Dupire Local Volatility Model

Local Vol

A deterministic volatility model where the instantaneous volatility depends on both spot price and time: σ_loc(S, t). Unlike stochastic volatility models, volatility is not random but state-dependent. The local volatility surface is derived from the implied volatility surface via Dupire's formula, ensuring the model exactly reproduces market option prices.

Mathematical Formulation

d S_{t} = μ S_{t} d t + σ_{loc} (S_{t}, t) S_{t} d W_{t}

σ_{loc}^{2} (K, T) = \frac{\partial w / \partial T}{\frac{1}{2} K ^{2} \partial ^{2} w / \partial K ^{2}}

w (K, T) = T \cdot σ_{imp}^{2} (K, T)

Parameters

Symbol	Description	Constraint
$S_{t}$	Asset price at time t	$S_{t} > 0$
$σ_{loc} (S, t)$	Local volatility function	$σ_{loc} > 0$
$w (K, T)$	Total implied variance surface	$w \geq 0$
$σ_{imp} (K, T)$	Market implied volatility surface	$(from options)$
$μ$	Drift rate (0 for risk-neutral)	$μ \in R$

Key Assumptions

•Deterministic volatility: σ is a function, not a process
•State-dependent: volatility varies with spot S and time t
•Smile-consistent: exactly reproduces any arbitrage-free IV surface
•No forward smile dynamics (smile flattens unrealistically)
•Different path-dependent risk than stochastic vol models

Reference

Dupire, B. (1994). "Pricing with a Smile." Risk Magazine, 7(1), 18-20.

Model Hierarchy

Complexity increases from left to right. More complex models capture additional market phenomena but require more parameters and may be harder to estimate.

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Dupire Local Volatility Model

Mathematical Formulation

Parameters

Key Assumptions

Reference

Model Hierarchy

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Empirical Context (SPY Returns)

What Dupire captures

What it cannot capture

Estimation Data

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