# Recitation 3 Note

*2021-07-22*

# 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,

\[ x, y \in \mathbb{R} \]

### 1.8.1 - Divisibility: fractional.

### 1.8.2 - Liquidity: any asset can be bought or sold on demand at the market price in arbitrary quantities.

## 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.

dealer A | buy | sell |
---|---|---|

1.0000 EUR | 1.0202 USD | 1.0284 USD |

1.0000 GBP | 1.5718 USD | 1.5844 USD |

dealer B | buy | sell |
---|---|---|

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)

We could borrow 1EUR and use A to change 1EUR into \(1×1.0202=1.0202\)USD

Use B to change 1.0202USD into \(1.0202×0.6299=0.6426\)GBP

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

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