Joint SPX–VIX Calibration With Gaussian Polynomial Volatility Models: Deep Pricing With Quantization Hints

The Quintic Ornstein-Uhlenbeck Volatility Model that Jointly Calibrates SPX & VIX Smiles

“…The particular parametrization with X means a = 0, b = αε −1 and c = ε α from (2.1). This model has shown to produce remarkable joint fits to both SPX and VIX implied volatility surfaces [6,7].…”

“…, where C t (S t , K, T ) denotes the European call option price with strike K and maturity T − t, k t := log(K/S t ) is the log-moneyness and φ(u) := F (t, x) the Fourier-Laplace transform of log(S T /S t ) by fixing g 1 ≡ iu and g 2 ≡ 0. We use the the representation of F (t, x) in (3.4) and compute the improper integral numerically via the Gauss-Laguerre quadrature, which has been demonstrated to be efficient, see [3,25].…”

“…More recently, Fourier techniques have also found applications in Signature volatility models in Abi Jaber and Gérard [3] and Cuchiero, Gazzani, Möller, and Svaluto-Ferro [12], where the volatility process is modeled as a linear functional of the path-signature of semi-martingales (e.g. a Brownian motion).…”

“…We demonstrate that Fourier techniques can be effectively extended to a broad class of flexible models previously considered infeasible as they fall beyond the conventional class of affine diffusions, including the celebrated one-factor Bergomi model (Bergomi [9], Dupire [15]) that has been shown to fit well to the SPX smiles and the recently introduced Quintic Ornstein-Uhlenbeck (OU) model of Abi Jaber, Illand, and Li [7], which has demonstrated remarkable capabilities in fitting jointly the SPX-VIX volatility surface for maturities between one week to three months supported by extensive empirical studies on more than 10 years of data in Abi Jaber and Li [4], Abi Jaber, Illand, and Li [6].…”