Transform the American cash-or-nothing call into a linear complementarity problem for the diffusion equation and show that the transformed payoff is
g(x,τ) = be^[(1/2)((k+1)^2)τ+(1/2)(k−1)x]H(x), ￼￼ where b = B/E. Since in this case the free boundary is always at x = 0, the problem can be solved explicitly:
(Hint: put u(x,τ)=be^[(1/2)((k+1)^2)τ]X(x)+w(x,τ) and choose X(x)appropriately. Alternatively, use Laplace transforms or Duhamel's theorem.