# ⚛  L1 Principle — Black-Scholes Option Pricing Calibration

**ID:** `L1-438` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Finance — *Option pricing*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** implied vol surface
> **📡 Carrier:** N/A · **🌫 Noise:** market bid ask gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555258

---

## 🧠 1. Introduction

**Black-Scholes Option Pricing Calibration** is a **parameter-estimation problem** whose unknown lives in **implied vol surface** space, within the **Option pricing** sub-domain of **Computational Finance**.

Measurements consist of N/A via a **option price calibration** sensing mechanism.

The forward operator applies, in order: applies a smooth nonlinear function element-wise; O · chi2 · market price operator; S · newton · root finding operator.

Observations are corrupted by market bid ask gaussian. Existence of the recovered implied_vol_surface 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); bid_ask_spread_percent dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Market bid ask gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Pointwise nonlinearity → O · chi2 · market price → S · newton · root finding → **y** (detector).

```
y = `S.newton.root_finding` `O.chi2.market_price` f(·) x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `N.pointwise` | Applies a smooth nonlinear function element-wise |
| `O.chi2.market_price` | O · chi2 · market price operator |
| `S.newton.root_finding` | S · newton · root finding operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Finance |
| Sub domain | Option pricing |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | implied_vol_surface |
| Noise model | market_bid_ask_gaussian |
| Integration axis | parameter_space |
| Difficulty delta | 3 |
| L dag | 2 |

## 📡 4. Measurement Model

Existence of the recovered implied_vol_surface 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); bid_ask_spread_percent dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Market bid ask gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | implied_vol_RMSE_bp |
| Secondary | price_RMSE_percent |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N strikes | — | 10 |
| N maturities | — | 5 |
| Moneyness range | — | 0.2 |
| Bid ask spread bp | — | 50 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N strikes | — | 3 – 50 |
| N maturities | — | 1 – 20 |
| Moneyness range | — | 0.05 – 0.5 |
| Bid ask spread bp | — | 5 – 500 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 50 implied_vol_RMSE_bp

| Metric | Range |
|---|---|
| Implied vol rmse bp | 5 – 300 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **implied_vol_RMSE_bp**, with κ = `200` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x33b2b64f770b9852cd5f8992910cd18cf894b536e1885a11382f9e0c0d01ec5d`
- **Chain tx hash:** `0xe1d1c50b015ba930d0a06caf6ece436cc45094a497c90366a489309d03263be3`
- **Chain block:** `41555258`

---

## File Mapping

This bundle consists of: `L1-438.md`, `L1-438.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-438.md` | Source of truth — edit this | Human or LLM |
| `L1-438.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 `L1-438`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._