A Volterra–Hawkes Model for American Option Pricing Under a Regularized Fractional Kernel

Rough volatility and jump clustering are empirically important features of equity dynamics, yet their joint treatment in American-option pricing remains computationally demanding. We develop a Volterra–Hawkes stochastic-volatility model in which a regularized weakly singular fractional kernel governs both rough diffusive memory and variance-jump propagation. Regularizing the kernel at an explicit resolution scale preserves complete monotonicity and a nonnegative Bernstein representation while replacing the unresolved zero-lag jump response by a finite plateau. A positive exponential-sum approximation then yields a finite-dimensional Ornstein–Uhlenbeck Markovian lift; we identify and correct a rank-one covariance defect in the naive shared-shock simulation of the lifted factors and combine the corrected scheme with least-squares Monte Carlo valuation for American puts. We assess the model by an ablation over a two-branch nested design—rough-Heston diffusion as the common base, a price-jump Hawkes channel and a variance-jump Hawkes channel as two parallel single-channel extensions, and the full model combining both—calibrated and evaluated out of sample on short-maturity puts for five underlyings (NVDA, TSLA, META, AAPL, MSFT). Pooled across assets, the full model attains the lowest per-date vega-weighted RMSE on 77.8% of out-of-sample dates and the lowest pooled error on four of five underlyings, with META the exception. The improvement is not claimed to be uniform, and the two jump channels are complementary rather than individually sufficient. The evidence in this short-maturity sample supports the presence of both channels through their baseline intensities; the identification of their self-exciting feedback is left to a longer-maturity panel.