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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Sample Solution

Sample solution

Dante Alighieri played a critical role in the literature world through his poem Divine Comedy that was written in the 14th century. The poem contains Inferno, Purgatorio, and Paradiso. The Inferno is a description of the nine circles of torment that are found on the earth. It depicts the realms of the people that have gone against the spiritual values and who, instead, have chosen bestial appetite, violence, or fraud and malice. The nine circles of hell are limbo, lust, gluttony, greed and wrath. Others are heresy, violence, fraud, and treachery. The purpose of this paper is to examine the Dante’s Inferno in the perspective of its portrayal of God’s image and the justification of hell. 

In this epic poem, God is portrayed as a super being guilty of multiple weaknesses including being egotistic, unjust, and hypocritical. Dante, in this poem, depicts God as being more human than divine by challenging God’s omnipotence. Additionally, the manner in which Dante describes Hell is in full contradiction to the morals of God as written in the Bible. When god arranges Hell to flatter Himself, He commits egotism, a sin that is common among human beings (Cheney, 2016). The weakness is depicted in Limbo and on the Gate of Hell where, for instance, God sends those who do not worship Him to Hell. This implies that failure to worship Him is a sin.

God is also depicted as lacking justice in His actions thus removing the godly image. The injustice is portrayed by the manner in which the sodomites and opportunists are treated. The opportunists are subjected to banner chasing in their lives after death followed by being stung by insects and maggots. They are known to having done neither good nor bad during their lifetimes and, therefore, justice could have demanded that they be granted a neutral punishment having lived a neutral life. The sodomites are also punished unfairly by God when Brunetto Lattini is condemned to hell despite being a good leader (Babor, T. F., McGovern, T., & Robaina, K. (2017). While he commited sodomy, God chooses to ignore all the other good deeds that Brunetto did.

Finally, God is also portrayed as being hypocritical in His actions, a sin that further diminishes His godliness and makes Him more human. A case in point is when God condemns the sin of egotism and goes ahead to commit it repeatedly. Proverbs 29:23 states that “arrogance will bring your downfall, but if you are humble, you will be respected.” When Slattery condemns Dante’s human state as being weak, doubtful, and limited, he is proving God’s hypocrisy because He is also human (Verdicchio, 2015). The actions of God in Hell as portrayed by Dante are inconsistent with the Biblical literature. Both Dante and God are prone to making mistakes, something common among human beings thus making God more human.

To wrap it up, Dante portrays God is more human since He commits the same sins that humans commit: egotism, hypocrisy, and injustice. Hell is justified as being a destination for victims of the mistakes committed by God. The Hell is presented as being a totally different place as compared to what is written about it in the Bible. As a result, reading through the text gives an image of God who is prone to the very mistakes common to humans thus ripping Him off His lofty status of divine and, instead, making Him a mere human. Whether or not Dante did it intentionally is subject to debate but one thing is clear in the poem: the misconstrued notion of God is revealed to future generations.

 

References

Babor, T. F., McGovern, T., & Robaina, K. (2017). Dante’s inferno: Seven deadly sins in scientific publishing and how to avoid them. Addiction Science: A Guide for the Perplexed, 267.

Cheney, L. D. G. (2016). Illustrations for Dante’s Inferno: A Comparative Study of Sandro Botticelli, Giovanni Stradano, and Federico Zuccaro. Cultural and Religious Studies4(8), 487.

Verdicchio, M. (2015). Irony and Desire in Dante’s” Inferno” 27. Italica, 285-297.