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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.

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Please, could you use latex for a mathematical question? There are some characters missing too. This is homework, right? –  Richard May 20 at 7:11

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