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ECO460 Assignment 1

Fall, 2022

I.    (22 points) Consider an economy with 3 states. Two assets are traded in this economy: A = (100, 100, 100) and B = (105, 100, 95).

a.   (6 points) Explain why the market is not complete. Explain with supporting calculations.

b.   ( 10 points) Can the market be completed by allowing for trading a forward contract on B with a forward price of 100? If not, what value of the forward price in the forward   contract can complete the market? Explain carefully with supporting calculations.        (Hint: assume a forward price of K in the forward contract).

c.   (6 points) Explain how the market can be completed by trading a call option on B with a strike price of 100 with supporting calculations.

II.   (24 points) Suppose that there are three assets traded in the economy: A = (1, 2, 3), B = (1, 1, 1), and C = (2.5, 4.5, 6.5).

a.   (6 points) Is the market complete? Explain with supporting calculations.

b.   (6 points) Suppose that P (A) = 2, P (B) = 1 and P(C) = 5. Explain why the principle of no arbitrage is violated in this economy. (Hint: Replicate stock C using A and B).

c.   (8 points) Assume that you can buy and/or sell integer multiples of the three assets. Can you design a trading strategy that will generate arbitrage profit?

d.   (4 points) Suppose shorting a stock will be charged a borrowing cost of $K per share shorted. Will this eliminate the arbitrage opportunities in this setting? Explain with   supporting calculations.

III. (16 points) Consider a scenario similar to the insurance example discussed in the lecture. A household has wealth of $35,000 and may contract a serious disease. The possible loss from the disease is $10,000. An insurance company offers health insurance with a compensation  of $K if the client contracts the disease, and charges an insurance premium of 0. 1K. The      cost of operation of the company is 0.01K1.06, which exists regardless of whether the            household contracts the disease or not.

a.   (4 points) The insurance company estimates the chance of the client contracting the disease is 8%. What is its expected profit from selling the contract? What is the      maximum amount of insurance (in terms of the amount of compensation) that the company is willing to provide? (Hint: Consider the break-even point)

b.   (5 points) Suppose the household only cares about her expected wealth (i.e., she is risk neutral), and she estimates that the chance of her getting the disease is 20%, how much insurance (in terms of the amount of compensation) would she purchase (demand) at    the current price? How much insurance would she demand if her estimated chance of  getting the disease is 10%?

c.   (4 points) Suppose the household knows her condition better. Her estimated chance of  20% of getting the disease is more accurate. She chooses to withhold such information  and purchase the insurance at her will. What would happen to the insurance company if it tries to maintain the price of 0. 1K? (Hint: can the insurance company fulfill the          demand of the household?)

d.   (3 points) Following the information from the previous part, suppose the insurance    company can change the price after viewing the demand from the household. What is the highest price it will charge that the client still accepts? Explain with supporting    calculations.

IV.   (18 points) Answer the following questions.

a.   (5 points) What is a put option? If on the expiration date, the spot price of the asset, ST , is $10, and the strike price, K, is $20. Find the buyer’s payoff and profit of 1 unit of a   put option on 1 share of the stock. Assume the buyer does not have the stock and the    option premium that she paid for the put option is $5. What is the payoff and profit of  the buyer of this put option if the spot price on the expiration date is $25?

b.   (6 points) On the expiration date, the spot price of an asset is ST . Consider a put option on the asset, with option premium, Pp, and strike price K. Use a mathematical equation to express the profits of a buyer and a writer of one such put option as a function of ST , Pp, and K.

c.   (3 points) Considering the same asset and put option in the previous part, use a              mathematical equation to express the profits of a buyer and a writer of “n” units of such a put option as a function of ST , Pp, K, and n.

d.   (4 points) On the same diagram, draw the profits of the following portfolios on the         expiration date. Suppose all options have the same expiration date. Set the spot price of the asset as the horizontal axis, and the profit as the vertical axis. Label your curves       clearly. Portfolio (1): buy (long) one call option with a strike price of $10, option           premium of 8; Portfolio (2): buy (long) one call option with a strike price of $30, option premium of $4; Portfolio (3): write (sell) 2 call options with a strike price of $20,

option premium of $4.5. Also, draw the combined profit ofPortfolio (1), (2), and (3) above.

V.    (20 points) Consider a forward contract on some asset A, with a delivery date of T years from now. Let t0 be the current date when the forward contract is entered. There is a risk- free bond, B, offering a rate of R per annum, continuous compounding. The current spot price of asset A is S0 .

a.   (6 points) Let K be the forward price of the asset which is negotiated at t0 . Show that K must be: K=S0 eRT . Explain carefully and show how arbitrage can be achieved if K is   not S0 eRT , and also explain how prices will adjust to restore K to S0 eRT .

b.   (8 points) Suppose the current price of asset A is S0 . T years from now, asset A will      have only two states, state 1 where the price ofA is S0D, and state 2, where the price of A is S0U, 0<D<1<eRT<U. Let b(1) the basic (A-D) security corresponding to state 1,     and b(2)  is the basic security corresponding to state 2. Find the price of b(1) , q(1), and    the price of b(2) , q(2). Show your calculations carefully.

c.   (6 points) What is the price/value of the forward contract at t0, evaluated from the perspective of a long position? Show your calculations carefully.