Chapter 23, Problem 26PS

a

Summary Introduction

To compute: The proceeds of stock short sales needed to earn arbitrage profits. If the current interest rate is 2.5%, dividend rate of the stock is 1.9%.

Introduction:

Arbitrage Profit: Arbitrage is an act of buying an asset in the market and at the same time, the selling part is handled in another market. Such a calculated act of buying and selling is done to earn a profit when there is an imbalance of prices is called arbitrage profit.

Answer to Problem 26PS

The fraction of the proceeds which helps in earning the arbitrage profit is 0.76.

Explanation of Solution

S&P 500 index=1950

June maturity contract F₀=1951

Current rate of interest=2.5%

Dividend=1.9%

Let us consider the S&P index value at S_0.

Spot index S₀=1950

Future index F₀=1951

Time T=1

Rate of interest r_f=2.5% when converted it becomes 0.025 (2.5/100=0.025)

Dividend=1.9% When converted in becomes 0.019(1.9/100=0.019)

To calculate arbitrage profits, proceeds of short sales is required. Let us assume the fraction of proceeds of short sales to be ‘p.’

We have to use the formula of spot-future parity here.

  $F_0=S_0{\left(1+r_f\times p\right)}^T-D$

Where,

F₀= Future index

S₀= Current index

R_f= Interest rate

P=Proceeds

T=Time

D= Dividend payment

To proceed further, we have to first calculate the dividend payment.

  $\begin{array}{l}Dividendpayment=d\times S_0\\ =0.019\times 1950=37.05\end{array}$

By substituting the values in the formula, we get

  $F_0=S_0{\left(1+r_f\times p\right)}^T-D$

  $1951=1950{(1+0.025\times p)}^1-37.05$

After simplifying the equation, we get

  $\begin{array}{l}1951+37.05=1950(1+0.025p)\\ 1988=1950(1+0.025p)\\ \frac{1988}{1950}=1+0.025p\\ 1.019=1+0.025p\\ 1.019-1=0.025p\end{array}$

By interchanging the values, we get

  $\begin{array}{l}0.025p=0.019\\ p=\frac{0.019}{0.025}=0.76\end{array}$

Therefore, the fraction of the proceeds which helps in earning the arbitrage profit is 0.76.

b.

Summary Introduction

To compute: The lower bound on the future prices which rules the arbitrage opportunities. Having 90% of the sales proceeds.

Introduction:

Arbitrage opportunity: It is an opportunity which can be availed to make a risk-free profit even in market fluctuations. The process of arbitrage involves buying of an asset in one market with a lesser price and sell it another market with a higher price.

Answer to Problem 26PS

The lower bound on the future prices which rules the arbitrage opportunities is $1956.83

Explanation of Solution

S&P 500 index=1950

June maturity contract F₀=1951

Current rate of interest=2.5%

Dividend=1.9%

We are told that the proceeds from short sales is 90%. So, p=90%

Therefore p=0.9 $\frac{90}{100}=0.90$

By substituting the values in the formula, we get

  $F_0=S_0{\left(1+r_f\times p\right)}^T-D$

  $\begin{array}{l}$ =1950{(1+0.025\times 0.9)}^1-37.05$=1950\times 1.0225-37.05\\ =1993.875-37.05\\ =1956.825\end{array}$

or 1956.83 (when rounded off)

Therefore, the lower bound on the future prices which rules the arbitrage opportunities is $1956.83

c.

Summary Introduction

To evaluate: The value of actual future price which falls below the no-arbitrage bound.

Introduction:

Arbitrage Profit: Arbitrage is an act of buying an asset in the market and at the same time, the selling part is handled in another market. Such a calculated act of buying and selling is done to earn a profit when there is an imbalance of prices is called arbitrage profit.

Answer to Problem 26PS

The fall of actual price fall below the no-arbitrage opportunities will be by 5.83.

Explanation of Solution

S&P 500 index=1950

June maturity contract F₀=1951

Current rate of interest=2.5%

Dividend=1.9%

From the above calculations, the values of lower bound on future price is $1956.83.

The calculations have to be done by using the formula:

  $\begin{array}{l}Potentialprofit=Lowerboundfutureprice-Actualfutureprice\\ 1956.83-1951=5.83\end{array}$

So, the value of $5.83 reflects the potential profit by using arbitrage strategy.

Therefore, the fall of actual price fall below the no-arbitrage opportunities will be by 5.83.

d.

Summary Introduction

To determine: The arbitrage strategy and profits earned by using it.

Introduction:

Arbitrage Profit: Arbitrage is an act of buying an asset in the market and at the same time, the selling part is handled in another market. Such a calculated act of buying and selling is done to earn a profit when there is an imbalance of prices is called arbitrage profit.

Answer to Problem 26PS

The profit per contract will be $1720 when the multiplier is $250.

Explanation of Solution

S&P 500 index=1950

June maturity contract F₀=1951

Current rate of interest=2.5%

Dividend=1.9%

There are many strategies used by the investor. When the investor wants to earn risk free profits, the choice of arbitrage strategy proves to be good. The fundamental in this strategy is very simple. Arbitrage is the act of buying an asset in the market and at the same time, the selling part is handled in another market. Expecting two markets to deal with the same price is impossible. When there is a mismatch of prices between the two markets, arbitrage takes place. Like all other strategies even arbitrage strategy works on some rules namely,

• When the actual prices are greater than the future price, then the investor should purchase the spot and fell the futures.
• When the actual price is found to be lower than the future price, the investor should purchase futures and sell spot.

Her in this situation, we find that actual price is less than the future price, so investor should short the stock. The profit earned if the investor sells at 90% of the sales as proceeds can be calculated as below:

Selling price of the stock=1950

Proceeds = 90% of the sale

  $\begin{array}{l}=1950\times 90\%\\ =1950\times \frac{90}{100}\\ =1950\times 0.9=1755\end{array}$

So, now we have to calculate the remaining proceeds.

  $=1950-1755=195$

So, 195 has to be kept in the margin account till the short position gets covered within 1 year. Therefore, the investor purchases future and lends 1755.

Particulars	Current cash flow	Cash flow after 1 year
Purchase futures	0	$1950-1951=-1$
Sale of shares	1950-195	$195-1950-37.05=-1792$
Lend	-1755	$1755\times 1.025=1798.87$
Total payoff	0	6.875

Let us now consider the multiplier of S&P 500 contract to be $250.

The profit from arbitrage is 6.875 or 6.88 (when rounded off)

So, profit per contract will be calculated as follows:

  $6.88\times \$250=\$1720$ .

The profit per contract will be $1720 when the multiplier is $250.