Local Volatility Calibration via the Andreasen–Huge Scheme

Calibration of a local-volatility surface on Euro Stoxx 50 (SX5E) option quotes using the fully-implicit one-step PDE scheme of Andreasen & Huge (2010), plus extrapolation of the surface to new maturities.

Submitted as Assignment 3 of Advanced Derivatives (Prof. Elena Perazzi, EPFL — Fall 2025).

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Overview

Standard Dupire-style local-volatility calibration differentiates the implied-volatility surface twice in strike and once in maturity, which amplifies any market noise. Andreasen & Huge (2010) sidestep this by propagating call prices forward one maturity at a time with a fully-implicit finite-difference step, and fitting a piecewise-constant local-volatility parameter per maturity layer by least squares. The result is a smooth, arbitrage-friendly surface obtained from a sequence of tiny convex problems instead of a single ill-posed differentiation.

This project implements the full pipeline:

1. Read the quoted IV grid and convert to Black–Scholes call prices on the $(K, T)$ nodes.
2. For each maturity step, calibrate the piecewise-constant parameter $\theta(K)$ by least squares against quoted prices and propagate with the implicit AH step.
3. Invert Black–Scholes at every node to recover the calibrated implied-volatility surface $\hat\sigma(K, T)$.
4. Extrapolate the surface to $T = 1$ and $T = 1.5$ by propagating from the nearest quoted expiry with no refit.
5. Visualise the resulting price and IV surfaces and the strike cross-sections.

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Method

Setup

Spot $S_0 = 100$, rates $r = q = 0$. Strikes are absolute, $K_i = S_0 \cdot k_i$ with $k_i$ the relative-strike grid (40% to 200% of spot). Only cells with a quote are usable, so a binary mask mask_valid = (implied_vols > 0) is kept throughout to avoid polluting the least-squares objective with missing data.

From IVs to call prices

At each quoted node, IVs are converted to Black–Scholes prices

$$C^\text{mkt}(K_i, T_j) = C_\text{BS}(S_0, K_i, T_j ; \sigma_{i,j}, r, q),$$

and the $t = 0$ column is set to the intrinsic value $\max(S_0 - K, 0)$.

The Andreasen–Huge step

Between two consecutive expiries $T_j \to T_{j+1}$, the AH fully-implicit scheme propagates one time step:

$$A_{j+1}(v) , C_{j+1} = C_j,$$

where $A_{j+1}(v)$ is the full one-step matrix (identity already included) parameterised by a piecewise-constant local-volatility vector $v$, with the partition defined by the midpoints of the quoted strike indices. Letting

$$z_i = \frac{\Delta t}{2, \Delta k^2}, \theta_{i+1}^2 \qquad (i = 1, \ldots, N-2),$$

the interior rows of $A_{j+1}(v)$ form the tridiagonal

$$A_{j+1}(v) = \text{tridiag}\big( -z_i,; 1 + 2z_i,; -z_i \big),$$

with identity boundary rows. The linear system is solved with a single tridiagonal pass.

Per-layer calibration

For each maturity step, $v$ is chosen by least squares on the available quoted strikes:

$$\min_v \sum_{i \in \text{quoted}(j)} \left( C^\text{model}_{i,j+1}(v) - C^\text{mkt}_{i,j+1} \right)^2,$$

with initial guess $v^{(0)} = 0.3, S_0$ (per parameter) and bounds $v \in [0, 2 S_0]$.

Implied-volatility inversion

Once the price grid is built, Black–Scholes is inverted at each node by Nelder–Mead from a seed of $0.5$,

$$\hat\sigma(K_i, T_j) = \arg\min_{\sigma \geq 0} \big( C_\text{BS}(S_0, K_i, T_j ; \sigma) - C_{i,j} \big)^2,$$

with $|\hat\sigma|$ recorded to guard against tiny negative steps from the unconstrained search.

Extrapolation

For target maturities $T_\text{new} \in {1, 1.5}$, the price surface is propagated from the nearest quoted expiry $T_j < T_\text{new}$ using the parameters already calibrated on that interval. No refit is done. BS is then re-inverted to obtain $\hat\sigma(K, T_\text{new})$.

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Results

On the original grid

The interpolated call-price surface is monotone in maturity and decreasing in strike, as expected. The implied-volatility surface reproduces the pronounced left skew at short maturities and flattens for longer tenors — the SX5E equity-index pattern.

Extrapolation to T = 1 and T = 1.5

The price surface remains monotone in $T$ (no calendar arbitrage). The extrapolated IVs extend smoothly from the nearest quoted terms and no spurious ripples appear at the junctions.

Strike cross-sections

The shortest maturity has the largest curvature and the steepest left-skew, with smiles flattening as $T$ grows. The extrapolated smiles at $T = 1$ and $T = 1.5$ sit naturally between their neighbouring quoted terms and preserve the overall term structure.

All figures are reproduced in andreasen_huge_report.pdf and rendered inline in the notebook.

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Implementation notes

• Tridiagonal solve. Each AH step amounts to one banded linear solve. The implementation embeds the band in a sparse/dense system; for a production-grade implementation, a dedicated Thomas algorithm would be the next step.
• Per-layer least squares. Calibration is local in maturity, which keeps each subproblem small and convex-friendly. Calendar arbitrage is structurally avoided by the forward-only propagation: every $C_{j+1}$ is derived from $C_j$ through a positive monotone operator.
• Unconstrained IV inversion with absolute value. A simple guard rather than a full constrained optimiser, justified by the small grid size. A constrained Brent or bracketed bisection would be more robust at scale.

Limitations and natural extensions

• No hard no-arbitrage constraints beyond the implicit scheme itself and visual checks.
• The IV inversion uses unconstrained Nelder–Mead; bracketed bisection would be safer.
• Possible improvements: (i) warm-starting $v$ from the previous tenor, (ii) adding light Tikhonov regularisation across $K$ to smooth the local-vol profile, (iii) testing alternative seeds/optimisers to quantify sensitivity.

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Repository structure

.
├── andreasen_huge_calibration.ipynb   # Jupyter notebook: full pipeline
├── andreasen_huge_report.pdf          # Method, figures and discussion
├── requirements.txt                   # Python dependencies
├── .gitignore
└── README.md

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How to run

git clone https://github.com/Theo2co/local-volatility-andreasen-huge.git
cd local-volatility-andreasen-huge
python -m venv .venv
source .venv/bin/activate            # Windows: .venv\Scripts\activate
pip install -r requirements.txt
jupyter lab                          # then open andreasen_huge_calibration.ipynb

Requires Python 3.10+.

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Reference

Andreasen, J. and Huge, B. (2010). Volatility Interpolation. Danske Markets working paper. SSRN: https://ssrn.com/abstract=1694972

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Authors

Group work submitted for Advanced Derivatives at EPFL:

• Théodore Decaux — @Theo2co (repository maintainer)
• Elias Bourgon
• Jason Santangelo

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License

Released under the MIT License.