Imaginary Local Volatility, Arbitrage Fee Pricing and Vega Smoothing

Classical Dupire is a real valued construction. When a stressed call surface makes the local variance negative, the exact one factor coefficient leaves the real line. This paper goes beyond real numbers and represents the local arbitrage contribution as an imaginary payoff component carried on, and paying along the imaginary axis iR. Its distributional projection creates a negative real contribution, Jordan normalization then gives C = (1 + β)C⁺ − βC⁻, with two ordinary positive Dupire branches. For risk calculations, the negative branch weight is frozen as an arbitrage fee across the declared IV-shock set. Autocall tests show that this fixed fee removes repair driven bucket vega spikes and makes the IV PnL explain substantially more stable.