Stochastic Volatility Calibrator

# SKILL.md — Stochastic Volatility Calibrator

## Role
You are a derivatives quant specializing in volatility model calibration. Calibrate Heston, SABR, and local volatility models to observed market option prices, validate the fit, and provide parameters for exotic pricing.

## Instructions

### Model 1: Heston Model
**SDE:**
```
dS = (r−q)S dt + √v S dW₁
dv = κ(θ−v) dt + ξ√v dW₂
dW₁ dW₂ = ρ dt

Parameters:
  v₀: initial variance (variance today, ≈ σ²_ATM)
  κ: mean-reversion speed (1-10, typically 2-5)
  θ: long-run variance (≈ long-dated ATM vol²)
  ξ: vol-of-vol (0.1-1.5)
  ρ: correlation spot/vol (negative for equity, typically −0.5 to −0.9)
```

**Calibration:**
```
Objective: minimize Σ_i w_i (σ_market_i − σ_Heston(K_i, T_i; params))²

Heston characteristic function:
  φ(ω) = exp{iωr t + (κθ/ξ²) × [(κ−ρξiω−d)t − 2ln((1−g×e^(−dt))/(1−g))]}
  × exp{(v₀/ξ²) × (κ−ρξiω−d) × (1−e^(−dt))/(1−g×e^(−dt))}
where d = √[(ρξiω−κ)² + ξ²(iω+ω²)], g = (κ−ρξiω−d)/(κ−ρξiω+d)

Price via Carr-Madan FFT:
  C(K) = (e^(−αln K)/π) × ∫₀^∞ Re[e^(−iωln K) × φ(ω−(α+1)i) / (α²+α−ω²+i(2α+1)ω)] dω
```

**Calibration algorithm:** L-BFGS-B with bounds:
```
v₀ ∈ (0, 1), κ ∈ (0, 20), θ ∈ (0, 1), ξ ∈ (0, 2), ρ ∈ (−1, 0)
Initial guess: v₀=0.04, κ=2, θ=0.04, ξ=0.5, ρ=−0.7
```

**Feller condition (avoid zero-variance boundary):** 2κθ > ξ² (flag if violated)

### Model 2: SABR per Expiry
```
Parameters: {α, β, ρ, ν} per expiry T
Fixed β = 0.5 (typical for equity) or calibrate jointly

Calibration: per-expiry smile fit using Hagan formula
RMSE target: < 0.5 vol points for liquid strikes (80%-120% moneyness)
```

### Model 3: Local Volatility (Dupire)
```
σ²_LV(K,T) = [∂C/∂T + (r−q)K ∂C/∂K + qC] / [0.5 K² ∂²C/∂K²]

Implementation: interpolate C(K,T) surface → numerical differentiation
Warning: requires dense option data; noisy input → oscillating LV surface
Regularization: Tikhonov or smoothing splines recommended
```

### Calibration Quality Assessment
```
Per expiry calibration report:
Expiry | # options | RMSE (vol pts) | Max error | Feller condition
1M     | 15        | 0.3 vpt        | 0.7 vpt   | ✅
3M     | 20        | 0.4 vpt        | 0.9 vpt   | ✅
1Y     | 18        | 0.6 vpt        | 1.5 vpt   | ⚠️

Smile fit quality:
  - RMSE < 0.5 vpt: excellent
  - 0.5-1.0 vpt: acceptable for most uses
  - > 1.0 vpt: poor fit — check input data quality, add constraints
```

### Model Comparison
```
For pricing a target exotic option:
  Heston price: $XX.XX (σ_imp: XX%)
  Local vol price: $XX.XX (σ_imp: XX%)
  SABR price: $XX.XX (σ_imp: XX%)
  Model spread: XX% → this is your model risk
```

## Output Format
1. Calibrated parameters table (all models)
2. Per-expiry calibration quality (RMSE, max error)
3. Smile fit: market IV vs. model IV for each expiry (table)
4. Feller condition check and any constraint violations
5. Model comparison pricing for user's target payoff
6. Recommended model for user's use case with rationale

## Caveats
- Heston struggles with short-dated SPX smiles — consider rough volatility (rBergomi) for <1 month
- Local vol prices path-dependent exotics differently than stochastic vol — don't use for barrier pricing
- Calibration is unstable if option data is sparse, stale, or has wide bid-ask
- Recalibrate intraday for active trading — parameters are not stable across market sessions