Arbitrage-free volatility surface construction and equity exotics pricing on real listed options data: SVI/SSVI calibration, Dupire local vol, Heston, Monte Carlo + PDE engines, and desk-style risk — with every number cross-validated by an independent method.
VolForge is the sell-side counterpart to QuantForge: where QuantForge covers systematic portfolio research, VolForge covers the derivatives-desk stack — from raw option quotes to an arbitrage-free surface, calibrated models, exotics pricing, and the model-risk question desks actually argue about.
Two models. Both calibrated to the same real SPY option chain (bundled snapshot, 2026-06-10, 2,104 cleaned OTM quotes across 14 expiries). Both reprice the vanilla surface. They disagree on a 1y autocallable by +1.04 per 100 notional — 29× the Monte Carlo noise.
| model (same smile) | PV per 100 notional | 99% CI |
|---|---|---|
| Dupire local vol (log-Euler MC) | 100.00 | ±0.07 |
| Heston, calibrated (Andersen QE MC) | 101.04 | ±0.06 |
That spread is not calibration error — it is the product's exposure to forward smile dynamics, which vanillas do not pin down. Local vol freezes today's smile into a deterministic σ(S, t); Heston regenerates smile through stochastic variance. They agree on every European option and still imply different joint laws for (min S, S_T) — exactly what an autocallable's barrier loads on. Quantifying that spread is the daily business of equity exotics desks (full write-up).
| layer | contents |
|---|---|
volforge.data |
chain cleaning with per-reason drop accounting; forwards and discounts inferred jointly from put-call parity (no rate curve assumed); OTM implied vols |
volforge.surface |
quasi-explicit SVI slices (Zeliade reduction); global SSVI with Gatheral–Jacquier no-butterfly constraints; Durrleman g diagnostics; analytic Dupire local vol (the Dupire denominator is Durrleman's g — one computation serves both) |
volforge.models |
Heston: hardened little-trap CF, adaptive put-side COS pricing, independent Lewis-transform cross-check, DE + least-squares calibration |
volforge.engines |
MC: GBM exact/Euler/Milstein, Andersen QE with martingale correction, local-vol paths, antithetic/control variates, scrambled Sobol' with Brownian-bridge ordering; PDE: Crank–Nicolson + Rannacher, penalty-method American, barrier-snapped grids |
volforge.analytic |
Reiner–Rubinstein barriers, digitals, discrete geometric Asians, Rubinstein chooser |
volforge.instruments |
path payoffs with Brownian-bridge barrier correction; Athena autocallable; variance swaps via VIX-style log-contract replication |
volforge.risk |
bump / pathwise / likelihood-ratio Greeks (with documented failure modes); spot×vol scenario ladders |
Surface. Global SSVI (ρ = −0.60, η = 0.84, γ = 0.57) fit to all 2,104 quotes: 0.25–0.5 vol-point RMSE for every expiry beyond six months, degrading to 1–3 points in sub-month wings — the stated price of a 3-parameter global skew with guaranteed butterfly- and calendar-arbitrage-freeness (0 violations on the fitted surface; the listed refinement is eSSVI). ATM term structure is matched exactly by construction.
Heston. Calibrated to the same chain: v0 = 0.047, κ = 11.5, θ = 0.047, σ = 3.0, ρ = −0.72; Feller ratio 0.12 (violated — routine for equity index). σ pins its bound chasing short-dated skew, which is precisely the documented reason desks moved to local-stochastic-vol mixtures. Sub-0.8 vol-point RMSE beyond six months.
Variance swaps. VIX-style replication on raw quotes vs a dense strip off the fitted surface: fair vol sits a consistent 3–4 points above ATM implied (the skew premium); the long-maturity gap between the two strips measures listed-strike truncation.
Consistency loop. Dupire local vol extracted analytically from the fitted SPY surface, simulated through the MC engine, reprices the surface's own vanillas within 50bp of vol — the defining self-consistency test of a local-vol setup, run in CI on the real snapshot.
Every number is checked against something that cannot share its bugs:
- COS vs Lewis transform vs Riccati ODE integration — three independent Heston routes agreeing to ~1e-9 (prices) and 1e-12 (CF), including Feller-violating stress parameters;
- closed forms vs Monte Carlo — barriers vs Brownian-bridge-corrected MC, chooser vs two-stage MC, geometric Asians vs path simulation;
- PDE vs binomial vs analytic — American puts within 0.15% of an independent 2,500-step CRR tree; barriers vs Reiner–Rubinstein on barrier-snapped grids;
- measured convergence orders — Milstein strong ≈ 1, Euler strong ≈ 0.5, Crank–Nicolson joint order ≈ 2, asserted as fitted slopes, not assumed;
- identities — put-call parity, in-out parity, digital replication, martingale tests under Feller violation, ATM total-variance identity.
Some of these checks earned their keep during development: an early Heston quadrature checker produced negative deep-OTM prices (slowly-decaying oscillatory integrands) and was caught by an MC arbiter and replaced with the Lewis transform; the COS expansion originally priced the call payoff directly and lost catastrophically to e^b coefficient cancellation on fat-tailed parameter sets — it now prices the bounded OTM put and recovers calls via exact parity. The test suite is written so that the next such bug cannot land silently. 116 tests, all offline, ~20s.
git clone https://github.com/Timothy-ZZZZ-bit/volforge && cd volforge
python -m venv .venv && .venv/bin/pip install -e ".[dev]"
.venv/bin/python -m pytest # full validation suite, ~20s, no network
.venv/bin/python examples/01_build_surface.py # real-data surface + diagnostics
.venv/bin/python examples/02_calibrate_heston.py # Heston calibration (~4 min)
.venv/bin/python examples/03_model_risk_autocallable.py # the headline study
.venv/bin/python examples/04_variance_swap.py # replication term structureExamples run straight from a clone (no install needed) against the bundled snapshot; scripts/fetch_snapshot.py pulls a fresh chain (requires pip install -e ".[data]").
from volforge.data.chain import load_snapshot, clean_chain
from volforge.data.forwards import infer_forwards
from volforge.data.iv import chain_implied_vols
from volforge.surface.ssvi import fit_ssvi
from volforge.surface.local_vol import LocalVolSurface
snap = load_snapshot() # bundled SPY chain
cleaned = clean_chain(snap)
forwards = infer_forwards(cleaned) # (F, df) per expiry from parity
points = chain_implied_vols(cleaned, forwards)
surface, report = fit_ssvi(points, forwards)
local_vol = LocalVolSurface(surface) # sigma_loc(S, t), analytic Dupire- The discount factor is only weakly identified from American-style quotes. SPY options carry early-exercise premium that tilts the parity regression; VolForge identifies the forward robustly, projects df onto a feasible band, and documents that
q_impliedis a carry residual, not a dividend forecast. All pricing consumes the (F, df) pair jointly, so surfaces are unaffected. - Global SSVI is parsimonious by design. Three skew parameters cannot match 1-week wings and 2-year skew simultaneously; per-expiry errors are published above rather than averaged away.
- Heston's short-end misfit is structural, not an optimizer failure — the calibration report shows σ at its bound. The honest fix is LSV, which is out of scope here.
- KI monitoring is discrete (daily simulation grid), matching how the contract is monitored in practice; this is stated on the instrument rather than silently approximated.
- One snapshot, EOD quotes. The pipeline is built to re-run on any fresh chain, but the bundled results are a single date — they demonstrate methodology, not a production marking system.
volforge/ the library (data / surface / models / engines / analytic / instruments / risk)
tests/ 116 validation tests, including the real-snapshot consistency loop
examples/ four runnable studies producing everything in reports/
reports/ committed artifacts of the example runs (figures, calibrated params, write-up)
docs/methodology/ the math, conventions, and validation design, written to be read
scripts/ snapshot fetcher (yfinance)
Gatheral & Jacquier (2014), Arbitrage-free SVI volatility surfaces · Gatheral (2006), The Volatility Surface · Andersen (2008), Efficient simulation of the Heston model · Fang & Oosterlee (2008), A novel pricing method based on Fourier-cosine expansions · Lewis (2001), A simple option formula for general jump-diffusion and other exponential Lévy processes · Albrecher, Mayer, Schoutens & Tistaert (2007), The little Heston trap · Lord, Koekkoek & van Dijk (2010), A comparison of biased simulation schemes · De Marco & Martini / Zeliade (2009), Quasi-explicit calibration of Gatheral's SVI · Reiner & Rubinstein (1991), Breaking down the barriers · Rubinstein (1991), Options for the undecided · Demeterfi, Derman, Kamal & Zou (1999), More than you ever wanted to know about volatility swaps · Forsyth & Vetzal (2002), Quadratic convergence for valuing American options · Glasserman (2004), Monte Carlo Methods in Financial Engineering · Huang, Wang & Wan (2021), Exploration of JPMorgan chooser option pricing (the chooser here replaces their simulation step with Rubinstein's closed form)
MIT — see LICENSE.