Suppose an American put option is only allowed to be exercised at N time instants between now and expiration. Let the current time be zero and denote the exercisable instants by the time vector t = (t₁ t₂ ··· t_N )^T. Let N_i(d_i; R_i) denote the i-dimensional multi-variate normal integral with upper limits of integration given by the i-dimensional vector d_i and correlation matrix R_i. Define the diagonal matrix D_i = diag (1, ··· 1, −1), and let d^∗_i = D_id_i and R^∗_i = D_iR_iD_i. Show that the value of the above American put with N exercisable instants is found to be (Bunch and Johnson, 1992)

and S^∗_tj is the optimal exercise price at t_j . Also, find the expression for the correlation matrix R_i.

When N = 3 and the exercisable instants are equally spaced, the correlation matrix R₃ is found to be

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where N N -rti P = XΣe ¹¹₁ N₁ (d₁; R) - S[Ni(d; R), i=1 i=1 di₁ di₂ = d₁₁0(√₁ √T₂ dj1 = (d₁1, d21, ..., di1), d = Did₁₁, ... In + (r+)1, ST₁ o √tj √ti), di₂ = Didi₂, 12 j = 1, ..., i,