chernoffpy 0.1.0

Financial derivatives pricing via Chernoff operator splitting with certified error bounds

ChernoffPy

Financial derivatives pricing via Chernoff operator splitting with certified error bounds.

ChernoffPy prices options by applying Chernoff product formulas to transformed pricing PDEs. The library includes European, barrier, double-barrier, American, Heston, and Bates pricing, plus calibration tools and certified error-bound utilities.

Installation

pip install chernoffpy
pip install "chernoffpy[fast]"
pip install "chernoffpy[gpu]"
pip install "chernoffpy[all]"

Quick Start

European Option

from chernoffpy import CrankNicolson
from chernoffpy.finance import EuropeanPricer, MarketParams

market = MarketParams(S=100, K=100, T=1.0, r=0.05, sigma=0.20)
pricer = EuropeanPricer(CrankNicolson())
result = pricer.price(market, n_steps=50, option_type="call")
print(result.price)

Barrier Option (DST)

from chernoffpy import CrankNicolson
from chernoffpy.finance import BarrierDSTPricer, BarrierParams, MarketParams

market = MarketParams(S=100, K=100, T=1.0, r=0.05, sigma=0.20)
params = BarrierParams(barrier=90, barrier_type="down_and_out")
result = BarrierDSTPricer(CrankNicolson()).price(market, params, n_steps=80, option_type="call")
print(result.price)

Certified Error Bound

from chernoffpy import CrankNicolson
from chernoffpy.finance import CertifiedEuropeanPricer, MarketParams

market = MarketParams(S=100, K=100, T=1.0, r=0.05, sigma=0.20)
res = CertifiedEuropeanPricer(CrankNicolson()).price_certified(market, n_steps=50, option_type="call")
print(res.price, res.certified_bound.bound)

Heston / Bates

from chernoffpy import CrankNicolson
from chernoffpy.finance import HestonFastPricer, BatesPricer, BatesParams
from chernoffpy.finance.heston_params import HestonParams

heston = HestonParams(S=100, K=100, T=1.0, r=0.05, v0=0.04, kappa=2.0, theta=0.04, xi=0.3, rho=-0.7)
print(HestonFastPricer(CrankNicolson()).price(heston, n_steps=50, option_type="call").price)

bates = BatesParams(S=100, K=100, T=1.0, r=0.05, v0=0.04, kappa=2.0, theta=0.04, xi=0.3, rho=-0.7,
                    lambda_j=0.5, mu_j=-0.1, sigma_j=0.2)
print(BatesPricer(CrankNicolson()).price(bates, n_steps=50, option_type="call").price)

Choosing the Right Pricer

Which pricer for which option type?

Option type	Recommended pricer	Notes
European call/put	EuropeanPricer	Fastest; use CertifiedEuropeanPricer for guaranteed error bounds
Single barrier (down/up, in/out)	BarrierDSTPricer	Recommended — DST avoids Gibbs artifacts near barrier
Single barrier (high performance)	BarrierPricer	FFT-based, slightly faster but possible Gibbs at barrier
Double barrier	DoubleBarrierDSTPricer	DST, no artifacts; DoubleBarrierPricer for FFT variant
American (no dividends)	AmericanPricer	Payoff projection method; use CrankNicolson() or PadeChernoff()
American (discrete dividends)	AmericanPricer + DividendSchedule	Pass dividends= argument to .price()
Stochastic volatility (Heston)	HestonFastPricer	Faster than HestonPricer; use HestonPricer only for custom grids
Jump-diffusion (Heston + jumps)	BatesPricer	Heston model with Merton jump component
Local volatility	LocalVolPricer	Dupire surface; use flat_vol, linear_skew, or custom LocalVolParams
With guaranteed error bound	CertifiedEuropeanPricer / CertifiedBarrierDSTPricer	Returns certified_bound alongside price

Which Chernoff function (scheme) to use?

Scheme	Order	Stability	When to use
BackwardEuler()	O(1/n)	A-stable ✓	Debugging, large time steps
CrankNicolson()	O(1/n²)	A-stable ✓	Default choice — best speed/accuracy trade-off
PadeChernoff(1, 2)	O(1/n³)	A-stable ✓	High accuracy, smooth payoffs
PadeChernoff(2, 2)	O(1/n⁴)	A-stable ✓	Maximum accuracy
PhysicalG()	O(1/n)	A-stable ✓	Explicit scheme, large grids
PhysicalS()	O(1/n²)	A-stable ✓	Explicit, 2nd order
PadeChernoff(2, 1)	O(1/n²)	⚠ NOT A-stable	Avoid — high-frequency divergence

Rule of thumb: start with CrankNicolson(). Switch to PadeChernoff(1, 2) if you need tighter certified bounds.

Quick decision flowchart

Vanilla option?       → EuropeanPricer
Needs error bound?    → CertifiedEuropeanPricer
Barrier present?      → BarrierDSTPricer  (single)
                      → DoubleBarrierDSTPricer  (double)
Early exercise?       → AmericanPricer
Discrete dividends?   → AmericanPricer + DividendSchedule
Stoch. volatility?    → HestonFastPricer
Stoch. vol + jumps?   → BatesPricer
Custom vol surface?   → LocalVolPricer

Features

• European pricing and Greeks
• Barrier and double-barrier options (FFT and DST)
• American options with early exercise and discrete dividends
• Local volatility and implied volatility
• Heston stochastic volatility and Bates jump-diffusion
• Calibration helpers (flat/skew/SVI)
• Certified bounds based on Chernoff convergence rates
• Optional acceleration via Numba ([fast])

Mathematical Basis

ChernoffPy uses the Chernoff product formula exp(tL) f = lim_{n->inf} F(t/n)^n f with practical schemes (Backward Euler, Crank-Nicolson, Padé). Certified-bound utilities are motivated by convergence-rate estimates in Galkin & Remizov (2025, Israel Journal of Mathematics).

Development

pip install -e ".[dev]"
pytest tests/ -q

References

1. Chernoff, P. (1968), J. Functional Analysis.
2. Galkin, O. & Remizov, I. (2025), Israel Journal of Mathematics.
3. Butko, Ya. (2020), Lecture Notes in Mathematics.

License

MIT. See LICENSE.