Bass local vol

01The problem

The Bass local-volatility model is a martingale whose law at each listed expiry is a prescribed marginal, here read off the SPX smile. Conze and Henry-Labordère show how to calibrate it by a fixed point and simulate it with one Gaussian draw per expiry, with no time stepping. I implemented their paper, Bass Construction with Multi-Marginals: Lightspeed Computation in a New Local Volatility Model (2021), on real option data as an MSc summer research project at the University of Vienna.

02What I built

• End to end. Put–call-parity forwards, OTM smiles, Breeden–Litzenberger marginals, the Theorem 2.1 fixed point, local volatility, exact large-step Monte Carlo, Asian and forward-start pricing, hat-bump vega Greeks (Eqs. 9–15) with a nine-call vega hedge, and the Bass2 general-kernel operator (Theorem 6.1).
• A numerical fix. The original notebook used 24-node Gauss–Hermite quadrature for every heat-kernel convolution. On the kinked maps of a real smile this broke the martingale property (conditional drift up to 1.2% of the forward). An exact convolution of the piecewise-linear maps restores it.
• Checks. Greeks agree with bump-and-revalue within 0.2% (Asian) and 1.3% (calls); the 9×9 vanilla vega matrix has full rank.

03Results

evidence

17.8 → 1.8max |z| of the conditional drift, quadrature vs exact convolution

count

10 and 12fixed-point iterations per interval, down from 254 and 499

count

55tests on synthetic inputs, including Black–Scholes recovery, in CI

One historical snapshot: SPX close of 15 Aug 2019 (spot 2847.60), expiries Sep 2019, Dec 2019 and Jun 2020; ± is one standard error.

With the original numerics the package reproduces the original notebook bit for bit. It is a replication for research, not a production pricer.

04Links

• RepositoryGitHub · public
  Open: Repository