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Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

2026-05-20 23:52:15 +09:00

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title

몬테카를로 시뮬레이션(Monte Carlo Simulation)

매 한 줄

"매 random sampling 의 deterministic answer 추정". Stanislaw Ulam (Manhattan Project, 1946)이 neutron diffusion 위해 고안. 매 closed-form solution 없는 high-dim integral, optimization, risk modeling 의 해법 — 매 large-N law of large numbers 의 convergence.

매 핵심

매 4단계

Define domain of possible inputs (probability distribution).
Sample randomly from domain (PRNG).
Compute deterministically for each sample.
Aggregate results (mean, variance, percentiles).

매 수학 기반

Law of Large Numbers: sample mean → expected value as N → ∞.
Convergence rate: O(1/√N) — 매 4× samples 의 2× accuracy.
Curse of dimensionality 의 escape: deterministic methods (grid) 매 d²/d³, MC 매 dimension-independent.

매 응용

금융: VaR, option pricing (Black-Scholes 외 path-dependent).
물리: particle transport, lattice QCD.
AI: Monte Carlo Tree Search (AlphaGo, MuZero).
Engineering: reliability analysis, sensitivity.
3D rendering: path tracing (Blender Cycles, Pixar RenderMan).

💻 패턴

1. π 추정 (canonical example)

import numpy as np

def estimate_pi(n: int = 1_000_000) -> float:
    pts = np.random.uniform(-1, 1, (n, 2))
    inside = np.sum(pts[:, 0]**2 + pts[:, 1]**2 <= 1)
    return 4 * inside / n

print(estimate_pi(10_000_000))  # ~3.14159

2. Option pricing (Black-Scholes via MC)

import numpy as np

def european_call_mc(S0=100, K=105, r=0.05, sigma=0.2, T=1.0, n_paths=1_000_000):
    Z = np.random.standard_normal(n_paths)
    ST = S0 * np.exp((r - 0.5 * sigma**2) * T + sigma * np.sqrt(T) * Z)
    payoff = np.maximum(ST - K, 0)
    return np.exp(-r * T) * payoff.mean()

price = european_call_mc()
print(f"Call price: {price:.4f}")

3. Value at Risk (VaR)

import numpy as np

def historical_var(returns: np.ndarray, alpha: float = 0.05) -> float:
    return np.quantile(returns, alpha)

def parametric_var(mu, sigma, alpha=0.05, n=100_000):
    samples = np.random.normal(mu, sigma, n)
    return np.quantile(samples, alpha)

# portfolio daily returns
returns = np.random.normal(0.001, 0.02, 10_000)
print(f"5% VaR: {historical_var(returns):.4f}")

4. Monte Carlo integration

import numpy as np

def mc_integrate(f, a, b, n=1_000_000):
    x = np.random.uniform(a, b, n)
    return (b - a) * np.mean(f(x))

# integrate sin(x) on [0, π] (true value = 2)
result = mc_integrate(np.sin, 0, np.pi)
print(result)  # ~2.0

5. MCTS (Monte Carlo Tree Search) — AI

import math, random

class Node:
    def __init__(self, state, parent=None):
        self.state = state
        self.parent = parent
        self.children = []
        self.visits = 0
        self.wins = 0

    def ucb1(self, c=1.41):
        if self.visits == 0:
            return float('inf')
        return self.wins / self.visits + c * math.sqrt(math.log(self.parent.visits) / self.visits)

def mcts(root, n_iter=10_000, get_actions=None, simulate=None):
    for _ in range(n_iter):
        # 1. select via UCB1
        node = root
        while node.children:
            node = max(node.children, key=lambda n: n.ucb1())
        # 2. expand
        for action in get_actions(node.state):
            node.children.append(Node(apply(node.state, action), node))
        # 3. simulate (random rollout)
        leaf = random.choice(node.children) if node.children else node
        result = simulate(leaf.state)
        # 4. backpropagate
        cur = leaf
        while cur:
            cur.visits += 1
            cur.wins += result
            cur = cur.parent
    return max(root.children, key=lambda n: n.visits)

6. Variance reduction: antithetic variates

import numpy as np

def mc_call_antithetic(S0=100, K=105, r=0.05, sigma=0.2, T=1.0, n=500_000):
    Z = np.random.standard_normal(n)
    ST_plus = S0 * np.exp((r - 0.5 * sigma**2) * T + sigma * np.sqrt(T) * Z)
    ST_minus = S0 * np.exp((r - 0.5 * sigma**2) * T - sigma * np.sqrt(T) * Z)
    payoff = (np.maximum(ST_plus - K, 0) + np.maximum(ST_minus - K, 0)) / 2
    return np.exp(-r * T) * payoff.mean()
# 매 same N 의 ~2× variance reduction

7. Reproducibility (seed)

import numpy as np
rng = np.random.default_rng(seed=42)
samples = rng.normal(0, 1, 1000)  # reproducible

매 결정 기준

상황	Approach
Low-dim (d ≤ 3), smooth integrand	quadrature (Gauss, Simpson) — faster.
High-dim integral	MC — escapes curse of dimensionality.
Path-dependent option	MC.
European option (closed form)	Black-Scholes formula — instant.
Need confidence interval	MC + bootstrap.
Game tree search	MCTS + UCB1.

기본값: NumPy MC 매 baseline, antithetic variates / control variates 매 variance reduction. JAX/CUDA 매 GPU acceleration.

🔗 Graph

Adjacent: Bayesian_Inference

🤖 LLM 활용

언제: high-dim integral, risk metrics, option pricing, game AI, sensitivity analysis. 언제 X: 매 closed-form 존재 매 (Black-Scholes European), low-dim quadrature 효율적인 case, deterministic answer 필요 매 (use seed).

❌ 안티패턴

Too few samples: O(1/√N) 매 slow — 매 1% accuracy 의 N=10000+.
No seed in production: non-reproducible bugs.
Bad PRNG: random.random() 매 OK, Math.random() (JS) 매 not crypto-safe — but fine 매 simulation.
No variance reduction: antithetic / control variates 매 free 2-10× speedup.
MC 의 deterministic 문제: 매 closed form 존재 의 use that.

🧪 검증 / 중복

Verified (Metropolis & Ulam 1949 original, Glasserman "Monte Carlo Methods in Financial Engineering" 2003, Kalos & Whitlock 2008).
신뢰도 A.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1
2026-05-10	Manual cleanup — full content with π/option/VaR/MCTS patterns

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