Pricing American-Style Options by Monte Carlo Simulation ...

C.5. Modified Matching Projection PursuitThe Modified Matching Projection Pursuit (MMPP) method constrains the magnitudeof the projection in each iteration to a small quantity rather than the magnitude thatwould minimize the L 2 norm of the residual vector.MMPP AlgorithmStep 1: Set up a dictionary with basis functions: {x j } N bj=1Step 2: Determine the cutoff c.Step 3: Determine the step size ɛ.Step 4: Among the dictionary directions, find the direction γ (1) that best describes thedata y (0) .{∣ 〈 〉∣ γ (1) y (0) , x h ∣ ∣ 〈 〉∣ }y (0) , x j ∣= h :≥ , ∀j‖x h ‖ 2‖x j ‖ 2〈y (0) , x γ (1)〉a 1 =∥∥∥ xγ ∥ 2 (1)2where 〈 y (0) , x h〉=∑ Li=1 y(0) i (x h ) iand ‖x h ‖ 2=√ ∑Li=1 (x h) 2 i .Step 5: Project along the best direction under the step size constraint and computethe residual vector y (1) .a ′ 1 =sign (a 1 ) × min {|a 1 | , ɛ}y (1) =y (0) − a ′ 1x γ (1)11