Questions

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 the present period (i.e., present apple), GA = good weather apple in the

next period, FA = fair weather apple in the next period, BA = bad weather apple in the

next period.

Suppose that an apple tree firm o↵ers for sale a bond and stock. The apple tree

produces 80 GA, 50 FA, and 25 BA. The bond pays 20 GA, 20 FA and 20 BA. The stock

pays 60 GA, 30 FA and 5 BA. The price of the bond is 18 PA, and the price of the stock

is 22 PA. In addition, security C, D, and E are traded for 31 PA, 11 PA, and 10 PA

respectively. Security C pays 70 GA, 40 FA, and 15 BA. Security D pays 30 GA, 15 FA,

and 2.5 BA. Security E pays 40 GA, 10 FA, and 0 BA.

1.1) Find the arbitrage-free price of the atomic securities.

1.2) Calculate the arbitrage-free price of an apple tree. Verify it equals the price implied

by the firm’s securities.

1.3) Calculate the discount factor and explain its economic interpretation. How is it

related to the risk-free interest rate?

1.4) An investor wants a security that will pay 30 GA, 30 FA, and 50 BA in period 1.

Construct such a security and determine its arbitrage-free price.

1.5) Compute the arbitrage-free price of a security that will pay 45 GA, 15 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 strike price of 15 in period 1.

1.6) Design a profitable arbitrage strategy if security C costs 32 PA instead.

2) Consider another world similar to the one considered in Question 1 except there is a

new set of atomic prices involving dealers.

2.1) Dealer I is willing to trade 0.15PA for 1GA (or vice versa), and dealer II is willing

to trade 1GA for 0.6FA (or vice versa) and dealer III is willing to trade 1FA for 0.5BA.

a) What is the arbitrage-free price of a BA in terms of PA? b) What is the arbitrage-free

discount factor?

2.2) In addition, dealer IV is willing to trade 1 PA for 4 BA (or vice versa). Are there

arbitrage opportunities? If so, design a profitable arbitrage strategy.

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2.3) Suppose now there are transaction costs. The four dealers have the following bid-ask

spreads:

Dealer I: sell 1PA for 7GA

buy 1PA for 5GA

Dealer II: sell 1GA for 0.7FA

buy 1GA for 0.5FA

Dealer III: sell 1FA for 0.6BA

buy 1FA for 0.4BA.

Dealer IV: sell 1PA for 5BA

buy 1PA for 3BA.

Are there arbitrage opportunities now? Explain.

2.4) Optional (Not graded): You have been hired by a financial investment company

to write a MATLAB script that detects whether there is an arbitrage opportunity here,

and if there is, tells the company of how to take advantage of it, and calculates the payo↵

in terms of present apples. The program should only need the bid and ask prices of each

of the dealers.

3) Explain the relationship between Arbitrage and the Law of One Price in financial markets.

Compare the latter with examples of Law of One Price in other areas of economics,

and discuss the limitations of this so-called “law”.

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