Spectral pricing and Greeks for double-barrier and rebate options via a bordered-banded Chebyshev tau method

We develop and analyse a barrier-aligned Chebyshev integral tau method for pricing double-barrier and rebate options under the Black–Scholes model. The two barriers define the computational corridor exactly, and the interior strike kink of the payoff is placed on an element interface; both are enforced through finite-rank tau rows rather than through interpolation. On each element the unknown is the Chebyshev coefficient vector of the second derivative together with two integration constants, so the spatial operator is a banded block plus a low-rank border. We give the construction explicitly and provide a complete analysis: a detailed proof of elementwise analytic coefficient decay; a stability theory comprising the sectoriality (dissipativity) of the discrete generator, the A - and L -stability of the time integrators, and a uniform-in-degree bound on the spectral radius of the discrete propagator; a semidiscrete spectral error bound; and a fully discrete second-order bound under Rannacher–Crank–Nicolson time stepping. The implicit systems are solved by a Sherman–Morrison–Woodbury elimination of the interface constants in O ( s + r 3 + r s ) work, with s the number of coefficients and r the number of interface constraints. Every result is tested against an exact Dirichlet eigenfunction benchmark, itself validated by Monte Carlo. Strike alignment lowers the payoff-projection error from 1.7 × 1 0 − 1 to 1.3 × 1 0 − 13 and the priced error from ∼ 1 0 − 3 to the temporal floor; Rannacher startup restores second-order time accuracy ( 1.00 → 2.03 ); deltas and gammas are recovered to ∼ 1 0 − 9 ; the spectrum of the discrete generator is shown to lie on the negative real axis; the propagator is uniformly stable in the degree; and the bordered solver is up to 26 × faster than a dense factorisation at the sizes tested.