Chapter 1 Two Asset Model

Suppose that two assets are traded: one risk-free and one risky security. The former can be thought of as a bank deposit or a bond issued by a government, a financial institution, or a company. The risky security will typically be some stock. It may also be a foreign currency, gold, a commodity or virtually any asset whose future price is unknown today.

Throughout the introduction we restrict the time scale to two instants only: today, t = 0, and some future time, say one year from now, t = 1. More refined and realistic situations will be studied in later chapters.

The position in risky securities can be specified as the number of shares of stock held by an investor. The price of one share at time t will be denoted by S(t). The current stock price S(0) is known to all investors, but the future price S(1) remains uncertain: it may go up as well as down. The difference S(1) − S(0) as a fraction of the initial value represents the so-called rate of return, or briefly return:

\[ K_S =\frac{S(1)-S(0)}{S(0)} \] The return on bonds is defined in a similar way as that on stock: \[ K_A =\frac{A(1)-A(0)}{A(0)} \] Our task is to build a mathematical model of a market of financial securities. A crucial first stage is concerned with the properties of the mathematical objects involved. This is done below by specifying a number of assumptions, the purpose of which is to find a compromise between the complexity of the real world and the limitations and simplifications of a mathematical model, imposed in order to make it tractable. The assumptions reflect our current position on this compromise and will be modified in the future.

1.1 Assumption 1.1 Randomness

The future stock price \(S(1)\) is a random variable with at least two different values. The future price \(A(1)\) of the risk-free security is a known number.

1.2 Assumption 1.2 Positivity of Prices

All stock and bond prices are strictly positive, \[ A(t) >0 \text{ and } S(t) >0, \forall t = 0, 1. \] The total wealth of an investor holding \(x\) stock shares and \(y\) bonds at a time instant \(t = 0,1\) is \[ V (t) = xS(t) + yA(t). \] The pair \((x, y)\) is called a portfolio, \(V(t)\) being the value of this portfolio or, in other words, the wealth of the investor at time \(t\). The jumps of asset prices between times 0 and 1 give rise to a change of the portfolio value:

\[ V (1) − V (0) = x(S(1) − S(0)) + y(A(1) − A(0)). \] This difference (which may be positive, zero, or negative) as a fraction of the initial value represents the return on the portfolio,

\[ K_V =\frac{V(1)-V(0)}{V(0)} \]

The returns on bonds or stock are particular cases of the return on a portfolio (with \(x = 0\) or \(y = 0\), respectively). Note that because \(S(1)\) is a random variable, so is \(V (1)\) as well as the corresponding returns \(K_S\) and \(K_V\). The return \(K_A\) on a risk-free investment is deterministic.

1.3 Assumption 1.3 Divisibility, Liquidity, and Short Selling

An investor may hold any number \(x\) and \(y\) of stock shares and bonds, whether integer or fractional, negative, positive or zero. In general, \[x,y\in \mathbb{R}\] The fact that one can hold a fraction of a share or bond is referred to as divisibility. Almost perfect divisibility is achieved in real world dealings whenever the volume of transactions is large as compared to the unit prices.

The fact that no bounds are imposed on x or y is related to another market attribute known as liquidity. It means that any asset can be bought or sold on demand at the market price in arbitrary quantities. This is clearly a mathematical idealisation because in practice there exist restrictions on the volume of trading.

If the number of securities of a particular kind held in a portfolio is positive, we say that the investor has a long position. Otherwise, we say that a short position is taken or that the asset is shorted. A short position in risk-free securities may involve issuing and selling bonds, but in practice the same financial effect is more easily achieved by borrowing cash, the interest rate being determined by the bond prices. Repaying the loan with interest is referred to as closing the short position. A short position in stock can be realised by short selling. This means that the investor borrows the stock, sells it, and uses the proceeds to make some other investment. The owner of the stock keeps all the rights to it. In particular, she is entitled to receive any dividends due and may wish to sell the stock at any time. Because of this, the investor must always have sufficient resources to fulfil the resulting obligations and, in particular, to close the short position in risky assets, that is, to repurchase the stock and return it to the owner. Similarly, the investor must always be able to close a short position in risk-free securities, by repaying the cash loan with interest. In view of this, we impose the following restriction.

1.4 Assumption 1.4 Solvency

The wealth of an investor must be non-negative at all times,

\[V (t) \geq 0, \text{for }t = 0, 1.\] A portfolio satisfying this condition is called admissible. In the real world the number of possible different prices is finite because they are quoted to within a specified number of decimal places and because there is only a certain final amount of money in the whole world, supplying an upper bound for all prices.

1.5 Assumption 1.5 Discrete Unite Prices

The future price \(S(1)\) of a share of stock is a random variable taking only finitely many values.