Numerical methods for solving PIDEs arising in swing option pricing under a two-factor mean-reverting model with jumps

This paper concerns the numerical valuation of swing options with discrete action times under a linear two-factor mean-reverting model with jumps. The resulting sequence of two-dimensional partial integro-differential equations (PIDEs) are convection-dominated and possess a nonlocal integral term due to the presence of jumps. Further, the initial function is nonsmooth. We propose various second-order numerical methods that can adequately handle these challenging features. The stability and convergence of these numerical methods are analysed theoretically. By ample numerical experiments, we confirm their second-order convergence behaviour.

💡 Research Summary

This paper addresses the numerical valuation of swing options in electricity markets by modelling the underlying spot price with a linear two‑factor mean‑reverting process that incorporates finite‑activity jumps. The spot price Sₜ is expressed as the sum of an Ornstein‑Uhlenbeck component Xₜ (mean reversion level μ, speed α, volatility σ) and a jump‑driven mean‑reverting component Yₜ (mean reversion speed β, Poisson intensity λ, jump‑size density f). This affine formulation allows for negative electricity prices, a phenomenon observed in recent day‑ahead markets.

A swing option is defined by a finite set of predetermined exercise dates {T₁,…,T_{Nₐ}}. At each date the holder may purchase up to L units at a fixed strike K, while the cumulative purchase over the whole contract cannot exceed a global bound M. The optimal control problem with multiple stopping times leads, via dynamic programming and the Feynman‑Kac theorem, to a sequence of two‑dimensional partial integro‑differential equations (PIDEs) for the value function v(x,y,t). Between two consecutive exercise dates the PIDE reads

∂ₜv = (σ²/2)∂ₓₓv + α(μ−x)∂ₓv − βy∂ᵧv − r v + λ∫