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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.

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A portfolio is simply the collection of all assets you own. So in all your three cases, you still have a portfolio. In a Sense, a trading strategy is a synonym for portfolio in maths finance since you only need to know how much you invest at a certain time in a certain asset. For instance, your first scenario may be described by $(0,100,0,0....)$, i.e. you have no money invested in the cash account, you hold 100 shares of the first asset and zero shares of all other assets. Of course, you can allow for negative position which simply translates to short selling/borrowing. Note that the portfolio (or trading strategy) may be random since it may depend on the evolution of the stock prices. If they rise, you may want to purchase more of them or such. However, the portfolio positions need to be adapted such that you cannot look into the future. Nonetheless, you can trade (change the portfolio) as much as you wish and alter the position sizes as much as you want. So at any time point, you can completely change everything and yet it remains a portfolio. The portfolio value is then just the amount of money invested in asset 1 times the price of asset 1 plus your position in asset 2 times the price of asset two etc.

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  • $\begingroup$ Okay. So are you saying that a portfolio not only specifies the quantities of each asset that I hold at one point in time, it also details the trading strategy I am using? Trading strategy effectively means that I specify the quantities of each asset not just at one point in time but as a continuous function of time for the duration under consideration, correct? $\endgroup$ – gjm24 Aug 12 at 7:19
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    $\begingroup$ Yes and no. A portfolio (= trading strategy) specifies only how many quantities of which asset you hold at time $t$. You don’t have to define this at time zero. Your portfolio may be random (but adapted), i.e. depend on the evolution of the stock prices. So you can change your trading strategy (= portfolio) at any time. $\endgroup$ – KeSchn Aug 12 at 10:23

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