# Expiration and concentrated Gamma

**DL-11 · Dealer Logic · Core**

**Lesson question:** Why is the near-strike region special near expiry?

Original fictional examples; educational writing, not live market data, orders or production approval.

## By the end

- Explain the mechanism in the lesson question.
- Reproduce the worked comparison and recognize its failure case.

**Prerequisites:** DL-01

As time runs down, finishing in or out of the money becomes increasingly sensitive to small price changes near the strike. The hedge ratio can change rapidly there. Far from the strike, the same option may instead become almost stock-like or nearly irrelevant.

## Understand the mechanism

ATM Gamma can rise as remaining time approaches zero in the smooth model. At exact expiry the payoff has a kink at strike; the ordinary smooth Gamma is not a normal finite number there.

Expiring inventory can disappear, be exercised or be replaced. Tomorrow's book is not simply today's book with a clock moved forward.

## Walk through the process

Separate mathematical concentration from settlement operations. Identify exercise style, settlement method and the consequences of short options near strike. Decide whether the hedge can be managed under realistic liquidity.

Gross 0DTE activity is not proof of net dealer short Gamma. Offsetting flows can leave a different retained inventory.

## Evidence boundary

The arithmetic is original conditional teaching material, not a measurement of current dealer positions. The cited Cboe research is historical product-specific evidence, not a live inventory feed. Model outputs should remain distinguishable from observed market facts.

## A worked example

Hold S=K=100 and sigma=25%, with zero rate and yield. Calculate Gamma at 30, 10, 3 and 1 day. The graph increases toward shorter time near this strike.

It deliberately stops above zero. A zero-time value is intrinsic, but the smooth derivative formulas need explicit boundary handling rather than an undisclosed tiny-time substitution.

## Graph: Expiration and concentrated Gamma—worked comparison

**Horizontal:** Calendar days remaining  
**Vertical:** ATM Gamma per USD

Fictional data from the written example. The chart organizes the comparison; its shape is not a forecast.

Complete numerical data are embedded in `scripts/dealer-logic/DL-11.js` and displayed as an accessible table in the HTML lesson.

## What would invalidate the reading?

Keeping yesterday's Delta near expiry can leave a rapidly changing exposure. Flooring exact zero time silently can also create misleading boundary outputs.

## Check your understanding

### Should smooth Gamma at exact expiry and strike be labeled an ordinary finite value?

**Answer:** No.

**Why:** The payoff kink requires explicit boundary treatment.

### What is the critical failure to avoid?

**Answer:** Keeping yesterday's Delta near expiry can leave a rapidly changing exposure. Flooring exact zero time silently can also create misleading boundary outputs.

**Why:** Explain this risk before treating the example as a decision rule. A correct calculation does not guarantee a favorable market outcome. **Critical risk check.**

## Takeaway

Return to the question: Why is the near-strike region special near expiry? State the observed facts, the assumptions and the failure condition separately.

## References and provenance

The cited references support definitions and conventions. The teaching prose, tickets and numerical scenarios are original; they are not attributed as historical market observations.

- [OIC Greeks](https://www.optionseducation.org/advancedconcepts/understanding-options-greeks)

- [Cboe 0DTE positioning study, May 2025](https://www.cboe.com/insights/posts/0-dt-es-decoded-positioning-trends-and-market-impact)