Adaptive COS Option Pricing via Per-Mode Convergence Rates

The COS method (Fang and Oosterlee, 2008) for option pricing via Fourier-cosine expansion uses a uniform number of backward steps \(T\) across all \(N\) Fourier modes. We prove that each mode \(k\) converges at its own rate \(\delta|\varphi(k)|\), where \(\delta\) is the per-step discount and \(\varphi(k)\) the characteristic function value. This per-mode rate is a direct consequence of a structural identity: the COS backward step is a per-mode Bellman equation, with characteristic function values playing the role of transition eigenvalues (Theorem 1). The multi-step error for mode \(k\) after \(t_k\) steps is bounded by \((\delta|\varphi(k)|)^{t_k} \cdot |c_{k,0}|\), decaying geometrically (Theorem 2). We derive an adaptive algorithm that allocates \(t_k = \lceil \log(\varepsilon/|c_{k,0}|) / \log(\delta|\varphi(k)|) \rceil\) steps to mode \(k\), achieving the same accuracy \(\varepsilon\) with total computation \(\sum_k t_k \ll N \cdot T\). The realized speedup depends on the characteristic function decay profile of the underlying process: for GBM (Black-Scholes) with daily exercise, \(|\varphi(k)|\) decays slowly and the speedup is approximately 3.7\(\times\); for stochastic volatility models (Heston) where the vol-of-vol component accelerates high-frequency decay, the speedup reaches 8--12\(\times\); for Bermudan products with few exercise dates and many Fourier modes, reductions of 10\(\times\) or more are achievable under favorable configurations. All theoretical results are machine-verified in Lean 4 (10 theorems, 0 sorry). The bounds apply directly to Bermudan options with fixed exercise schedules; for American options with endogenous exercise boundaries, we establish that the adaptive bound remains a valid upper bound via a mode-wise domination argument (Section 8.3).

Keywords: COS method, option pricing, adaptive algorithm, spectral convergence, Lean 4