---
title: "How the options math works"
description: "The one primer for the whole suite: how dealer hedging moves price, the greeks and GEX math, and how dealer inventory is estimated from the trade tape."
---

# How the options math works

Every indicator in the options suite is built from the same three ingredients: the live options board (every listed contract with its open interest and greeks), a small amount of standard options math, and an answer to who is actually holding each position: estimated from real trade flow where the evidence supports it, and a stated assumption where it does not. This page explains each once, so the indicator pages can stay short.

## Why dealer hedging moves price

When you buy a call option, a market maker (a "dealer") usually sells it to you. The dealer does not want directional risk, so they hedge: they buy some of the underlying. How much they buy is the option's **delta**. As price moves, delta changes, and the rate of that change is **gamma**. Gamma is what forces dealers to keep trading:

- **Dealers long gamma** (they own options): as price rises their delta grows too long, so they sell into strength; as price falls they buy weakness. Their hedging **dampens** moves. Price tends to grind and mean-revert.
- **Dealers short gamma** (they are short options): the same logic flips. They must buy strength and sell weakness, **amplifying** moves. Price tends to trend and overshoot.

![How dealer hedging dampens or amplifies price moves](/learn/options/dealer-hedging-loop.svg)

Knowing which regime you are in, and where it flips, is the single most useful thing options data tells a spot or futures trader.

## The greeks we use

The venue publishes delta, gamma, and vega directly for every live contract. Wherever a greek is not published, or the suite is reconstructing a historical day (the heatmap's per-day metrics are rebuilt from stored implied volatility, not live greeks), it comes from one Black-Scholes evaluation instead. Using $S$ for the underlying price, $K$ for strike, $\sigma$ for implied volatility as a fraction (0.65 for 65%), and $T$ for time to expiry in years:

$$d_1 = \frac{\ln(S/K) + \tfrac{1}{2}\sigma^2 T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}$$

$$\Delta = \begin{cases} N(d_1) & \text{call} \\ N(d_1) - 1 & \text{put} \end{cases}$$

$$\Gamma = \frac{\varphi(d_1)}{S\,\sigma\sqrt{T}}$$

$$\mathrm{Vega} = \frac{S\,\varphi(d_1)\sqrt{T}}{100}$$

$\varphi$ is the standard normal density and $N$ its cumulative distribution (a standard polynomial approximation, accurate enough for exposure surfaces). Gamma comes out as delta change per dollar of underlying, matching Deribit's own convention. Vega is scaled to **per 1 IV point** (a 0.01 move in $\sigma$), the same units the venue quotes, so a modeled vega and a published vega compare directly.

**The simplification.** The model treats the risk-free rate as zero and carries no dividend term (coins do not pay one), matching how Deribit itself prices these instruments closely enough for exposure work. Checked against Deribit's own published greeks, the reconstruction runs about 7% off in aggregate, and most of that gap comes from Deribit rounding its own greeks to 5 decimal places, not from the model itself.

The same $d_1$ and $d_2$ also drive vanna (how much delta shifts per IV point) and charm (how much delta decays per day), used in the [heatmap's](/learn/options/gex-heatmap) alternative metrics. Both are held at zero inside the last 15 minutes before expiry, where $d_1$ stops meaning anything numerically.

## GEX: turning the board into dollars of hedging pressure

**Gamma exposure (GEX)** aggregates gamma across every open contract into a dollar number you can compare across strikes and days. For each contract:

$$\mathrm{GEX} = \Gamma \times \mathrm{OI} \times S^2 \times 0.01$$

where $\Gamma$ is the option's gamma, $\mathrm{OI}$ its open interest (in coin units), and $S$ the underlying price. The $S^2 \times 0.01$ term converts "delta change per dollar" into **dollars of hedging flow per 1% move**, which is the unit every GEX panel displays.

**Signing the exposure.** The formula above gives the size of each contract's hedging pressure; something still has to decide its sign, positive when dealer hedging dampens price moves, negative when it amplifies them. Nobody can observe dealers' actual inventory directly, so that sign is itself built on an estimate. The shipped default, **dealer flow**, reconstructs each contract's dealer position from the public trade tape and signs its gamma by that estimate (see [Estimating dealer inventory from flow](#estimating-dealer-inventory-from-flow) below). Where the estimate is not reliable enough for a given contract, that one contract falls back to the classic industry assumption instead: dealers are long calls and short puts (traders mostly buy puts for protection and sell calls for yield, dealers take the other side), so calls contribute positive GEX and puts negative. An explicit **Open interest** setting applies that assumption everywhere, uniformly, which is also the default most of the options-analytics industry uses, so those levels line up with what other desks are watching. Delta exposure (DEX) always keeps natural delta signs, regardless of which basis GEX uses.

Three strike-level landmarks fall out of the GEX profile:

- **Call wall**: the strike with the largest call open interest above spot. Dealer hedging tends to slow rallies there.
- **Put wall**: the same for puts below spot, often acting as support.
- **Gamma flip**: the price where cumulative net GEX crosses zero. Above it dealers dampen; below it they amplify. Crossing the flip changes the market's character, not just its level.

Walls and the flip are computed from whichever basis is active. Under **Open interest**, walls are literally the largest call or put open interest by strike, matching the definitions above. Under **dealer flow**, walls follow signed dealer gamma instead (the largest positive net GEX for the call wall, the largest negative for the put wall), and the flip is still just the zero crossing of whichever net GEX is on screen. Max pain stays open-interest-based either way; it is a settlement concept, not a gamma one.

**Max pain** is related but different: the strike where the total value paid out to option holders at expiry would be smallest,

$$\text{max pain} = \arg\min_K \sum_{\text{contracts}} \mathrm{OI} \times \text{payout}(K)$$

Price often gravitates toward it into large expiries because that is where hedging pressure and position-unwinding balance out.

## Estimating dealer inventory from flow

Under the dealer-flow basis, the suite does not assume dealers are short every position. It reconstructs each contract's dealer (maker) inventory from Deribit's own public trade tape, then signs that contract's gamma by the actual estimate, long or short, in the same GEX formula above with the estimated inventory standing in for open interest. A dealer holding a large long put position, for instance, contributes positive gamma exposure under dealer flow even though the open-interest convention would have signed it negative.

**The fold.** Every options trade carries a taker side. When the taker buys, the market maker on the other side of the trade sold, so the maker's inventory moves short; when the taker sells, the maker's inventory moves long. Each trade contributes:

$$\text{contribution} = \begin{cases} -v & \text{taker buy} \\ +v & \text{taker sell} \end{cases}$$

where $v$ is the trade's volume in coin units. Summed over every trade an instrument has seen, this running total is the estimated inventory: how many contracts dealers are net long or short.

**Block-trade downweight.** Large, privately negotiated blocks and multi-leg combos cross off-book, so the taker flag on those prints is a weak signal of real intent. A trade bucket whose volume is more than 25 times the instrument's trailing median bucket volume is treated as a probable block and contributes at reduced confidence instead of full weight (at least 4 comparable buckets are required before a bucket can even be judged this way; otherwise every bucket counts fully):

$$\text{confidence} = \begin{cases} 0.25 & v > 25 \times \tilde v_{\text{trailing}} \\ 1 & \text{otherwise} \end{cases}$$

so the fold is really a confidence-weighted sum, not a plain one.

**Open-interest reconciliation.** A dealer's position can never be larger than the instrument's own open interest, so the folded estimate is clamped:

$$\mathrm{inventory} = \begin{cases} \text{folded estimate} & |\text{folded estimate}| \le \mathrm{OI} \\ \mathrm{OI} \times \operatorname{sign}(\text{folded estimate}) & \text{otherwise} \end{cases}$$

A clamp that fires is itself informative: it means the trade tape disagrees with open interest, so the instrument's reconciliation quality, $\mathrm{OI} / |\text{folded estimate}|$ when clamped (1 otherwise), feeds directly into the coverage score below.

**Settlement.** Deribit options are European and cash-settled at 08:00 UTC on their expiry date. The moment that clock passes, the instrument's contribution zeroes out: a settled book carries no gamma.

**Coverage and the fallback gate.** Every instrument gets a coverage score between 0 and 1: how much of the instrument's life the trade tape actually covers, times reconciliation quality. An instrument first traded comfortably inside the retained tape counts as fully covered; one already trading at the tape's edge is only as covered as the tape is deep, bounded by the longest an option could realistically have been listed before its expiry (400 days, comfortably longer than Deribit lists anything). Only instruments clearing the gate use the estimate:

$$\text{coverage} \ge 0.6 \implies \text{dealer flow}, \qquad \text{coverage} < 0.6 \implies \text{open interest (fallback)}$$

This is a **per-instrument** decision, never a blend: a single contract is signed wholly by its estimated inventory or wholly by the open-interest convention (calls positive OI, puts negative OI), never something in between. An instrument with open interest but no trade history at all is fallback by definition; there is no tape to estimate from.

**The disclosed split.** Because coverage varies contract by contract, most boards run mixed: some contracts on dealer flow, others on the fallback. Every dealer-flow payload discloses exactly how mixed, weighted by each instrument's unsigned open-interest gamma notional so the split describes the board's structure rather than the estimate it is describing:

$$\text{dealer\%} = 100 \times \frac{\sum_{\text{dealer-flow instruments}} \left|\Gamma \times \mathrm{OI} \times S^2 \times 0.01\right|}{\sum_{\text{all instruments}} \left|\Gamma \times \mathrm{OI} \times S^2 \times 0.01\right|}, \qquad \text{naive\%} = 100 - \text{dealer\%}$$

That is the "Dealer flow · N% naive fallback" line stamped on the GEX profile, curve, and heatmap legends, where N is the share still running on the fallback. In the settings dialog the toggle is called **GEX basis**, with values **Dealer flow** (the default) and **Open interest** (the fallback convention, always available as an explicit opt-out); on the heatmap it only appears while the Metric is Net GEX, the only cell metric with a dealer-flow variant.

## Assumptions and limitations

The dealer-flow estimate is genuinely useful, and it is still an estimate. Read it with these in mind:

- **It is inferred, not observed.** Nobody publishes dealer books. The estimate is built entirely from public trade prints, and it can be wrong for any single instrument, which is exactly why the coverage gate and the disclosed fallback percentage exist.
- **Coverage depends on how much trade history has been retained.** The suite seeds up to 185 days of daily trade history per coin in the background, walking from the newest closed day backward. Recently listed instruments reach full coverage fastest; instruments that have traded for a while take longer, because their early history sits further back in the seeding queue. An under-covered instrument just runs on the open-interest fallback until its coverage catches up: nothing is lost in the meantime.
- **The live estimate updates on its own timer, not tick by tick.** Today's running inventory re-derives roughly every 25 seconds, tied to the same poll that refreshes the rest of the board.
- **Open interest can lag the trade tape.** The clamp uses the open interest known at the time: the latest polled snapshot for live inventory, that day's closing aggregate for history. A very recently closed day can still revise within its first hour or so as the backend backfills, so its reconciliation is provisional too.
- **Deribit only.** The estimate is built entirely from Deribit's own trade tape. Flow on other venues, and any dealer hedging happening there, is invisible to it.
- **Both product lines feed the tape.** Coin-settled (inverse) and USDC-settled (linear) options are both included, matching how open interest itself is counted everywhere else in the suite.
- **Block trades and multi-leg combos are a weak signal, not a solved problem.** The downweight reduces their influence on the estimate; it cannot recover what a negotiated block was actually for.
- **A position built up before the retained tape began is invisible to the fold**, even once seeding finishes. The open-interest clamp bounds the error, but this is what coverage protects against: an instrument in that situation should stay on the fallback rather than trust an under-counted estimate.
- **Today's inventory is live and provisional.** It is never cached and recomputes continuously through the day. It can revise as more trades print, and a UTC-day rollover discards the day's incomplete running total and refetches it whole before treating it as settled history.

## The volatility ingredients

**Implied volatility (IV)** is the market's priced-in expectation of movement, quoted in annualized percent. The suite never invents IV; it reads each contract's mark IV from the venue and combines them carefully:

- **The forward, per expiry.** Options on a December expiry hedge against December futures, not today's spot. Every at-the-money calculation anchors to that expiry's own forward price.
- **OTM stitching.** Below the forward the suite reads puts, above it calls (the out-of-the-money side of each). OTM options carry the cleanest pricing because they are the liquid side. The [IV smile](/learn/options/iv-smile) is exactly this stitched curve.

![OTM stitching: puts below the forward, calls above](/learn/options/otm-stitch.svg)

- **ATM IV** interpolates the stitched curve at the forward. It is the reference "how much vol" number per expiry.
- **25-delta risk reversal (RR25)**: $\mathrm{IV}_{25\Delta\,\mathrm{put}} - \mathrm{IV}_{25\Delta\,\mathrm{call}}$. Positive means downside options are bid over upside, in other words the market pays up for puts (protection). (Some platforms quote the same number with the opposite sign; check the sign convention before comparing.)
- **25-delta butterfly (FLY25)**: $\tfrac{1}{2}(\mathrm{IV}_{25\Delta\,\mathrm{call}} + \mathrm{IV}_{25\Delta\,\mathrm{put}}) - \mathrm{IV}_{\mathrm{ATM}}$, how expensive the wings are versus the middle, a tail-risk gauge.
- **Implied move**: what an ATM vol number means in plain terms, $\mathrm{move} = \mathrm{IV}_{\mathrm{ATM}} \times \sqrt{\mathrm{DTE}/365}$, the one-standard-deviation move priced in by that expiry.

Where a greek is not published by the venue, the suite derives it with the same Black-Scholes model described in [The greeks we use](#the-greeks-we-use) above. That covers vega, vanna (how delta shifts when vol moves), and charm (how delta decays with time) in the [heatmap's](/learn/options/gex-heatmap) alternative metrics.

## Units and conventions, stated once

- **Open interest and traded volume are in coin units** (BTC, ETH, ...), matching how Deribit quotes size. Dollar panels say "USD" explicitly and convert at the current underlying price.
- **Net GEX defaults to the dealer-flow estimate** (see [Estimating dealer inventory from flow](#estimating-dealer-inventory-from-flow)), with **Open interest** (dealer-naive: calls positive, puts negative) as the explicit fallback and opt-out. **VEX, charm, and vanna are always dealer-naive; DEX always uses natural delta signs,** regardless of which GEX basis is selected.
- **Both product lines are counted.** BTC and ETH trade inverse (coin-settled) and USDC-settled linear options; open-interest metrics sum both, while IV curves read the deeper inverse book only so mixed pricing never distorts the smile.
- **Freshness**: panels poll on their own timer, default 60 seconds (configurable 30 to 600 per indicator). The GEX heatmap is the exception; it reconstructs history and refreshes with the chart. Historical depth varies by metric and is stated on each page.

That is the whole toolbox. Every indicator page tells you what question it answers, how to read it, and exactly which of these formulas it applies.