2.1 Financial Basics

Cashflow, trading, portfolio, simple examples, not so simple examples. Forward contracts, pricing.

We will begin by modelling the cash-flow of an investor in discrete time. The basic idea is that an investor will have a set of \(K\) assets (for example, shares in all the companies in the FTSE 100, commodities, etc.), which, at the end of each day, they can buy and sell. There will also be one ‘special’ asset, which we will think of as (interchangeably), cash, a bank account, or a bond.

In particular, we will suppose that at time \(n\), the price of the \(K\) assets are \(S_n^1, S_n^2, \dots, S_n^K\). Typically we will suppose that these are strictly positive random variables. In addition, we have our ‘special’ asset (the bank account), which we think about as the value of 1 invested in a bank account. The special feature of a bank account is that we never lose money invested in the bank, and in fact, we gain an amount of interest which is known at time \(n\). This means if I have \(x\) invested at time \(n\) in the bank, then at time \(n+1\) I will have \(\widetilde{x}\), where \(\widetilde{x} > x\). So we can write \(\widetilde{x} = (1+r_n)x\), where \(r_n>0\) is the interest rate.

For the rest of the course, we will think of \(r_n\) as being a fixed number, \(r\), known at the outset. Although note that, in practice, the interest may change, and it will certainly depend on the time period.

Example 2. Suppose there are two assets (and the bank account), such that: \[S_0^1 = 100, S_0^2 = 50, S_1^1 = 120, S_1^2 = 40, S_2^1 = 110, S_2^2 = 60, r = 0.1 (=10\%).\]

An investor begins with 1,000, with which they buy 5 units of asset 1, and 4 units of asset 2 (the remaining money staying in the bank account). At time 1, they sell 2 units of the second asset, and buy one more unit of the first asset. How much is their portfolio worth at time 2, if they do not spend/earn any additional money?

Solution: At time 0:

  • 5 units of asset 1 = 500

  • 4 units of asset 2 = 200

  • remaining cash = 300

  • total value of portfolio = 1000

At time 1:

  • new values: \(S_1^1 = 120, S_1^2 = 40\)

  • portfolio: \(5 S_1^1+4S_1^2+\text{ cash }\times (1+r)=1090\)

  • sell 2 units \(S_1^2\), buy 1 unit \(S_1^1\): new cash = \(330+2\times 40-120=290\)

  • old portfolio (at time 1): (cash, units asset 1, units asset 2) = \((330,5,4)=((1+r)\phi_0^0,\phi_0^1,\phi_0^2)\)

  • new portfolio: (cash, units asset 1, units asset 2) = \((290,6,2)=(\phi_1^0,\phi_1^1,\phi_1^2)\)

Note that value of old portfolio \(= 330 + 5 \times 120 + 4 \times 40 = 1090\) and value of new portfolio \(= 290 + 6 \times 120 + 2 \times 40 = 1090\).

At time 2:

  • new values: \(S_2^1 = 110, S_2^2 = 60\)

  • portfolio: \((1.1\times 290,6,2)\)

  • value of portfolio: \((319+660+120)=1099\)

\(\square\)

Definition 3. A portfolio of assets (and cash) is a sequence of vectors, \[\boldsymbol{\phi}_n = (\phi_n^0, \phi_n^1, \phi_n^2, \dots \phi_n^K)\] where \(\phi_n^0\) is the cash held at time \(n\), and \(\phi_n^i\) is the number of units of asset \(i\) held at time \(n\), for \(i = 1, \dots, K\). The value of the portfolio at time \(n\) is then: \[V_n = \phi_{n}^0 + \sum_{i = 1}^K \phi_{n}^i S_{n}^i.\]

Definition 4. A portfolio is self-financing if: \[\phi_{n+1}^0 + \sum_{i = 1}^K \phi_{n+1}^i S_{n+1}^i = \phi_{n}^0 (1+r) + \sum_{i = 1}^K \phi_{n}^i S_{n+1}^i.\] That is, the value of the portfolio at time \(n+1\) is equal to the value at time \(n+1\) of the portfolio purchased at time \(n\).

If I buy additional units of an asset at time \(n+1\), I have to find this money from somewhere, either by selling other assets, or using cash from the bank account.

We need one more financial observation: we suppose that we can short sell the assets — that is, sell units of the asset that we do not hold, on the basis that we must buy them back at a later date. In terms of the portfolio we hold, this corresponds to negative values of \(\phi_n^i\). We can also borrow money from the bank (at the interest rate \(r\)), which corresponds to \(\phi_n^0\) being negative.

If a trader has \(\phi_n^i >0\), we say that the trader is long the \(i^{\text{th}}\) asset. If \(\phi_n^i <0\), we say that the trader is short the \(i^{\text{th}}\) asset.

Short selling is the financial equivalent of holding a negative number of stocks. The best way of thinking about this is in terms of the cash flow. If I buy a stock, I need to hand over some money to buy the stock now, then at some point in the future, I choose to sell the stock, and I receive the value of the stock at that later time. If I short sell a stock, the reverse happens: I receive cash equal to the current value of the stock when I enter into the agreement, but at a later point, I need to pay back the value of the asset at this future date. Note also that, when I buy an asset, I benefit if the price rises, and lose if the price falls, if I short sell, I benefit when the price falls (I have to pay less at the end), but lose if the price rises. Short selling can be though of as a bet against a particular asset. In practice, the way it works is the following: suppose I wish to short sell one unit of a stock. I phone my banker, and tell him that I wish to do this. The banker finds another of his clients who owns the stock, takes this from his account, sells it on the market, and gives me the proceeds, leaving an IOU from me in the other customer’s account. At a future date, if I wish to close the position (end the short sale), I go to the market, and buy a unit of the stock, and the banker puts this back in his account.3

Example 5. Suppose there are two assets (and the bank account), such that: \[S_0^1 = 100, S_0^2 = 50, S_1^1 = 120, S_1^2 = 40, S_2^1 = 110, S_2^2 = 60, r = 0.1 (=10\%).\] An investor begins with 1,000, with which they buy 20 units of asset 1, and short sell 5 units of asset 2 (the remaining money staying in the bank account). At time 1, they sell 40 (or sell 20, and short sell a further 20) units of the first asset, and buy 20 units of the second asset, to give a total holding of 15 units of the second asset. How much is their portfolio worth at time 2, assuming the portfolio is self-financing?

Solution:

At time 0:

  • portfolio: \((\phi_0^0,20,-5)\)

  • value \(V_0 =1000=\phi_0^0+20 \times 100-5 \times 50=\phi_0^0+1750\)

  • \(\implies\phi_0^0=-750\), i.e., borrow 750, long 20 units of asset 1, and short 5 units of asset 2

At time 1:

  • new values: \(S_1^1 = 120, S_1^2 = 40\)

  • self-financing: \[\begin{align} \text{value} = V_1 = & \phi_0^0(1+r)+\phi_0^1S_1^1+\phi_0^2S_1^2\\ =&-750\times 1.1+20\times 120 + (-5) \times 40\\ =&1375\end{align}\]

  • Now sell 40 units of asset 1, buy 20 units of asset 2

  • cashflow: \(40\times 120-20\times 40=4000\) \(\implies \phi_1^0=-825+4000=3175\)

  • new portfolio \(\boldsymbol{\phi}_1 = (3175,-20,15)\)

Or equivalently, since the portfolio is self-financing: \[\begin{align} \phi_1^0 + \phi_1^1 S_1^1 + \phi_1^2 S_1^2 & = \phi_0^0 (1+r) + \phi_0^0 (1+r) \phi_0^1 S_1^1 + \phi_0^2 S_1^2\\ & = 1375\end{align}\] Which implies \[\begin{align} \phi_1^0 & = 1375 - (-20) \times 120 - 15 \times 40\\ & = 3175.\end{align}\]

At time 2:

  • new values: \(S_2^1 = 110, S_2^2 = 60\)

  • self-financing (old portfolio \((3175,-20,15)\)): \[\begin{align} V_2 & = \phi_1^0(1+r)+\phi_1^1S_2^1+\phi_1^2S_2^2\\ & = 3175\times 1.1+(-20)\times 110 +15\times 60\\ & = 2192.5\end{align}\]

\(\square\)

We also make a number of assumptions that will simplify reality a little bit. On a small-to-medium scale, for a large institutional investor (a bank or hedge fund) none of these are serious, but on a large scale, they can be hard to justify:

  1. Agents are price-takers

    • Any trades that we make will not move the market (i.e. alter the price of the asset)

  2. Assets are perfectly divisible

    • We can always buy or sell \(\frac{1}{2}, \frac{1}{3}, \frac{1}{\sqrt{2}},\dots\) units of an asset

  3. Short selling is permitted

    • We can hold ‘negative’ amounts of an asset

  4. There are no transaction costs or taxes:

    • We can buy and sell for the same price, and will not accrue any additional costs for buying or selling.

A market which satisfies these assumptions is said to be ‘perfect and frictionless’.We will often also assume that the assets we consider do not pay dividends, nor do they have storage costs, unless otherwise stated. E.g. Gold can be bought and sold on the market, but there are costs associated with keeping Gold as an asset, since you need to keep it safe (e.g. in a bank vault). Note that we also only consider assets which are fungible — that is, different instances are essentially exchangeable: I can swap one share of Vodafone for any other, and there is no substantial difference. Similarly, for a given level of purity, I can swap 1kg of Gold or Platinum for another. However, I could not easily swap one diamond of a given size for another, since there are many other qualities of a diamond that determine how valuable it is.

Definition 6. A forward contract is a contract which obliges the first party to purchase an asset from the second party at a future date, at a fixed price. The price is known as the forward price.

For example, I agree to purchase 100 Barclays shares from you in 12 months time, at 3 per share. The forward price is then: 3.

Lemma 7. The fair forward price for an asset which is worth \(S_0\) today, with delivery date \(N\) is: \((1+r)^NS_0\).

Shortly, we will introduce a more explicit notion of fair, for now, just consider the following argument:

Proof. Suppose I can enter into either side of a forward contract, with forward price \(K\), and delivery date \(N\). Consider the following:

  • If \(S_0(1+r)^N < K\): I ‘sell’ the contract, so I must deliver the asset at time \(N\). At the outset, I also buy the asset, borrowing \(S_0\) from the bank to do so. At time \(N\), I own the asset, which I am contracted to sell to the other party for \(K\), and owe the bank \(S_0(1+r)^N\). Hence I am left with \((K-S_0 (1+r)^N)\), which is greater than \(0\), and so I am better off.

  • If \(S_0(1+r)^N > K\): I ‘buy’ the contract, so I must purchase the asset at time \(N\). At the outset, I also short-sell the asset, investing the resulting \(S_0\) in the bank. At time \(N\), I buy the asset from the other party for \(K\), and hold in the bank \(S_0(1+r)^N\). Hence I am left with \((S_0 (1+r)^N-K)\), which is greater than \(0\), and so I am better off.

The only ‘fair price’ — that is, the only price at which either the buyer or seller cannot make money, is therefore the forward price \(S_0(1+r)^N\).

Arguments such as this one will form the foundation of many of our conclusions about derivatives pricing. To get a bit further, however, we need to introduce a bit more background.

  1. There are a few other details that need to be added to the picture — if the banker’s other client chooses to sell his asset, then either I will be forced into buying at whatever the prevailing price is, or I could replace this with a copy of the share held by another of the banker’s customers. In practice, I must always be cautious of the risk that my short sale may be closed down by the bank, and the bank will probably demand some sort of collateral to ensure that I will pay my future debts.↩︎