In-Family Arbitrage-Free Interpolation of Mixture Densities Across Expirations

Abstract: Given risk-neutral densities of a tradeable forward, fitted as $N$-component mixtures at a finite set of expiration pillars, we look for a continuous-time interpolation that (i) stays inside the mixture family (it remains a mixture of the same kernel, though generically with more components than either pillar), and (ii) is the marginal flow of a Markov martingale, equivalently carries a non-negative Dupire local volatility. The second requirement is the peacock (convex-order) property. For full-support kernels (Gaussian, lognormal) a peacock corresponds to a unique continuous local-volatility diffusion (Lowther). We give a constructive interpolation that stays in a fixed $2N$-component family, note as an open question whether $N$ components suffice, and describe the main practical difficulty: in strongly bimodal regimes the local volatility stays finite but becomes badly conditioned.