# ⚛  L1 Principle — Heston Stochastic Volatility Calibration

**ID:** `L1-439` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Finance — *Stochastic volatility*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** stochastic vol parameter vector
> **📡 Carrier:** N/A · **🌫 Noise:** market bid ask gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555258

---

## 🧠 1. Introduction

**Heston Stochastic Volatility Calibration** is a **nonlinear inverse problem** whose unknown lives in **stochastic vol parameter vector** space, within the **Stochastic volatility** sub-domain of **Computational Finance**.

Measurements consist of N/A via a **vol surface calibration heston** sensing mechanism.

The forward operator applies, in order: applies a smooth nonlinear function element-wise; F · fourier · carr madan operator; O · chi2 · vol surface operator.

Observations are corrupted by market bid ask gaussian. Existence of the recovered stochastic_vol_parameter_vector 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 ~= 500); data_sparsity_OTM_options 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 → F · fourier · carr madan → O · chi2 · vol surface → **y** (detector).

```
y = `O.chi2.vol_surface` `F.fourier.carr_madan` f(·) x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `N.pointwise` | Applies a smooth nonlinear function element-wise |
| `F.fourier.carr_madan` | F · fourier · carr madan operator |
| `O.chi2.vol_surface` | O · chi2 · vol surface operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Finance |
| Sub domain | Stochastic volatility |
| Carrier | N/A |
| Problem class | nonlinear_inverse |
| Solution space | stochastic_vol_parameter_vector |
| Noise model | market_bid_ask_gaussian |
| Integration axis | parameter_space |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Existence of the recovered stochastic_vol_parameter_vector 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 ~= 500); data_sparsity_OTM_options 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 | vol_surface_RMSE_bp |
| Secondary | Heston_price_RMSE_percent |

## 📏 5. Operating Range (Ω)

**Center problem class:** `nonlinear_inverse` · **Forward operator:** `vol_surface_calibration_heston`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N strikes | — | 15 |
| N maturities | — | 8 |
| Data noise bp | — | 30 |
| Max maturity yr | — | 2 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N strikes | — | 5 – 50 |
| N maturities | — | 3 – 30 |
| Data noise bp | — | 10 – 200 |
| Max maturity yr | — | 0.5 – 5.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 60 vol_surface_RMSE_bp

| Metric | Range |
|---|---|
| Vol surface rmse bp | 10 – 500 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **vol_surface_RMSE_bp**, with κ = `10000.0` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x69f3e2451637b97f6c2a1cdd10bb56024ec7ca0ba74f929baa3c0aec3e689c33`
- **Chain tx hash:** `0x8ba9e1e8a4ae53459689d357eff7b7f7578dce5df9ae361c6e2b07c1ccb8444c`
- **Chain block:** `41555258`

---

## File Mapping

This bundle consists of: `L1-439.md`, `L1-439.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-439.md` | Source of truth — edit this | Human or LLM |
| `L1-439.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 `L1-439`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._