sabr-replication-demo 0.1.3

Monte Carlo replication scaffold for efficient SABR simulation.

Demo notebook

Open the main branch version:

SABR Simulation Paper Replication

This workspace contains a Python replication scaffold for:

Choi, Hu, Kwok, "Efficient and accurate simulation of the stochastic-alpha-beta-rho model"

The current scaffold focuses on reproducing the paper's core Monte Carlo machinery:

• Algorithm 1: shifted-lognormal sampling of conditional integrated variance
• Algorithm 2: martingale-preserving CEV approximation for the conditional forward
• Algorithm 3: exact sampling of a CEV transition via a shifted-Poisson mixture gamma law
• Algorithm 4: full SABR terminal simulation over a time grid

Presentation overview

The paper studies efficient Monte Carlo simulation of the SABR model

dF_t = \sigma_t F_t^\beta\,dW_t,\qquad
\frac{d\sigma_t}{\sigma_t} = \nu\,dZ_t,\qquad
dW_t\,dZ_t = \rho\,dt.

The volatility step is sampled exactly:

\sigma_{t+h}
=
\sigma_t
\exp\left(\nu\sqrt{h}\,X_\sigma-\frac{1}{2}\nu^2h\right),
\qquad X_\sigma\sim N(0,1).

The first difficult quantity is the normalized conditional average variance

I_t^h
=
\frac{1}{\sigma_t^2h}
\int_t^{t+h}\sigma_s^2\,ds
\;\Bigg|\;\sigma_{t+h}.

Algorithm 1 approximates this with a shifted lognormal random variable:

I_t^h
\approx
\mu
\left[
\frac{1}{6}
+
\frac{5}{6}
\exp\left(aX-\frac{1}{2}a^2\right)
\right],
\qquad X\sim N(0,1),

where

\mu=\mathbb{E}[I_t^h\mid\sigma_{t+h}],\qquad
a=\sqrt{\log\left(1+\frac{36}{25}v^2\right)},\qquad
v=
\frac{\sqrt{\mathrm{Var}(I_t^h\mid\sigma_{t+h})}}
{\mathbb{E}[I_t^h\mid\sigma_{t+h}]}.

The second difficult quantity is the conditional forward price. For 0 < beta < 1, Algorithm 2 uses a martingale-preserving CEV approximation:

F_{t+h}\mid\sigma_{t+h},I_t^h
\approx
\mathrm{CEV}_\beta
\left(
\bar F_t^h,
(\rho^*)^2\sigma_t^2hI_t^h
\right),
\qquad
\rho^*=\sqrt{1-\rho^2},

with conditional mean

\bar F_t^h
=
F_t
\exp\left(
\frac{\rho(\sigma_{t+h}-\sigma_t)}{\nu F_t^{1-\beta}}
-
\frac{\rho^2\sigma_t^2hI_t^h}{2F_t^{2(1-\beta)}}
\right).

This choice is designed so that the forward process remains close to a martingale:

\mathbb{E}[F_{t+h}]\approx F_t.

Algorithm 3 then samples the CEV transition exactly using a Gamma-Poisson-Gamma construction. Algorithm 4 combines the steps: exact volatility step, shifted-lognormal average variance, martingale-preserving CEV approximation, exact CEV sampling, and repetition over the full time grid.

Files

• sabr_replicate.py: core implementation
• run_experiments.py: CLI entrypoint for tables, figures, and validation
• notebooks/paper_reproduction_walkthrough.ipynb: presentation-oriented walkthrough with formulas, sanity checks, paper tables, figure datasets, and validation
• notebooks/replication_sanity_checks.ipynb: notebook with model-level sanity checks and validation summaries
• requirements.txt: lightweight dependency list for the replication environment
• notebooks/asian_option_extension.ipynb: path-dependent extension notebook that uses the replicated SABR Monte Carlo simulator to price arithmetic-average Asian call options and compare them with European calls.

What is implemented

• Exact volatility stepping
• PyFeng-backed conditional moment formulas for the normalized integrated variance
• PyFeng-backed shifted-lognormal parameter fitting with fixed shift lambda = 5/6
• Exact CEV sampling for 0 < beta < 1
• Special handling for beta = 1 and |rho| = 1
• European call pricing from Monte Carlo samples
• Arithmetic-average Asian call extension using the replicated SABR Monte Carlo path simulation
• Table 3 parameter presets as named cases
• Direct experiment entrypoints for Tables 1, 2, 4, 5, 6, 7
• Figure 1 moment-comparison dataset
• Figure 2 / Table 7 convergence dataset
• Figure 3 comparison dataset between the paper scheme and Islah's approximation
• PyFeng analytic approximation rows where the package already provides them, plus paper-reference rows for the remaining baselines
• Group 16 ADI finite-difference benchmark integration through the mafn-finite-difference package, with explicit domain settings for stressed SABR cases
• CLI support for switching benchmark sources with --benchmark-source paper|fdm|mc|none
• A regression test covering the rho = 1 Islah edge case
• A notebook with extra sanity checks outside the paper tables:
  1. nu = 0 CEV limit
  2. beta = 1, nu = 0 Black-Scholes limit
  3. martingale checks across maturities
  4. |rho| = 1 Islah stability
  5. quick validation summary

What is not implemented yet

• Direct reimplementation of competing baselines such as Euler, low-bias, PSE, Hagan, ZC Map, or Hyb ZC Map
• Variance reduction and full performance tuning

Current status

Paper-scale validation with the ADI-FD benchmark currently reports:

• overall conclusion: Implementation consistent with paper
• replication conclusion: Replication likely successful
• martingale test: noise-dominated with no evidence of systematic drift
• Table 2: PASS
• Table 1: WARNING, with a small residual discrepancy (max relative error about 0.43%)

This means the main replication target now looks successful, while Table 1 still shows a mild benchmark-level gap worth documenting.

Run

Install the small dependency set first:

python -m pip install -r .\requirements.txt

Then run the CLI:

python .\run_experiments.py --experiment table1 --paper-scale
python .\run_experiments.py --experiment table1 --paper-scale --benchmark-source fdm
python .\run_experiments.py --experiment table4 --paper-scale
python .\run_experiments.py --experiment table7 --n-paths 20000 --repeats 3 --benchmark-source fdm
python .\run_experiments.py --experiment figure3 --n-paths 50000 --repeats 5 --benchmark-source fdm
python .\run_experiments.py --experiment validate --quick --benchmark-source fdm

You can also save any tabular output:

python .\run_experiments.py --experiment table7 --output-csv .\outputs\table7.csv

Current paper-scale reproducibility commands:

python .\run_experiments.py --experiment table1 --paper-scale --benchmark-source fdm --output-csv .\outputs\table1_paper_scale_fdm.csv
python .\run_experiments.py --experiment table2 --paper-scale --benchmark-source fdm --output-csv .\outputs\table2_paper_scale_fdm.csv
python .\run_experiments.py --experiment validate --paper-scale --benchmark-source fdm --output-csv .\outputs\validation
pytest -q .\tests\test_islah_rho1.py

For an explanation-oriented walkthrough rather than a test log, open:

.\notebooks\paper_reproduction_with_asian.ipynb

For a shorter sanity-check-only notebook, open:

.\notebooks\replication_sanity_checks.ipynb

For the path-dependent Asian option extension, open:

.\notebooks\asian_option_extension.ipynb

Notes

• The paper's formulas were transcribed from the PDF and implemented directly.
• We now rely on PyFENG for building blocks it already exposes well: conditional average-variance moments, shifted-lognormal moment fitting, and several analytic SABR approximation models.
• FDM benchmarks are delegated to Group 16's mafn-finite-difference package, which implements a Douglas-scheme ADI SABR solver and cites von Sydow et al. (2010), "ADI finite difference schemes for option pricing in the Heston model with correlation."
• When calling the Group 16 ADI pricer, this project passes wider default asset and volatility domains than the package defaults (S_max = 8 * max(F0, K), v_max = max(10 * sigma0, 1)). This avoids upper-boundary truncation in long-maturity / high-vol-of-vol cases such as Table 7 / Figure 2 Case IV.
• The project directly reproduces the paper's proposed simulation method. Several competing baseline rows are included from PyFeng or paper-reference values rather than being fully reimplemented from scratch.
• --benchmark-source paper uses the tabulated paper benchmarks when they are available.
• --benchmark-source fdm recomputes benchmark prices with Group 16's ADI finite-difference package.
• table7 / figure3 fall back to the internal high-resolution Monte Carlo benchmark unless --benchmark-source fdm is requested.
• This is now a stronger reproduction scaffold with direct table/figure entrypoints and an external ADI-FD benchmark integration, but it is still not a finished paper-grade reproduction package.
• The saved validation CSVs are written as validation_table1_df.csv, validation_table2_df.csv, and validation_martingale_df.csv inside the output directory.
• The fastest path from here is:
  1. replace paper-reference baseline rows with actual baseline implementations,
  2. tune variance reduction and runtime for paper-scale sweeps,
  3. compare the new PDE benchmarks against an independent reference implementation.