I have very recently started studying quantitative finance on my own through a book called An Introduction To Quantitative Finance by Stephen Blythe.
In chapter 6 of his book, he sets out to prove what he calls the 'monotonicity theorem' using the assumption of no-arbitrage. This prove has made me very confused about what is a portfolio. He uses V^A(t) to denote the value of a portfolio A at time t. I am confused about how a portfolio is defined.
I think my confusion is best illustrated with an example. Let us say initially at t=0 I have a portfolio, A, which consists of 100 units of stocks of a company. After one year, t=1, these stocks pay a cash dividend. Consider these three scenarios:
Scenario 1: I immediately use all of this cash dividend and buy, maybe, 4 extra units of this stock so that I end up with 104 units.
Scenario 2: I place this cash dividend in a bank's timed deposit to earn interest.
Scenario 3: I decide to sell all my 100 stocks. I use the proceeds from the sale, together with the cash dividends and buy, maybe, 200 stocks of another company with any remaining cash put in a timed deposit.
In each of these scenarios, is my final portfolio still considered as portfolio A? How is a portfolio defined? Is it meaningful to talk about V^A(t=1) for each of these scenarios? How about V^A(t=2) assuming I did not perform any market transactions in the second year. Is it still meaningful to talk about the value of portfolio A at a later time when I have completely changed the assets that I am holding?
I apologise if my question seems unclear. Any help would be greatly appreciated. I need to understand this because he uses the monotonicity theorem to prove a lot of different things.