# Fgdgdfvdfv Essay

3414 words - 14 pages

SEMINARS 3 AND 4CHAPTERS 4-51. - Final Exam December 2009 - Question 2:Consider an economy that lasts only one period (two dates: t = 0 and t = 1). In t = 1 three different states of nature may occur: the economy may be in a boom state, the economy may be in a recession state or the economy may be in a crisis state. In this economy 3 assets exist with the following prices and payoffs:
Assets

Price in t=0

Payoffs in t=1

States

Boom (s=1)

Recession (s=2)

Crisis (s=3)

Asset 1

140

200

150

100

Asset 2

95

100

100

100

Asset 3

45

100

50

0

Show that one of the assets is redundant and that as a result markets are not complete.Suppose we add another asset to this market: a call option whose underlying is asset 3, with a strike price K = € 80, which matures in t=1 and with a cost (price or premium) of c0 = € 5. Are markets complete in this case? If they are, then calculate the price of the Arrow-Debreu securities associated with each state of nature.What is the no-arbitrage price in t = 0 of a basic bond that matures in t = 1? What is the interest rate associated with that bond?What is the no-arbitrage forward price of Asset 1 agreed in t = 0 for delivery in t = 1?If a new asset appears in the market that pays 0 in t = 1 if the economy is in a boom state, € 50 if the economy is in a recession state, and 0 if the economy is in a crisis state, what is its no-arbitrage price in t = 0?Imagine that the price at which that asset is offered to the market is € 10. Design an arbitrage strategy to benefit from that opportunity (limit yourself to using asset 1, asset 3, the call and the new asset), and define the positions that you would take in each security.SOLUTION:a) Any of the 3 assets can be obtained as a linear combination of the other two assets. For example, asset 3 = asset 1 - asset 2 (that is, asset 3 can be replicated by a portfolio consisting of a long position in one unit of asset 1 and a short position in one unit of asset 2). Likewise, asset 1 = asset 2 + asset 3, and asset 2 = asset 1 - asset 3.Given that I can replicate any asset with a combination of the other two assets, I do not need to have the three assets, two are enough and one of them (any one of them) is "redundant".b) The payoffs associated to the call option are:
Assets

Price in t=0

Payoffs in t=1

States

Boom (s=1)

Recession (s=2)

Crisis (s=3)

Asset 1

140

200

150

100

Asset 2

95

100

100

100

Asset 3

45

100

50

0

Call option on A3

5

20

0

0

Yes, now markets are complete. There are three ways one can explain that markets are complete now. One is by arguing that the new asset, the call option, cannot be replicated by any combination of the other assets. Remember we can get rid of one of the three original assets so to show if markets are complete all we have to do is show that it is impossible to replicate the call option by any two assets. Suppose we got rid of A1, you could...

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