Closed-form solutions and fast calibration & simulation for SABR-based models with mean-reverting stochastic volatility
The seminal SABR model (2002) offered a fast closed-form solution for stochastic-volatility models.
However, the assumption of unbounded volatility diffusion renders the model unrealistic for many markets, particularly equities.
Several recently developed SABR-based models, which extend the standard SABR model to include mean-reverting stochastic volatility, address this shortcoming.
Previous research on such extensions has provided only semi-analytical solutions in the form of deeply nested integrals, except for a few restrictive special cases. Numerical evaluation of these integrals makes implementations slow and technically challenging in many settings.
This repository derives, applies, and visualizes fast closed-form solutions (in elementary functions) for the mean-reverting SABR extensions.
Thanks to their evaluation speed and robustness, the closed-form solutions become suitable e.g. for Monte Carlo simulations of entire implied-volatility surfaces, such as in market-risk VaR scenarios. The 3D-surface below is an example output of a model variant:
For transparency and performance, this repository is implemented in Python. However, the resulting closed-form solutions are simple enough to be translated into Excel formulas, enabling model fitting (via Solver) and option pricing even in basic Excel environments without VBA.
The code of this repository is based on the following paper:
Perederiy, V., Mean-Reverting SABR Models: Closed-form Surfaces and Calibration for Equities, 2024
which is referenced in the code as this PAPER and is accessible at:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5171071
Other relevant papers (previous research), as referenced in the code, are:
hSABR PAPER: Hagan, P. S., Lesniewski, A., Woodward D. 2018. Implied volatility formulas for Heston models. Wilmott Magazine, November 2018, 44–57
mrSABR PAPER: Hagan, P. S., Lesniewski, A., Woodward D. 2020. Implied volatilities for mean reverting SABR models. Wilmott Magazine, July 2020, 62–77
mrZABR PAPER: Felpel, M., Kienitz J., McWalter T. A. 2021. Effective stochastic volatility: Applications to ZABR-type models. Quantitative Finance, 21(5), 837–852
The mean-reverting stochastic-volatility models generally have 5 parameters:
- α: instantaneous volatility
- θ: long-term / equilibrium volatility
- ν: volatility of volatility
- λ: speed of mean reversion
- ρ: correlation
This extends the standard SABR model, which has only 3 parameters (α, ν, ρ), by capturing the mean reversion in the additional parameters (θ and λ).
This repository deals with the following three mean-reverting variants (and their SABR-based approximations), each defined by a different stochastic process for the volatility σ:
-
Heston model approximation (hSABR) with CIR process of the stochastic variance, as in the hSABR PAPER:
dσ² = λ (θ² - σ²) dt + σ dW -
Mean-reverting SABR (mrSABR) with lognormal mean-reverting process of the stochastic volatility, as in the mrSABR PAPER:
dσ = λ (θ - σ) dt + σ dW -
Mean-reverting ZABR model (mrZABR), which generalizes mrSABR to mean-reverting CEV processes of the stochastic volatility, as in the mrZABR PAPER:
dσ = λ (θ - σ) dt + σ γ dW
To arrive at the closed-form solutions, we build on the semi-analytical solutions from the previous research and apply SymPy symbolic integration, together with additional approximations for mrZABR (using suitable convergent series expansions).
The resulting closed-form expressions are the so-called effective coefficients. These can then be easily translated into 3 standardized-SABR parameters (αstd, νstd and ρstd) of an approximating standard-SABR model.
Note that the standardized parameters (αstd, νstd and ρstd) depend not only on the original parameters (α, θ, λ, ν, ρ), but also on the option expiry.
The standard SABR formula (2002) can then be applied on the standardized-SABR parameters to infer the implied volatilities of options for desired expiry / moneyness levels.
Due to the solutions remaining closed-form, they are very fast in Python:
After the SymPy effective coefficients are lambdified into Python functions (which is needed only once),
the computation time is only about 0.1-1 millisecond per smile slice of the surface (i.e. for each standard-SABR parameter set αstd, νstd and ρstd as needed for each option expiry).
This is about 100-1000 times faster than numerical integration of the semi-analytical solutions. The main advantage stems from triple-nested integrals present in the semi-analytical solutions, which are computationally intensive to evaluate numerically.
With this speed advantage, calibrating the 5 parameters to empirical volatility quotes — even with several hundred gradient-descent evaluations — takes under 100 milliseconds, even for more complex model variants.
Given the parameters, producing a smooth implied-volatility surface on a 100 × 100 moneyness-expiry grid takes less than 50 milliseconds.
Derives the closed-form solutions in the form of the effective coefficients as SymPy symbolic expressions and saves them in pickle files.
For one code execution, the hSABR, mrSABR and mrZABR coefficients are generated
(the latter for the particular mrZABR configuration set at the beginning of the code).
The code also rigorously verifies the closed-form results by comparing them to numerical integrations (in the general case α ≠ θ), and by comparing to the closed-form expressions derived in the previous research for a simple special case (α=θ).
For a chosen model variant, loads the result expressions (effective coefficients) as generated/saved by derive_validate_expressions.py from pickle files.
The code then translates these into the standardized-SABR parameters, and applies the standard-SABR model to infer the implied lognormal (Black & Scholes) volatilities.
The code demonstrates how to calibrate the original mean-reverting SABR-based parameters (α, θ, λ, ν, ρ) using SciPy to best fit an empirical dataset of implied volatility quotes.
Finally, the code generates an interactive 3D surface that interpolates and extrapolates the empirical quotes.
Numeric integration routines.
Used in this repository to benchmark the resulting symbolic expressions for correctness and performance.
Symbolic integration routines for a special case of quadratic polynomials.
Provides graphical demonstrations of the convergence (and potential divergence to be avoided) in Taylor/binomial expansions used to approximate the mrZABR integrands.
Loads all closed-form results (effective coefficients) from the pickle files as generated/saved by derive_validate_expressions.py.
From these expressions, generates equivalent formulas in Excel format (to be used in Excel cells without VBA), and in html/MathML format (for viewing in a browser).
OUTPUT FILES (RESULTS)
These pickle files are generated by derive_validate_expressions.py and contain the closed-form results (effective coefficients) as SymPy expressions.
Interactive 3D surface as fitted/generated by fit_display_surface.py (open the HTML file in a browser to view).
These are the closed-form results (effective coefficients) as transformed into html/MathML format by output_excel_html.py (open the HTML file in a browser to view).
Python version >= 3.9
sympy
mpmath
numpy
scipy
plotly
matplotlib
Vlad Perederiy
Perederiy Consulting
53119 Bonn, Germany
https://perederiy-consulting.de/
This software is provided for evaluation and testing purposes, including within a business environment.
Production use or day-to-day business operations require a separate commercial license; for details, please contact the author via the website listed above.
This software is provided "as is", without any warranty.
The author is not liable for any damages arising from its use.