a) What is the meaning of “complete markets”? What is an Arrow security? Briefly

Question

a) What is the meaning of “complete markets”? What is an Arrow security? Briefly explain how they work, in theory, to handle risk.

b) Suppose you are considering buying a share with price $100. Two states of the world are possible tomorrow: good and bad. You know that if tomorrow the state is good the share price will be $105; if the state tomorrow is bad the share price will be $98. What do you need to know in order to decide whether $100 is a fair price or not?

c) Two Arrow securities are available, the first paying one in the good state and zero in the bad state, tomorrow, while the second pays zero in the good and one in the bad state, again tomorrow. The current prices of the two Arrow securities are 40 cents and 60 cents respectively. What does this new information tell you and what can you say about the current share price?

Explanation / Answer

1

Arrow security: Instrument with fixed payout of 1 unit in specified state & no payout in others

Complete markets: If all contingent claims in model can be replicated using primitive assets. That is , price for each asset in each possible state of world. It can be tested using uniqueness of risk-neutral probability

We can handle risk as any contingent claim may be written as portfolio of Arrow securities

2

I need to know the probability of each state so that if probability of good state is p then porbability of bad state is 1-p so, share price=p*105+(1-p)*98

Hence, 7p+98 must be greater than or equal to 100 for me to decide whether 100 is a fair price or not

3

Lets say the probability of good state is p and hence probability of bad state is 1-p

So, 1st one: p*1+0*(1-p)=p given p=0.4 so probability of good state is 0.4 or 40%..Hence, probability of bad state is 1-p=1-0.4=0.6 or 60%

We can verify using the second one: p*0+1*(1-p)=1-p=1-0.4=0.6..which is the price of the second..SO, current share price of both is correct.

Market is complete.

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