MATH 3589 Lecture Notes - Lecture 4: Random Variable, Adapted Process, Fair Coin

• School: Ohio State University
• Department: Mathematics
• Course: MATH 3589
• Professor: John Holmes

Document Summary

Homework assignment #4 solutions: consider the n-period binomial model. , n 1 be any adapted portfolio process. Therefore, at time t = k + 1 we have. Xk+1 = ksk+1 + (1 + r)(xk ksk). Prove that the discounted portfolio under the risk-neutral probability measure. 1 (1+r)n xn, n = 0, 1, . To prove that risk-neutral probability measure we calculate (1+r)n xn, n = 0, 1, . Using the property of linearity of expectation, we obtain. Now we use sk+1 = usk with probability p and sk+1 = dsk with probability. Qd u(1 + r) du + ud d(1 + r) Pu (1 + r)k+1 + (1 + r)k , and therefore, after substituting back into the previous equation, we nd (1 + r)k+1 = (u d)(1 + r)k+1. Since k is arbitrary, we may conclude the discounted expected value of our wealth process is a martingale under the risk-neutral probability measure. (cid:3)