Chapter 22, Problem 19PS

A

Summary Introduction

To explain: Calculate the total cash flow in strategy at different time intervals and as per the given information.

Introduction: An arbitrage strategy is given, first future contract price $F(T_1)$ at a time period of $T_1$ , now consider the next time period $T_2$ with future price $F(T_2)$ .

Answer to Problem 19PS

Thus the final cash after $T_2$ years will be $F(T_2)-F(T_1)\times {\left(1+rf\right)}^{\left(T_2-T_1\right)}$ .

Explanation of Solution

Given information: Future contract price $F(T_1)$ at a time interval of $T_1$ , next future price is $F(T_2)$ with time period $T_2$ .

There are two future prices with definite time period, initially the cash was zero, after one year it will be $P_1-F(T_1)$ and after two year it will be zero again. Short selling of future contract after two year will be $F(T_2)-P_2$ . Number of assets at $T_1$ will be - $P_1$ and sell asset will be + $P_2$ . Now to borrow future price at $T_1$ will be $F(T_1)$ and at $T_2$ will be $-F(T_1)\times {\left(1+rf\right)}^{T_2-T_1}$ . Now the final cash flow after $T_2$ years will be $F(T_2)-F(T_1)\times {\left(1+rf\right)}^{\left(T_2-T_1\right)}$ .

B

Summary Introduction

To explain: If arbitrage opportunity is none then why profit will be zero at time $T_2$ .

Introduction: An arbitrage strategy is given, first future contract price $F(T_1)$ at a time period of $T_1$ , now consider the next time period $T_2$ with future price $F(T_2)$ .

Answer to Problem 19PS

Thus investment at this period is zero and also it is risk free. Hence the total profit also is null when there is no arbitrage opportunity.

Explanation of Solution

Given information: Future contract price $F(T_1)$ at a time interval of $T_1$ , next future price is $F(T_2)$ with time period $T_2$ .

In the absence of the arbitrage opportunity the profit will be zero because investment in time $T_2$ is risk free and net investment is also zero in this time period. Hence the profit will also be zero at this condition.

C

Summary Introduction

To explain: Establish the relation between $F(T_1)$ and $F(T_2)$ for the profit at T2 to be equal to zero.

Introduction: An arbitrage strategy is given, first future contract price $F(T_1)$ at a time period of $T_1$ , now consider the next time period $T_2$ with future price $F(T_2)$ .

Answer to Problem 19PS

Hence relation between $F(T_1)$ and $F(T_2)$ is given by this equation $F(T_2)=F(T_1)\times {\left(1+rf\right)}^{\left(T_2-T_1\right)}$ .

Explanation of Solution

Given information: Future contract price $F(T_1)$ at a time interval of $T_1$ , next future price is $F(T_2)$ with time period $T_2$ .

At last stage for profit should be zero at time $T_2$ , the below relation should be followed by $T_1$ and $T_2$

  $F(T_2)=F(T_1)\times {\left(1+rf\right)}^{\left(T_2-T_1\right)}$