This paper makes four contributions to the analysis of two-asset digital barrier options via wedge heat kernels. First, the stationary drift-diffusion exit problem on a wedge (López and Pérez Sinusía, 2004) is extended to the finite-horizon parabolic setting. A one-parameter family of contracts indexed by κ ∈ 0, r is introduced: κ = r prices deferred payment (holder receives S 1 at expiry if the paying barrier is hit first), κ = 0 prices immediate payment (holder receives S 1 at the hitting time), and intermediate κ interpolates between the two. Symmetry and a no-arbitrage identity reduce pricing to a single basis problem VB^ (κ). Second, a boundary-driven reformulation (R_κ) is derived: after a decorrelating coordinate change and drift removal, the remaining non-trivial time dependence of VB^ (κ) is shown to be carried entirely by the paying sector boundary. In the general wedge, the resulting Duhamel representation does not collapse to a single Sommerfeld-type contour integral. Third, for uncorrelated assets (ρ = 0), the two-dimensional Poisson kernel factorises exactly, yielding a unified single-integral formula valid for all κ ∈ 0, r. The integrand is the product of a first-passage density (Asset 1 to its barrier) and a survival probability (Asset 2 above its barrier), weighted by e^ (- (r-κ) σ): deferred payment (κ = r) corresponds to the bare first-passage probability; immediate payment (κ = 0) to its Laplace transform at rate r. Fourth, the framework is extended to asset-or-nothing digitals via a change of numeraire: a single Girsanov drift shift yields a finite-horizon closed-form formula (AB) of the same single-integral structure, together with its perpetual limit, thereby extending the framework in two directions not previously treated in closed finite-horizon form.
J.M. de Jong (Sat,) studied this question.