Hi, Regarding Step 5 on the 5-step method in discrete time. I am little confused on the derivation of øt(j), see below (please refer to page 23 of the acted combined notes package 2021 set): We are told that: øt(j)=change Et(j)\change Dt(j) (this is fine, I understand this) =Et(2j-1)-Et-1(j)\Dt(2j-1)-Dt-1(j) (how can we say this? Is it because ratio d,squiggle(t,Yt)/d(t,Yt) = u,squiggle(t, Xt)/u(t,Xt), page15 of notes) =Et(2j-1)-(qt-1(j)Et(2j-1)+(1-qt-1(j))Et(2j))\Dt(2j-1)-(qt-1(j)Dt(2j-1)+(1-qt-1(j))Dt(2j)) (please can someone explain this line?) = Et(2j-1)-Et(2j)\Dt(2j-1)-Dt(2j) (please can someone explain this line?) = Vt(2j-1)-Vt(2j)/St-1(j)[ut-1 -dt-1(j)] (please can someone explain this line?) I follow what one is trying to derive just would appriciate an explanation for some of the steps. Many thanks, Darragh
Yes. The earlier work on the martingale representation theorem shows that the ratio of the changes in the two martingales is the same irrespective of whether they increase or decrease. The RHS of this line focuses on the "up" movements.
This line rewrites Et-1 in terms of Et. Given the definition of the martingale E, its current value (Et-1) is equal to its expected future value. The algebra in this line is calculating the expected value of the process at time t.
In the numerator and denominator of the previous line both have (1-qt-1(j)) as a factor. These cancel to leave: Et(2j-1)-Et(2j)\Dt(2j-1)-Dt(2j).
