**Assignment Task**

**Task **

**Section A**

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Get Help Now!**Question 1 **

For each of the following statements, state if they are TRUE or FALSE, fully justifying your answer. Simply answering True or False will be awarded no marks.**a**. Consider the market with basis assets whose payoff matrix is

This market is complete.**b**. Consider the static model we have discussed in class. In particular, consider a complete market without redundant assets. In this case, a state price vector certainly exists.**c**. A decision involves risk if it leads to consequences that are not precisely predictable, but follow a known probability distribution.**d**. According to the consumption CAPM, the more impatient investors are, the higher the state prices are.

**Section B**

**Question 1 **

Suppose that there are three market scenarios, and there are three basis assets with payoff matrix and price vector as follows:

- Perform Gaussian elimination (showing all steps) to determine how many redundant asset there are in this market. Is the market complete?
- Find all state prices which are consistent with the Law of One Price, and based on this finding determine whether there are any arbitrage opportunities. If there is an arbitrage opportunity, state what type of arbitrage it is.
- c. Calculate the risk-free return and the risk-neutral probabilities.
- d. A focus asset with payoff h 1 2 0i0 is introduced. What are the possible no-arbitrage prices of the focus asset?

**Question 3 **

Please answer the following questions, justifying your answers.**a**. Consider the following statement: In any market, there are multiple portfolios that replicate an arbitrary payoff. Is this true?**b**. What is the relative risk aversion of the utility function u (x) = e ??x, where ? is

a constant?**c**. Consider the consumption CAPM model we have discussed in class. Is it true that the risk-free return is increasing in the time discount factor ??**d**. Consider using the binomial model to price a derivative. Under what circumstances is it safe to use the recombining binomial tree to price the derivative?

**Question 4**

Consider a three-period binomial model, each period corresponding to a month. There is a bond worth er in one year, where r = 0.05, and interest is continuously compounded. There is also a stock, worth 100 at t = 0, with two possible returns at each node: 1.058 or 0.95.

Consider an American barrier put option on the stock, expiring in 3 periods, with strike price 100. You can exercise the option at any time, as long as the stock price has not reached or exceeded the barrier 105 at any time.

**Please answer the following questions (remember to write all intermediate steps rather than simply writing the results):****a**. For each one-period model, calculate the risk-neutral probability of a stock price increase.**b**. Draw a tree indicating the stock price at each node. Also write the payoff ofthe option on the terminal nodes of the tree, assuming it is not exercised early.**c**. At t = 2, at which (if any) nodes would you want to exercise the option? So what is the value of the option at each node at t = 2?**d**. Consider the bottom node at t = 2. If you have determined that you would want to exercise early, under what circumstance would you not want to exercise early? And vice versa, if you have determined that you would not want to exercise early, under what circumstance would you want to exercise early?**e**. Compute the option’s no-arbitrage price at t = 0. What is the amount deposited in (borrowed from) the bank and the number of shares that need to be bought/sold according to the self-financing portfolio that perfectly replicates the payoff of this option?

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