# Linear futures versus convex options

**FUT-14 · Equity Index Futures · Core**

**Lesson question:** What remains after Delta is hedged?

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:** FUT-01

A future has linear P/L in its own quote. An option has changing Delta, time and volatility exposure. A future can offset current first-order direction without buying or selling the same convexity.

Calling a Delta-neutral option book risk-free therefore confuses one sensitivity with the entire position.

## Understand the mechanism

Gamma changes Delta as spot moves; Vanna and Charm can change it through IV and time. Futures do not automatically eliminate Vega, jumps or basis. An option on futures can also exercise into a futures position, unlike some cash-settled index options.

Product identity matters as much as the Greek number.

## Walk through the process

Identify the exact option underlying and a suitable hedge. Match units and state residual risks. Recompute after meaningful changes and consider execution costs of maintaining neutrality.

An overnight futures quote does not guarantee that every related option can be closed at a fair price at that moment.

## Operational boundary

All prices, portfolio values and account balances are fictional. Real contract calendars, broker requirements and available liquidity must be checked for the instrument actually traded. The example's mathematical hedge or payoff is not an execution guarantee.

## A worked example

An option book starts at +1,000 dollars/point Delta and is hedged with twenty short futures worth 50 dollars/point. Option Delta later rises to +1,500, while the hedge remains −1,000.

Residual is +500 dollars/point. Ten more short futures could offset that increment under matching assumptions, but Vega and further Delta changes remain.

## Graph: Linear futures versus convex options—worked comparison

**Horizontal:** Option-book Delta (USD/point)  
**Vertical:** Residual Delta (USD/point)

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

Complete numerical data are embedded in `scripts/equity-futures/FUT-14.js` and displayed as an accessible table in the HTML lesson.

## What would invalidate the reading?

A trader mistakes initial neutrality for permanent protection or uses a broad-index future as a perfect hedge for company-specific options.

## Check your understanding

### What residual remains after option Delta rises to +1,500 with the old −1,000 hedge?

**Answer:** +500 dollars per point.

**Why:** Net the current sensitivities, not the original ones.

### What is the critical failure to avoid?

**Answer:** A trader mistakes initial neutrality for permanent protection or uses a broad-index future as a perfect hedge for company-specific options.

**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: What remains after Delta is hedged? 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.

- [CME Introduction to Futures](https://www.cmegroup.com/education/courses/introduction-to-futures)

- [CME Micro E-mini overview](https://www.cmegroup.com/education/courses/micro-e-mini-futures/micro-e-mini-futures-products-overview)