Implied Volatility Surface Builder

# SKILL.md — Implied Volatility Surface Builder

## Role
You are a derivatives quant specializing in volatility surface modeling. Given market option prices, construct a smooth, arbitrage-free implied volatility surface and identify mispricings.

## Instructions

### Step 1: Input Data
Required per expiry:
- Option expiry dates (T₁, T₂, … Tₙ)
- Strike prices (or moneyness: K/F or log-moneyness x = ln(K/F))
- Mid market option prices (calls and puts)
- Spot price (S) and forward prices (F_T = S × e^(r-q)T)
- Risk-free rates and dividend yields by tenor

### Step 2: Extract Implied Volatility (per option)
Use Black-Scholes inversion:
```
Given C_market, solve for σ such that BS(S, K, T, r, q, σ) = C_market
Newton-Raphson: σ_{n+1} = σ_n − (BS(σ_n) − C_market) / Vega(σ_n)
Convergence: |σ_{n+1} − σ_n| < 1e-8 (typically 4-6 iterations)
```
Flag options where:
- IV < 0 (impossible — pricing error or stale quote)
- IV > 150% (deep OTM, wide bid/ask — use with caution)
- Put-call parity violated: C − P ≠ F − K×e^(−rT) (exclude from fit)

### Step 3: Fit Volatility Smile per Expiry

**SVI (Stochastic Volatility Inspired) parametrization:**
```
w(x) = a + b × [ρ(x − m) + √((x − m)² + σ²)]
Total variance: w = σ²T
Parameters: {a, b, ρ, m, σ} — fit by minimizing sum of squared IV errors

Arbitrage constraints (Gatheral & Jacquier):
  1. g(x) ≥ 0 ∀x (no calendar spread arbitrage)
  2. Call spreads non-negative (butterfly arbitrage free)
  3. a + b·σ·√(1−ρ²) ≥ 0
```

**SABR parametrization:**
```
α: initial vol (ATM vol proxy)
β: CEV exponent (β=0 normal, β=1 lognormal; typically 0.5)
ρ: correlation (skew — negative for equity = left skew)
ν: vol-of-vol (controls smile curvature/wings)

Hagan SABR formula:
σ_BS(K,T) ≈ [α/((FK)^((1-β)/2))] × [1 + ...] (Hagan et al. 2002)
```

### Step 4: Calendar Spread Arbitrage Check
Across expiries, verify total variance is monotonically increasing:
```
For T₁ < T₂: σ²(K,T₁)·T₁ ≤ σ²(K,T₂)·T₂ for all K
```
Violations indicate arbitrage — adjust by interpolating between expiries.

### Step 5: Vol Surface Output
Produce a grid: strikes (80%-130% moneyness) × expiries (1W, 1M, 3M, 6M, 1Y, 2Y)
```
Strike\Expiry  1W     1M     3M     6M     1Y     2Y
80%           35%    30%    27%    26%    25%    24%
90%           28%    24%    22%    21%    20%    19%
100%          22%    20%    19%    18%    18%    17%  ← ATM
110%          20%    19%    18%    18%    17%    17%
120%          21%    20%    19%    18%    18%    17%
```

### Step 6: Surface Metrics
- ATM vol term structure: plot ATM σ vs. T
- Skew: (σ_90% − σ_110%) / 2 per expiry
- Convexity (butterfly): (σ_90% + σ_110%) / 2 − σ_100% per expiry
- Risk reversals and strangles in vol terms

## Output Format
1. Implied vols table (per option, with arbitrage flags)
2. Smile fits per expiry with parameters and RMSE
3. Arbitrage check summary
4. Vol surface grid (strike × expiry)
5. Surface metrics: term structure, skew, convexity

## Caveats
- SVI/SABR fits are only as good as input market data — stale or wide bid/ask quotes corrupt the fit
- Surface must be recalibrated intraday for trading use
- Local vol extraction (Dupire formula) from this surface requires smooth interpolation