... =12;r =14, and payoff function5 , Sk+.Theriskneutral probabilities are~p =12,~q =12, and thusSk= S0uMk;CHAPTER 8. Random Walks103whereMkis a symmetric random walk ... we say that IPandfIPare equivalent. IPandfIPare equivalent if and only ifIP A=0exactly whenfIP A=0; 8A2F:If IPandfIPare equivalent and Zis the Radon-Nikodym derivative offIPw.r.t. ... non-random initial wealth, and Xk+1=kSk+1+1+rkXk, kSk;k=0;::: ;n , 1:Then the following processes are martingales underfIP:1MkSknk=0 and 1MkXknk=0;and...