options-pricing

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Option pricing, Greeks and hedging in Python, CPU-optimised and validated against QuantLib and market data.

European, Asian and American options (with discrete dividends): closed-form formulas, Monte Carlo with variance reduction, quasi-Monte Carlo, Longstaff–Schwartz and binomial tree. Arbitrage-free SVI smile calibration on real option chains, and backtesting of hedging strategies. Code vectorised with NumPy, compiled and parallelised with Numba.

The numerical methods follow G. Pagès, Numerical Probability (Springer, 2018); the hedging part follows Bouchard & Chassagneux, Fundamentals and Advanced Techniques in Derivatives Hedging (Springer, 2016).

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Contents

• At a glance
• Features
• Results
• Project structure
• Installation
• Quick start
• Numerical methods
• Tests
• External validation
• Known limitations
• References

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At a glance

European call, quasi-Monte Carlo	10.45055 ± 0.00006 vs. an exact value of 10.45058
Price + 5 Greeks from a single simulation	2M paths, error vs. closed form < 0.1%
Real AAPL option chain, American pricer + SVI smile	94.3% of options repriced within the bid-ask spread
Hedging, unknown realised vol between 15% and 25%	P&L standard deviation divided by 8 (17% → 2% of the premium) with a gamma-neutral hedge
Test suite	233 pytest tests, including property-based tests with Hypothesis

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Features

Option	Price	Greeks (Δ, Γ, Vega, ρ, Θ)
European	Closed form, Monte Carlo (antithetic + control variate), quasi-Monte Carlo (scrambled Sobol)	Closed form; Monte Carlo pathwise and likelihood ratio
Arithmetic Asian	Numba Monte Carlo + geometric control variate	Pathwise / likelihood ratio, with control variate
Geometric Asian	Closed form	Derivatives of the closed form
American	Longstaff–Schwartz (Numba), CRR binomial tree	CRR tree; finite differences with common random numbers
American with discrete dividends	CRR tree, "spot" and "escrowed" models	CRR tree

Also included:

• 95% confidence interval on every Monte Carlo estimate (MCResult).
• SVI volatility smile (Gatheral) with no-arbitrage checks: butterfly (Durrleman condition), calendar, Roger Lee bound on the wings.
• Market data validation: quote cleaning, implied borrow cost calibrated per expiry, implied vol inversion, repricing within the bid-ask spread.
• Hedging backtest: discrete delta hedging, misspecified vol, uncertain vol (Black–Scholes–Barenblatt), gamma-neutral hedging, P&L CVaR.

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Results

Measured on a call with S = K = 100, r = 5%, σ = 20%, T = 1 year (Linux, 2 cores). Timings are indicative: rerun bench.py and bench_greeks.py on your own machine.

Variance reduction

Monte Carlo error decreases as σ/√N: dividing the standard deviation by 10 is equivalent to dividing the number of simulations needed by 100.

Method (≈ 2M draws)	Time	Price ± 95% CI half-width
Plain Monte Carlo	0.064 s	10.4556 ± 0.0204
+ antithetic + control variate	0.047 s	10.4480 ± 0.0038
Quasi-Monte Carlo (scrambled Sobol)	0.091 s	10.45055 ± 0.00006
Closed form		10.45058

For the arithmetic Asian option (52 dates), the geometric control variate shrinks the confidence interval by a factor of 36 (± 0.022 → ± 0.0006) at equal computing time.

Monte Carlo Greeks vs. exact formulas

A single simulation yields the price and all 5 Greeks (2M paths):

Greek	Monte Carlo ± 95% CI	Exact
Delta	0.63697 ± 0.00028	0.63683
Gamma	0.01874 ± 0.00005	0.01876
Vega	37.486 ± 0.091	37.524
Rho	53.251 ± 0.030	53.232
Theta (per year)	−6.411 ± 0.009	−6.414

American put

Method	Price
Longstaff–Schwartz, 200,000 paths × 50 dates	6.0803 ± 0.0316
CRR tree, 2,000 steps	6.0900
European put (lower bound)	5.5735

Volatility smile on market data

valider_marche.py inverts the American pricer (discrete dividends) on every quote, fits an SVI smile per expiry and measures the share of options repriced within the bid-ask spread.

• Synthetic chain priced by QuantLib (independent engine), 6 expiries from 17 to 290 days: 100% of the 322 retained options repriced within the spread, median smile − market gap of 0.04 vol points, borrow cost recovered (0.30% target, 0.21% to 0.39% fitted depending on expiry), no butterfly or calendar arbitrage.
• Real AAPL chain (Yahoo Finance, delayed data, 7 October 2026 during market hours): 94.3% of the 350 retained options repriced within the spread, across 6 expiries from 7 days to 14 months; median smile − market gap of 0.15 vol points, median call/put vol gap at the same strike of 0.94 points, no butterfly or calendar arbitrage.

Hedging

6-month ATM call sold at the Black–Scholes price with 20% vol, delta-hedged 52 times. The realised vol is unknown, drawn uniformly between 15% and 25% on each path (100,000 paths):

Strategy	Mean P&L	P&L std. dev.	5% CVaR
Delta only	+0.004 ± 0.010	1.091 (17.1% of the premium)	−2.57
Delta + gamma-neutral (second option)	+0.000 ± 0.001	0.136 (2.1% of the premium)	−0.29

The script also checks the theory point by point: zero mean P&L under the correct model, standard deviation scaling as 1/√N (measured log-log slope −0.49, 1 to 4% off the first-order theory), the formula ½ ∫ e^{−rt}(σ̃² − σ²) S² Γ dt for a misspecified vol, and that hedging at σ_max never loses on average when the vol is uncertain.

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Project structure

File names are in French; here is what each one does.

options-pricing/
├── pricer.py               Prices: Black–Scholes, MC, QMC, Asian, American (LSM)
├── greeks.py               Greeks: closed form, pathwise, LR, finite differences, CRR
├── dividendes.py           American CRR tree with discrete dividends (spot / escrowed)
├── svi.py                  SVI smile: fitting, butterfly and calendar arbitrage checks
├── couverture.py           Hedging backtest: delta hedging, gamma-neutral, P&L
├── bench.py                Price benchmark
├── bench_greeks.py         Greeks benchmark
├── valider_pricer.py       crr_american vs QuantLib, theoretical properties
├── adaptateur_crr.py       Adapter for crr_american used by valider_pricer.py
├── valider_lsm.py          Longstaff–Schwartz vs QuantLib
├── valider_dividendes.py   crr_american_div vs QuantLib
├── valider_marche.py       Test on a real option chain (Yahoo Finance or CSV)
├── valider_couverture.py   Validation of the hedging backtest against theory
├── generer_chaine_test.py  Synthetic chain (QuantLib) for offline testing
├── docs/                   README charts
├── pyproject.toml          Project and pytest configuration
└── tests/
    ├── conftest.py
    ├── test_closed_form.py     Closed forms, parities, Black–Scholes PDE
    ├── test_monte_carlo.py     MC estimators, variance reduction, LSM
    ├── test_greeks_mc.py       MC Greeks, finite differences, CRR tree
    ├── test_proprietes.py      Arbitrage properties and cross-method consistency
    ├── test_dividendes.py      Discrete-dividend tree vs QuantLib
    ├── test_svi.py             SVI fitting and arbitrage checks
    └── test_valider_marche.py  Time measurement, borrow calibration, implied vols

flowchart LR
    P[pricer.py] --> G[greeks.py]
    G --> D[dividendes.py]
    P --> C[couverture.py]
    S[svi.py] --> M[valider_marche.py]
    D --> M
    P --> B["bench*.py"]
    G --> B
    G --> V["valider_pricer / lsm / dividendes<br/>(QuantLib)"]
    D --> V
    C --> VC[valider_couverture.py]

Loading

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Installation

Python 3.10 or later.

git clone https://github.com/matthieu-briche/python-options-pricing.git
cd python-options-pricing
python -m pip install numpy scipy numba pytest pytest-cov hypothesis

For external validation (optional):

python -m pip install QuantLib pandas matplotlib yfinance

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Quick start

Prices

from pricer import BlackScholes, bs_price, mc_european, rqmc_european, mc_asian_arithmetic, lsm_american_put

m = BlackScholes(s0=100, r=0.05, sigma=0.2)      # continuous dividend q = 0 by default

print(bs_price(m, K=100, T=1))                   # 10.4506  (closed form)
print(mc_european(m, K=100, T=1))                # 10.447964 ± 0.003823  (95% CI [...], N=2,000,000)
print(rqmc_european(m, K=100, T=1))              # 10.450554 ± 0.000055
print(mc_asian_arithmetic(m, K=100, T=1, n_steps=52))
print(lsm_american_put(m, K=100, T=1))           # 6.080280 ± 0.031559

bs_price is vectorised: you can pass an array of strikes or maturities.

Greeks

from greeks import bs_greeks, mc_european_greeks, crr_american

exact = bs_greeks(m, K=100, T=1)                 # closed form
mc    = mc_european_greeks(m, K=100, T=1)        # price + 5 Greeks, with CI
print(mc["delta"])                               # 0.636969 ± 0.000282
amer  = crr_american(m, K=100, T=1, kind="put")  # American, CRR tree

American option with discrete dividends

from dividendes import crr_american_div

divs = [(0.25, 1.5), (0.50, 1.5), (0.75, 1.5)]   # (date in years, amount)
print(crr_american_div(m, K=100, T=1, kind="put", dividends=divs, model="escrowed"))

Hedging

import numpy as np
from pricer import BlackScholes
from couverture import simuler_couverture, OptionCouverture

m = BlackScholes(s0=100, r=0.03, sigma=0.20)                 # hedging vol
sigma_reelle = np.random.default_rng(0).uniform(0.15, 0.25, 100_000)   # realised vol per path

delta  = simuler_couverture(m, K=100, T=0.5, n_reb=52, sigma_reelle=sigma_reelle)
gamma0 = simuler_couverture(m, K=100, T=0.5, n_reb=52, sigma_reelle=sigma_reelle,
                            couverture_gamma=OptionCouverture(K=100, T=1.0))
print(delta.resume()["ecart_type_rel"], gamma0.resume()["ecart_type_rel"])   # ≈ 0.17  0.02

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Numerical methods

Method	Function	Pagès, chapter
Exact simulation of S_T (no discretisation bias)	mc_european	2
Confidence interval, online block-wise standard error	MCResult, _RunningStats	2.1
Antithetic variates	mc_european, mc_european_greeks	3.1.2
Control variate with optimal β	mc_european, mc_asian_arithmetic, mc_asian_greeks	3.2
Randomised quasi-Monte Carlo (scrambled Sobol)	rqmc_european	4.4
Pathwise Greeks (tangent process)	mc_european_greeks, mc_asian_greeks	2.2.4, 10
Gamma: mixed pathwise / likelihood ratio estimator	same	2.2.3, 10
Finite differences with common random numbers	fd_greeks_crn	10
Longstaff–Schwartz (polynomial regression)	lsm_american_put	12
Cox–Ross–Rubinstein binomial tree	crr_american, crr_american_div	12

Why a mixed estimator for Gamma? The pathwise Delta of a call, e^{−rT} 1{S_T > K} S_T / S_0, is discontinuous at K: it cannot be differentiated a second time path by path. The likelihood ratio method is therefore applied to the pathwise Delta, which amounts to weighting it by Z/(σ√T) − 1.

Why common random numbers? A finite difference (V(S+h) − V(S−h)) / 2h computed with two independent simulations is drowned in Monte Carlo noise. Using the same seed for both prices makes the noise cancel out.

Why SVI rather than a polynomial? Wings linear in variance (consistent with Roger Lee's theorem), sound extrapolation beyond quoted strikes, interpretable parameters and verifiable no-arbitrage conditions.

CPU optimisation

Technique	Effect
Variance reduction	up to ×36 on the confidence interval at equal time
One simulation for the price and all 5 Greeks	avoids re-pricing by finite differences
NumPy vectorisation	no Python loop over paths
Numba @njit(parallel=True) kernels	Asian, Asian Greeks, LSM in multi-core machine code
Block computation + online statistics	bounded memory, data stays in CPU cache
PCG64 generator + SeedSequence	reproducible results

The first call to a Numba function compiles it (a few seconds); the compiled code is then cached on disk.

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Tests

python -m pytest                   # 233 tests (≈ 1 min)
python -m pytest -m "not slow"     # 220 tests, without the heavy Monte Carlo runs

File	Tests	What is checked
test_closed_form.py	42	Reference values, put-call parity, arbitrage bounds, Black–Scholes PDE (Θ + ½σ²S²Γ + (r−q)SΔ − rV = 0)
test_monte_carlo.py	31	Unbiasedness, 1/√N error, actual coverage of the 95% CI, reproducibility
test_greeks_mc.py	43	Each MC estimator vs. the exact formula, finite differences, CRR tree
test_proprietes.py	59	Properties holding for any parameter set (Hypothesis): monotonicity, convexity in strike, bounds, tree convergence
test_dividendes.py	13	Discrete-dividend tree vs. QuantLib reference values
test_svi.py	18	SVI fitting, butterfly and calendar conditions
test_valider_marche.py	27	Time measurement (time zone, ACT/365), borrow calibration, vol inversion

Monte Carlo tests use fixed seeds: an estimate passes if its distance to the reference is less than 4 standard errors.

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External validation

python valider_pricer.py --module adaptateur_crr     # CRR tree vs QuantLib
python valider_lsm.py                                # Longstaff–Schwartz vs QuantLib
python valider_dividendes.py                         # discrete dividends vs QuantLib
python valider_couverture.py                         # hedging vs theory (≈ 20 s)

# Real chain (Yahoo Finance, delayed data)
python valider_marche.py --ticker AAPL --taux 0.04   # set today's risk-free rate

# Offline: synthetic chain priced by QuantLib
python generer_chaine_test.py
python valider_marche.py --chaine chaine_test.csv --dividendes dividendes_test.csv \
                         --spot 227.5 --date 2026-10-06 --taux 0.04

Output files (rapport_*.csv, resultats_marche_*.csv, smiles_*.png…) can be regenerated and are not version-controlled.

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Known limitations

• Black–Scholes pricer: constant volatility and rates. The SVI smile is used for market validation (implied vol per strike); there is no local or stochastic volatility in the pricer.
• Longstaff–Schwartz gives a lower bound on the price (estimated exercise policy); for vanilla American options, the CRR tree is more accurate.
• Time measurement in valider_marche.py: ACT/365 to the second, without an exchange calendar (weekends, holidays); AM-settled index options are out of scope.
• Hedging: no transaction costs or market impact.

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References

• G. Pagès, Numerical Probability: An Introduction with Applications to Finance, Universitext, Springer, 2018.
• B. Bouchard, J.-F. Chassagneux, Fundamentals and Advanced Techniques in Derivatives Hedging, Universitext, Springer, 2016.
• J. Gatheral, "A parsimonious arbitrage-free implied volatility parameterization", Global Derivatives, 2004.
• R. Lee, "The Moment Formula for Implied Volatility at Extreme Strikes", Mathematical Finance, 14(3), 2004.
• F. Longstaff, E. Schwartz, "Valuing American Options by Simulation: A Simple Least-Squares Approach", Review of Financial Studies, 14(1), 2001.
• J. Cox, S. Ross, M. Rubinstein, "Option Pricing: A Simplified Approach", Journal of Financial Economics, 7(3), 1979.
• P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2003.

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Author

Matthieu Briche · LinkedIn