This paper develops a simple network approach to American exotic option valuation under Lévy processes using the fast Fourier transform (FFT). The forward shooting grid (FSG) technique of the lattice approach is then generalized to expand the FFT-network to accommodate path-dependent variables. This network pricing approach is applicable to all Lévy processes for which the characteristic function is readily available. In other words, the log-value of the underlying asset can follow finite-activity or infinite-activity Lévy processes. With the powerful computation of FFT, the proposed network has a negligible additional computational burden compared to the binomial tree approach. The early exercise policy and option values in the continuation region are determined in a way very similar to that of the lattice approach. Numerical examples using American-style barrier, lookback, and Asian options demonstrate that the FFT-network is accurate and efficient.

Empirical evidence indicates that commodity prices are mean reverting and exhibit jumps. As some commodity option payoffs involve the arithmetic average of historical commodity prices, we derive an analytical solution to arithmetic Asian options under a mean reverting jump diffusion process. The analytical solution is implemented with the fast Fourier transform based on the joint characteristic function of the terminal asset price and the realized average value. We also examine the accuracy and computational efficiency of the proposed method through numerical studies.

We propose a novel 4/2 rough and smooth stochastic volatility model by combining the rough Heston (rough 1/2) and smooth 3/2 models through a convex specification. This parsimonious two-factor model admits semi-closed-form pricing formulas for equity and volatility index (VIX) derivatives, while capturing key stylized facts documented in empirical studies. The model flexibly generates elasticity of variance estimates consistent with empirical findings from equity markets and produces realistic variance distributions. Although the rough 1/2 component carries a small weight, our numerical experiments confirm a degree of roughness comparable with that obtained with the rough Heston model. Empirical analysis using S&P 500 and VIX option data shows that the model outperforms benchmark specifications both in- and out-of-sample. We further provide insights into how rough volatility modeling influences the estimation of risk-neutral return moments and variance risk premia.