Small contracts: three experiments

The small option contract is not a new idea. It has been listed three times in the United States, and the record is precise enough to say what distinguished the one that worked.

5.12013: the mini options

On 18 March 2013 the US options exchanges (CBOE and C2, ISE, NYSE Amex and Arca, Nasdaq, BATS, MIAX) listed options with a 10-share multiplier on five high-priced names: AAPL, AMZN, GOOG, GLD and SPY [14]. Six months later the minis were 0.5% of SPY's standard-contract volume and 3.8% of AMZN's and GOOG's. On the September 2013 expiry the standard SPY 170 call was quoted $0.65 bid at $0.67 offered with 20,830 contracts traded; the mini was quoted $0.56 at $0.76 with 369 traded, ten times the spread for one-tenth of the exposure [15]. Commissions were charged per contract, so a mini cost ten times as much per share to trade as the standard. Market makers had no fungibility between the mini and the standard contract, quoted wide, volume stayed low, and the exchanges delisted the minis through 2014 [16].

5.22019: the Micro E-mini

On 6 May 2019 CME Group listed futures at one-tenth the size of its E-mini contracts on the S&P 500, Nasdaq-100, Russell 2000 and Dow. They traded a million contracts in their first four days [17]. By 12 November 2021 they had traded a billion, at an average of 2,060,344 contracts a day in 2021, across 240,000 unique accounts and 340 firms, with 21% of volume outside US trading hours; Micro E-mini options followed in August 2020 [18]. CME describes it as one of the most successful launches in its history.

5.32022: the Nanos

On 14 March 2022 Cboe listed Nanos, a cash-settled S&P 500 index option with a multiplier of 1, about one-hundredth of the notional of a standard SPX contract, with weekly expiries, aimed explicitly at retail [19]. The product is still listed. On 28 October 2026 it traded 29 contracts and carried 158 contracts of open interest [20].

5.4What distinguished the one that worked

Figure 5 puts the three on one axis. The difference was not the idea; it was what arrived with it (Table 6). The Micro E-mini was a tenth of a contract that already had the deepest book in the world, quoted by the same market makers who hedged it one-for-ten against the E-mini, so its spread was tight from the first minute. It traded 23 hours a day, and a fifth of its volume came from outside US hours. It arrived in the year retail brokerage commissions went to zero and the year before the largest retail trading wave in history, on the index that every futures broker already distributed. The minis of 2013 arrived with none of that: a separate book with no linkage to the standard contract, per-contract commissions that made them ten times as expensive per share, five names, partial broker support, and a retail options market a quarter of its present size, four years before fractional shares and nine years before daily expiries. The Nanos arrived in a commission-free world, but as an index-only product on a weekly ladder, without a broker push, in the year retail attention went to single stocks and to 0DTE spreads on the full-size contract.

The lesson is specific. A small contract works when it arrives with its liquidity, its hours and its distribution at once, on the thing people already trade. It fails when the unit is the only thing that changed. None of the 2013 conditions hold now: the retail options market is three and a half times larger, commissions are zero, fractional shares are standard at every broker, daily expiries exist on the most traded underlyings, and, as the next section describes, there is now an underlying that trades at every hour and a settlement layer whose cost does not scale with notional.

6The rails

Three things that did not exist in 2013, or in 2019, now do.

6.1An underlying that trades at every hour

Perpetual futures have been the retail derivative of the on-chain market since 2020: a single instrument per underlying, no expiry, funding payments that tie it to spot, leverage in one click. In 2025 on-chain perpetual venues traded about $7.9 trillion, almost three times their 2024 volume, and the on-chain share of global perpetual volume tripled from 2.5% to 7.8% in a year [25]. The largest venue, Hyperliquid, traded $1.29 trillion in the first half of 2026 with $9.1 billion of open interest, 54% of all on-chain perpetual open interest and 10% of all crypto perpetual open interest; 1.73 million addresses have traded on it and about 221,000 are active in a month [22].

In October 2025 Hyperliquid opened its deployment layer, HIP-3, to equities, indices and commodities. Figure 6 shows the first twelve months: $0.8 billion in October 2025, $25 billion in January 2026, $69 billion in March, $115 billion in July, the record, $61 billion in September, and about $665 billion cumulative to 19 October 2026 [21]. Open interest reached $4.4 billion in August and stood at $3.9 billion in October, 57% of it in stocks and pre-IPO names, 21% in indices and ETFs and 21% in commodities, across 162 live markets. The record day was $9.16 billion on 29 July 2026, 68% of all Hyperliquid perpetual volume that day; the SpaceX perpetual traded $1.4 billion on the day of its IPO [24]. Sixty percent of equity-perpetual volume trades outside US market hours [23], though only 5% of stock-perpetual volume trades at weekends against the 29% that an even spread over the week would imply, which is to say that the perpetual trades when there is news, not merely when it can.

For the first time there is a liquid instrument on NVDA, on the S&P 500, on gold and on a company that has not gone public that trades, and can be traded against, at three in the morning on a Sunday. An option built on it inherits the hours.

6.2A dollar that settles anywhere in minutes

A trader in Seoul, São Paulo or Lagos who wants a dollar position in a US stock needs, in the listed world, a US brokerage account, a wire, a tax form, and a broker willing to take the account. In the stablecoin world they need a wallet or a sign-in. Table 7 and Figure 7 give the size of that base: about $312 billion of dollar stablecoins outstanding in June 2026, $184 billion of it Tether's USDT and $74 billion Circle's USDC; $1.79 trillion of adjusted on-chain transfer volume in June 2026 alone, 125% more than a year earlier [26]. The GENIUS Act of July 2025 gave payment stablecoins a federal framework in the United States, with compliance enforceable from 2027 [27]. Hyperliquid alone holds about $5.4 billion of USDC as trading collateral [22]. The collateral for a fractional option is already in the hands of the people the listed market does not reach.

6.3Markets that do not close

Crypto has always traded 24/7. US equities are following, slowly, and the record of that transition is itself evidence about where the demand for off-hours trading went (Table 8). Overnight trading of US stocks, available since 2021 through the Blue Ocean alternative trading system and since 2023 through Robinhood's 24/5 offering, was under 0.11% of US share volume in 2025 [28]. The first SEC-approved 23-hour exchange, 24X National Exchange, opened in October 2025 [29]; Nasdaq has announced 24-hour weekday trading for the second half of 2026; NYSE Arca plans 22 hours a day; DTCC has targeted 24-by-5 clearing for the second quarter of 2026 [28]. On 19 October 2026 Cboe opened two extra sessions, 7:30 to 9:25 and 16:00 to 16:15 ET, for options on 24 stocks and 5 ETFs [30]. Tokenised stocks traded $13.7 billion on decentralised exchanges in the twelve months to mid-2026, and Robinhood's own chain went live in July 2026 [31].

Figure 8 puts two numbers side by side. Overnight trading is 0.11% of US stock volume; 60% of equity-perpetual volume is outside US hours. The demand for trading when the news lands did not wait for the exchanges. It went where the liquidity was. The first panel gives the reason an option market must follow it: an option can be quoted at any hour, but it can only be hedged when the instrument it references is open. The hedge for a listed single-stock option is open 32.5 hours a week. The hedge for an index ETF option quoted around the clock, index futures, is open about 115 hours and closed from Friday evening to Sunday evening. The hedge for an option on a perpetual future is the perpetual itself, open 168 hours. Section 8 and Appendix A give the arithmetic of why that matters to the price.

7A market without a country

The listed options market is American. It requires an American brokerage account, it keeps American hours, and the 100-share contract is denominated in the prices of American stocks. The people who want it increasingly are not American, and the walls between them and it are of three kinds: regulatory, operational and the unit.

The clearest case is Korea. Korean investors held $200.4 billion of US stocks on 26 May 2026, up 22% from $163.6 billion at the end of 2025, from $44.2 billion at the end of 2022 and about $37 billion in the pandemic years (Figure 9) [32, 33]. A fifth of a trillion dollars of US equity is now owned by households whose waking hours are New York's night. Korea's own options market is built around the KOSPI 200 index; a Korean saver who wants to write a covered call on a US holding needs a US options account, a broker that offers one, and New York's hours. Samsung Electronics and SK hynix, the two largest Korean stocks, now have perpetual futures too, and with them the first single-stock option chains on those names that trade around the clock.

Table 9 gives the other large cases. India has 210 million dematerialised securities accounts [34], and its central bank's remittance scheme does not permit remittances for margins or margin calls to overseas exchanges, so overseas derivatives are closed to residents [35]. EU retail investors have been unable to buy US-domiciled ETFs such as SPY and QQQ since the PRIIPs regulation of 2018, because US issuers do not produce the required key information document [36]. Chinese residents reach overseas securities only through quota schemes. Everywhere else, the question is whether a US broker will open the account, and then the hours and the unit are the same as for everyone.

The SEBI figures in Section 3 belong here too: ten million individuals traded index options in a single country over three years, and nine in ten lost. The demand for options outside the United States is not hypothetical. It is already the largest options market in the world by contract count, and it is served almost entirely by index products, in local hours, on terms that the data say are not working for the people trading them.

8The option as the hedge for the perpetual

The perpetual future is a good product. It has one instrument per underlying, no expiry, no strike grid and leverage in a click. What it cannot do is cap a loss. A long perpetual at leverage $L$ is liquidated when the underlying falls by about $1/L$, and because that is a barrier rather than a terminal condition, what matters is whether the price touches the level at any time before the horizon, not where it finishes. For a driftless lognormal price with volatility $\sigma$ over horizon $T$, the reflection principle gives

which ignores funding, fees and the maintenance buffer, all of which make it worse. Figure 10 evaluates it at 5×, 10× and 20× and across the range; its horizon can be changed.

The record supplies the empirical version. On Friday 10 October 2025 a single tariff headline liquidated $19.3 billion of positions across the crypto market, 1.66 million traders, 87% of them long; $10.3 billion of it, 53%, was on Hyperliquid, and Bitcoin fell 14.5% in the day [37]. The previous single-day record was about $2 billion. The traders did not need to be wrong about the year. They were wrong about an afternoon.

A put caps the loss at the premium plus the distance to the strike, however far the price falls, and it does so as a terminal condition rather than a barrier: the position survives the afternoon. The two instruments are not substitutes. Every payoff a perpetual can deliver, a pair of options can deliver (a long call and a short put at the same strike is a forward, and a perpetual is a forward rolled continuously), and the converse is false: no linear position reproduces a straddle. Options span perpetuals; perpetuals do not span options.

The mechanism that turns this into a product is cross-margining. A margin rule that charges a portfolio's worst loss, rather than each leg's, is subadditive: the collateral for a long perpetual plus a long put is at most the pair's maximum loss, which is the premium plus the distance from entry to strike, while the perpetual alone posts margin against a fraction of its full notional. On a venue where both instruments settle to one ledger, the hedged pair is cheaper to hold than the perpetual by itself. That is the only durable reason for a trader to hold the hedge, and for a market maker to carry the inventory that makes the hedge available. A trader with 1.66 million counterparts liquidated in an afternoon is the clearest statement of demand for a defined-risk instrument that trades at the hour the headline arrives.

Appendix A gives the market maker's side of the same argument: a hedger's error variance scales with the sum of squared rebalancing intervals, so a hedger who can trade through a weekend carries a fraction of the risk of one who must take it as a single 66-hour gap, and that reduction is available only when the hedge instrument trades. The perpetual is the first underlying on stocks for which it does.

9What a one-share option changes, and what it does not

9.1What it changes

1. The seller's side opens. A $12,500 account can write a covered call on one share of NVDA, or a cash-secured put, and collect the premium that Section 3 shows the institutions collect, sized at 1.6% of the account instead of 163%. At a premium of about 1.5% of spot a month for a call 5% out of the money, a 10% NVDA sleeve of $1,250, six shares, earns about $18 a month: small in dollars, identical in yield to what a $204,000 account earns on the listed contract.
2. Risk is sized by the budget, not the contract. By equation (1), a $125 risk budget buys $125 of premium, or runs a spread at $125 of risk. The defined-risk format that 95% of 0DTE traders already choose becomes available at the size a sensible rule would set.
3. The perpetual trader gets a cap. The put that turns a liquidation into a bounded loss sits in the same account, margined with the perpetual, at any hour.
4. The market is not a country. The collateral is a stablecoin, the sign-in is an account, the hours are all of them. A Korean saver, a Brazilian trader and a European who cannot buy SPY see the same chain.
5. Private companies get a surface. Anthropic and OpenAI have perpetual futures; an option chain on them is the first market-implied distribution of a private company's value that anyone can trade. No listed market can offer it.

9.2What it does not change

1. Buying short-dated options is, on average, a losing trade, in the United States ($4 billion of retail losses in twenty months [10]) and in India (Figure 11). A smaller contract does not change the expectancy of a bad strategy. It changes who can run the good ones. The honest statement is that the premium is on the selling side, selling has tail risk, and the right size for either is a fraction of the account; the small contract makes that size reachable.
2. Options are not simpler than perpetuals. They are safer in one specific way: the loss is known at entry. The venue's job is to make the ticket say what that loss is, in a broker's terms, and not to teach anyone to trade.
3. A perpetual is not the stock. An option on a perpetual settles on the perpetual, which can trade away from the stock, most of all at night. That is also the reason it can be hedged at night, and the trade-off should be stated to every user before the first order.
4. A venue has responsibilities the contract does not discharge. No uncovered short options for anyone, since the naked short is the one format whose loss is unbounded; published results from any house that quotes; and real money only where it is permitted.

10Design implications and the venue

The evidence above fixes most of the design. The contract is one share, cash-settled and European, with the listed market's expiry ladder from dailies to LEAPS, so that a trader's existing mental model transfers. The reference price is the perpetual future, because it is the one instrument on a stock that trades, and can be hedged, at every hour; settlement is on a time-weighted average of the perpetual's mark price over the final thirty minutes before expiry, so that no single print decides a contract. The perpetuals trade in the same account, cross-margined with the options, so that the hedge is priced as a hedge. Short options are covered or capped only. The collateral is a dollar stablecoin and the sign-in is an ordinary account, because the chain is the rails, not the interface.

Liquidity is the open question for any new exchange, and an options venue has the hard version of it: one perpetual per underlying, hundreds of series on each. A central order book with no flow needs paid market makers from the first minute, and their quotes need a hedge. The history in Section 5 says the unit must arrive with its liquidity. Our answer is a dealer first: the house quotes both sides of every series from the first minute, priced from the perpetual and hedged in it, with spreads set in volatility points that widen with risk, and the house's results after hedging published. A dealer is not the long-term model; it is the way to open with a full chain rather than an empty one. Resting orders, a central order book and outside market makers follow once there is flow to fill them.

Iridium is that venue. It is live in public test with mock collateral, every hour of the week, on 54 markets and about 35,000 series: 39 stocks, ETFs and indices (2 of them Korean, on the Korea Exchange calendar), 6 commodities, 6 crypto assets and 3 pre-IPO names. Real-money trading comes only where it is permitted; dollar-sized orders, the order book and outside market makers follow in that order.

11Conclusion

The small option contract has been tried three times. This time it comes with its hours, its liquidity and its rails. The demand is documented: fifteen billion contracts a year in the United States, ten million individual derivatives traders in India alone, a fifth of a trillion dollars of US stock in Korean hands, 1.66 million perpetual positions liquidated in one afternoon, 60% of equity-perpetual volume outside the hours the listed market keeps. The premium is documented too, and it flows to the side of the market that the 100-share unit reserves for those who already have $20,000 to post per contract. One share at a time is not a smaller version of the same product. It is the version of the product that the people trading it can use.

AThe hedger's side of the hours

A market maker who sells an option takes no view on direction. They hedge direction away in the underlying and are left with a position on how much the price moves between one rebalance and the next. Write $V(S,t)$ for the option value, $\Delta = V_S$, $\Gamma = V_{SS}$ and $\Theta = V_t$. A hedger long the option and short $\Delta_i$ units of the underlying, rebalancing only at times $t_i$, has a profit over $[t_i, t_{i+1}]$ equal to the change in option value less the hedge's. Expanding to second order, the first-order term is cancelled by the hedge, and with the Black–Scholes relation $\Theta = -\tfrac12\sigma^2 S^2 \Gamma$ (zero rates) the remainder is

the gap between realised and implied variance over the interval, scaled by dollar gamma [38, 39]. Its expectation is zero when the sold volatility is right. If returns are Gaussian, $\pi_i = \tfrac12\Gamma_i S_i^2\sigma^2\Delta t_i\,(Z_i^2-1)$ with independent standard normal $Z_i$, so $\operatorname{Var}(\pi_i) = \tfrac12\left(\Gamma_i S_i^2 \sigma^2 \Delta t_i\right)^2$ and, over a horizon with $\Gamma$ and $S$ roughly constant,

Total time $\sum_i \Delta t_i = T$ is fixed by the calendar; $\sum_i \Delta t_i^2$ is set by how often the hedger can trade. A 66-hour weekend taken as one interval gives $\sum \Delta t^2 = 66^2 = 4{,}356$ hour²; the same weekend as 66 hourly intervals gives 66. If the weekend's variance arrives evenly, the hedger who can rebalance hourly carries one sixty-sixth of the variance and about one-eighth of the standard deviation; if it arrives in a burst, the reduction is smaller but still positive, since a sum of squares never exceeds the square of the sum. With time-varying variance the sum becomes $\sum_i (\sigma_i^2 \Delta t_i)^2$, and the comparison should use the weekend's actual variance, which for stock perpetuals is a fraction of a weekday's.

The condition that is usually skipped: an interval exists in equation (4) only if the hedge instrument trades in it. For a listed single-stock option the hedge is the stock, open 32.5 hours a week. For an option on a stock quoted around the clock, the hedge between sessions is a proxy (index futures for an index ETF, nothing liquid for a single stock), and at the weekend there is none. For an option on a perpetual future, the hedge is the perpetual, on the same book, at every hour. The reduction in (4) is a reduction in the market maker's cost of carrying short-dated inventory, which is the one durable reason to quote tighter on one venue than another, and it is available only where the underlying trades.

BData notes

• OCC annual volumes for 2010 to 2019 are from OCC historical releases, rounded to 0.01 billion; 2024 is derived from the 2025 total and the stated 24.4% increase.
• The $12,500 mean account is $355 billion of platform assets over 28.5 million funded customers (Robinhood, July 2026). The median is not disclosed.
• The long-dated NVDA call in Section 4 and Table 4 is the January 2028 at-the-money call's ask on Iridium on 31 October 2026, about 20% of spot. With the live feed, Table 4 shows the current ask on the long-dated expiry nearest eighteen months.
• Hours per week for index futures (about 115) are CME equity index futures hours, Sunday 18:00 to Friday 17:00 ET with a daily one-hour break; overnight equity sessions on alternative trading systems add thin hours for single stocks and none at weekends.
• The 2025 on-chain perpetual volume (about $7.9 trillion, almost three times 2024's) is DefiLlama data as reported in [25].
• The liquidation probabilities use equation (2) with $T$ in years (30/365 by default) and no drift, funding, fees or maintenance buffer; Figure 10 computes them rather than quoting them.
• Rupee conversions use Rs 85 per dollar.
• Prices of NVDA ($204) and LLY ($1,419) in the text are perpetual-future marks on 31 October 2026. Tables 3 and 4 and the live check in Section 9 use the marks and quotes at the time you read them.