Duffie-Pan-Singleton Affine Jump-Diffusion Model

Jump-Diffusion

Extends Bates with state-dependent jump intensity

The most comprehensive model: combines stochastic volatility with state-dependent jump intensity. Higher variance leads to more frequent jumps, capturing the empirical observation that crashes cluster in high-volatility regimes.

Mathematical Formulation

\frac{d S _{t}}{S _{t^{-}}} = (μ - λ_{t} k) d t + v_{t} d W_{t}^{S} + (J - 1) d N_{t}

d v_{t} = κ (θ - v_{t}) d t + ξ v_{t} d W_{t}^{v}

λ_{t} = λ_{0} + λ_{1} v_{t}

Parameters

Symbol	Description	Constraint
$v_{t}$	Instantaneous variance	$v_{t} \geq 0$
$κ$	Variance mean reversion speed	$κ > 0$
$θ$	Long-run variance	$θ > 0$
$ξ$	Vol-of-vol	$ξ > 0$
$ρ$	Price-variance correlation	$ρ \in [- 1, 1]$
$λ_{0}$	Baseline jump intensity	$λ_{0} \geq 0$
$λ_{1}$	Variance-dependent intensity coefficient	$λ_{1} \geq 0$
$μ_{J}$	Mean log-jump size	$μ_{J} \in R$
$σ_{J}$	Jump size volatility	$σ_{J} > 0$

Key Assumptions

•State-dependent jump intensity (more jumps when vol is high)
•Nests Heston (λ₀ = λ₁ = 0), Bates (λ₁ = 0), and Merton
•Captures volatility and crash clustering
•Affine structure enables semi-analytical pricing

Reference

Duffie, D., Pan, J., Singleton, K. (2000). "Transform Analysis and Asset Pricing for Affine Jump-Diffusions." Econometrica.

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Duffie-Pan-Singleton Affine Jump-Diffusion Model

Mathematical Formulation

Parameters

Key Assumptions

Reference

Model Hierarchy

Bates

Heston

GBM

Empirical Context (SPY Returns)

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Estimation Data

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Bates

Heston

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