# Chapter 1 Chapter 1

## 1.1 Two Assets Model

• one risk-free: bond
• one risky security: stock

restrict the time scale to two instants only:

• today, $$t = 0$$,
• future time, say one year from now, $$t = 1$$

### 1.1.1 Stock

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

### 1.1.2 Bond

$$A(t)$$ and $$A(0)$$

## 1.2 Rate of Return (Return)

• The difference between intial value and current value (for stock):

$S(t)-S(0)$

The return is defined as:

$K_S = \frac{S(t)-S(0)}{S(0)}, t= 1$

Is the return $$K_S$$ a fixed value or a random value?

Similar for return of bond:

$K_A = \frac{A(t)-A(0)}{A(0)}, t = 1$

Is the return $$K_A$$ a fixed value or a random value?

## 1.3 Assumption

### 1.3.1 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.

### 1.3.2 2. Positivity of Prices

All stock and bond prices are strictly positive,

$A(t) >0 \text{ and }S(t) >0 \text{, for }t = 0, 1.$

## 1.4 Portfolio

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 (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:

$V (1) − V (0) = x(S(1) − S(0)) + y(A(1) − A(0)).$

## 1.5 Return on the Portfolio

The difference (which may be positive, zero, or negative) is $$V(1)-V(0)$$, hence return is:

$K_V = \frac{V(t)-V(0)}{V(0)}, t = 1$

Is the return $$K_V$$ a fixed value or a random value?

## 1.6 Exercise 1.1

Let $$A(0) = 90, A(1) = 100, S(0) = 25$$ dollars and let $$S(1) = 30$$ with prob $$p$$ and $$S(1) = 20$$ with prob $$1-p$$. For a portfolio with $x = 10$ shares and $$y = 15$$ bonds calculate $$V (0)$$, $$V (1)$$ and $$K_V$$.

$V(0)=xS(0)+yA(0)=10×25+15×90=1600$

$V(1)= \begin{cases}xS(1)_{high}+yA(1)=10×30+15×100=1800,& \text{ with probability }p \\ xS(1)_{low}+yA(1)=10×20+15×100=1700,&\text{ with probability }1-p\end{cases}$

$K_V=\begin{cases} \frac{V(1)_{high}-V(0)}{V(0)} = \frac{1800-1600}{1600} = 0.125, &\text{ with probability }p\\ \frac{V(1)_{low}-V(0)}{V(0)} = \frac{1800-1600}{1600} = 0.0625, &\text{ with probability }1-p \end{cases}$

## 1.7 Exercise 1.2

Given the same bond and stock prices as in Exercise 1.1, find a portfolio whose value at time 1 is $$V(1) = 1160$$ if is high and $$V(1)=1040$$ if is low. What is the value of this portfolio at time 0?

$\begin{cases} 30x+100y = 1160 \\ 20x+100y = 1040 \end{cases}\longrightarrow \begin{cases} x = 12, \\ y = 8 \end{cases}$

$V(0) = xS(0) + yA(0) = 12\times 25 + 8\times 90 = 1020$

## 1.8 Assumption

An investor may hold any number $$x$$ and $$y$$ of stock shares and bonds, whether integer or fractional, negative, positive or zero. In general,

$x, y \in \mathbb{R}$

## 1.9 Assumption

### 1.9.1 - Solvency

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

$V(t)\geq 0$

A portfolio satisfying this condition is called admissible.

### 1.9.2 - Discrete Unit Prices

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

## 1.10 No-Arbitrage Principle

In brief, we shall assume that the market does not allow for risk-free profits with no initial investment.

1.0000 EUR 1.0202 USD 1.0284 USD
1.0000 GBP 1.5718 USD 1.5844 USD
1.0000 EUR 0.6324 GBP 0.6401 GBP
1.0000 USD 0.6299 GBP 0.6375 GBP

## 1.11 Solution

euros (EUR), British pounds (GBP) and US dollars (USD)

1. We could borrow 1EUR and use A to change 1EUR into $$1×1.0202=1.0202$$USD

2. Use B to change 1.0202USD into $$1.0202×0.6299=0.6426$$GBP

3. Use B to change 0.6426GBP into $$0.64260.6401=1.00394$$EUR

The arbitrage gain will be $$0.00394$$EUR.

## 1.12 No-Arbitrage Principle

There is no admissible portfolio with initial value $$V (0) = 0$$ such that $$V (1) > 0$$ with non-zero probability.

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

$V(t)\geq 0$

A portfolio satisfying this condition is called admissible.

If the initial value of an admissible portfolio is zero, $$V (0) = 0$$, then what is the probability that $$V (1) = 0$$?

If a portfolio violating this principle did exist, we would say that an arbitrage opportunity was available.

## 1.13 Risk and Return

$$A(0) = 100$$ and $$A(1) = 110$$ dollars, as before, but $$S(0) = 80$$ dollars and $$S(1) = 100$$ with probability 0.8 and $$S(1) = 60$$ with probability 0.2.

Buy $$x = 50$$ shares, $$y = 60$$. Then:

$V(1) =\begin{cases} 11600 &\text{ if stocks goes up}\\ 9600 &\text{ if stocks goes down} \end{cases}, K_V =\begin{cases} 0.16, \\ -.04 \end{cases}$

The expected return:

$\mathbb{E}(K_V) = 0.16\times 0.8 − 0.04\times0.2 = 0.12,$

The risk of this investment is defined to be the standard deviation of the random variable $$K_V$$ :

$\sigma_V = \sqrt{(0.16 − 0.12)^2\times0.8 + (−0.04 − 0.12)^2\times0.2 }= 0.08,$

## 1.14 Exercise 1.4

For the previous stock and bond prices, design a portfolio with initial wealth of \$10000 split fifty-fifty between stock and bonds. Compute the expected return and risk as measured by standard deviation.

$x80 = 5000 \rightarrow x = 62.5\\ y100 = 5000 \rightarrow y = 50$

$V(1) = \begin{cases} 62.5\times 100 + 50\times 110 = 11750&\text{ if stocks goes up}\\ 62.5\times 60 + 50\times 110 = 9250&\text{ if stocks goes down} \end{cases}\\ K_V=\begin{cases} \frac{V(1)_{high}-V(0)}{V(0)} = \frac{11750-10000}{10000} = 0.175, &\text{ if stocks goes up}\\ \frac{V(1)_{low}-V(0)}{V(0)} = \frac{9250-10000}{1000} = -0.075, &\text{ if stocks goes down} \end{cases}$

## 1.15 Exercise 1.4

\mathbb{E}(K_V) = 0.175\times 0.8 - 0.075\times 0.2 = 0.125\\ \begin{align} \sigma_V &= \sqrt{(0.175-0.125)^2\times 0.8+(-0.075-0.125)^2\times0.2} \\ &= 0.1\end{align}