# ⚛  L1 Principle — Monte Carlo Option Pricing

**ID:** `L1-441` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Finance — *Monte Carlo simulation*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** option price scalar
> **📡 Carrier:** N/A · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555259

---

## 🧠 1. Introduction

**Monte Carlo Option Pricing** is a **parameter-estimation problem** whose unknown lives in **option price scalar** space, within the **Monte Carlo simulation** sub-domain of **Computational Finance**.

Measurements consist of N/A via a **monte carlo path simulation** sensing mechanism.

The forward operator applies, in order: S · mc · path generation operator; O · discounting · expectation operator; adds a prior term that biases the solution toward smoothness/sparsity.

Observations are corrupted by additive Gaussian noise. Existence of the recovered option_price_scalar is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 10); discretization_error_Euler_Milstein dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → S · mc · path generation → O · discounting · expectation → **y** (detector).

```
y = `O.discounting.expectation` `S.mc.path_generation` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.mc.path_generation` | S · mc · path generation operator |
| `O.discounting.expectation` | O · discounting · expectation operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `O.regularize` | Adds a prior term that biases the solution toward smoothness/sparsity |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Finance |
| Sub domain | Monte Carlo simulation |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | option_price_scalar |
| Noise model | gaussian |
| Integration axis | simulation_paths |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 4. Measurement Model

Existence of the recovered option_price_scalar is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 10); discretization_error_Euler_Milstein dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | price_standard_error_percent |
| Secondary | price_bias_percent |

## 📏 5. Operating Range (Ω)

**Center problem class:** `parameter_estimation` · **Forward operator:** `monte_carlo_path_simulation`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N paths | — | 10000 |
| Sigma vol | — | 0.2 |
| N timesteps | — | 100 |
| T maturity yr | — | 1 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N paths | — | 1000 – 10000000.0 |
| Sigma vol | — | 0.05 – 0.8 |
| N timesteps | — | 10 – 1000 |
| T maturity yr | — | 0.1 – 10.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.5 price_standard_error_percent

| Metric | Range |
|---|---|
| Price standard error percent | 0.01 – 5.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **price_standard_error_percent**, with κ = `100` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x5ecc3f0ab38af6c3204ae3d349c809268ffa2bf1527f6761e6b57f9b3198c25a`
- **Chain tx hash:** `0xf571c624f15786b36078fbe26f1277ec0bf4e58f63802ff20436469a4a3553c7`
- **Chain block:** `41555259`

---

## File Mapping

This bundle consists of: `L1-441.md`, `L1-441.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-441.md` | Source of truth — edit this | Human or LLM |
| `L1-441.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-441`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._