Monte Carlo Derivatives Pricing & Numerical Methods

A quantitative finance research package studying Monte Carlo option pricing, variance reduction, convergence rates, Greeks estimation, and optimal stopping for early-exercise derivatives.

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Key Results

Pricing Accuracy & Convergence

• European MC converges to Black–Scholes at empirical rate O(N^−1/2) ✓ verified across 1K–100K paths
• 100K paths: $6.0394 (MC) vs $6.0401 (BS) | error within 1 standard error
• Terminal sampling: Efficient direct sampling without storing full paths

Variance Reduction Achievements

Method	SE @ 50K paths	Reduction Factor	Interpretation
Standard MC	$0.0562	baseline	—
Antithetic Variates	$0.0439	1.28×	SE reduced by 22%
Control Variates	$0.0289	1.94×	SE reduced by 49%

Reduction Factor: How many times smaller the standard error is relative to Standard MC.
Arithmetic Asian with geometric control: ~80× variance reduction (geometric exactly matches log-normal distribution under GBM)

Greeks & Risk Management

• MC Greeks vs analytical: Delta ±0.002, Gamma ±0.0003, Vega ±0.001 | all with common random numbers
• Newton-Raphson IV solver: Converges in 3 iterations | recovers 20% vol from price to 0.0002 error

American Options (Longstaff–Schwartz)

• LSM vs CRR binomial: 0.3–1.0% error across 5K–100K paths
• Basis functions: Polynomial, normalized, Laguerre — all converge within tolerance
• Parameter sensitivity: Robust across moneyness (0.8–1.2), volatility (10–40%), maturity (0.25–2.0y)

Exotic Derivatives

• Arithmetic Asian pricing: MC essential (no closed form)
• Geometric control variate: ~80× variance reduction without extra paths

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Empirical Validation

Monte Carlo Convergence: O(N^-1/2) Verification

Theoretical and empirical RMSE decline on log-log axes. All three variance-reduction methods follow the same convergence rate (parallel slopes), with variance reduction lowering the constant factor.

Variance Reduction Factor by Method

Antithetic Variates achieve ~1.2-1.5× reduction; Control Variates achieve ~1.8-2.3× reduction. The flat profiles show that reduction factors are stable across path counts.

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Installation

pip install -e .

Or install with development dependencies:

pip install -e ".[dev]"

Then run tests:

pytest

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Quick Start

European Option Pricing

from mcoptions import price_european_option_mc_terminal, black_scholes_call

# Efficient terminal sampling
result = price_european_option_mc_terminal(
    S0=100, K=110, T=1.0, r=0.05, sigma=0.20,
    num_simulations=100_000, seed=42
)
print(f"MC Price: ${result['price']:.4f}")
print(f"95% CI: {result['confidence_interval']}")

# Validate against analytical
bs_price = black_scholes_call(100, 110, 1.0, 0.05, 0.20)
print(f"BS Price: ${bs_price:.4f}")

Variance Reduction

from mcoptions import price_european_option_mc_control_variate

# Control variate reduces SE by ~45%
result = price_european_option_mc_control_variate(
    S0=100, K=110, T=1.0, r=0.05, sigma=0.20,
    num_simulations=50_000, seed=42
)
print(f"Price: ${result['price']:.4f}")
print(f"Variance reduction factor: {result['variance_reduction_factor']:.1f}×")

Greeks Estimation

from mcoptions import mc_delta, mc_gamma, delta_call, gamma

# Monte Carlo Greeks with common random numbers
S0, K, T, r, sigma = 100, 110, 1.0, 0.05, 0.20

delta_mc = mc_delta(S0, K, T, r, sigma, num_simulations=100_000)
print(f"MC Delta: {delta_mc['delta_mc']:.6f}")
print(f"BS Delta: {delta_mc['delta_bs']:.6f}")
print(f"Error: {delta_mc['error']:.6f}")

American Options

from mcoptions import price_american_put_lsm, price_american_put_binomial

# Longstaff–Schwartz Monte Carlo
lsm_price, paths, cashflows = price_american_put_lsm(
    S0=100, K=100, T=1.0, r=0.05, sigma=0.20,
    steps=100, num_simulations=50_000, seed=42
)

# Independent binomial validation
crr_price = price_american_put_binomial(
    S0=100, K=100, T=1.0, r=0.05, sigma=0.20, steps=1000
)

print(f"LSM:     ${lsm_price:.4f}")
print(f"CRR:     ${crr_price:.4f}")
print(f"Error:   {abs(lsm_price - crr_price):.4f}")

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Core Features

Pricing Engines

• European options: Direct terminal sampling (300× faster)
• American options: Longstaff–Schwartz least-squares regression
• Exotic options: Arithmetic Asian with geometric control variate
• Implied volatility: Newton-Raphson inversion

Variance Reduction

• Antithetic variates: ±0 correlation, O(N) cost
• Control variates: Optimal β calculation, 40–90% reduction
• Common random numbers: For Greeks, multi-leg strategies

Numerical Methods

• Convergence verification: O(N^−1/2) empirically confirmed
• RMSE analysis: 100 independent trials per configuration
• Parameter sensitivity: Across moneyness, volatility, maturity

Risk Management

• 5 analytical Greeks: Delta, Gamma, Vega, Theta, Rho
• MC Greeks: Finite-difference bump-and-revalue with CRN
• Independent validation: Black-Scholes benchmarks

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Module Architecture

mcoptions/
├── black_scholes.py          # Analytical pricing & Greeks
├── monte_carlo.py            # European option pricing
├── variance_reduction.py      # Antithetic, control variates
├── mc_greeks.py              # Monte Carlo Greeks (CRN)
├── american_option.py        # Longstaff–Schwartz
├── binomial.py               # CRR binomial tree (validation)
├── exotic_options.py         # Asian, path-dependent
├── implied_volatility.py      # IV inversion (NR, Brent)
├── gbm.py                    # GBM simulation
├── convergence_analysis.py   # Empirical convergence
├── lsm_analysis.py           # LSM validation suite
└── __init__.py               # Public API

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Design Philosophy

Numerical Methods First

• Focus on variance reduction, convergence rates, and estimator efficiency rather than feature breadth
• Verify all claims empirically (O(N^−1/2), variance ratios, error bounds)
• Compare methods on variance, runtime, and implementation complexity

Independent Validation

• Each pricing method has a ground-truth benchmark
  • MC ↔ Black-Scholes (European)
  • LSM ↔ CRR binomial (American)
  • Arithmetic ↔ Geometric (Asian)
• Tests verify invariants, not just function execution

Production-Ready Implementation

• Type hints on all public functions (Python 3.9+)
• Comprehensive test suite (29 tests, all passing)
• GitHub Actions CI (Python 3.10, 3.11, 3.12)
• Installable package via pip install -e .
• Configurable: seeds, observation dates, basis functions

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Testing

Run the full suite:

pytest -v

Test coverage includes:

• Pricing fundamentals: Payoff correctness, GBM paths, Black-Scholes put-call parity
• Convergence & variance reduction: MC convergence rate, antithetic/control-variate efficacy
• Greeks: Delta, gamma, vega, theta, rho (analytical vs. Monte Carlo)
• American options: LSM convergence, basis function comparison (polynomial/normalized/Laguerre), exercise boundary sanity
• Exotic derivatives: Geometric Asian analytical validation, Asian control variate effectiveness
• Risk management: Implied volatility recovery (Newton-Raphson & Brent), dividend-adjusted parity

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Mathematical Foundation

Risk-Neutral Pricing

Under the risk-neutral measure: $$dS_t = (r - q) S_t dt + \sigma S_t dW_t$$

$$S_T = S_0 \exp\left[(r - q - \tfrac{1}{2}\sigma^2)T + \sigma\sqrt{T}Z\right]$$

Monte Carlo estimate: $$V_0 \approx e^{-rT} \frac{1}{N} \sum_{i=1}^{N} \text{Payoff}(S_T^{(i)})$$

Convergence

Standard error decreases at rate: $$SE \propto N^{-1/2}$$

Verified empirically: 10× improvement when increasing paths 100×.

Variance Reduction

Control variate adjustment: $$Y_{CV} = Y - \beta^*\bigl(X - E[X]\bigr)$$

where $\beta^* = \operatorname{Cov}(Y,X) / \operatorname{Var}(X)$ reduces variance by up to 90%.

Greeks (Risk Sensitivities)

Analytical (Black–Scholes):

• Δ = ∂V/∂S (delta)
• Γ = ∂²V/∂S² (gamma)
• ν = ∂V/∂σ (vega)
• Θ = −∂V/∂t (theta)
• ρ = ∂V/∂r (rho)

Monte Carlo (finite-difference with CRN):

• Same pricing framework
• Shared random draws for bump-and-revalue
• ~1% error to analytical on 100K paths

American Options (Longstaff–Schwartz)

Backward induction with continuation-value regression:

At each time step t ∈ {T − Δt, ..., Δt}:

1. Estimate continuation value via least-squares regression
2. Compare to intrinsic value (exercise payoff)
3. Optimal exercise = max(intrinsic, continuation)

Basis functions: 1, S, S², optionally S³, ...

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Academic References

1. Black, F., Scholes, M. (1973). "The pricing of options and corporate liabilities." Journal of Political Economy, 81(3), 637–654.

2. Longstaff, F. A., Schwartz, E. S. (2001). "Valuing American options by simulation: A simple least-squares approach." Review of Financial Studies, 14(1), 113–147.

3. Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer-Verlag.

4. Kemna, A. G., Vorst, A. C. (1990). "A pricing method for options based on average asset values." Journal of Banking & Finance, 14(1), 113–129.

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Author & Citation

Samarth Uday (samarthuday.202@gmail.com)

@software{monte_carlo_options,
  author = {Uday, Samarth},
  title = {Monte Carlo Derivatives Pricing \& Numerical Methods},
  year = {2026},
  url = {https://github.com/Samarthuday/monte-carlo-options}
}

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License

MIT License. See LICENSE for details.