# Volatility term structure

**ADV-22 · Advanced Options · Core**

**Lesson question:** How much event risk belongs to each maturity?

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:** OPT-F01, OPT-F02, OPT-F03, OPT-F04

Term structure describes IV across maturities under a comparable strike or moneyness convention. Calendars and diagonals buy and sell different portions of that structure. A high front IV can reflect a near event rather than an inexpensive short-volatility opportunity.

Calendar means different expiries at the same strike; changing strikes as well creates a diagonal.

## Understand the mechanism

Volatility itself is not additive through time. Under a simple integrated-variance comparison, sigma squared times T is the relevant total quantity. A high short-dated annualized IV can still represent less total variance than a lower longer-dated IV.

Forward variance calculations require matched conventions and do not guarantee the future realized volatility.

## Walk through the process

Identify where events fall and compare total variance. Stress front and back expiries independently. A long calendar can be damaged if back-month IV falls or spot leaves the area where its structure is useful.

Do not use a same-expiry terminal payoff engine to value a still-live far leg.

## Model and implementation boundary

These are complete fictional reading examples, not live opportunities. Terminal identities assume the stated common settlement. Multi-expiry values require a surviving-leg model. Costs, assignment and financing can alter realized outcomes; a mathematical bound alone does not guarantee the same operational loss path.

## A worked example

Fictional seven-day IV is 60% and thirty-day IV 35%. Total variance is 0.006904110 and 0.010068493. The implied forward variance rate over the remaining 23 days is (0.010068493−0.006904110)/(23/365)=0.050217391.

Its square root is about 22.409% annualized. This is a model-consistent comparison, not a promise about the next 23 days. The graph shows total variance rather than averaging the two quoted IVs.

## Calculation popup: Forward variance comparison

```text
v_forward=(sigma2²T2−sigma1²T1)/(T2−T1)
```

Total variance, not quoted IV, is additive under the declared comparison.

1. T1=7/365; sigma1=.60; total variance=0.006904109589.

2. T2=30/365; sigma2=.35; total variance=0.010068493151.

3. Forward variance rate=0.050217391304; square root=0.224092372258.

**Result:** Approximately22.409% annualized forward IV under assumptions.

## Graph: Volatility term structure—worked comparison

**Horizontal:** Calendar days remaining  
**Vertical:** Total variance (annual decimal volatility squared × years)

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

Complete numerical data are embedded in `scripts/advanced-options/ADV-22.js` and displayed as an accessible table in the HTML lesson.

## What would invalidate the reading?

A trader averages IV60 and35 directly or describes a calendar as having a fixed maximum payoff at the first expiry. Both confuse distinct time concepts.

## Check your understanding

### What quantity is subtracted to estimate the forward variance here?

**Answer:** Total variance sigma squared times maturity.

**Why:** Then divide by the remaining interval before taking a square root.

### What is the critical failure to avoid?

**Answer:** A trader averages IV60 and35 directly or describes a calendar as having a fixed maximum payoff at the first expiry. Both confuse distinct time concepts.

**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: How much event risk belongs to each maturity? 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 Option Calendar Spreads](https://www.cmegroup.com/education/courses/option-strategies/option-calendar-spreads)

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