How Andreasen–Huge Local-Volatility Algorithms Calibrate Option Surfaces One Maturity at a Time: Implicit Forward PDEs, Piecewise Local Variance, Repricing and Arbitrage Diagnostics

Reader question: Dupire local volatility is theoretically determined by derivatives of an arbitrage-free option-price surface. But market quotes are sparse and noisy. Can we calibrate a local-volatility surface without numerically differentiating a fragile implied-volatility surface?

The Andreasen–Huge method answers yes by working forward in maturity, one expiry interval at a time. Instead of first smoothing the entire implied-volatility surface and then differentiating it, the algorithm applies a one-step implicit finite-difference version of the forward option-pricing equation and solves for a piecewise local-variance function that reprices the next maturity’s options.

This article owns one precise computational job: sequential one-step calibration of a local-volatility/interpolation surface through an implicit forward PDE. It does not own the Dupire identity itself, SVI total-variance fitting, SABR dynamics, or ordinary backward finite-difference option pricing.

This is public mathematical and computational education. It is not financial advice, a volatility trade recommendation or a guarantee that fitting European option prices makes exotic-option dynamics correct.

1. Why direct Dupire differentiation is difficult

Under simplified zero-rate notation, Dupire’s forward equation for call prices can be written:

∂C(K,T)/∂T = ½ σ_loc²(K,T) K² ∂²C(K,T)/∂K².

Rearranging gives:

σ_loc²(K,T) = 2 C_T / (K² C_KK).

In full market conventions, rates/dividends/forwards add terms and coordinate choices matter.

The numerical difficulty is obvious: market option prices are sparse and noisy, yet the formula asks for a maturity derivative and a second strike derivative. Differentiation amplifies noise.

2. The Andreasen–Huge reversal

Instead of:

1. fit a global smooth surface;
2. differentiate it;
3. obtain local volatility;

Andreasen–Huge effectively does:

1. start from known option prices at one maturity;
2. choose a parameterised local-variance function for the next maturity interval;
3. propagate prices forward by one implicit PDE step;
4. adjust local variance until next-expiry market prices are matched;
5. repeat maturity by maturity.

The local variance is therefore calibrated inside the discrete pricing operator rather than recovered by unstable standalone differentiation.

3. Work in option price space

Market quotes may arrive as implied volatilities, but the PDE evolves option values.

A robust pipeline therefore converts each quote into a call price under the same:

• forward;
• discount factor;
• strike convention;
• option style;
• volatility convention.

Calibration residuals can be measured in price space, implied-vol space, vega-normalised price space or another documented metric.

4. Forward versus spot/strike coordinates

Many implementations transform to normalised coordinates such as log-moneyness or forward moneyness to simplify rates/dividends.

For example:

k = K/F(T)

or:

x = ln(K/F(T)).

The exact PDE coefficients depend on that choice.

Falsifier: derive the discrete operator from the chosen coordinates and conventions. Copying a matrix formula from another coordinate system is a silent but serious error.

5. One maturity interval at a time

Let T_j and T_j+1 be consecutive market expiries.

Assume local variance is represented over that time slab:

v(K,t) ≈ v_j(K), T_j < t ≤ T_j+1.

The algorithm uses the option-price vector at T_j as the initial condition and propagates it once to T_j+1.

This is the “one-step” idea.

6. The implicit finite-difference step

After discretising strike, the forward PDE becomes a matrix ODE:

dC/dT = L(v) C.

A one-step implicit Euler update is:

[I − ΔT L(v)] C_j+1 = C_j.

Depending on sign conventions and coordinate form, implementations may write the matrix differently, but the computational structure is a sparse linear system whose coefficients depend on local variance.

7. Why implicit stepping is attractive

Implicit schemes are generally much more stable than explicit finite differences for diffusion equations.

That matters because each maturity interval can be relatively large: the algorithm is not necessarily taking hundreds of tiny timesteps between quoted expiries during calibration.

Stability, however, is not the same as accuracy. Grid and one-step discretisation error still require validation.

8. Piecewise local variance across strike

At each maturity slab, local variance can be represented by values at selected strike/moneyness nodes and interpolated between them.

Common practical choices are piecewise constant or piecewise linear parameterisations.

This gives enough flexibility to fit a smile while keeping the per-expiry calibration relatively small.

9. Sequential calibration

A stylised layer calibration is:

1. take C_j from the previously calibrated maturity;
2. guess v_j(K);
3. solve the implicit system for C_j+1;
4. compare C_j+1 at quoted strikes with market prices;
5. update v_j;
6. repeat until residuals pass;
7. freeze the layer and move to T_j+2.

The entire surface is built forward, maturity by maturity.

10. Why the previous layer matters

Each later expiry begins from the price distribution produced by earlier calibrated layers.

So calibration errors or arbitrage defects can propagate forward.

Diagnostic: do not validate each expiry independently. Reprice every maturity from time zero through the complete calibrated sequence.

11. Initial condition at T=0

For a forward-normalised call, the time-zero payoff is the intrinsic shape:

C(K,0) = (S₀−K)⁺

or its normalised equivalent.

This payoff has a kink. The discrete forward operator smooths it as maturity increases.

Grid placement around the forward/ATM region therefore matters especially for short expiries.

12. Static-arbitrage structure in price space

For standard calls at fixed maturity, an arbitrage-consistent price curve should satisfy:

• non-negative prices;
• non-increasing price in strike;
• convexity in strike;
• appropriate boundary limits.

Across maturity, after correct forward/discount normalisation, calendar consistency imposes its own monotonicity structure.

The forward PDE representation helps preserve probability-density structure when local variance is non-negative and the discretisation/operator is built correctly.

13. Non-negative local variance is necessary

A negative local variance is not physically meaningful for the diffusion model.

Optimisation or interpolation should enforce:

v(K,T) ≥ 0.

Parameterising log-variance, variance squares or applying explicit bounds are common approaches.

14. Exact repricing versus noisy quotes

If there are as many flexible local-volatility parameters as quotes, the method can fit market prices extremely closely.

But exact fit can become overfit when:

• quotes contain stale points;
• bid–ask noise is large;
• one strike is erroneous;
• very short-dated wings are illiquid.

Research comparing arbitrage-free interpolation methods notes that highly flexible Andreasen–Huge-style fits can produce noisy implied densities when they chase every quote.

Public lesson: exact repricing is not automatically the right objective when input quotes are noisy.

15. Bid–ask-aware calibration

A useful alternative to hard exact fit is to penalise deviations only when model prices leave bid–ask intervals, or to weight residuals by liquidity/vega.

This avoids forcing the local-volatility surface to explain microstructure noise that is not economically identifiable.

16. Strike-grid design

The computational strike grid usually extends beyond quoted strikes.

It must be wide enough that boundary conditions do not contaminate the quoted region.

Useful diagnostics include:

• doubling wing width;
• halving strike spacing;
• adding more nodes near ATM;
• checking boundary-value sensitivity.

17. Wing extrapolation

Market quotes stop at finite strikes, but the PDE needs boundary behaviour.

Any wing extrapolation of local variance or price shape becomes a model assumption.

Falsifier: vary the wing rule and measure its effect on quoted-region repricing, density and exotic prices.

18. Maturity extrapolation

If a target maturity lies beyond the last calibrated expiry, one may propagate using the last calibrated local-variance layer or another documented extrapolation.

That is not market calibration. It is an assumption.

Keep interpolation and extrapolation diagnostics separate.

19. Inputs and outputs

Inputs can include:

• option strikes and expiries;
• bid/ask or mid implied vols/prices;
• forwards and discount factors;
• coordinate convention;
• strike grid;
• local-variance node parameterisation;
• boundary conditions;
• optimiser/bounds;
• residual weighting;
• maturity extrapolation rule.

Outputs can include:

• calibrated local-variance/local-volatility layers;
• model call-price grid;
• recovered implied-volatility grid;
• repricing residuals;
• density/convexity diagnostics;
• calendar-arbitrage diagnostics;
• grid-refinement sensitivities;
• wing/extrapolation sensitivities;
• Greeks/exotic benchmark results.

20. Evidence polarity

Evidence for confidence includes:

• quoted options reprice within intended tolerance;
• price curves are monotone and convex in strike;
• risk-neutral density is non-negative up to numerical tolerance;
• calendar consistency holds under the chosen normalisation;
• local variance stays non-negative;
• grid refinement changes prices/Greeks negligibly;
• independent Dupire/PDE benchmarks agree on smooth synthetic surfaces;
• held-out strikes interpolate sensibly.

Evidence against confidence includes:

• local-volatility spikes tied to single quotes;
• negative density or butterfly violations;
• large wing sensitivity;
• calibration depends strongly on initial guesses;
• later maturities inherit earlier-layer errors;
• exotics/Greeks move materially under minor quote perturbations;
• exact fit produces oscillatory density.

21. Counterexample: a bad quote creates a local-vol spike

Suppose one OTM option is misquoted several volatility points above its neighbours.

A highly flexible layer can force local variance to jump sharply near that strike to hit the bad price.

Falsifier: remove or move the quote within bid–ask and recalibrate. If the entire local-volatility neighbourhood changes, the surface is quote-noise sensitive.

22. Counterexample: exact fit violates held-out smoothness

A surface can fit every quoted strike yet interpolate implausibly between them.

Falsifier: withhold selected liquid strikes, calibrate without them, then compare interpolated model prices to the held-out quotes.

23. Counterexample: coarse grid hides arbitrage

A price curve can look convex at calibration nodes but violate convexity between them.

Falsifier: evaluate prices/density on a much denser audit grid than the calibration grid.

24. Counterexample: one-step discretisation error

A very long expiry gap may make one implicit step stable but insufficiently accurate.

Falsifier: split the same calibrated time slab into two or more numerical propagation steps using consistent local variance and compare prices.

25. Counterexample: local volatility fits Europeans but misses exotic dynamics

Local volatility is constructed to reproduce the European surface under its diffusion assumptions.

Two models can fit the same vanilla surface while producing different pathwise volatility dynamics and barrier/autocall prices.

Falsifier: benchmark relevant exotics under stochastic-volatility/local-stochastic-volatility alternatives rather than using vanilla fit as proof.

26. Relation to Dupire

Dupire local-volatility algorithms own the continuous inverse relationship between an arbitrage-free option-price surface and local variance.

Andreasen–Huge owns a discrete calibration/interpolation approach designed to avoid directly differentiating noisy market surfaces.

27. Relation to finite differences

Finite-difference option pricing owns general grid discretisation, stability and boundary-condition principles.

Andreasen–Huge uses those numerical ideas inside a forward calibration operator.

28. Relation to SVI

SVI algorithms own a parametric representation of implied total variance across log-moneyness.

SVI is often used to create a smooth static-arbitrage-aware implied-volatility surface. Andreasen–Huge instead builds prices/local variance through the forward PDE. They solve different representation problems.

29. Alternatives

Smoothed Dupire: fit a smooth price/vol surface, differentiate carefully, then compute local variance.

SVI/eSSVI: fit arbitrage-aware implied total variance, then optionally derive local volatility.

Stochastic volatility: fit vanillas with a dynamic volatility factor rather than deterministic local volatility.

Local-stochastic volatility: combine local fit with stochastic-volatility dynamics at higher calibration complexity.

30. Weak links

• coordinate/PDE mismatch;
• bad forward or discount input;
• negative local variance;
• single bad quote overfit;
• wing boundaries too close;
• coarse strike grid;
• large maturity gaps with one-step error;
• calibration residual metric ignoring bid–ask/liquidity;
• vanilla fit mistaken for dynamic truth.

31. What would falsify confidence?

Confidence should be withdrawn if quoted instruments fail repricing tests; if dense-grid convexity/density checks fail; if local variance becomes negative or spiky; if grid/maturity refinement changes prices materially; if held-out interpolation is poor; or if tiny quote perturbations cause large exotic/Greek changes.

32. Verification and update triggers

Preserve the exact quote set, forwards/discounts, coordinate transform, strike grid, boundary conditions, maturity slabs, parameterisation, optimiser, residual weighting and all arbitrage/refinement diagnostics.

Revalidate when:

• quote vendors/conventions change;
• new strikes/expiries appear;
• forward/discounting framework changes;
• negative-rate/normal-vol conventions become relevant;
• grid or PDE scheme changes;
• surface wings move materially;
• exotic-pricing use expands.

33. Primary and high-quality references

• Jesper Andreasen and Brian Huge, Volatility Interpolation, original working paper.
• Dupire’s local-volatility framework, for the continuous forward-PDE inversion that motivates the method.
• QuantLib’s Andreasen–Huge volatility interpolation/calibration implementation, for a widely inspected open-source reference.
• Research comparing arbitrage-free interpolation methods, including Model-Free Stochastic Collocation for an Arbitrage-Free Implied Volatility.

Educational boundary: Andreasen–Huge stabilises local-volatility calibration by embedding it in a discrete forward pricing step. It does not eliminate input-noise risk, discretisation risk or the fundamental limitation that vanilla-surface fit does not uniquely determine exotic dynamics.