asian-option-pricer 0.1.0

Efficient numerical pricing methods for arithmetic Asian options under GBM

asian-option-pricer

Fast, accurate pricing of arithmetic Asian options under GBM — with variance reduction.

MATH 5030 — Numerical Methods in Finance — Columbia University, Spring 2025

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Pricing an arithmetic Asian option requires simulation because the average of lognormal prices has no closed-form distribution. This package implements two analytic benchmarks and six Monte Carlo / Quasi-Monte Carlo estimators, and shows that combining a geometric-Asian control variate with Brownian-bridge Sobol QMC reduces pricing error by up to 82× versus plain Monte Carlo at the same computational cost.

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Installation

pip install asian-option-pricer

Or from source:

git clone https://github.com/ZixuLiAdrian/MATH5030-Numerical-Methods-in-Finance
cd MATH5030-Numerical-Methods-in-Finance
pip install -e ".[test]"

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Quick start

from asian_option_pricer import AsianOptionParams, antithetic_cv_price, rqmc_sobol_price

params = AsianOptionParams(S0=100, K=100, r=0.05, sigma=0.30, T=1.0, N=50)

# Antithetic + control variate (~25x variance reduction)
print(antithetic_cv_price(params, n_paths=100_000))
# {'price': 8.0726, 'std_err': 0.0005, 'runtime_s': 0.04, 'n_paths': 100000, 'beta': 1.03, 'var_reduction': 586.0}

# Randomised QMC with Brownian bridge (~82x RMSE reduction)
print(rqmc_sobol_price(params, n_paths=131_072, n_replications=16, path_method="brownian_bridge"))
# {'price': 8.0727, 'std_err': 0.0002, 'runtime_s': 0.12, 'n_paths': 131072, ...}

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API reference

AsianOptionParams

Frozen dataclass holding all option parameters. Validated automatically by every pricing function.

Parameter	Type	Description
S0	float	Initial stock price
K	float	Strike price (> 0)
r	float	Risk-free rate
sigma	float	Volatility (> 0)
T	float	Time to maturity in years (> 0)
N	int	Number of monitoring dates (≥ 1)
option_type	str	"call" (default) or "put"

Analytic benchmarks

Function	Returns	Description
geometric_asian_call_price(params)	float	Exact Kemna–Vorst (1990) closed form for the discrete geometric Asian call.
levy_approx_call_price(params)	float	Levy (1992) lognormal moment-matching approximation. Accurate for sigma ≲ 0.3.

MC / QMC estimators

All estimators return a dict with at minimum {price, std_err, runtime_s, n_paths}.

Function	Extra keys	Description
standard_mc_price(params, n_paths, seed, path_method)	—	Plain Monte Carlo.
antithetic_mc_price(params, n_paths, seed, path_method)	—	Antithetic variates; pairs each draw z with −z.
control_variate_price(params, n_paths, seed, path_method)	beta, var_reduction	Geometric-Asian control variate with sample-estimated beta.
antithetic_cv_price(params, n_paths, seed, path_method)	beta, var_reduction	Antithetic + control variate combined.
sobol_qmc_price(params, n_paths, seed, path_method)	—	Single scrambled Sobol replication. std_err is a proxy only.
rqmc_sobol_price(params, n_paths, seed, n_replications, path_method)	n_replications	Multiple scrambles; reports an honest cross-replication SE.

path_method — "incremental" (default) or "brownian_bridge". The bridge construction concentrates path variance on the best-equidistributed Sobol coordinates and is recommended for QMC estimators.

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License

MIT — see LICENSE. Authors: Zixu Li, Zhanpeng Jiang, Kaifan Yang, Terry Zhang, Jack Jia.

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Results at a glance

Reference: S0 = 100, K = 100, r = 5%, sigma = 30%, T = 1 year, N = 50. RMSE measured across 12 independent replications; see experiments/run_efficiency.py.

Method	RMSE @ 524k paths	Gain over plain MC
Standard Monte Carlo	1.9 × 10⁻²	1.0 ×
Antithetic variates	8.2 × 10⁻³	≈ 2.6 ×
Sobol QMC — incremental	1.8 × 10⁻³	≈ 10 ×
Control variate (geometric Asian)	7.3 × 10⁻⁴	≈ 26 ×
Antithetic + Control variate	7.8 × 10⁻⁴	≈ 24 ×
Randomised QMC — Brownian bridge	3.5 × 10⁻⁴	≈ 54 ×
Sobol QMC — Brownian bridge	2.3 × 10⁻⁴	≈ 82 ×

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Mathematical setup

Under the risk-neutral measure the underlying follows geometric Brownian motion

$$ \frac{dS_t}{S_t} = r,dt + \sigma,dW_t, \qquad S_0 \text{ given}. $$

For equally-spaced monitoring dates $t_i = i,\Delta t,; i = 1, \dots, N,;\Delta t = T/N$ the arithmetic average and Asian call payoff are

$$ A_N ;=; \frac{1}{N}\sum_{i=1}^N S(t_i), \qquad \text{payoff} ;=; \bigl(A_N - K\bigr)^+. $$

The arbitrage-free price is $C = e^{-rT},\mathbb{E}\bigl[(A_N - K)^+\bigr]$. Because $A_N$ is a sum of correlated lognormals, $C$ has no elementary closed form.

The auxiliary geometric average $G_N = \bigl(\prod_i S(t_i)\bigr)^{1/N}$ is lognormal and admits a closed form (Kemna–Vorst 1990). We use that price as a benchmark and as the control mean for variance reduction.

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Validation

Geometric Asian: closed form vs. Monte Carlo

Kemna–Vorst closed form versus antithetic MC at 200k pairs. Maximum $|z|$-score across a 12-scenario (sigma, K) grid is 0.86 — every closed-form price within one standard error of its MC estimate.

sigma	K	closed form	MC price	|z|-score
0.10	90	11.9127	11.9136	0.74
0.10	100	3.6355	3.6400	0.85
0.20	100	5.6411	5.6516	0.86
0.30	100	7.6225	7.6394	0.86
0.50	100	11.3296	11.3590	0.82

Arithmetic Asian: method error vs. high-precision reference

Errors at S0 = 100, K = 100, T = 1, N = 50 (reference SE ≤ 3 × 10⁻⁴):

Method	sigma = 0.10	sigma = 0.20	sigma = 0.30	sigma = 0.50
Levy approximation	+0.0059	+0.0192	+0.0460	+0.1664
Standard MC	+0.0046	+0.0114	+0.0200	+0.0401
Antithetic	+0.0048	+0.0099	+0.0161	+0.0311
Control variate	+0.0001	+0.0002	+0.0001	+0.0006
Antithetic + CV	−0.0002	−0.0008	−0.0015	−0.0035
Sobol (bridge)	−0.0001	−0.0000	+0.0002	+0.0010
RQMC (bridge)	+0.0002	+0.0005	+0.0008	+0.0014

Continuous-limit cross-validation (PyFENG BsmAsianJsu)

As $N \to \infty$ discrete prices converge to the continuous arithmetic Asian price. Cross-validated against PyFENG's BsmAsianJsu (Johnson's SU approximation):

N	Our discrete price	PyFENG JSU	Gap
52	5.8541	5.7630	+0.0912
250	5.7824	5.7630	+0.0194
500	5.7719	5.7630	+0.0090
1000	5.7679	5.7630	+0.0049

Gap shrinks at the expected $O(1/N)$ rate.

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Robustness

Random parameter sweep

200 scenarios across sigma ∈ [0.05, 0.80], T ∈ [0.25, 3.0], moneyness K/S0 ∈ [0.7, 1.3], r ∈ [0.0, 0.10], N ∈ {12, 26, 52, 100, 250}:

Method	NaNs	Out-of-bounds	Mean SE
Standard MC	0	≤ 1	0.142
Antithetic	0	≤ 1	0.113
Control variate	0	≤ 1	0.017
Antithetic + CV	0	≤ 1	0.017
Sobol bridge	0	≤ 1	0.142 (proxy SE)
RQMC bridge	0	≤ 1	0.007

Zero NaNs, zero crashes. The rare out-of-bounds count (≤ 1 per method) reflects Monte Carlo noise on a tight geometric-Asian lower bound — not a pricing error.

Monotonicity

Strike grid $K \in [70, 130]$ and volatility grid $\sigma \in [0.05, 0.80]$ (31 points each). Every variance-reduced method is weakly decreasing in K and weakly increasing in σ — no violations.

Edge cases

• sigma → 0: all MC/QMC estimators converge to the deterministic intrinsic value (2.46511). The geometric closed-form correctly gives a different value (2.45495) by Jensen's inequality.
• Deep OTM (K ∈ {200, 400}): every estimator returns exactly 0.0.
• Dense monitoring (N = 500): variance-reduced estimators (CV, A+CV, Sobol bridge, RQMC) agree within 2 bp of each other.

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Repository layout

MATH5030-Numerical-Methods-in-Finance/
├─ src/asian_option_pricer/          # core package
│  ├─ analytic.py                    # Kemna-Vorst + Levy
│  ├─ control_variate.py             # CV and antithetic+CV estimators
│  ├─ monte_carlo.py                 # plain + antithetic MC
│  ├─ paths.py                       # incremental + Brownian-bridge construction
│  ├─ qmc.py                         # Sobol QMC + randomised QMC
│  ├─ models.py, utils.py
│  └─ benchmarks.py
├─ experiments/
│  ├─ run_benchmarks.py              # smoke test
│  ├─ run_validation.py              # validation tables
│  ├─ run_robustness.py              # robustness tables
│  └─ run_efficiency.py              # efficiency table + figure
├─ tests/                            # 47 unit tests
├─ notebooks/demo.ipynb              # interactive walkthrough
├─ results/
│  ├─ tables/                        # CSVs from experiments
│  └─ figures/                       # PNGs from experiments
├─ pyproject.toml
├─ LICENSE
└─ README.md

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References

1. Kemna, A. G. Z., & Vorst, A. C. F. (1990). A pricing method for options based on average asset values. Journal of Banking and Finance, 14(1), 113–129.
2. Levy, E. (1992). Pricing European average rate currency options. Journal of International Money and Finance, 11(5), 474–491.
3. Turnbull, S. M., & Wakeman, L. M. (1991). A quick algorithm for pricing European average options. Journal of Financial and Quantitative Analysis, 26(3), 377–389.
4. Caflisch, R. E., Morokoff, W. J., & Owen, A. B. (1997). Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension. Journal of Computational Finance, 1(1), 27–46.
5. Owen, A. B. (1997). Scrambled net variance for integrals of smooth functions. Annals of Statistics, 25(4), 1541–1562.
6. Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer.
7. Choi, J., & Kwok, Y. K. (2022). Moments of the continuous arithmetic Asian option under GBM. SIAM Journal on Financial Mathematics. (Implemented as BsmAsianJsu in PyFENG.)