Quant Academy Math · Probability · Quant Finance
Phase 15 - Lesson 15.5

The Volatility Surface: Dynamics, Arbitrage Constraints, and Calibration

The whole object a desk trades - and the no-arbitrage rules any fitted surface must obey.

⏱ 55 min● Advanced🔗 Prereqs: 15.2–15.4
↖ Phase 15 hub
Builds on: 15.2–15.4 built implied vol and models that fit slices; here we treat the whole \((K,T)\) object.
Leads to: Phase 16 turns to the microstructure and execution of the trades that hedge these books.

Learning Objectives

Click a status chip to cycle: Not started → In progress → Studied → Practiced → Needs review → Mastered.

Key Vocabulary

Volatility surface
The map \((K,T)\mapsto\sigma_{\text{imp}}(K,T)\) of implied vols across all strikes and maturities.
Total variance
\(w(k,T)=\sigma_{\text{imp}}^2(k,T)\,T\) in log-moneyness \(k=\ln(K/F)\); the natural no-arbitrage coordinate.
Calendar arbitrage
A violation where total variance decreases with maturity for fixed moneyness; forbidden.
Butterfly arbitrage
A violation where the implied risk-neutral density goes negative; forbidden.
SVI
Stochastic Volatility Inspired parameterization of a total-variance slice with five parameters.
Sticky strike
Dynamics in which implied vol at a fixed strike stays put as spot moves.
Sticky delta (moneyness)
Dynamics in which the smile is a fixed function of moneyness and rides with the spot.

Intuition & Motivation

Intuition
A single option gives one implied vol; the market quotes a whole two-dimensional surface \(\sigma_{\text{imp}}(K,T)\). Not every surface is admissible: prices must respect no-arbitrage, which in vol coordinates becomes two clean rules - total variance must increase with maturity (no calendar arbitrage) and each smile slice must correspond to a non-negative probability density (no butterfly arbitrage). Fitting a surface is therefore a constrained problem: smooth the noisy quotes with an arbitrage-free parameterization (like SVI) first, then calibrate a model to it. And because a desk re-hedges as the market moves, how the surface moves - sticky-strike versus sticky-delta - changes the delta you should actually use.

The surface and its coordinates

Collecting implied vols across strikes and maturities gives the surface \(\sigma_{\text{imp}}(K,T)\). Analysis is cleanest in log-moneyness \(k=\ln(K/F)\) and total implied variance

\[w(k,T)=\sigma_{\text{imp}}^2(k,T)\,T,\] (15.13)

because the two no-arbitrage conditions and the Dupire formula (15.3) take their simplest form in \(w\).

Static no-arbitrage constraints

Proposition - Calendar-spread condition
For fixed log-moneyness \(k\), total variance must be non-decreasing in maturity: \(\partial_T w(k,T)\ge 0\). Equivalently, longer-dated options may not be cheaper (in variance terms) than shorter-dated ones - otherwise a calendar spread is a free lunch.
Proposition - Butterfly condition (non-negative density)
Each maturity slice must define a valid risk-neutral density \(q(K)=e^{rT}\partial_{KK}C\ge0\). In total-variance coordinates this is the Gatheral condition \(g(k)\ge0\) where \(g(k)=\big(1-\tfrac{k\,w'}{2w}\big)^2-\tfrac{w'^2}{4}\big(\tfrac1w+\tfrac14\big)+\tfrac{w''}{2}\). A negative density means a butterfly spread has negative cost - arbitrage.
Key Idea
These two conditions are exactly the positivity of the numerator and denominator in the Dupire formula. An arbitrage-free surface is precisely one from which a valid (positive) local volatility can be extracted.

Discrete version you can test

On a discrete grid of strikes the density non-negativity becomes a butterfly check on call prices: for equally spaced strikes \(K-\Delta,K,K+\Delta\),

\[C(K-\Delta)-2\,C(K)+C(K+\Delta)\ \ge\ 0,\] (15.14)

since the left side is \(\Delta^2\,\partial_{KK}C\ge0\) to leading order. Monotonicity \(-1\le \partial_K C\le 0\) (a vertical-spread condition) must also hold.

Worked Example - Spotting a butterfly arbitrage in three quotes
1
Three one-year calls: \(C(95)=8.0\), \(C(100)=5.0\), \(C(105)=3.5\), equally spaced (\(\Delta=5\)).
2
Butterfly value \(=C(95)-2C(100)+C(105)=8.0-10.0+3.5=1.5\gt 0\) - fine here.
3
Now suppose the middle quote were \(C(100)=6.5\) instead: \(8.0-13.0+3.5=-1.5\lt 0\) - a negative-cost butterfly, i.e. a risk-free profit and a negative implied density.
4
Lesson: before extracting local vol or calibrating Heston, screen the raw surface with (15.14); reject or smooth any slice that fails.

Parameterization: SVI

The raw-SVI slice models total variance as

\[w(k)=a+b\Big(\rho(k-\mu)+\sqrt{(k-\mu)^2+\sigma^2}\Big),\] (15.15)

five parameters per maturity with an intuitive geometry (level \(a\), angle/slope \(b,\rho\), smoothness \(\sigma\), shift \(\mu\)). SVI is popular because there are explicit conditions on its parameters guaranteeing no butterfly arbitrage within a slice and, with care across slices, no calendar arbitrage - giving a smooth, arbitrage-free surface to differentiate for Dupire or to calibrate Heston against.

Surface dynamics and hedging

How the surface moves as spot moves changes the correct hedge:

RegimeAssumptionHedging implication
Sticky strikeVol fixed at each strikeUse the Black–Scholes delta at that strike
Sticky delta/moneynessSmile rides with spotAdd a skew term: delta picks up \(\partial\sigma/\partial k\)
Local volSkew moves ~twice the implied slopePredicts a specific (often too-fast) smile move

Empirically index markets are between sticky-strike and sticky-delta and are regime-dependent; picking the wrong convention biases the delta and leaks P&L through the skew.

Interactive: butterfly & calendar no-arbitrage checker

Common Mistakes to Avoid
  • Calibrating to raw quotes without an arbitrage-free fit: the resulting local vol or exotic prices inherit the arbitrage.
  • Checking butterfly but forgetting calendar (or vice versa): both are needed for a valid surface.
  • Assuming sticky-strike when the market is sticky-delta: your delta is then systematically wrong by a skew term.
  • Interpreting the surface as a forecast of future vols rather than a set of risk-neutral, premium-laden quotes.
Quant Practitioner Tips
  • Fit each slice with SVI (or an equivalent) and enforce the slice + cross-slice no-arbitrage conditions, then differentiate.
  • Diagnose with the density: a dip below zero in \(\partial_{KK}C\) localizes exactly where the surface is arbitrageable.
  • Test your realized delta against both sticky-strike and sticky-delta on historical data to learn your market’s regime.
  • Keep total variance monotone in \(T\) at every moneyness - the cheapest, most common arbitrage to accidentally introduce when interpolating in time.

Knowledge Check

Q1 Medium
In total-variance coordinates, the calendar no-arbitrage condition is:
\(\partial_k w\ge0\)
\(\partial_T w(k,T)\ge0\) for every fixed moneyness
\(w\le1\)
\(\partial_{kk}w=0\)
Q2 Medium
A butterfly-spread value \(C(K-\Delta)-2C(K)+C(K+\Delta)\) that is negative signals:
A steep but valid skew
A negative implied risk-neutral density - butterfly arbitrage
High interest rates
Correct convexity
Q3 Hard
Why must arbitrage-free smoothing (e.g. SVI) precede calibrating a local-vol or Heston model?
To reduce the number of strikes
Because Dupire’s derivatives amplify noise and any surface arbitrage yields negative local variance or nonsensical fits
Because SVI is faster to evaluate than Black–Scholes
It is not necessary

Practical Exercise

You are handed a raw one-maturity smile as call prices on strikes \(90,95,100,105,110\): \([12.0, 8.0, 5.2, 3.0, 1.6]\). (a) Run the discrete butterfly test on each interior strike. (b) Is the slice free of butterfly arbitrage? (c) Separately, the desk assumes sticky-strike but the market turns out sticky-delta after a 3% sell-off. Qualitatively, was the delta used too high or too low for an OTM put, and why?

▶ Show full solution

(a) Second differences \(C_{i-1}-2C_i+C_{i+1}\): at 95: \(12.0-16.0+5.2=1.2\); at 100: \(8.0-10.4+3.0=0.6\); at 105: \(5.2-6.0+1.6=0.8\). All positive.

(b) Yes - every interior second difference is \(\ge0\), so the implied density is non-negative and the slice has no butterfly arbitrage. (One would still need monotonicity \(-1\le\partial_K C\le0\) and a calendar check against neighboring maturities for the full surface.)

(c) Under sticky-delta the smile rides down with the spot after the sell-off, so the implied vol attached to the (now more OTM) put is higher than the sticky-strike assumption predicted. The sticky-strike delta ignores the extra \(\partial\sigma/\partial k\) skew term, so for the OTM put the desk’s hedge delta was effectively too small in magnitude (under-hedged) - it under-accounted for the vol increase that accompanies the drop.

After the reveal, answer for yourself: Which no-arbitrage violation is easiest to introduce by accident when interpolating a surface in the time direction?

Lesson Summary

The volatility surface is the full \((K,T)\) map of implied vols; in total-variance/log-moneyness coordinates its admissibility reduces to two rules - calendar (\(\partial_T w\ge0\)) and butterfly (non-negative density) - which are exactly the positivity of Dupire’s numerator and denominator. Practical work fits an arbitrage-free parameterization such as SVI before differentiating or calibrating, and hedging depends on whether the surface moves sticky-strike or sticky-delta.

Formula Sheet Additions

Total variance & calendar rule
\[w(k,T)=\sigma_{\text{imp}}^2 T,\qquad \partial_T w(k,T)\ge 0\]
No calendar arbitrage: variance grows with maturity.
Butterfly (discrete)
\[C(K-\Delta)-2C(K)+C(K+\Delta)\ge 0\]
Non-negative risk-neutral density: no butterfly arbitrage.
Error Log Checklist
  • Did I check BOTH calendar and butterfly conditions?
  • Did I fit an arbitrage-free surface before differentiating for Dupire?
  • Did I state my surface-dynamics assumption (sticky-strike vs sticky-delta)?
  • Did I treat implied vols as risk-neutral quotes, not forecasts?

Retrieval Practice

Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.

▶ Show retrieval prompts & answers
Q: State the two static no-arbitrage conditions on a volatility surface.
A: Calendar: total variance \(w(k,T)\) is non-decreasing in \(T\) at fixed moneyness. Butterfly: each slice yields a non-negative risk-neutral density (\(\partial_{KK}C\ge0\)).
Q: How does a negative butterfly value \(C(K-\Delta)-2C(K)+C(K+\Delta)\) reveal arbitrage?
A: It approximates \(\Delta^2\partial_{KK}C\propto\) the risk-neutral density; negative means a negative density, so the butterfly spread has negative cost - a free lunch.
Q: Why fit SVI (an arbitrage-free surface) before running Dupire or calibrating Heston?
A: Dupire uses surface derivatives that amplify noise; any calendar/butterfly violation gives negative local variance or unstable fits, so a smooth arbitrage-free input is required.

Flashcards

Click to flip. These feed the site-wide spaced-repetition queue.

Volatility surface
The map \((K,T)\mapsto\sigma_{imp}\); analyzed via total variance \(w=\sigma_{imp}^2T\) in log-moneyness.
No-arbitrage rules
Calendar: \(\partial_T w\ge0\). Butterfly: non-negative density \(\partial_{KK}C\ge0\).
Sticky-strike vs sticky-delta
Whether vol stays at each strike or rides with moneyness; changes the correct hedge delta.

Completion Checklist

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Source References

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