Research · October 23, 2026

Realized Variance in Prediction Markets

A yes-or-no contract has to move a predictable amount before it settles. This piece explains why from scratch, with no statistics assumed, and then measures how much 5,658 Kalshi contracts moved.

Vertex Point Capital · about 15 minutes

Contracts that pay one dollar or nothing

An event contract pays $1 if something happens and $0 if it does not. “Will the high temperature in Chicago reach 80°F tomorrow?” “Will the Fed cut rates in September?” Until the answer is known the contract trades at a price between 0 and 100 cents. A price of 63¢ means the market, taken as a whole, treats the event as about 63% likely.[1]

News moves the price. A warm morning pushes the temperature contract up. A soft jobs report pushes the rate-cut contract up. By the deadline the price has to land on exactly $1 or exactly $0. This piece asks how much a price should move on its way there. The question has a precise answer, and it needs one idea from statistics. We build that idea, variance, from nothing.

Average and distance from the average

Suppose your train was 3, 5, 6, 8 and 13 minutes late on five days. The average delay is the total divided by the number of days: 35 minutes over 5 days, or 7 minutes. No single day was exactly 7 minutes late. Each day sits some distance from the average: the first day 4 minutes below it, the last day 6 minutes above it.

We want one number for how spread out the days are. Averaging the distances fails because the days below the average cancel the days above it, and the result is always zero. Variance fixes this by squaring each distance first.

Variance as an area

Squaring a number is the same as drawing a square with that side and measuring its area. A day 6 minutes above average becomes a square 6 minutes on a side, with an area of 36 square minutes. A day 1 minute away becomes a square of area 1. Variance is the average area of these squares. For the five days the squares are 16, 4, 1, 1 and 36, and their average is 11.6 square minutes.

Two things follow from the picture. Every square has a positive area, and nothing cancels. A large miss counts far more than a small one: the 6-minute day contributes 36 times as much as a 1-minute day. The side of a square whose area equals the variance is the standard deviation, a typical distance from the average in the original units. Here it is about 3.4 minutes.

Drag a dot, or focus it and use the arrow keys.

Squares: 16.0 + 4.0 + 1.0 + 1.0 + 36.0Average square (variance): 11.6 square minutesSide of that square: 3.4 minutes

Figure 1. Variance as the average area of squares. Each dot is a day; each square is built on that day’s distance from the average. Drag the dots and watch the squares, the average and the variance change together. The outlined square has the average area. Its side is the standard deviation.

Variance of a yes-or-no contract

A contract has only two possible endings: $1 or $0. Suppose it trades at 63¢ and the price is a fair forecast: the event happens 63% of the time. On average the contract then pays 63¢, its price. If the event happens, the payout of $1 lands 37¢ above that average, and its square is 0.37 × 0.37 = 0.137. If it does not, the payout of $0 lands 63¢ below, and its square is 0.63 × 0.63 = 0.397.

Variance is the average square, weighted by how often each ending happens: 63% of the time the small square, 37% of the time the big one. That average works out to 0.233, exactly 0.63 × 0.37. The same thing happens at every price, and it gives the one formula this piece needs. For a contract priced at $p$ dollars,

\text{variance} = p\,(1-p).

(1)

In pictures, the variance is the area of a rectangle $p$ wide and $1-p$ tall inside a square one dollar on a side. The rectangle is largest at 50¢, where its area is 0.25. It shrinks to zero as the price approaches either end. A contract priced at 99¢ has almost no variance left. Its outcome is close to decided. Prices here are in dollars and variance is in squared dollars. No contract can have more than 0.25.

Price63¢

If yes, the dollar lands 0.37 above the price; square 0.137, weight 0.63If no, zero lands 0.63 below; square 0.397, weight 0.37Weighted average: 0.233 = 0.63 × 0.37

Figure 2. Variance of a yes-or-no contract. Left: the rectangle with sides equal to the price and to one minus the price; its area is the variance. Right: the same area at every price, an arch that peaks at 0.25 at 50¢. Move the slider to change the price.

Prices that move as news arrives

Real prices do not sit still until the deadline. Each piece of news moves them a little. Take the price at one moment and again a little later. The change between the two looks is a move. Square every move and add up the squares. The total is the realized variance: the variance that has already shown up in the price. Quantitative traders measure realized variance in stocks and currencies the same way.[2][3]

Figure 3 runs one simulated contract through 40 news updates. Each step adds a square to the pile of realized variance. Watch the bar under the chart as you step through it.

Realized and remaining variance

At any moment the rectangle from Figure 2, drawn at the current price, measures how much variance is still to come. Call it the remaining variance; traders sometimes say unrealized. At the start nothing has been realized and everything remains. At settlement the price is $1 or $0, the rectangle has collapsed to nothing, and everything has been realized.

A ball rolling down a hill is a useful picture. As it rolls it trades height for speed. Height is stored, or potential, energy. Speed is kinetic energy. Their total stays the same. Remaining variance is the height. Realized variance is the speed the ball has picked up. A contract starts with a fixed amount of stored variance, the rectangle at its first price, and by the deadline all of it has turned into price moves.

The picture fits on average, and only on average. A single ball conserves its energy exactly. A single contract can move more or less than its starting rectangle. Average over many contracts and the total comes back to the starting amount.

Starting price50¢

Each move squared, at true size:

Realized so far 0.100Remaining 0.173 = 0.78 × 0.22Total 0.273 against a start of 0.250

Figure 3. One simulated contract. Top: the price after each news update. Middle: each move squared, at true size; the newest square is darker. Bottom: realized variance so far (blue) and remaining variance at the current price (orange), against the starting rectangle (black mark). Press Play, or step through with Next update.

Suppose every price is a fair forecast: on average the next price equals the current one. Then realized plus remaining variance stays at the starting rectangle on average at every moment. By the deadline, when nothing remains, the realized variance of a contract that started at price $p$ averages exactly

\text{average realized variance} = p\,(1-p).

(2)

Two features make this useful. The rule needs no model of the event, and weather, elections and interest rates all obey it. It also does not depend on how often you look at the price. Checking every minute or every hour gives the same average, because the many small squares of frequent checks add up to the few large squares of rare ones. Augenblick and Rabin proved the same result for beliefs in general, built a statistical test on it and applied the test to the beliefs of individuals, algorithms and markets.[4] The proof takes a few lines and sits in the section for readers who want the math, below.

Contracts:Starting price50¢

realized, averageremaining, averagerealized + remaining, averagestarting rectangle 0.250

At settlement, average realized 0.242 against a start of 0.250Thin gray lines: realized + remaining for single contracts

Figure 4. Many simulated contracts, all starting at the same price. Blue: realized variance so far, averaged. Orange: remaining variance, averaged. Black: their sum, which stays on the slate starting line. Thin gray lines: realized plus remaining for single contracts, which wander. Switch between 1 and 1,000 contracts to see the average settle.

The rule turns into a test of a market. Add up the realized variance of many contracts and divide by the sum of their starting rectangles. We call this the variance ratio. A ratio of 1 is what fair prices produce. A ratio above 1 means prices moved more than fair prices would. They swung toward one outcome and back, or sat closer to $0 or $1 than the odds justified and had to jump a long way at the end. A ratio below 1 means prices moved less: they sat too close to the middle and reached the outcome late.

Measuring it on Kalshi

We took every contract in four Kalshi groups that closed between July 1 and September 30, 2026 and settled yes or no. The data are Kalshi’s public market lists and minute-by-minute price history.[5]

Weather: daily high temperature in seven cities, 3,845 contracts.
Crypto: hourly bitcoin and ether price levels, a fixed one-in-200 sample, 255 contracts.
Sports: baseball and WNBA game winners, one side per game, 1,364 contracts.
Economics: Fed decisions, inflation, payrolls, GDP and unemployment, 194 contracts.

For the price we use the midpoint between the best offer to buy (the bid) and the best offer to sell (the ask), minute by minute. Each contract starts at the first minute both offers sit within 10¢ of each other; 5,658 of the 10,120 listed contracts reach that point, and nearly all of the rest are hourly crypto contracts with strikes far from the price. Each contract is followed over at most its last 30 days and ends at its settled value of $1 or $0. Figure 5 shows one contract from each group. The black line in the lower panel is realized plus remaining variance: it starts at the rectangle and ends at the contract’s realized variance.

KXHIGHCHI-26JUN30-B96.5settled nopredicted 0.192, realized 0.175

realized so farremainingrealized + remainingstarting rectangle 0.192

Figure 5. One real contract from each group: among contracts that start between 25¢ and 75¢, the one with the median trading volume. Top: the price. Bottom: realized variance so far (blue), remaining variance (orange) and their sum (black), against the starting rectangle (slate). Use the buttons to switch groups; hover or use the arrow keys for values.

How often to check the price

In theory the checking interval does not matter. In a real market it does. Between trades the midpoint jitters as offers are posted and pulled, and a check every minute records that jitter as real moves. The extra squares inflate realized variance. Plotting realized variance against the checking interval shows the effect directly. Traders call the chart a volatility signature plot; volatility is their word for the standard deviation of price moves.[6][7][8]

WeatherCryptoSportsEconomicsfair-price prediction

Show the numbers as a table

Vertical	1 min	5 min	15 min	1 h	4 h	1 day
Weather	1.00 (0.94 to 1.05)	0.99 (0.93 to 1.04)	0.96 (0.90 to 1.02)	0.92 (0.87 to 0.98)	0.93 (0.88 to 0.98)	0.94 (0.90 to 0.97)
Crypto	0.96 (0.75 to 1.18)	0.91 (0.71 to 1.14)	0.93 (0.70 to 1.19)	0.91 (0.69 to 1.15)	0.90 (0.68 to 1.13)	0.88 (0.66 to 1.13)
Sports	0.96 (0.91 to 1.00)	0.95 (0.90 to 0.99)	0.95 (0.90 to 0.99)	0.97 (0.92 to 1.02)	0.97 (0.94 to 1.01)	0.98 (0.94 to 1.01)
Economics	2.56 (2.09 to 3.06)	1.96 (1.53 to 2.33)	1.77 (1.35 to 2.12)	1.60 (1.23 to 1.97)	1.42 (1.03 to 1.80)	1.26 (0.81 to 1.67)
All	1.02 (0.98 to 1.06)	0.99 (0.95 to 1.04)	0.97 (0.94 to 1.01)	0.96 (0.92 to 0.99)	0.96 (0.93 to 1.00)	0.96 (0.93 to 0.99)

Figure 6. Variance ratio, realized over predicted, against the time between price checks, with 95% ranges from resampling whole events. The slate line at 1 is what fair prices produce. Press a group’s button to hide or show it; hiding economics rescales the chart to the other three. Hover or use the arrow keys for values; the table lists every number.

Checked every five minutes, the ratio across all 5,658 contracts is 0.99, with a 95% range of 0.95 to 1.04. Taken together, Kalshi prices moved almost exactly as much as fair prices must. By group the ratio is 0.99 for weather, 0.95 for sports, 0.91 for crypto and 1.96 for economics.

Economics is the exception, and the signature plot shows why. Its ratio is 2.56 with checks every minute and falls steadily to 1.26 with daily checks, where the 95% range (0.81 to 1.67) includes 1. These contracts trade for up to 30 days, and a check every minute counts tens of thousands of small jitters between bid and ask. Weather and sports stay close to 1 at every interval. Crypto contracts last an hour and few of them have tight quotes. The crypto range is therefore wide.

Variance near the deadline

The rule says how much a price moves in total. It says nothing about when. Figure 7 shows when it happens: the share of each contract group’s realized variance already done at each point before the close. Across all contracts, 31% of the realized variance arrives in the last hour, and 15% comes from the final jump to $1 or $0 alone.

The groups differ sharply. In economics, 51% of the realized variance is the final jump: these contracts stop trading shortly before the data release or rate decision that settles them, and the price waits for the number. Sports contracts realize 48% of their variance in the last hour, as games end. Weather contracts realize 87% within the last day and 16% in the final jump. Hourly crypto contracts live for one hour, and nearly all of their movement falls inside it.

WeatherCryptoSportsEconomics

Show the numbers as a table

Vertical	Settlement jump	Last hour	Last day	Last week
Weather	16.1%	16.1%	87.4%	100.0%
Crypto	19.0%	89.3%	98.8%	100.0%
Sports	6.9%	47.7%	96.6%	100.0%
Economics	50.7%	51.3%	54.5%	62.4%
All	14.5%	31.2%	89.4%	98.1%

Figure 7. Share of each group’s realized variance already done at each point before the close, checked every five minutes; time runs left to right toward settlement. A line that stays low and jumps at the right edge means the movement is concentrated at the deadline. The table separates the final jump to the settled value.

Summary

Variance is the average area of squares built on distances from the average.
A yes-or-no contract priced at $p$ has variance $p(1-p)$ , the area of a rectangle.
Realized variance adds up the squared price moves. Remaining variance is the rectangle at today’s price. On average their sum stays fixed. A contract with fair prices therefore realizes its starting rectangle by the deadline.
On Kalshi, checked every five minutes, 5,658 contracts realized 0.99 times their starting rectangles. Weather, sports and crypto sit between 0.91 and 0.99. Economics realized 1.96 times, and most of the excess is bid-ask jitter that builds up over thousands of checks.
Movement crowds toward the deadline: 31% of all realized variance arrives in the last hour.

For readers who want the math

This section restates the article in the language of probability and measure. Realized variance becomes a measure on the time axis and remaining variance a supermartingale whose compensator is that measure. The squares of Figure 3 become the corner squares of an exact area decomposition. The last two results explain why movement is measured with squares: the graph of a fair price has infinite length and zero area.

Prices as conditional probabilities

Let $(\Omega,\mathcal F,\mathbb P)$ have a filtration $(\mathcal F_t)_{t\in[t_0,T]}$ for the information available over time, and let $Y\in\{0,1\}$ be the payoff, measurable with respect to $\mathcal F_T$ . A fair price is the conditional probability $p_t=\mathbb E[Y\mid\mathcal F_t]$ . Such a price is a martingale with values in $[0,1]$ that ends at $p_T=Y$ .[9] We take its right-continuous version with left limits. That version allows jumps such as a data release.[10] The remaining variance is $V_t=p_t(1-p_t)$ , the conditional variance of $Y$ given $\mathcal F_t$ . Every statement below conditions on $\mathcal F_{t_0}$ . By optional sampling it applies unchanged when $t_0$ is a stopping time, such as the first minute with a tight quote.

Realized variance as a measure on time

Definition 1 (realized-variance measure)

For a partition $\pi=\{t_0<t_1<\dots<t_n=T\}$ with increments $\Delta_k=p_{t_k}-p_{t_{k-1}}$ , the realized-variance measure is the finite Borel measure on $(t_0,T]$

\mu_\pi=\sum_{k=1}^{n}\Delta_k^{2}\,\delta_{t_k},

(3)

where $\delta_t$ is the unit point mass at $t$ . Its total mass $\mathrm{RV}_\pi=\mu_\pi((t_0,T])$ is the realized variance.

Each check places the area of its square at the time of the check. Figure 7 plots the cumulative distribution of this measure, summed over contracts.

Proposition 1 (expected mass of an interval)

For checking times $s<t$ in $\pi$ ,

\mathbb E\bigl[\mu_\pi((s,t])\,\big|\,\mathcal F_s\bigr]=V_s-\mathbb E\bigl[V_t\,\big|\,\mathcal F_s\bigr].

(4)

The realized variance expected over a stretch of time equals the remaining variance expected to disappear over it, whatever the checking times in between.

Proof.

Martingale increments are orthogonal. For checking times $s\le t_{j-1}<t_j\le t_{k-1}<t_k$ ,

\begin{aligned}\mathbb E[\Delta_j\Delta_k\mid\mathcal F_s]&=\mathbb E\bigl[\Delta_j\,\mathbb E[\Delta_k\mid\mathcal F_{t_{k-1}}]\,\big|\,\mathcal F_s\bigr]\\&=0.\end{aligned}

Expanding the square of $p_t-p_s=\sum_{s<t_k\le t}\Delta_k$ therefore leaves only the diagonal terms:

\begin{aligned}\mathbb E[\mu_\pi((s,t])\mid\mathcal F_s]&=\mathbb E[(p_t-p_s)^2\mid\mathcal F_s]\\&=\mathbb E[p_t^2\mid\mathcal F_s]-p_s^2.\end{aligned}

Since $\mathbb E[p_t\mid\mathcal F_s]=p_s$ , the right side of (4) is

\begin{aligned}p_s-p_s^2-\mathbb E[p_t-p_t^2\mid\mathcal F_s]&=\mathbb E[p_t^2\mid\mathcal F_s]\\&\quad-p_s^2 .\end{aligned}

∎

Corollary 2 (realized plus remaining)

$\mathbb E[\mathrm{RV}_\pi\mid\mathcal F_{t_0}]=V_{t_0}$ , and for every checking time $t$ , $\mathbb E\bigl[\mu_\pi((t_0,t])+V_t\,\big|\,\mathcal F_{t_0}\bigr]=V_{t_0}$ .

Proof.

Take $s=t_0$ in (4). For the first claim take $t=T$ , where $V_T=Y(1-Y)=0$ .

∎

The first claim is equation (2) of the main text and the second is the conservation in Figure 4. Proposition 1 also fixes the expected measure. The function $F(t)=\mathbb E[V_{t_0}-V_t]$ is nondecreasing by (4), and it is right-continuous because $p$ is right-continuous and bounded. Its Lebesgue–Stieltjes measure $\nu$ on $(t_0,T]$ has total mass $\mathbb E[V_{t_0}]$ , and every partition satisfies $\mathbb E\,\mu_\pi=\nu$ on the intervals it generates. Figure 6 tests the total mass of $\nu$ . Figure 7 estimates its shape in time-to-close coordinates.

Remaining variance and its compensator

Proposition 2 (Doob–Meyer decomposition of remaining variance)

Let $[p]$ be the quadratic variation of $p$ . For $t\in[t_0,T]$ ,

\begin{aligned}V_t&=V_{t_0}+\int_{(t_0,t]}\bigl(1-2p_{s-}\bigr)\,dp_s\\&\quad-\bigl([p]_t-[p]_{t_0}\bigr),\end{aligned}

(5)

and the stochastic integral is a martingale. Remaining variance is a supermartingale, and its compensator, the increasing process that makes $V+[p]$ a martingale, is the quadratic variation.

Proof.

Apply Itô’s formula for semimartingales with jumps to $f(x)=x-x^2$ .[10] The second derivative is the constant $-2$ , and the continuous second-order term is $-([p]^c_t-[p]^c_{t_0})$ . Each jump contributes exactly

\begin{aligned}&f(p_s)-f(p_{s-})-f'(p_{s-})\,\Delta p_s\\&\qquad=-(\Delta p_s)^2,\end{aligned}

and the two together are $-([p]_t-[p]_{t_0})$ . The integrand $1-2p_{s-}$ is predictable and bounded by 1, and $p$ is a square-integrable martingale. The integral is therefore a martingale.

∎

The sums $\sum_k\Delta_k^2$ over $(t_0,t]$ converge to $[p]_t-[p]_{t_0}$ uniformly in $t$ , in probability, as the mesh of $\pi$ goes to zero.[10] Along a subsequence of partitions the measures $\mu_\pi$ then converge weakly to $d[p]$ path by path. Föllmer showed that Itô’s formula holds for each such path without probability.[11] Realized variance is the discrete measure and quadratic variation is its limit.

Lemma 3 (pathwise excess)

For any path that ends at 0 or 1, whether or not prices are fair,

\mathrm{RV}_\pi-V_{t_0}=\sum_{k=1}^{n}\bigl(1-2p_{t_{k-1}}\bigr)\Delta_k.

(6)

Proof.

Write $\Delta_k^2=p_{t_k}^2-p_{t_{k-1}}^2-2p_{t_{k-1}}\Delta_k$ and sum over $k$ :

\mathrm{RV}_\pi=Y^2-p_{t_0}^2-2\sum_k p_{t_{k-1}}\Delta_k .

Then use $Y^2=Y$ and $Y-p_{t_0}=\sum_k\Delta_k$ .

∎

Equation (6) is the left-point Riemann sum of (5) from $t_0$ to $T$ . Its right side adds up each move, signed by whether it went toward 50¢ or away from it and weighted by the distance of the price from 50¢. Under fair prices each term has mean zero. Positive excess means that price steps lean toward 50¢, the pattern of a market that overshoots and corrects. Negative excess means steps lean toward the nearer end. The study checks (6) on every contract at every checking interval as an exact identity.

Realized variance as an area

The squares of Figure 3 are exact. Draw the square $[0,p_{t_k}]^2$ for each checking time. Between consecutive squares lies an L-shaped region, a gnomon, of signed area $p_{t_k}^2-p_{t_{k-1}}^2$ .

Proposition 3 (gnomon decomposition)

Each gnomon splits into two rectangles of area $p_{t_{k-1}}\Delta_k$ and one corner square of area $\Delta_k^2$ . Summed over the path,

Y^{2}-p_{t_0}^{2}=2\sum_{k=1}^{n}p_{t_{k-1}}\Delta_k+\sum_{k=1}^{n}\Delta_k^{2},

(7)

and realized variance is the total area of the corner squares.

Proof.

The square of side $p_{t_{k-1}}+\Delta_k$ consists of the square of side $p_{t_{k-1}}$ , two rectangles $p_{t_{k-1}}$ by $\Delta_k$ and a square of side $\Delta_k$ :

(p_{t_{k-1}}+\Delta_k)^2=p_{t_{k-1}}^2+2p_{t_{k-1}}\Delta_k+\Delta_k^2 .

For a step down $\Delta_k<0$ and the rectangles count with a negative sign. Summing over $k$ telescopes the left side to $Y^2-p_{t_0}^2$ .

∎

Corollary 4 (left points and trapezoids)

For every partition,

\begin{aligned}&\sum_{k=1}^{n}\tfrac12\bigl(p_{t_{k-1}}+p_{t_k}\bigr)\Delta_k-\sum_{k=1}^{n}p_{t_{k-1}}\Delta_k\\&\qquad=\tfrac12\,\mathrm{RV}_\pi .\end{aligned}

(8)

Proof.

Term by term, $\tfrac12(p_{t_{k-1}}+p_{t_k})\Delta_k-p_{t_{k-1}}\Delta_k=\tfrac12\Delta_k^2$ .

∎

Both sums approximate the area $\int x\,dx$ under the diagonal of the plane between $p_{t_0}$ and $Y$ . The trapezoid rule is exact for a straight line and gives $\tfrac12(Y^2-p_{t_0}^2)$ . The left-point staircase falls short by one right triangle of area $\tfrac12\Delta_k^2$ per step, up or down, and realized variance is twice the area it misses. In the continuous limit the left-point sum is the Itô integral and the trapezoid sum the Stratonovich integral. They differ by half the quadratic variation.[10]

Why squares and not lengths

Proposition 4 (no finite length)

Suppose $p$ is continuous on $[s,t]\subset[t_0,T)$ . On the event $[p]_t>[p]_s$ , the sums $\sum_k|\Delta_k|$ over partitions of $[s,t]$ tend to infinity in probability as the mesh goes to zero, and the graph $\{(u,p_u):u\in[s,t]\}$ has infinite length. Its one-dimensional Hausdorff measure $\mathcal H^1$ is infinite.

Proof.

For every partition of $[s,t]$ ,

\sum_k\Delta_k^2\le\Bigl(\max_k|\Delta_k|\Bigr)\sum_k|\Delta_k| .

The maximum tends to zero because $p$ is uniformly continuous on $[s,t]$ , while the left side tends to $[p]_t-[p]_s>0$ in probability. Hence $\sum_k|\Delta_k|\to\infty$ in probability. The polygon through the points $(t_k,p_{t_k})$ has length at least $\sum_k|\Delta_k|$ . The graph therefore has length above every bound with probability tending to one, and its length is infinite almost surely on the event. The graph is the image of the continuous injective curve $u\mapsto(u,p_u)$ , and for such a curve the length equals $\mathcal H^1$ of the image.[12]

∎

Proposition 5 (zero area)

If $p$ is Borel measurable on $[s,t]$ , as every right-continuous path is, its graph $G=\{(u,p_u):u\in[s,t]\}$ has planar Lebesgue measure zero, and its two-dimensional Hausdorff measure $\mathcal H^2(G)$ is zero.

Proof.

The graph of a Borel function is a Borel set. Each vertical line $\{u\}\times\mathbb R$ meets $G$ in the single point $(u,p_u)$ , and a single point has one-dimensional Lebesgue measure zero. By Fubini’s theorem the planar Lebesgue measure of $G$ is the integral of these slice measures over $[s,t]$ , and that integral is zero. On the plane $\mathcal H^2$ is a constant multiple of Lebesgue measure.

∎

The graph of a fair price therefore has infinite length and zero area, and neither measures how much the price moves. Quadratic variation, the limit of the total area of the squares, is finite and positive whenever the price moves at all, and realized variance estimates it.

Jitter

Suppose the observed midpoint is $q_k=p_{t_k}+e_k$ , with jitter $e_k$ independent of the price and of each other, of mean zero and variance $\eta^2$ , and the settled value is observed exactly. Then

\begin{aligned}\mathbb E\sum_{k=1}^{n}(q_k-q_{k-1})^2&=p_{t_0}(1-p_{t_0})\\&\quad+(2n-1)\,\eta^2,\end{aligned}

(9)

which grows with the number of checks $n$ . That growth produces the rise at short intervals in a signature plot.[7][8]

Estimator and selection

For contract $i$ the start $t_0^{(i)}$ is the first minute in its last 30 days with both offers posted and at most 10¢ apart. That rule uses only information available at the time, and $t_0^{(i)}$ is therefore a stopping time. Between quotes the last midpoint stands. The ratio is

\mathrm{VR}_\Delta=\frac{\sum_i \mathrm{RV}_i^{(\Delta)}}{\sum_i p^{(i)}_{t_0}\bigl(1-p^{(i)}_{t_0}\bigr)},

(10)

with checks every $\Delta$ minutes. Ranges are 2.5% and 97.5% quantiles of 2,000 bootstrap draws that resample whole events, since the temperature bins of one day, or the strikes of one hour, settle on the same outcome. The midpoint is a price with fees and a bid-ask spread around it, and a buyer may demand a premium. A ratio away from 1 can reflect those as well as over- or underreaction.

References

J. Wolfers and E. Zitzewitz, “Prediction Markets,” Journal of Economic Perspectives 18(2), 107–126, 2004. doi:10.1257/0895330041371321
T. G. Andersen, T. Bollerslev, F. X. Diebold and P. Labys, “The Distribution of Realized Exchange Rate Volatility,” Journal of the American Statistical Association 96(453), 42–55, 2001. doi:10.1198/016214501750332965
O. E. Barndorff-Nielsen and N. Shephard, “Econometric Analysis of Realized Volatility and its Use in Estimating Stochastic Volatility Models,” Journal of the Royal Statistical Society Series B 64(2), 253–280, 2002. doi:10.1111/1467-9868.00336
N. Augenblick and M. Rabin, “Belief Movement, Uncertainty Reduction, and Rational Updating,” Quarterly Journal of Economics 136(2), 933–985, 2021. doi:10.1093/qje/qjaa043
Kalshi, Trade API v2 documentation: market lists and one-minute candlesticks, public endpoints. docs.kalshi.com
T. G. Andersen, T. Bollerslev, F. X. Diebold and P. Labys, “Great Realisations,” Risk 13, 105–108, 2000. Kellogg listing
L. Zhang, P. A. Mykland and Y. Aït-Sahalia, “A Tale of Two Time Scales,” Journal of the American Statistical Association 100(472), 1394–1411, 2005. doi:10.1198/016214505000000169
P. R. Hansen and A. Lunde, “Realized Variance and Market Microstructure Noise,” Journal of Business & Economic Statistics 24(2), 127–161, 2006. doi:10.1198/073500106000000071
D. Williams, Probability with Martingales, Cambridge University Press, 1991. doi:10.1017/CBO9780511813658
P. E. Protter, Stochastic Integration and Differential Equations, 2nd ed., Springer, 2005. doi:10.1007/978-3-662-10061-5
H. Föllmer, “Calcul d’Itô sans probabilités,” Séminaire de Probabilités XV, Lecture Notes in Mathematics 850, 143–150, Springer, 1981. doi:10.1007/BFb0088364
E. Le Donne, Metric Lie Groups: Carnot-Carathéodory Spaces from the Homogeneous Viewpoint, book project, 2024, Proposition 2.1.40. arXiv:2410.07291