There is a misprint in the question. The payoff should be :

Transform the American cash-or-nothing call into a linear complemen- tarity problem for the diffusion equation and show that the transformed payoff is g(x, 7) = bet(4+1) ++{(k-1)=H(x), (E7.1) T where b= B/E. Since in this case the free boundary is always at x = 0, the problem can be solved explicitly: do this. (Hint: put u(x,7) be (k+1)2+ X(X) + W(x,T) and choose X (x) appropriately, then use images, as discussed in Chapter 12. Alternatively, use Laplace transforms or Duhamel's theorem.) B g(x, y) = Be = 4(k+1)2++{(k-1)2H(x) E Transform the American cash-or-nothing call into a linear complemen- tarity problem for the diffusion equation and show that the transformed payoff is g(x, 7) = bet(4+1) ++{(k-1)=H(x), (E7.1) T where b= B/E. Since in this case the free boundary is always at x = 0, the problem can be solved explicitly: do this. (Hint: put u(x,7) be (k+1)2+ X(X) + W(x,T) and choose X (x) appropriately, then use images, as discussed in Chapter 12. Alternatively, use Laplace transforms or Duhamel's theorem.) B g(x, y) = Be = 4(k+1)2++{(k-1)2H(x) E