From 9361d70b387180fa2703d7a404762fa4c73613b5 Mon Sep 17 00:00:00 2001 From: Luca Date: Sun, 26 Jul 2026 19:23:31 +0100 Subject: [PATCH 1/4] Add ssvi --- .github/copilot-instructions.md | 4 +- .github/instructions/makefile.instructions.md | 4 + .github/instructions/release.instructions.md | 4 + app/api/cointegration.py | 9 +- app/api/volatility.py | 2 +- app/utils/paths.py | 38 +- docs/api/options/ssvi.md | 3 + docs/bib2md.py | 2 +- docs/bibliography.md | 96 + docs/contributing.md | 2 +- docs/references.bib | 20 + docs/stylesheets/bibliography.css | 17 + mkdocs.yml | 6 + pyproject.toml | 6 +- quantflow/options/ssvi.py | 246 +++ quantflow/options/surface.py | 14 +- quantflow/options/svi.py | 12 +- quantflow/ta/paths.py | 2 +- quantflow_tests/test_ssvi.py | 208 ++ uv.lock | 1743 +++++++++-------- 20 files changed, 1612 insertions(+), 826 deletions(-) create mode 100644 docs/api/options/ssvi.md create mode 100644 docs/stylesheets/bibliography.css create mode 100644 quantflow/options/ssvi.py create mode 100644 quantflow_tests/test_ssvi.py diff --git a/.github/copilot-instructions.md b/.github/copilot-instructions.md index 0fb0657e..4336990a 100644 --- a/.github/copilot-instructions.md +++ b/.github/copilot-instructions.md @@ -25,7 +25,7 @@ applyTo: '/**' ## Run Tests -* To run all tests use `make test` — runs all tests in the `tests/` directory using pytest +* To run all tests use `make tests` — runs all tests in the `tests/` directory using pytest * To run a specific test file, use `uv run pytest tests/path/to/test_file.py` ## Docker @@ -56,7 +56,7 @@ applyTo: '/**' * Do not repeat concept definitions inline in tutorials or docstrings, link to the glossary instead using a relative markdown link (e.g. `[moneyness](../glossary.md#moneyness)`). * Use relative links for all mkdocs page links (e.g. `[Option Pricing](../theory/option_pricing.md)`) — prefer relative over absolute URLs to keep links shorter and portable. * Prefer mkdocstrings relative cross-references whenever the target is visible from the current scope: write `[label][.member]` (same class) or `[label][..Sibling]` (same module) instead of repeating the fully-qualified path. Use the full path only when the target lives in a different module than the current docstring. -* To rebuild doc examples run `uv run ./dev/build-examples` — runs all scripts in `docs/examples/` and writes their output to `docs/examples_output/` +* To rebuild doc examples run `uv run ./dev/build-examples` — runs all scripts in `docs/examples/` and writes their output to `docs/examples/output/` ## Pydantic models diff --git a/.github/instructions/makefile.instructions.md b/.github/instructions/makefile.instructions.md index 002d45a2..31fc0172 100644 --- a/.github/instructions/makefile.instructions.md +++ b/.github/instructions/makefile.instructions.md @@ -1,3 +1,7 @@ +--- +applyTo: 'Makefile' +--- + # Makefile Conventions - Keep all targets sorted alphabetically. diff --git a/.github/instructions/release.instructions.md b/.github/instructions/release.instructions.md index 1843cd6e..7603a7c5 100644 --- a/.github/instructions/release.instructions.md +++ b/.github/instructions/release.instructions.md @@ -1,3 +1,7 @@ +--- +applyTo: 'pyproject.toml,docs/release-notes.md,.github/workflows/release.yml,Makefile' +--- + # Release Instructions Releases are driven by `v*` git tags. Pushing a tag triggers diff --git a/app/api/cointegration.py b/app/api/cointegration.py index 6664f8a0..31adf360 100644 --- a/app/api/cointegration.py +++ b/app/api/cointegration.py @@ -1,3 +1,4 @@ +import warnings from functools import partial import numpy as np @@ -62,8 +63,12 @@ async def _cointegration(fmp: FMP, frequency: FMP.freq) -> CointegrationResponse log_prices_3 = np.log(prices_3) std = log_prices_3.std() scaled = log_prices_3 / std - johansen_result = coint_johansen(scaled, det_order=0, k_ar_diff=1) - deltas = johansen_result.evec[:, 0] / std.values + with warnings.catch_warnings(): + # statsmodels assigns complex eigenvalue statistics into real arrays, + # discarding noise-level imaginary parts + warnings.simplefilter("ignore", np.exceptions.ComplexWarning) + johansen_result = coint_johansen(scaled, det_order=0, k_ar_diff=1) + deltas = np.real_if_close(johansen_result.evec[:, 0]).real / std.values deltas = deltas / np.linalg.norm(deltas) residuals = log_prices_3.dot(deltas) diff --git a/app/api/volatility.py b/app/api/volatility.py index b9ad96b1..aeb0134e 100644 --- a/app/api/volatility.py +++ b/app/api/volatility.py @@ -119,7 +119,7 @@ async def _volatility_surface(asset: str) -> VolSurfaceResponse: options = [op.info() for op in surface.option_prices(converged=True)] max_ttm = max(float(op.ttm) for op in options) if options else 1.0 - ttm_grid = list(np.linspace(1 / 365, max_ttm, 50)) + ttm_grid = [float(t) for t in np.linspace(1 / 365, max_ttm, 50)] return VolSurfaceResponse( inputs=inputs, diff --git a/app/utils/paths.py b/app/utils/paths.py index c1243083..0477a04d 100644 --- a/app/utils/paths.py +++ b/app/utils/paths.py @@ -1,3 +1,4 @@ +from html import escape from pathlib import Path from typing import Any @@ -8,7 +9,42 @@ def head_snippet(path: Path) -> str: return (path / "assets" / "logos" / "head-snippet.html").read_text() +def social_meta(page: Any, config: Any) -> str: + site_name = escape(str(config.get("site_name", ""))) + site_description = escape(str(config.get("site_description", ""))) + site_url = str(config.get("site_url", "")).rstrip("/") + title = escape(f"{page.title} | {site_name}" if page.title else site_name) + page_url = escape(str(page.canonical_url)) + image = escape(f"{site_url}/assets/logos/png/quantflow-github-social-1280x640.png") + image_alt = "QuantFlow logo and tagline" + return "\n".join( + ( + '', + f'', + f'', + f'', + f'', + f'', + f'', + '', + '', + '', + f'', + '', + f'', + f'', + f'', + f'', + ) + ) + + def on_post_page(output: str, page: Any, config: Any) -> str: """Hook to inject custom HTML into the head of each page in mkdocs build.""" - snippet = head_snippet(APP_PATH.parent / "docs") + snippet = "\n".join( + ( + head_snippet(APP_PATH.parent / "docs"), + social_meta(page, config), + ) + ) return output.replace("", f"{snippet}", 1) diff --git a/docs/api/options/ssvi.md b/docs/api/options/ssvi.md new file mode 100644 index 00000000..3651d1cd --- /dev/null +++ b/docs/api/options/ssvi.md @@ -0,0 +1,3 @@ +# SSVI Volatility Surface + +::: quantflow.options.ssvi.SSVI diff --git a/docs/bib2md.py b/docs/bib2md.py index deb05924..3764f31a 100644 --- a/docs/bib2md.py +++ b/docs/bib2md.py @@ -147,7 +147,7 @@ def format_entry(entry: dict[str, str]) -> str: if suffix: body = f"{body}, {suffix}" - return f"#### {key}\n\n{body}\n" + return f'
\n\n#### {key}\n\n{body}\n\n
\n' # --------------------------------------------------------------------------- diff --git a/docs/bibliography.md b/docs/bibliography.md index e3627639..462a0f4d 100644 --- a/docs/bibliography.md +++ b/docs/bibliography.md @@ -4,82 +4,178 @@ The raw BibTeX source for all entries is available in [references.bib](reference --- +
+ #### bertoin J. Bertoin. (1996) Lévy Processes, Cambridge University Press +
+ +
+ #### bns O.E. Barndorff-Nielsen, N. Shephard. (2001) [Non-Gaussian OU based models and some of their uses in financial economics](https://drive.google.com/file/d/11TnGl2ER_-QzEu_h7ZtGGass2rF8_u1Y/view){target="_blank" rel="noopener"}, Journal of the Royal Statistical Society, 63(2) +
+ +
+ #### carr_madan P. Carr, D. Madan. (1999) [Option valuation using the fast Fourier transform](http://faculty.baruch.cuny.edu/lwu/890/CarrMadan99.pdf){target="_blank" rel="noopener"}, Journal of Computational Finance, 3:463-520 +
+ +
+ #### carr_wu P. Carr, L. Wu. (2002) [Time-changed Lévy processes and option pricing](https://engineering.nyu.edu/sites/default/files/2019-03/Carr-time-changed-levy-processes-option-pricing.pdf){target="_blank" rel="noopener"}, Journal of Financial Economics, 7:113-141 +
+ +
+ #### cgmy Carr P., Geman H., Madan D.B., Yor M. (2003) [Stochastic Volatility for Lévy processes](https://engineering.nyu.edu/sites/default/files/2019-03/Carr-stochastic-volatility-levy-processes.pdf){target="_blank" rel="noopener"}, Mathematical Finance, 13(3) +
+ +
+ #### chourdakis K. Chourdakis. (2004) [Option Pricing Using the Fractional FFT](https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=6bdf4696312d37427eda2740137650c09deacda7){target="_blank" rel="noopener"}, Journal of Computational Finance, 8:1-18 +
+ +
+ #### cos F. Fang, K. Oosterlee. (2008) [A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions](https://mpra.ub.uni-muenchen.de/8914/4/MPRA_paper_8914.pdf){target="_blank" rel="noopener"} +
+ +
+ #### dsp Unknown. (2017) [Doubly Stochastic Poisson Processes with Affine Intensities](https://editorialexpress.com/cgi-bin/conference/download.cgi?db_name=sbe35&paper_id=179){target="_blank" rel="noopener"}, internet +
+ +
+ #### durbin_koopman J. Durbin, S. J. Koopman. (2012) Time Series Analysis by State Space Methods, Oxford University Press +
+ +
+ #### ekf Wang X., He X., Zhao Y., Zuo Z. (2017) [Parameter Estimations of Heston Model Based on Consistent Extended Kalman Filter](https://www.sciencedirect.com/science/article/pii/S2405896317324758){target="_blank" rel="noopener"}, internet +
+ +
+ #### gamma-ou P. Sabino, C. Petroni. (2021) [Gamma Related Ornstein-Uhlenbeck Processes and their Simulation](https://doi.org/10.1080/00949655.2020.1842408){target="_blank" rel="noopener"}, Journal of Statistical Computation and Simulation, 91(6) +
+ +
+ +#### gatheral_jacquier + +Jim Gatheral, Antoine Jacquier. (2014) [Arbitrage-free SVI volatility surfaces](https://doi.org/10.1080/14697688.2013.819986){target="_blank" rel="noopener"}, Quantitative Finance, 14(1):59-71 + +
+ +
+ +#### gatheral_svi + +Jim Gatheral. (2004) [A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives](https://faculty.baruch.cuny.edu/jgatheral/madrid2004.pdf){target="_blank" rel="noopener"} + +
+ +
+ #### heston-calibration Milan Mrázek, Jan Pospíšil. (2017) [Calibration and simulation of Heston model](https://doi.org/10.1515/math-2017-0058){target="_blank" rel="noopener"}, Open Mathematics, 15(1):679-704 +
+ +
+ #### heston-simulation Leif B.G. Andersen. (2008) [Efficient Simulation of the Heston Stochastic Volatility Model](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=946405){target="_blank" rel="noopener"}, Journal of Computational Finance, 11(3) +
+ +
+ #### julier_uhlmann S. J. Julier, J. K. Uhlmann. (1997) [New extension of the Kalman filter to nonlinear systems](https://doi.org/10.1117/12.280797){target="_blank" rel="noopener"}, Proceedings of SPIE, 3068:182-193 +
+ +
+ #### kalman R. E. Kalman. (1960) [A New Approach to Linear Filtering and Prediction Problems](https://doi.org/10.1115/1.3662552){target="_blank" rel="noopener"}, Journal of Basic Engineering, 82(1):35-45 +
+ +
+ #### lee_option R. W. Lee. (2004) [Option Pricing by Transform Methods: Extensions, Unification, and Error Control](https://www.math.uchicago.edu/~rl/dft.pdf){target="_blank" rel="noopener"} +
+ +
+ #### lewis A. L. Lewis. (2001) [A Simple Option Formula for General Jump-Diffusion and other Exponential Lévy Processes](https://drive.google.com/file/d/1JiYfyOU7lUKHrMqTRPbBCdasctwYDc4Z/view){target="_blank" rel="noopener"} +
+ +
+ #### molnar Peter Molnar. (2020) [Volatility modeling and forecasting: utilization of realized volatility, implied volatility and the highest and lowest price of the day](https://drive.google.com/file/d/1zCU1OZyrKQLpxaypPv9U5UPbReBDXcMf/view){target="_blank" rel="noopener"}, University of Economics in Prague +
+ +
+ #### saez G. K. G. Saez. (2014) [Fourier Transform Methods for Option Pricing: An Application to extended Heston-type Models](https://www.uv.es/bfc/TFM2014/008-014.pdf){target="_blank" rel="noopener"}, Universidad del Pais Vasco +
+ +
+ #### ukf Merwe. (2014) [The Unscented Kalman Filter for Nonlinear Estimation](https://groups.seas.harvard.edu/courses/cs281/papers/unscented.pdf){target="_blank" rel="noopener"}, internet + +
diff --git a/docs/contributing.md b/docs/contributing.md index c2007bba..3f3d42c6 100644 --- a/docs/contributing.md +++ b/docs/contributing.md @@ -41,7 +41,7 @@ Example scripts can be rebuilt with `make docs-examples`. AI assisted contributions are welcome: using AI tools to draft, review, or improve documentation is perfectly fine. -However, you are the author of your PR. You must understand every change you submit, be able to explain it, and respond to review feedback yourself. Please do not submit unreviewed AI output: if you cannot vouch for the content, do not open the PR. +However, you are the author of your PR. You must understand every change you submit, be able to explain it, and respond to review feedback yourself. Please do not submit unreviewed AI output: if you cannot take responsibility for the content, do not open the PR. ## Code of Conduct diff --git a/docs/references.bib b/docs/references.bib index c6447e2d..1afe76f1 100644 --- a/docs/references.bib +++ b/docs/references.bib @@ -100,6 +100,26 @@ @article{gamma-ou url={https://doi.org/10.1080/00949655.2020.1842408}, } +@article{gatheral_jacquier, + title={Arbitrage-free SVI volatility surfaces}, + author={Jim Gatheral and Antoine Jacquier}, + journal={Quantitative Finance}, + year={2014}, + volume={14}, + number={1}, + pages={59--71}, + url={https://doi.org/10.1080/14697688.2013.819986}, +} + +@misc{gatheral_svi, + title={A parsimonious arbitrage-free implied volatility parameterization + with application to the valuation of volatility derivatives}, + author={Jim Gatheral}, + howpublished={Presentation at Global Derivatives \& Risk Management, Madrid}, + year={2004}, + url={https://faculty.baruch.cuny.edu/jgatheral/madrid2004.pdf}, +} + @article{heston-calibration, title={Calibration and simulation of Heston model}, author={Milan Mrázek and Jan Pospíšil}, diff --git a/docs/stylesheets/bibliography.css b/docs/stylesheets/bibliography.css new file mode 100644 index 00000000..e2cc1e76 --- /dev/null +++ b/docs/stylesheets/bibliography.css @@ -0,0 +1,17 @@ +/* Highlight the whole bibliography entry (title and body) when its + anchor is the URL fragment, e.g. /bibliography/#carr_madan */ +.bib-entry { + display: flow-root; + padding: 0 0.6rem; + margin: 0 -0.6rem; + border-radius: 0.2rem; + border-bottom: 1px solid var(--md-default-fg-color--lightest); +} + +.bib-entry:last-of-type { + border-bottom: none; +} + +.bib-entry:has(:target) { + background-color: var(--md-accent-fg-color--transparent); +} diff --git a/mkdocs.yml b/mkdocs.yml index 254548e0..01727a3e 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -1,4 +1,5 @@ site_name: QuantFlow +site_description: Quantitative analysis and pricing tools. site_url: https://quantflow.quantmind.com site_dir: app/docs repo_name: quantmind/quantflow @@ -82,6 +83,7 @@ nav: - Moneyness: api/options/moneyness.md - Put-Call Parity: api/options/parity.md - Pricer: api/options/pricer.md + - SSVI Surface: api/options/ssvi.md - SVI Smile: api/options/svi.md - Volatility Surface: api/options/vol_surface.md - Stochastic Processes: @@ -145,6 +147,7 @@ nav: markdown_extensions: - admonition - attr_list + - md_in_html - tables - pymdownx.arithmatex: generic: true @@ -158,6 +161,9 @@ markdown_extensions: - toc: permalink: true +extra_css: + - stylesheets/bibliography.css + extra_javascript: - javascripts/mathjax.js - https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js diff --git a/pyproject.toml b/pyproject.toml index 82d71f62..2233172e 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -33,7 +33,7 @@ readme = "readme.md" requires-python = ">=3.11,<3.15" dependencies = [ "scipy>=1.14.1", - "pandas>=2.0.0", + "pandas>=2.3.0", "pydantic>=2.0.2", "ccy>=2.0.0", "statsmodels>=0.14.6,<0.15.0", @@ -59,11 +59,11 @@ dev = [ "ghp-import>=2.0.2", "hypothesis>=6.152.2", "isort>=8.0.0", - "mypy>=1.14.1", + "mypy>=2.3.0", "pytest-asyncio>=1.0.0", "pytest-cov>=7.0.0", "python-dotenv>=1.0.1", - "ruff>=0.15.4", + "ruff>=0.16.0", "types-python-dateutil>=2.9.0.20251115", ] docs = [ diff --git a/quantflow/options/ssvi.py b/quantflow/options/ssvi.py new file mode 100644 index 00000000..7333aeff --- /dev/null +++ b/quantflow/options/ssvi.py @@ -0,0 +1,246 @@ +from __future__ import annotations + +from collections.abc import Sequence +from decimal import Decimal + +import numpy as np +from numpy.typing import ArrayLike +from pydantic import BaseModel, Field +from scipy.optimize import least_squares +from typing_extensions import Annotated, Doc + +from quantflow.utils.numbers import ONE, ZERO, to_decimal + + +class SSVI(BaseModel, extra="forbid"): + r"""Surface SVI (SSVI) parametrisation of the implied volatility smile, + introduced by + [Gatheral and Jacquier (2014)](../../bibliography.md#gatheral_jacquier). + + The SSVI parametrisation expresses the total implied variance + $w(k) = \sigma^2(k) \cdot \tau$ as a function of log-strike + $k = \log(K/F)$ and the at-the-money total variance $\theta$: + + \begin{equation} + w(k) = \frac{\theta}{2}\left[1 + \rho \varphi(\theta) k + + \sqrt{\left(\varphi(\theta) k + \rho\right)^2 + 1 - \rho^2}\right] + \end{equation} + + The shape function $\varphi$ uses the power law form: + + \begin{equation} + \varphi(\theta) = \frac{\eta}{\theta^\gamma (1 + \theta)^{1 - \gamma}} + \end{equation} + + Each instance represents a single maturity slice through its $\theta$ + parameter, while $\rho$, $\eta$ and $\gamma$ are global parameters shared + across the whole surface. + + Use [fit][.fit] to calibrate a single slice, or [fit_surface][.fit_surface] + to jointly calibrate the global parameters across several maturities. A + surface built this way is free of calendar spread arbitrage provided the + fitted $\theta$ are non decreasing in maturity. + """ + + rho: Decimal = Field( + gt=-ONE, + lt=ONE, + description=( + "Correlation parameter controlling the skew of the smile. " + "Negative values produce a left-skewed smile (typical for equities), " + "positive values produce a right skew. Must satisfy $|\\rho| < 1$." + ), + ) + eta: Decimal = Field( + gt=ZERO, + description=( + "Level of the power law shape function $\\varphi$. " + "Larger values steepen the smile away from the money. " + "Must be strictly positive." + ), + ) + gamma: Decimal = Field( + gt=ZERO, + le=ONE, + description=( + "Exponent of the power law shape function $\\varphi$, controlling " + "how the smile curvature decays as the at-the-money total variance " + "grows. Must lie in $(0, 1]$." + ), + ) + theta: Decimal = Field( + gt=ZERO, + description=( + "At-the-money total implied variance of the slice, " + "$\\theta = \\sigma_{ATM}^2 \\cdot \\tau$. This is the only " + "maturity-dependent parameter. Must be strictly positive." + ), + ) + + def phi(self) -> float: + r"""Power law shape function $\varphi(\theta)$ evaluated at the slice + at-the-money total variance $\theta$.""" + theta = float(self.theta) + eta = float(self.eta) + gamma = float(self.gamma) + return eta / (theta**gamma * (1 + theta) ** (1 - gamma)) + + def total_variance( + self, + k: Annotated[ArrayLike, Doc("Log-moneyness log(K/F), scalar or array")], + ) -> np.ndarray: + r"""Total implied variance $w(k)$. + + Returns an array of the same shape as $k$. + """ + k_arr = np.asarray(k, dtype=float) + theta = float(self.theta) + rho = float(self.rho) + phi = self.phi() + pk = phi * k_arr + return 0.5 * theta * (1 + rho * pk + np.sqrt((pk + rho) ** 2 + 1 - rho**2)) + + def iv( + self, + k: Annotated[ArrayLike, Doc("Log-moneyness log(K/F), scalar or array")], + ttm: Annotated[float, Doc("Time to maturity in years")], + ) -> np.ndarray: + r"""Implied volatility $\sigma(k) = \sqrt{w(k) / \tau}$. + + Returns an array of the same shape as $k$. The SSVI total variance is + strictly positive for $|\rho| < 1$, so no clipping is required. + """ + return np.sqrt(self.total_variance(k) / ttm) + + def no_butterfly_arbitrage(self) -> bool: + r"""True if the slice satisfies the sufficient conditions for absence + of butterfly arbitrage. + + The conditions, from Theorem 4.2 of + [Gatheral and Jacquier (2014)](../../bibliography.md#gatheral_jacquier), are: + + \begin{equation} + \begin{aligned} + \theta \varphi(\theta) (1 + |\rho|) &< 4 \\ + \theta \varphi(\theta)^2 (1 + |\rho|) &\leq 4 + \end{aligned} + \end{equation} + """ + theta = float(self.theta) + rho = abs(float(self.rho)) + phi = self.phi() + return theta * phi * (1 + rho) < 4 and theta * phi * phi * (1 + rho) <= 4 + + @classmethod + def fit( + cls, + k: Annotated[ArrayLike, Doc("Log-moneyness log(K/F) for each option")], + iv: Annotated[ArrayLike, Doc("Observed implied volatilities")], + ttm: Annotated[float, Doc("Time to maturity in years")], + gamma: Annotated[ + float, + Doc( + "Fixed power law exponent. A single slice only identifies the " + "value of the shape function, not the split between eta and " + "gamma, so gamma is held fixed" + ), + ] = 0.5, + ) -> SSVI: + """Fit an SSVI slice to observed implied volatilities via non-linear + least squares. + + Minimises the sum of squared differences between observed and model + total variances, fitting rho, eta and theta with gamma held fixed. + """ + k_arr = np.asarray(k, dtype=float) + iv_arr = np.asarray(iv, dtype=float) + w_obs = iv_arr**2 * ttm + + atm_var = float(np.interp(0.0, k_arr, w_obs)) if k_arr.size else w_obs.mean() + x0 = [0.0, 1.0, max(atm_var, 1e-4)] + + def residuals(x: list[float]) -> np.ndarray: + rho, eta, theta = x + phi = eta / (theta**gamma * (1 + theta) ** (1 - gamma)) + pk = phi * k_arr + w_fit = 0.5 * theta * (1 + rho * pk + np.sqrt((pk + rho) ** 2 + 1 - rho**2)) + return w_fit - w_obs + + result = least_squares( + residuals, + x0, + bounds=( + [-1.0 + 1e-6, 1e-6, 1e-8], + [1.0 - 1e-6, np.inf, np.inf], + ), + ) + rho, eta, theta = result.x + return cls( + rho=to_decimal(round(rho, 10)), + eta=to_decimal(round(eta, 10)), + gamma=to_decimal(round(gamma, 10)), + theta=to_decimal(round(theta, 10)), + ) + + @classmethod + def fit_surface( + cls, + slices: Annotated[ + Sequence[tuple[ArrayLike, ArrayLike, float]], + Doc( + "One (log-moneyness, implied volatilities, time to maturity) " + "tuple per maturity slice" + ), + ], + ) -> list[SSVI]: + """Jointly fit the SSVI surface to several maturity slices via + non-linear least squares. + + The global parameters rho, eta and gamma are shared across all slices, + while each slice has its own at-the-money total variance theta. + Returns one [SSVI][quantflow.options.ssvi.SSVI] instance per input + slice, in the same order. + """ + data = [] + thetas0 = [] + for k, iv, ttm in slices: + k_arr = np.asarray(k, dtype=float) + iv_arr = np.asarray(iv, dtype=float) + w_obs = iv_arr**2 * ttm + data.append((k_arr, w_obs)) + atm = float(np.interp(0.0, k_arr, w_obs)) if k_arr.size else w_obs.mean() + thetas0.append(max(atm, 1e-4)) + + x0 = [0.0, 1.0, 0.5, *thetas0] + + def residuals(x: np.ndarray) -> np.ndarray: + rho, eta, gamma = x[0], x[1], x[2] + res = [] + for (k_arr, w_obs), theta in zip(data, x[3:]): + phi = eta / (theta**gamma * (1 + theta) ** (1 - gamma)) + pk = phi * k_arr + w_fit = ( + 0.5 * theta * (1 + rho * pk + np.sqrt((pk + rho) ** 2 + 1 - rho**2)) + ) + res.append(w_fit - w_obs) + return np.concatenate(res) + + n = len(data) + result = least_squares( + residuals, + x0, + bounds=( + [-1.0 + 1e-6, 1e-6, 1e-6] + [1e-8] * n, + [1.0 - 1e-6, np.inf, 1.0] + [np.inf] * n, + ), + ) + rho, eta, gamma = result.x[:3] + return [ + cls( + rho=to_decimal(round(rho, 10)), + eta=to_decimal(round(eta, 10)), + gamma=to_decimal(round(gamma, 10)), + theta=to_decimal(round(theta, 10)), + ) + for theta in result.x[3:] + ] diff --git a/quantflow/options/surface.py b/quantflow/options/surface.py index bb3feae6..7142cceb 100644 --- a/quantflow/options/surface.py +++ b/quantflow/options/surface.py @@ -235,12 +235,14 @@ def is_in_the_money(self, forward: Decimal) -> bool: def calculate_price(self) -> Self: price = Decimal( sigfig( - black_price( - np.asarray(self.log_strike), - self.iv, - self.ttm, - 1 if self.option_type.is_call() else -1, - ).sum(), + float( + black_price( + np.asarray(self.log_strike), + self.iv, + self.ttm, + 1 if self.option_type.is_call() else -1, + ).sum() + ), 8, ) ) diff --git a/quantflow/options/svi.py b/quantflow/options/svi.py index e8aeb6b9..4a1e96da 100644 --- a/quantflow/options/svi.py +++ b/quantflow/options/svi.py @@ -13,7 +13,9 @@ class SVI(BaseModel, extra="forbid"): r"""Gatheral's Stochastic Volatility Inspired (SVI) parametrisation of the - implied volatility smile. + implied volatility smile, introduced in + [Gatheral (2004)](../../bibliography.md#gatheral_svi) and further analysed + in [Gatheral and Jacquier (2014)](../../bibliography.md#gatheral_jacquier). The raw SVI parametrisation expresses the total implied variance $w(k) = \sigma^2(k) \cdot \tau$ as a function of log-strike @@ -22,14 +24,6 @@ class SVI(BaseModel, extra="forbid"): \begin{equation} w(k) = a + b \left[\rho (k - m) + \sqrt{(k - m)^2 + \theta^2}\right] \end{equation} - - References: - Gatheral, J. (2004). A parsimonious arbitrage-free implied volatility - parametrization with application to the valuation of volatility - derivatives. Global Derivatives, Madrid. - - Gatheral, J. and Jacquier, A. (2014). Arbitrage-free SVI volatility - surfaces. Quantitative Finance, 14(1), 59-71. """ a: Decimal = Field( diff --git a/quantflow/ta/paths.py b/quantflow/ta/paths.py index 8f3d8147..6a8372ce 100755 --- a/quantflow/ta/paths.py +++ b/quantflow/ta/paths.py @@ -142,7 +142,7 @@ def hurst_exponent( ts = self.time_steps // 2 n = min(steps or ts, 100) lags = [] - tau = [] + tau: list[float] = [] for lag in range(2, n): variances = np.var(self.data[lag:, :] - self.data[:-lag, :], axis=0) tau.extend(variances) diff --git a/quantflow_tests/test_ssvi.py b/quantflow_tests/test_ssvi.py new file mode 100644 index 00000000..b767eec5 --- /dev/null +++ b/quantflow_tests/test_ssvi.py @@ -0,0 +1,208 @@ +from __future__ import annotations + +from decimal import Decimal + +import numpy as np +import pytest + +from quantflow.options.ssvi import SSVI + +TTM = 1.0 +K = np.linspace(-0.5, 0.5, 21) + + +def make_ssvi(**kwargs: float) -> SSVI: + defaults = dict(rho=-0.3, eta=1.0, gamma=0.5, theta=0.04) + defaults.update(kwargs) + return SSVI(**{k: Decimal(str(v)) for k, v in defaults.items()}) + + +# --------------------------------------------------------------------------- +# phi +# --------------------------------------------------------------------------- + + +def test_phi_power_law() -> None: + ssvi = make_ssvi(eta=1.0, gamma=0.5, theta=0.04) + expected = 1.0 / (0.04**0.5 * 1.04**0.5) + assert ssvi.phi() == pytest.approx(expected) + + +def test_phi_decreases_with_theta() -> None: + phi_short = make_ssvi(theta=0.01).phi() + phi_long = make_ssvi(theta=0.25).phi() + assert phi_short > phi_long + + +# --------------------------------------------------------------------------- +# total_variance +# --------------------------------------------------------------------------- + + +def test_total_variance_is_positive() -> None: + ssvi = make_ssvi() + w = ssvi.total_variance(K) + assert np.all(w > 0) + + +def test_total_variance_atm_equals_theta() -> None: + ssvi = make_ssvi() + assert ssvi.total_variance(0.0) == pytest.approx(float(ssvi.theta)) + + +def test_total_variance_symmetric_when_rho_zero() -> None: + ssvi = make_ssvi(rho=0.0) + w = ssvi.total_variance(K) + assert np.allclose(w, w[::-1]) + + +def test_total_variance_skew_sign() -> None: + # negative rho => put wing (k < 0) above call wing (k > 0) + ssvi = make_ssvi(rho=-0.5) + w = ssvi.total_variance(K) + assert w[0] > w[-1] + + +def test_total_variance_scalar_input() -> None: + ssvi = make_ssvi() + w = ssvi.total_variance(0.0) + assert w.shape == (1,) or w.ndim == 0 + + +# --------------------------------------------------------------------------- +# iv +# --------------------------------------------------------------------------- + + +def test_iv_positive() -> None: + ssvi = make_ssvi() + iv = ssvi.iv(K, TTM) + assert np.all(iv > 0) + + +def test_iv_consistent_with_total_variance() -> None: + ssvi = make_ssvi() + iv = ssvi.iv(K, TTM) + w = ssvi.total_variance(K) + assert np.allclose(iv**2 * TTM, w) + + +def test_iv_scales_with_ttm() -> None: + ssvi = make_ssvi() + iv1 = ssvi.iv(K, 1.0) + iv2 = ssvi.iv(K, 0.25) + # same total variance, shorter ttm => higher iv + assert np.all(iv2 > iv1) + + +# --------------------------------------------------------------------------- +# no_butterfly_arbitrage +# --------------------------------------------------------------------------- + + +def test_no_butterfly_arbitrage_typical_params() -> None: + assert make_ssvi().no_butterfly_arbitrage() + + +def test_butterfly_arbitrage_violated_for_large_eta() -> None: + assert not make_ssvi(eta=15.0, theta=1.0).no_butterfly_arbitrage() + + +# --------------------------------------------------------------------------- +# fit +# --------------------------------------------------------------------------- + + +def test_fit_recovers_parameters() -> None: + true_ssvi = make_ssvi(rho=-0.2, eta=0.8, gamma=0.5, theta=0.05) + iv_obs = true_ssvi.iv(K, TTM) + fitted = SSVI.fit(K, iv_obs, TTM) + assert float(fitted.rho) == pytest.approx(float(true_ssvi.rho), abs=1e-4) + assert float(fitted.eta) == pytest.approx(float(true_ssvi.eta), abs=1e-4) + assert float(fitted.theta) == pytest.approx(float(true_ssvi.theta), abs=1e-4) + + +def test_fit_reproduces_ivs() -> None: + true_ssvi = make_ssvi() + iv_obs = true_ssvi.iv(K, TTM) + fitted = SSVI.fit(K, iv_obs, TTM) + iv_fit = fitted.iv(K, TTM) + assert np.allclose(iv_fit, iv_obs, atol=1e-5) + + +def test_fit_respects_bounds() -> None: + true_ssvi = make_ssvi() + iv_obs = true_ssvi.iv(K, TTM) + fitted = SSVI.fit(K, iv_obs, TTM) + assert -1 < float(fitted.rho) < 1 + assert float(fitted.eta) > 0 + assert float(fitted.theta) > 0 + + +def test_fit_fixed_gamma() -> None: + true_ssvi = make_ssvi() + iv_obs = true_ssvi.iv(K, TTM) + fitted = SSVI.fit(K, iv_obs, TTM, gamma=0.4) + assert float(fitted.gamma) == pytest.approx(0.4) + + +def test_fit_skewed_smile() -> None: + true_ssvi = make_ssvi(rho=-0.6) + iv_obs = true_ssvi.iv(K, TTM) + fitted = SSVI.fit(K, iv_obs, TTM) + iv_fit = fitted.iv(K, TTM) + assert np.allclose(iv_fit, iv_obs, atol=1e-5) + + +# --------------------------------------------------------------------------- +# fit_surface +# --------------------------------------------------------------------------- + + +def _synthetic_surface( + rho: float, eta: float, gamma: float, thetas: list[float], ttms: list[float] +) -> list[tuple[np.ndarray, np.ndarray, float]]: + slices = [] + for theta, ttm in zip(thetas, ttms): + ssvi = make_ssvi(rho=rho, eta=eta, gamma=gamma, theta=theta) + slices.append((K, ssvi.iv(K, ttm), ttm)) + return slices + + +def test_fit_surface_recovers_parameters() -> None: + thetas = [0.02, 0.05, 0.09] + ttms = [0.25, 0.5, 1.0] + slices = _synthetic_surface(-0.3, 0.9, 0.45, thetas, ttms) + fitted = SSVI.fit_surface(slices) + assert len(fitted) == len(slices) + for ssvi in fitted: + assert float(ssvi.rho) == pytest.approx(-0.3, abs=1e-3) + assert float(ssvi.eta) == pytest.approx(0.9, abs=1e-2) + assert float(ssvi.gamma) == pytest.approx(0.45, abs=1e-2) + for ssvi, theta in zip(fitted, thetas): + assert float(ssvi.theta) == pytest.approx(theta, abs=1e-4) + + +def test_fit_surface_shares_global_parameters() -> None: + slices = _synthetic_surface(-0.2, 1.1, 0.5, [0.03, 0.06], [0.5, 1.0]) + fitted = SSVI.fit_surface(slices) + assert fitted[0].rho == fitted[1].rho + assert fitted[0].eta == fitted[1].eta + assert fitted[0].gamma == fitted[1].gamma + + +def test_fit_surface_reproduces_ivs() -> None: + slices = _synthetic_surface(-0.4, 0.8, 0.5, [0.02, 0.05, 0.09], [0.25, 0.5, 1.0]) + fitted = SSVI.fit_surface(slices) + for ssvi, (k, iv_obs, ttm) in zip(fitted, slices): + assert np.allclose(ssvi.iv(k, ttm), iv_obs, atol=1e-5) + + +def test_fit_surface_thetas_increasing() -> None: + # increasing ATM total variance => no calendar spread arbitrage + slices = _synthetic_surface(-0.3, 0.9, 0.5, [0.02, 0.05, 0.09], [0.25, 0.5, 1.0]) + fitted = SSVI.fit_surface(slices) + thetas = [float(ssvi.theta) for ssvi in fitted] + assert thetas == sorted(thetas) + for ssvi in 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quantflow.options.ssvi.SSVI diff --git a/quantflow/options/ssvi.py b/quantflow/options/ssvi.py index 7333aeff..0ccfd3fc 100644 --- a/quantflow/options/ssvi.py +++ b/quantflow/options/ssvi.py @@ -2,44 +2,132 @@ from collections.abc import Sequence from decimal import Decimal +from typing import Self import numpy as np from numpy.typing import ArrayLike -from pydantic import BaseModel, Field +from pydantic import BaseModel, Field, PrivateAttr +from scipy.interpolate import PchipInterpolator from scipy.optimize import least_squares from typing_extensions import Annotated, Doc -from quantflow.utils.numbers import ONE, ZERO, to_decimal +from quantflow.utils.numbers import ONE, ZERO, DecimalNumber, to_decimal +from quantflow.utils.types import FloatArray, FloatArrayLike, maybe_float + + +class VarianceCurve(BaseModel, extra="forbid"): + r"""Monotone interpolated total ATM variance curve. + + The curve stores total ATM variances $\theta_i = w(0, \tau_i)$ at strictly + increasing maturities $\tau_i$. Static arbitrage requires the total ATM + variance curve to be non decreasing: + + \begin{equation} + \frac{d\theta}{d\tau} \geq 0 + \end{equation} + + The nodes are validated to satisfy this condition, then interpolated with a + shape preserving cubic Hermite spline (PCHIP). For non decreasing nodes, + PCHIP is monotone between nodes and therefore preserves the condition. + """ + + ttm: list[DecimalNumber] = Field( + description="Times to maturity in years, strictly increasing", + ) + theta: list[DecimalNumber] = Field( + description=( + "Total ATM variances at each maturity, same length as ttm, positive " + "and non decreasing" + ), + ) + + _ttm: FloatArray = PrivateAttr(default_factory=lambda: np.empty(0)) + _theta: FloatArray = PrivateAttr(default_factory=lambda: np.empty(0)) + + def model_post_init(self, context: object) -> None: + """Cache nodes and validate the no calendar arbitrage condition.""" + if len(self.ttm) != len(self.theta): + raise ValueError("ttm and theta must have equal length") + if not self.ttm: + raise ValueError("at least one variance curve node is required") + ttm = np.array([float(t) for t in self.ttm], dtype=float) + theta = np.array([float(t) for t in self.theta], dtype=float) + if np.any(ttm <= 0): + raise ValueError("ttm nodes must be strictly positive") + if np.any(np.diff(ttm) <= 0): + raise ValueError("ttm nodes must be strictly increasing") + if np.any(theta <= 0): + raise ValueError("theta nodes must be strictly positive") + if np.any(np.diff(theta) < 0): + raise ValueError("theta nodes must be non decreasing") + self._ttm = ttm + self._theta = theta + + def total_variance( + self, + ttm: Annotated[ArrayLike, Doc("Time to maturity in years, scalar or array")], + ) -> FloatArrayLike: + r"""Interpolated total ATM variance $\theta(\tau)$.""" + tau = np.maximum(np.asarray(ttm, dtype=float), 0.0) + if self._ttm.size < 2: + theta = np.full(np.shape(tau), self._theta[0]) + else: + clamped = np.clip(tau, self._ttm[0], self._ttm[-1]) + theta = PchipInterpolator(self._ttm, self._theta)(clamped) + return maybe_float(np.asarray(theta, dtype=float)) + + def derivative( + self, + ttm: Annotated[ArrayLike, Doc("Time to maturity in years, scalar or array")], + ) -> FloatArrayLike: + r"""Derivative $d\theta / d\tau$ of the interpolated variance curve.""" + tau = np.maximum(np.asarray(ttm, dtype=float), 0.0) + if self._ttm.size < 2: + derivative = np.zeros_like(tau) + else: + clamped = np.clip(tau, self._ttm[0], self._ttm[-1]) + inside = (tau >= self._ttm[0]) & (tau <= self._ttm[-1]) + derivative = PchipInterpolator(self._ttm, self._theta).derivative()(clamped) + derivative = np.where(inside, derivative, 0.0) + return maybe_float(np.asarray(derivative, dtype=float)) + + def is_non_decreasing(self) -> bool: + r"""True if the variance nodes satisfy $d\theta / d\tau \geq 0$.""" + return bool(np.all(np.diff(self._theta) >= 0)) class SSVI(BaseModel, extra="forbid"): - r"""Surface SVI (SSVI) parametrisation of the implied volatility smile, + r"""Surface SVI (SSVI) parametrisation of the implied volatility surface, introduced by [Gatheral and Jacquier (2014)](../../bibliography.md#gatheral_jacquier). The SSVI parametrisation expresses the total implied variance $w(k) = \sigma^2(k) \cdot \tau$ as a function of log-strike - $k = \log(K/F)$ and the at-the-money total variance $\theta$: + $k = \log(K/F)$ and the total ATM variance $\theta_\tau = w(0, \tau)$: \begin{equation} - w(k) = \frac{\theta}{2}\left[1 + \rho \varphi(\theta) k - + \sqrt{\left(\varphi(\theta) k + \rho\right)^2 + 1 - \rho^2}\right] + w(k, \tau) = \frac{\theta_\tau}{2}\left[1 + \rho \varphi(\theta_\tau) k + + \sqrt{\left(\varphi(\theta_\tau) k + \rho\right)^2 + 1 - \rho^2}\right] \end{equation} The shape function $\varphi$ uses the power law form: \begin{equation} - \varphi(\theta) = \frac{\eta}{\theta^\gamma (1 + \theta)^{1 - \gamma}} + \varphi(\theta_\tau) = + \frac{\eta}{\theta_\tau^\gamma (1 + \theta_\tau)^{1 - \gamma}} \end{equation} - Each instance represents a single maturity slice through its $\theta$ - parameter, while $\rho$, $\eta$ and $\gamma$ are global parameters shared - across the whole surface. + Each instance represents the surface through shared parameters $\rho$, + $\eta$ and $\gamma$ together with a monotone total ATM variance curve + $\theta(\tau)$. A single maturity slice is represented by a one node + [VarianceCurve][.VarianceCurve]. Use [fit][.fit] to calibrate a single slice, or [fit_surface][.fit_surface] - to jointly calibrate the global parameters across several maturities. A - surface built this way is free of calendar spread arbitrage provided the - fitted $\theta$ are non decreasing in maturity. + to jointly calibrate the global parameters and variance curve across + several maturities. A surface built this way is free of static arbitrage + provided each slice satisfies + [no_butterfly_arbitrage][.no_butterfly_arbitrage] and the ATM variance + curve is non decreasing in maturity, $d\theta / d\tau \geq 0$. """ rho: Decimal = Field( @@ -68,51 +156,66 @@ class SSVI(BaseModel, extra="forbid"): "grows. Must lie in $(0, 1]$." ), ) - theta: Decimal = Field( - gt=ZERO, + variance_curve: VarianceCurve = Field( description=( - "At-the-money total implied variance of the slice, " - "$\\theta = \\sigma_{ATM}^2 \\cdot \\tau$. This is the only " - "maturity-dependent parameter. Must be strictly positive." + "Monotone interpolated total ATM variance curve " + "$\\theta(\\tau) = \\sigma_{ATM}^2 \\cdot \\tau$." ), ) - def phi(self) -> float: - r"""Power law shape function $\varphi(\theta)$ evaluated at the slice - at-the-money total variance $\theta$.""" - theta = float(self.theta) + @property + def theta(self) -> Decimal: + """Total ATM variance of the first curve node.""" + return self.variance_curve.theta[0] + + def phi( + self, + ttm: Annotated[ArrayLike, Doc("Time to maturity in years, scalar or array")], + ) -> FloatArrayLike: + r"""Power law shape function $\varphi(\theta(\tau))$.""" + theta = np.asarray(self.variance_curve.total_variance(ttm), dtype=float) eta = float(self.eta) gamma = float(self.gamma) - return eta / (theta**gamma * (1 + theta) ** (1 - gamma)) + phi = eta / (theta**gamma * (1 + theta) ** (1 - gamma)) + return maybe_float(np.asarray(phi, dtype=float)) def total_variance( self, k: Annotated[ArrayLike, Doc("Log-moneyness log(K/F), scalar or array")], - ) -> np.ndarray: - r"""Total implied variance $w(k)$. + ttm: Annotated[ArrayLike, Doc("Time to maturity in years, scalar or array")], + ) -> FloatArrayLike: + r"""Total implied variance $w(k, \tau)$. - Returns an array of the same shape as $k$. + Returns an array broadcast from the shapes of $k$ and $\tau$. """ k_arr = np.asarray(k, dtype=float) - theta = float(self.theta) + theta = np.asarray(self.variance_curve.total_variance(ttm), dtype=float) rho = float(self.rho) - phi = self.phi() + phi = np.asarray(self.phi(ttm), dtype=float) pk = phi * k_arr - return 0.5 * theta * (1 + rho * pk + np.sqrt((pk + rho) ** 2 + 1 - rho**2)) + w = 0.5 * theta * (1 + rho * pk + np.sqrt((pk + rho) ** 2 + 1 - rho**2)) + return maybe_float(np.asarray(w, dtype=float)) def iv( self, k: Annotated[ArrayLike, Doc("Log-moneyness log(K/F), scalar or array")], - ttm: Annotated[float, Doc("Time to maturity in years")], - ) -> np.ndarray: - r"""Implied volatility $\sigma(k) = \sqrt{w(k) / \tau}$. + ttm: Annotated[ArrayLike, Doc("Time to maturity in years, scalar or array")], + ) -> FloatArrayLike: + r"""Implied volatility $\sigma(k, \tau) = \sqrt{w(k, \tau) / \tau}$. Returns an array of the same shape as $k$. The SSVI total variance is strictly positive for $|\rho| < 1$, so no clipping is required. """ - return np.sqrt(self.total_variance(k) / ttm) + tau = np.asarray(ttm, dtype=float) + return maybe_float(np.asarray(np.sqrt(self.total_variance(k, ttm) / tau))) - def no_butterfly_arbitrage(self) -> bool: + def no_butterfly_arbitrage( + self, + ttm: Annotated[ + float | None, + Doc("Optional maturity to check. All curve nodes are checked when omitted"), + ] = None, + ) -> bool: r"""True if the slice satisfies the sufficient conditions for absence of butterfly arbitrage. @@ -126,10 +229,18 @@ def no_butterfly_arbitrage(self) -> bool: \end{aligned} \end{equation} """ - theta = float(self.theta) rho = abs(float(self.rho)) - phi = self.phi() - return theta * phi * (1 + rho) < 4 and theta * phi * phi * (1 + rho) <= 4 + ttms = [ttm] if ttm is not None else [float(t) for t in self.variance_curve.ttm] + for tau in ttms: + theta = float(self.variance_curve.total_variance(tau)) + phi = float(self.phi(tau)) + if not (theta * phi * (1 + rho) < 4 and theta * phi * phi * (1 + rho) <= 4): + return False + return True + + def no_static_arbitrage(self) -> bool: + """True if the surface satisfies the implemented static checks.""" + return self.variance_curve.is_non_decreasing() and self.no_butterfly_arbitrage() @classmethod def fit( @@ -145,7 +256,7 @@ def fit( "gamma, so gamma is held fixed" ), ] = 0.5, - ) -> SSVI: + ) -> Self: """Fit an SSVI slice to observed implied volatilities via non-linear least squares. @@ -179,7 +290,10 @@ def residuals(x: list[float]) -> np.ndarray: rho=to_decimal(round(rho, 10)), eta=to_decimal(round(eta, 10)), gamma=to_decimal(round(gamma, 10)), - theta=to_decimal(round(theta, 10)), + variance_curve=VarianceCurve( + ttm=[to_decimal(round(ttm, 10))], + theta=[to_decimal(round(theta, 10))], + ), ) @classmethod @@ -192,31 +306,49 @@ def fit_surface( "tuple per maturity slice" ), ], - ) -> list[SSVI]: - """Jointly fit the SSVI surface to several maturity slices via + ) -> Self: + r"""Jointly fit the SSVI surface to several maturity slices via non-linear least squares. The global parameters rho, eta and gamma are shared across all slices, - while each slice has its own at-the-money total variance theta. - Returns one [SSVI][quantflow.options.ssvi.SSVI] instance per input - slice, in the same order. + while the maturity dependent total ATM variance is represented by a + monotone [VarianceCurve][quantflow.options.ssvi.VarianceCurve]. The + fitted variables are the shared parameters $\rho$, $\eta$ and $\gamma$, + plus positive increments for the total ATM variance curve. The + increment parameterisation guarantees $d\theta / d\tau \geq 0$. """ + if not slices: + raise ValueError("at least one maturity slice is required") data = [] thetas0 = [] for k, iv, ttm in slices: k_arr = np.asarray(k, dtype=float) iv_arr = np.asarray(iv, dtype=float) w_obs = iv_arr**2 * ttm - data.append((k_arr, w_obs)) + data.append((k_arr, w_obs, float(ttm))) atm = float(np.interp(0.0, k_arr, w_obs)) if k_arr.size else w_obs.mean() thetas0.append(max(atm, 1e-4)) + order = np.argsort([ttm for _, _, ttm in data]) + data = [data[i] for i in order] + theta_nodes = np.array([thetas0[i] for i in order], dtype=float) + theta0 = theta_nodes[0] + increments0 = np.maximum(np.diff(theta_nodes), 0.0) + ttms = [data[i][2] for i in range(len(data))] + + x0 = [0.0, 1.0, 0.5, theta0, *increments0] - x0 = [0.0, 1.0, 0.5, *thetas0] + def theta_from_params(x: np.ndarray) -> np.ndarray: + theta = np.empty(len(data), dtype=float) + theta[0] = x[3] + if theta.size > 1: + theta[1:] = theta[0] + np.cumsum(x[4:]) + return theta def residuals(x: np.ndarray) -> np.ndarray: rho, eta, gamma = x[0], x[1], x[2] + theta_nodes = theta_from_params(x) res = [] - for (k_arr, w_obs), theta in zip(data, x[3:]): + for (k_arr, w_obs, _), theta in zip(data, theta_nodes): phi = eta / (theta**gamma * (1 + theta) ** (1 - gamma)) pk = phi * k_arr w_fit = ( @@ -230,17 +362,18 @@ def residuals(x: np.ndarray) -> np.ndarray: residuals, x0, bounds=( - [-1.0 + 1e-6, 1e-6, 1e-6] + [1e-8] * n, - [1.0 - 1e-6, np.inf, 1.0] + [np.inf] * n, + [-1.0 + 1e-6, 1e-6, 1e-6, 1e-8] + [0.0] * (n - 1), + [1.0 - 1e-6, np.inf, 1.0, np.inf] + [np.inf] * (n - 1), ), ) rho, eta, gamma = result.x[:3] - return [ - cls( - rho=to_decimal(round(rho, 10)), - eta=to_decimal(round(eta, 10)), - gamma=to_decimal(round(gamma, 10)), - theta=to_decimal(round(theta, 10)), - ) - for theta in result.x[3:] - ] + theta = theta_from_params(result.x) + return cls( + rho=to_decimal(round(rho, 10)), + eta=to_decimal(round(eta, 10)), + gamma=to_decimal(round(gamma, 10)), + variance_curve=VarianceCurve( + ttm=[to_decimal(round(ttm, 10)) for ttm in ttms], + theta=[to_decimal(round(value, 10)) for value in theta], + ), + ) diff --git a/quantflow/options/svi.py b/quantflow/options/svi.py index 4a1e96da..db2e8f50 100644 --- a/quantflow/options/svi.py +++ b/quantflow/options/svi.py @@ -18,8 +18,8 @@ class SVI(BaseModel, extra="forbid"): in [Gatheral and Jacquier (2014)](../../bibliography.md#gatheral_jacquier). The raw SVI parametrisation expresses the total implied variance - $w(k) = \sigma^2(k) \cdot \tau$ as a function of log-strike - $k = \log(K/F)$: + $w(k) = \sigma^2(k) \cdot \tau$ as a function of + [log-strike](../../glossary.md#log-strike) $k = \log(K/F)$: \begin{equation} w(k) = a + b \left[\rho (k - m) + \sqrt{(k - m)^2 + \theta^2}\right] diff --git a/quantflow_tests/test_ssvi.py b/quantflow_tests/test_ssvi.py index b767eec5..890a18e6 100644 --- a/quantflow_tests/test_ssvi.py +++ b/quantflow_tests/test_ssvi.py @@ -4,8 +4,9 @@ import numpy as np import pytest +from pydantic import ValidationError -from quantflow.options.ssvi import SSVI +from quantflow.options.ssvi import SSVI, VarianceCurve TTM = 1.0 K = np.linspace(-0.5, 0.5, 21) @@ -14,7 +15,14 @@ def make_ssvi(**kwargs: float) -> SSVI: defaults = dict(rho=-0.3, eta=1.0, gamma=0.5, theta=0.04) defaults.update(kwargs) - return SSVI(**{k: Decimal(str(v)) for k, v in defaults.items()}) + theta = defaults.pop("theta") + return SSVI( + **{k: Decimal(str(v)) for k, v in defaults.items()}, + variance_curve=VarianceCurve( + ttm=[Decimal(str(TTM))], + theta=[Decimal(str(theta))], + ), + ) # --------------------------------------------------------------------------- @@ -25,15 +33,47 @@ def make_ssvi(**kwargs: float) -> SSVI: def test_phi_power_law() -> None: ssvi = make_ssvi(eta=1.0, gamma=0.5, theta=0.04) expected = 1.0 / (0.04**0.5 * 1.04**0.5) - assert ssvi.phi() == pytest.approx(expected) + assert ssvi.phi(TTM) == pytest.approx(expected) def test_phi_decreases_with_theta() -> None: - phi_short = make_ssvi(theta=0.01).phi() - phi_long = make_ssvi(theta=0.25).phi() + phi_short = make_ssvi(theta=0.01).phi(TTM) + phi_long = make_ssvi(theta=0.25).phi(TTM) assert phi_short > phi_long +# --------------------------------------------------------------------------- +# variance curve +# --------------------------------------------------------------------------- + + +def test_variance_curve_interpolates_nodes() -> None: + curve = VarianceCurve( + ttm=[Decimal("0.25"), Decimal("0.5"), Decimal("1.0")], + theta=[Decimal("0.02"), Decimal("0.05"), Decimal("0.09")], + ) + assert curve.total_variance(0.5) == pytest.approx(0.05) + + +def test_variance_curve_is_monotone_between_nodes() -> None: + curve = VarianceCurve( + ttm=[Decimal("0.25"), Decimal("0.5"), Decimal("1.0")], + theta=[Decimal("0.02"), Decimal("0.05"), Decimal("0.09")], + ) + grid = np.linspace(0.25, 1.0, 101) + theta = np.asarray(curve.total_variance(grid), dtype=float) + assert np.all(np.diff(theta) >= 0) + assert np.all(np.asarray(curve.derivative(grid), dtype=float) >= -1e-12) + + +def test_variance_curve_rejects_decreasing_theta() -> None: + with pytest.raises(ValidationError): + VarianceCurve( + ttm=[Decimal("0.25"), Decimal("0.5"), Decimal("1.0")], + theta=[Decimal("0.02"), Decimal("0.05"), Decimal("0.04")], + ) + + # --------------------------------------------------------------------------- # total_variance # --------------------------------------------------------------------------- @@ -41,32 +81,32 @@ def test_phi_decreases_with_theta() -> None: def test_total_variance_is_positive() -> None: ssvi = make_ssvi() - w = ssvi.total_variance(K) + w = ssvi.total_variance(K, TTM) assert np.all(w > 0) def test_total_variance_atm_equals_theta() -> None: ssvi = make_ssvi() - assert ssvi.total_variance(0.0) == pytest.approx(float(ssvi.theta)) + assert ssvi.total_variance(0.0, TTM) == pytest.approx(float(ssvi.theta)) def test_total_variance_symmetric_when_rho_zero() -> None: ssvi = make_ssvi(rho=0.0) - w = ssvi.total_variance(K) + w = np.asarray(ssvi.total_variance(K, TTM), dtype=float) assert np.allclose(w, w[::-1]) def test_total_variance_skew_sign() -> None: # negative rho => put wing (k < 0) above call wing (k > 0) ssvi = make_ssvi(rho=-0.5) - w = ssvi.total_variance(K) + w = np.asarray(ssvi.total_variance(K, TTM), dtype=float) assert w[0] > w[-1] def test_total_variance_scalar_input() -> None: ssvi = make_ssvi() - w = ssvi.total_variance(0.0) - assert w.shape == (1,) or w.ndim == 0 + w = ssvi.total_variance(0.0, TTM) + assert isinstance(w, float) # --------------------------------------------------------------------------- @@ -83,7 +123,7 @@ def test_iv_positive() -> None: def test_iv_consistent_with_total_variance() -> None: ssvi = make_ssvi() iv = ssvi.iv(K, TTM) - w = ssvi.total_variance(K) + w = ssvi.total_variance(K, TTM) assert np.allclose(iv**2 * TTM, w) @@ -165,7 +205,7 @@ def _synthetic_surface( slices = [] for theta, ttm in zip(thetas, ttms): ssvi = make_ssvi(rho=rho, eta=eta, gamma=gamma, theta=theta) - slices.append((K, ssvi.iv(K, ttm), ttm)) + slices.append((K, np.asarray(ssvi.iv(K, ttm), dtype=float), ttm)) return slices @@ -174,35 +214,42 @@ def test_fit_surface_recovers_parameters() -> None: ttms = [0.25, 0.5, 1.0] slices = _synthetic_surface(-0.3, 0.9, 0.45, thetas, ttms) fitted = SSVI.fit_surface(slices) - assert len(fitted) == len(slices) - for ssvi in fitted: - assert float(ssvi.rho) == pytest.approx(-0.3, abs=1e-3) - assert float(ssvi.eta) == pytest.approx(0.9, abs=1e-2) - assert float(ssvi.gamma) == pytest.approx(0.45, abs=1e-2) - for ssvi, theta in zip(fitted, thetas): - assert float(ssvi.theta) == pytest.approx(theta, abs=1e-4) + assert float(fitted.rho) == pytest.approx(-0.3, abs=1e-3) + assert float(fitted.eta) == pytest.approx(0.9, abs=1e-2) + assert float(fitted.gamma) == pytest.approx(0.45, abs=1e-2) + for ttm, theta in zip(ttms, thetas): + assert fitted.variance_curve.total_variance(ttm) == pytest.approx( + theta, abs=1e-4 + ) def test_fit_surface_shares_global_parameters() -> None: slices = _synthetic_surface(-0.2, 1.1, 0.5, [0.03, 0.06], [0.5, 1.0]) fitted = SSVI.fit_surface(slices) - assert fitted[0].rho == fitted[1].rho - assert fitted[0].eta == fitted[1].eta - assert fitted[0].gamma == fitted[1].gamma + assert len(set(fitted.variance_curve.theta)) == 2 + assert float(fitted.rho) == pytest.approx(-0.2, abs=1e-3) + assert float(fitted.eta) == pytest.approx(1.1, abs=1e-2) + assert float(fitted.gamma) == pytest.approx(0.5, abs=1e-2) def test_fit_surface_reproduces_ivs() -> None: slices = _synthetic_surface(-0.4, 0.8, 0.5, [0.02, 0.05, 0.09], [0.25, 0.5, 1.0]) fitted = SSVI.fit_surface(slices) - for ssvi, (k, iv_obs, ttm) in zip(fitted, slices): - assert np.allclose(ssvi.iv(k, ttm), iv_obs, atol=1e-5) + for k, iv_obs, ttm in slices: + assert np.allclose(fitted.iv(k, ttm), iv_obs, atol=1e-5) def test_fit_surface_thetas_increasing() -> None: # increasing ATM total variance => no calendar spread arbitrage slices = _synthetic_surface(-0.3, 0.9, 0.5, [0.02, 0.05, 0.09], [0.25, 0.5, 1.0]) fitted = SSVI.fit_surface(slices) - thetas = [float(ssvi.theta) for ssvi in fitted] + thetas = [float(theta) for theta in fitted.variance_curve.theta] + assert thetas == sorted(thetas) + assert fitted.no_static_arbitrage() + + +def test_fit_surface_enforces_monotone_variance_curve() -> None: + slices = _synthetic_surface(-0.3, 0.9, 0.5, [0.02, 0.015, 0.09], [0.25, 0.5, 1.0]) + fitted = SSVI.fit_surface(slices) + thetas = [float(theta) for theta in fitted.variance_curve.theta] assert thetas == sorted(thetas) - for ssvi in fitted: - assert ssvi.no_butterfly_arbitrage() From 59dc34744cb9b62143bdfcef85755b8a745c4b6e Mon Sep 17 00:00:00 2001 From: Luca Date: Mon, 27 Jul 2026 21:45:43 +0100 Subject: [PATCH 3/4] Better SSVI fit --- app/api/cointegration.py | 2 +- app/api/volatility.py | 3 + frontend/src/volatility-surface.md | 59 ++++++++++++++++ quantflow/options/ssvi.py | 105 ++++++++++++----------------- quantflow_tests/test_ssvi.py | 61 +++++------------ 5 files changed, 122 insertions(+), 108 deletions(-) diff --git a/app/api/cointegration.py b/app/api/cointegration.py index 31adf360..02df9907 100644 --- a/app/api/cointegration.py +++ b/app/api/cointegration.py @@ -68,7 +68,7 @@ async def _cointegration(fmp: FMP, frequency: FMP.freq) -> CointegrationResponse # discarding noise-level imaginary parts warnings.simplefilter("ignore", np.exceptions.ComplexWarning) johansen_result = coint_johansen(scaled, det_order=0, k_ar_diff=1) - deltas = np.real_if_close(johansen_result.evec[:, 0]).real / std.values + deltas = np.real(johansen_result.evec[:, 0]) / std.values deltas = deltas / np.linalg.norm(deltas) residuals = log_prices_3.dot(deltas) diff --git a/app/api/volatility.py b/app/api/volatility.py index aeb0134e..c2c94e4b 100644 --- a/app/api/volatility.py +++ b/app/api/volatility.py @@ -9,6 +9,7 @@ from quantflow.data.deribit import Deribit from quantflow.data.yahoo import Yahoo from quantflow.options.inputs import VolSurfaceInputs +from quantflow.options.ssvi import SSVI from quantflow.options.surface import OptionInfo, VolSurfaceLoader from quantflow.rates.cir import CIRCurve from quantflow.rates.nelson_siegel import NelsonSiegelCurve @@ -37,6 +38,7 @@ class ForwardCurveResponse(BaseModel): class VolSurfaceResponse(BaseModel): + ssvi: SSVI = Field(description="SSVI model fitted to the volatility surface") inputs: VolSurfaceInputs = Field(description="Volatility surface inputs") options: list[OptionInfo] = Field( description="List of option info with implied volatilities" @@ -122,6 +124,7 @@ async def _volatility_surface(asset: str) -> VolSurfaceResponse: ttm_grid = [float(t) for t in np.linspace(1 / 365, max_ttm, 50)] return VolSurfaceResponse( + ssvi=SSVI.fit_vol_surface(surface), inputs=inputs, options=options, quote_curve=_curve_response(surface.quote_curve, max_ttm), diff --git a/frontend/src/volatility-surface.md b/frontend/src/volatility-surface.md index c6738684..a7c51ac7 100644 --- a/frontend/src/volatility-surface.md +++ b/frontend/src/volatility-surface.md @@ -96,11 +96,63 @@ display(html`
bool: """True if the surface satisfies the implemented static checks.""" return self.variance_curve.is_non_decreasing() and self.no_butterfly_arbitrage() - @classmethod - def fit( - cls, - k: Annotated[ArrayLike, Doc("Log-moneyness log(K/F) for each option")], - iv: Annotated[ArrayLike, Doc("Observed implied volatilities")], - ttm: Annotated[float, Doc("Time to maturity in years")], - gamma: Annotated[ - float, - Doc( - "Fixed power law exponent. A single slice only identifies the " - "value of the shape function, not the split between eta and " - "gamma, so gamma is held fixed" - ), - ] = 0.5, - ) -> Self: - """Fit an SSVI slice to observed implied volatilities via non-linear - least squares. - - Minimises the sum of squared differences between observed and model - total variances, fitting rho, eta and theta with gamma held fixed. - """ - k_arr = np.asarray(k, dtype=float) - iv_arr = np.asarray(iv, dtype=float) - w_obs = iv_arr**2 * ttm - - atm_var = float(np.interp(0.0, k_arr, w_obs)) if k_arr.size else w_obs.mean() - x0 = [0.0, 1.0, max(atm_var, 1e-4)] - - def residuals(x: list[float]) -> np.ndarray: - rho, eta, theta = x - phi = eta / (theta**gamma * (1 + theta) ** (1 - gamma)) - pk = phi * k_arr - w_fit = 0.5 * theta * (1 + rho * pk + np.sqrt((pk + rho) ** 2 + 1 - rho**2)) - return w_fit - w_obs - - result = least_squares( - residuals, - x0, - bounds=( - [-1.0 + 1e-6, 1e-6, 1e-8], - [1.0 - 1e-6, np.inf, np.inf], - ), - ) - rho, eta, theta = result.x - return cls( - rho=to_decimal(round(rho, 10)), - eta=to_decimal(round(eta, 10)), - gamma=to_decimal(round(gamma, 10)), - variance_curve=VarianceCurve( - ttm=[to_decimal(round(ttm, 10))], - theta=[to_decimal(round(theta, 10))], - ), - ) - @classmethod def fit_surface( cls, @@ -324,9 +271,13 @@ def fit_surface( for k, iv, ttm in slices: k_arr = np.asarray(k, dtype=float) iv_arr = np.asarray(iv, dtype=float) + if k_arr.size == 0 or iv_arr.size == 0: + raise ValueError("k and iv must contain at least one quote") + if k_arr.shape != iv_arr.shape: + raise ValueError("k and iv must have the same shape") w_obs = iv_arr**2 * ttm data.append((k_arr, w_obs, float(ttm))) - atm = float(np.interp(0.0, k_arr, w_obs)) if k_arr.size else w_obs.mean() + atm = float(np.interp(0.0, k_arr, w_obs)) thetas0.append(max(atm, 1e-4)) order = np.argsort([ttm for _, _, ttm in data]) data = [data[i] for i in order] @@ -348,13 +299,16 @@ def residuals(x: np.ndarray) -> np.ndarray: rho, eta, gamma = x[0], x[1], x[2] theta_nodes = theta_from_params(x) res = [] - for (k_arr, w_obs, _), theta in zip(data, theta_nodes): + for (k_arr, w_obs, ttm), theta in zip(data, theta_nodes): phi = eta / (theta**gamma * (1 + theta) ** (1 - gamma)) pk = phi * k_arr w_fit = ( 0.5 * theta * (1 + rho * pk + np.sqrt((pk + rho) ** 2 + 1 - rho**2)) ) - res.append(w_fit - w_obs) + # normalise by ttm so residuals are in variance (sigma^2) space, + # otherwise short maturities carry tiny total variance and are + # outvoted by long maturities in the joint least squares + res.append((w_fit - w_obs) / ttm) return np.concatenate(res) n = len(data) @@ -366,6 +320,7 @@ def residuals(x: np.ndarray) -> np.ndarray: [1.0 - 1e-6, np.inf, 1.0, np.inf] + [np.inf] * (n - 1), ), ) + rho, eta, gamma = result.x[:3] theta = theta_from_params(result.x) return cls( @@ -377,3 +332,31 @@ def residuals(x: np.ndarray) -> np.ndarray: theta=[to_decimal(round(value, 10)) for value in theta], ), ) + + @classmethod + def fit_vol_surface( + cls, + surface: Annotated[ + VolSurface[Any], + Doc( + "Volatility surface with calculated implied volatilities and " + "converged options" + ), + ], + ) -> Self: + """Fit an SSVI model to a volatility surface.""" + slices = [] + for index, maturity in enumerate(surface.maturities): + options = list(surface.option_prices(index=index, converged=True)) + if not options: + continue + log_strike = np.array([option.log_strike for option in options]) + order = np.argsort(log_strike) + slices.append( + ( + log_strike[order], + np.array([option.iv for option in options])[order], + maturity.ttm(surface.ref_date), + ) + ) + return cls.fit_surface(slices) diff --git a/quantflow_tests/test_ssvi.py b/quantflow_tests/test_ssvi.py index 890a18e6..6c6f81b5 100644 --- a/quantflow_tests/test_ssvi.py +++ b/quantflow_tests/test_ssvi.py @@ -148,52 +148,6 @@ def test_butterfly_arbitrage_violated_for_large_eta() -> None: assert not make_ssvi(eta=15.0, theta=1.0).no_butterfly_arbitrage() -# --------------------------------------------------------------------------- -# fit -# --------------------------------------------------------------------------- - - -def test_fit_recovers_parameters() -> None: - true_ssvi = make_ssvi(rho=-0.2, eta=0.8, gamma=0.5, theta=0.05) - iv_obs = true_ssvi.iv(K, TTM) - fitted = SSVI.fit(K, iv_obs, TTM) - assert float(fitted.rho) == pytest.approx(float(true_ssvi.rho), abs=1e-4) - assert float(fitted.eta) == pytest.approx(float(true_ssvi.eta), abs=1e-4) - assert float(fitted.theta) == pytest.approx(float(true_ssvi.theta), abs=1e-4) - - -def test_fit_reproduces_ivs() -> None: - true_ssvi = make_ssvi() - iv_obs = true_ssvi.iv(K, TTM) - fitted = SSVI.fit(K, iv_obs, TTM) - iv_fit = fitted.iv(K, TTM) - assert np.allclose(iv_fit, iv_obs, atol=1e-5) - - -def test_fit_respects_bounds() -> None: - true_ssvi = make_ssvi() - iv_obs = true_ssvi.iv(K, TTM) - fitted = SSVI.fit(K, iv_obs, TTM) - assert -1 < float(fitted.rho) < 1 - assert float(fitted.eta) > 0 - assert float(fitted.theta) > 0 - - -def test_fit_fixed_gamma() -> None: - true_ssvi = make_ssvi() - iv_obs = true_ssvi.iv(K, TTM) - fitted = SSVI.fit(K, iv_obs, TTM, gamma=0.4) - assert float(fitted.gamma) == pytest.approx(0.4) - - -def test_fit_skewed_smile() -> None: - true_ssvi = make_ssvi(rho=-0.6) - iv_obs = true_ssvi.iv(K, TTM) - fitted = SSVI.fit(K, iv_obs, TTM) - iv_fit = fitted.iv(K, TTM) - assert np.allclose(iv_fit, iv_obs, atol=1e-5) - - # --------------------------------------------------------------------------- # fit_surface # --------------------------------------------------------------------------- @@ -253,3 +207,18 @@ def test_fit_surface_enforces_monotone_variance_curve() -> None: fitted = SSVI.fit_surface(slices) thetas = [float(theta) for theta in fitted.variance_curve.theta] assert thetas == sorted(thetas) + + +def test_fit_surface_requires_slices() -> None: + with pytest.raises(ValueError, match="at least one maturity slice"): + SSVI.fit_surface([]) + + +def test_fit_surface_rejects_empty_slice() -> None: + with pytest.raises(ValueError, match="at least one quote"): + SSVI.fit_surface([(np.array([]), np.array([]), 0.5)]) + + +def test_fit_surface_rejects_shape_mismatch() -> None: + with pytest.raises(ValueError, match="same shape"): + SSVI.fit_surface([(K, np.zeros(len(K) + 1), 0.5)]) From 0d70158731d6c273d645ce8213b99b726144e1c0 Mon Sep 17 00:00:00 2001 From: Luca Date: Mon, 27 Jul 2026 21:59:41 +0100 Subject: [PATCH 4/4] Better plot --- frontend/src/volatility-surface.md | 8 ++++++-- 1 file changed, 6 insertions(+), 2 deletions(-) diff --git a/frontend/src/volatility-surface.md b/frontend/src/volatility-surface.md index a7c51ac7..a44ef89f 100644 --- a/frontend/src/volatility-surface.md +++ b/frontend/src/volatility-surface.md @@ -155,6 +155,11 @@ const ssviSmile = selectedMaturity === null const xLabel = {moneyness: "Moneyness (log(K/F) / √T)", log_strike: "Log Strike (log K/F)", strike: "Strike"}[xAxis]; +const ivValues = smileData.map(d => d.iv).filter(v => v > 0); +const ivMin = ivValues.length ? d3.min(ivValues) : 0; +const ivMax = ivValues.length ? d3.max(ivValues) : 1; +const ivPad = (ivMax - ivMin) * 0.1 || 0.05; + display(Plot.plot({ width: 800, height: 450, @@ -162,7 +167,7 @@ display(Plot.plot({ marginBottom: 50, style: {background: "transparent"}, x: {label: xLabel}, - y: {label: "Implied Volatility", percent: true}, + y: {label: "Implied Volatility", tickFormat: d3.format(".0%"), domain: [Math.max(0, ivMin - ivPad), ivMax + ivPad]}, color: { type: "ordinal", domain: maturities, @@ -188,7 +193,6 @@ display(Plot.plot({ stroke: "maturity", strokeWidth: 1 }), - Plot.ruleY([0]), ...(xAxis === "moneyness" ? [Plot.ruleX([0], {stroke: "var(--theme-foreground-muted)", strokeDasharray: "4,4"})] : []) ] }));