Affine Volterra processes

By Eduardo Abi Jaber, Martin Larsson, Sergio Pulido. The Annals of Applied Probability, 2017

184 citations recorded by the source corpus.

We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier-Laplace functional in terms of the solution of an associated system of deterministic integral equations of convolution type, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. Our arguments avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. Our findings generalize and clarify recent results in the literature on rough volatility models in finance.

Subfields: Demography, Finance

View Paper: https://www.semanticscholar.org/paper/d5fdeffff513fa26e43b37c55e626040938c2537

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Citations in the corpus

Lifting the Heston model (2018, background): “Indeed, it follows from [3, 18] that the Fourier-Laplace transform of the log-price in the rough Heston model (1.”
Perfect hedging in rough Heston models (2017, background): “Actually, this result could probably be understood in a more general framework of affine processes, see [1, 8].”
Markovian structure of the Volterra Heston model (2018, methodology): “Finally, for the convenience of the reader we recall in Appendix B the framework and notations regarding stochastic convolutions as in [2].”
The Quadratic Rough Heston Model and the Joint S&P 500/VIX Smile Calibration Problem (2020, background): “In addition derivatives hedging is fully understood as shown in [15, 16], see also [1, 9].”
Affine forward variance models (2018, unknown): “We further show that the CGF can be obtained as the unique global solution of a convolution Riccati equation closely related to the Volterra–Riccati equations of [2].”
A weak solution theory for stochastic Volterra equations of convolution type (2019, methodology): “This motivated the work of Abi Jaber et al. (2017), although their results only apply in the path-continuous case.”
Generalized Feller processes and Markovian lifts of stochastic Volterra processes: the affine case (2018, background): “, [4, 21, 15, 16]), stochastic Volterra processes have been studied extensively, for instance in [24, 3, 17] and the references therein.”
Joint calibration to SPX and VIX options with signature‐based models (2023, methodology): “4) and for instance also the ‘quintic Ornstein-Uhlenbeck volatility model’, recently proposed by Abi Jaber et al. (2022b), which – with an additional input curve – is shown to fit SPX and VIX smiles well.”