Assignment 1 ECON3107/ECON5106 – Economics of Finance
Note: This assignment should be handed in to the lecturer before Tuesday, 9 April 2013 6pm. This assignment will count for 6 percent of the final mark. You do not have to type it, but write legibly. Warning: it is in your interest to do the assignment yourself since it is the best way to learn and prepare for the exam. 1. Consider a world in which there are only two periods: period 0 and period 1 and three possible states of the world in period 1 (a good weather state, a fair weather state, and a bad weather state). Also, apples are the only product produced in this world, and they cannot be stored from one period to the next. The following abbreviations will be used: PA = apple in first period (i.e., present apple), GA = good weather apple, FA = fair weather apple, BA = bad weather apple. Suppose that an apple tree firm offers for sale a bond and stock. An apple tree produces 80GA, 50 FA, and 25BA. The bond pays 20GA, 20FA and 20BA. The stock pays 60GA, 30FA and 5BA. The price of the bond is 18PA, and the price of the stock is 19PA. In addition, a security C is traded for 8PA. The security C pays 45GA, 5 FA, and 0 BA. (i) Find the arbitrage-free price of the atomic securities. (ii) Calculate the arbitrage-free price of an apple tree. (iii) Calculate the discount factor and explain its economic interpretation. (iv) An investor wants a security that will pay 30GA, 30FA, and 50BA in period 1. Construct such a security and determine its arbitrage-free price. (v) Compute the arbitrage-free price of a security that will pay 35GA, 5 FA, and 0 BA in period 1. This is not necessary to solve the problem, but note that this security is equivalent to a European call option to buy the stock at a price of 25 in period 1. 2. Consider a three period binomial time-state model in which there are two securities, a bond and a stock. The payments made by these securities in each state are shown in the trees below:
1
Stock
1.00
gg
1 1.69 g 1.30 PP * gb PP q 1.17 P
HH HH
j H
b 0.90
1 1.17 bg PP PP q P 0.81 bb
Bond
gg
1 1.1025 g 1.05 PPP * gb PP q 1.1025 1.00 H HH 1 1.1025 b HH bg j 1.05 PP PP q 1.1025 P bb
(i) White down the Q matrix and corresponding price vector (ps) derived from the following elemental payment combinations: B0: Buy a Bond in period 0, sell it in the end of the next period; S0: Buy a Stock in period 0, sell it in the end of the next period; Bg: In period 1, if the state is g, buy a Bond, sell it in the end of the next period; Sg: In period 1, if the state is g, buy a Stock, sell it in the end of the next period; Bb: In period 1, if the state is b, buy a Bond, sell it in the end of the next period; Sb: In period 1, if the state is b, buy a Stock, sell it in the end of the next period. (ii) Compute the atomic security prices (i.e., the price of one dollar in each of the six future time-states: g, b, gg, gb, bg, bb). Suppose you are offered a security that will pay 0.64 if gg, 0.12 if gb, 0.12 if bg and 0 if bb in period 2 (and nothing in period 1). This is not necessary to solve the problem, but note that this security is equivalent to a European call option to buy the stock at a price of 1.05 in the second period. (iii) Compute the arbitrage-free price of this security. Explain how you arrived at your answer. 2
Suppose you are offered a security that will pay 0.24 if bb and 0 in all other states (and nothing in period 1). This is not necessary to solve the problem, but note that this security is equivalent to a European put option to sell the stock at a price of 1.05 in the second period. (iv) Compute the arbitrage-free price of this security. Explain how you arrived at your answer.
3
