Affine Volterra processes

Abstract

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, 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.

The Quadratic Rough Heston Model and the Joint S&P 500/VIX Smile Calibration Problem

“…So fast pricing of European options is possible, see [6,19]. In addition derivatives hedging is fully understood as shown in [15,16], see also [1,9].…”

Deep PPDEs for Rough Local Stochastic Volatility

“…In fact, this condition alone ensures that the process V is an affine Volterra process, and by [2, Theorem 3.3], Assumption 2.1(i) ensures that the SDE for V admits a continuous weak solution, with sup t≥0 E[|V t | p ] finite for all p ≥ 2. This in particular implies [2,Theorem 4.3] that, under suitable integrability conditions,…”

“…The form of the kernel K ensures that the variance process is stationary. We borrow Condition (ii) from [2] so that, in the purely stochastic volatility case L(·, ·) ≡ 1, taking ς(v) = √ v, we are exactly in the setting of an affine Volterra system (log(S), V ). In fact, this condition alone ensures that the process V is an affine Volterra process, and by [2, Theorem 3.3], Assumption 2.1(i) ensures that the SDE for V admits a continuous weak solution, with sup t≥0 E[|V t | p ] finite for all p ≥ 2.…”

“…with boundary conditions ψ(0) = u, φ(0) = 0. Assumption 2.1(i) is again borrowed from [2], and refer the interested reader to this paper for examples of kernels satisfying this condition. Most examples so far in quantitative finance, such as power-law kernel K(t) ≡ t H−1/2 and Gamma kernel K(t) ≡ t H−1/2 e −λt , for H ∈ (0, 1), λ > 0, fit into this framework.…”

Markovian lifts of positive semidefinite affine Volterra-type processes

“…Note that this is not of standard Volterra form, as e.g. in [2], since Y (t, s) or E[X t |F s ] respectively cannot be expressed as a function of V t . By moving to a Brownian field analogous to (1.4) it could however be expressed as a path functional of (V s ) s≤t .…”