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Term structure through total and forward variance

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Start with the idea

Variance accumulates over time; comparing annualized volatilities alone can hide the amount priced between two maturities.

Symbols, units & horizon
  • T₁
  • σ(T): compatible annualized implied volatility fraction
  • w(T): dimensionless total implied variance
  • f₁₂: average annualized implied variance over the interval, fraction squared per year

When and why to use this

Compare maturity exposures and identify basic term-structure inconsistencies.

Variance accumulates over time; comparing annualized volatilities alone can hide the amount priced between two maturities.

For a compatible total-variance curve, subtract the shorter maturity’s total variance from the longer and divide by the interval length. This is an implied forward-variance accounting calculation; it is not a guaranteed future realized variance.

For simple surface diagnostics, hold the moneyness convention fixed and state rate/dividend assumptions. Comparing different strikes or inconsistent forwards can manufacture apparent calendar violations. Full absence of arbitrage requires more than this elementary nonnegativity check.

w(T)=σ(T)2T,f12=w(T2)−w(T1)T2−T1
Total-variance accounting and implied interval construction

Term structure through total and forward variance

  1. Square each volatility and multiply by its own years to maturity.
  2. Subtract shorter total variance from longer total variance.
  3. Divide by the year difference; only take a square root if forward variance is nonnegative.
Work it by hand

At T₁=.5, σ₁=.2, total variance is .02. At T₂=1, σ₂=.25, total variance is .0625. Forward variance=.0425/.5=.085, volatility≈.291548.

Apply it in a strategy

  • Freeze inputs at the stated decision time and record their units.
  • Compare maturity exposures and identify basic term-structure inconsistencies.
  • Recompute the example, then change the material assumption and explain the difference.

Research deliverable

Term structure through total and forward variance: produce the worked calculation, a timestamped input record and a written decision addressing this limitation: Different moneyness conventions or a malformed surface make the inferred interval variance misleading.

These are synthetic mechanics examples, not historical performance or paper replications. Module evidence and research boundaries record the 12 September 2026 review.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

# Python 3.10+; standard library and NumPy only.
# Synthetic teaching inputs; conventions and units are defined in the notation above.
from math import sqrt

def forward_variance(short_vol,short_time,long_vol,long_time):
    if min(short_vol,long_vol)<0 or not 0<=short_time<long_time: raise ValueError('Invalid volatilities or maturities')
    value=(long_vol**2*long_time-short_vol**2*short_time)/(long_time-short_time)
    return value,None if value<0 else sqrt(value)

assert abs(forward_variance(.2,.5,.25,1)[0]-.085)<1e-12
print(forward_variance(.2,.5,.25,1))

Continue learning

Volatility: Measurement, Surfaces & Variance Risk — all lessons
  1. Realized variance starts with squared returns
  2. EWMA as a causal variance baseline
  3. A multi-horizon realized-variance forecast
  4. Implied volatility is a model inversion
  5. Term structure through total and forward variance
  6. Strike convexity and a butterfly consistency check
  7. Vega requires a volatility-unit convention
  8. Variance exposure and the difference from arbitrage

Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations