A European call option is a contract with the following conditions: At a prescribed time in the future, known as the expiration date, the holder of the option has the right, but not the obligation to purchase a prescribed asset, known as the underlying asset ,r'security, for a prescribed amount, known as the strike price or exercise rice. For example, suppose an investor purchases a call option on stock XYZ with a T50 strike price. At expiration, say a month fmm the time of purchase, the spot price of stock XYZ is T75. In this case, the owner of the call option has the right to purchase the stock at T50 and exercises the option, making T25, or [T75 - T50), per share. However, in this scenario, if the spot price of stock XYZ is T30 at expiration, it does not make sense to exercise the option to purchase the stock at T50 when the same stock could be purchased in the spot market for T 30. in this case, the payoff is T0. Note the paxoff and prot are different. To calculate the prot from the option, the cost of the contract mUst be subtracted from the payoff. In this sense, the most an investor in the option can lose is the premium price paid for the option. In general, if S is the spot price of stock XYZ on the expiration date and K is the strike price of the European call option, then the call option payoff = max {0, {S IQ} and the prot = payoff-option price. Consider the following problem: You have T20,000 to invest. Stock XYZ sells at T20 per share today. A European call option to buy 100 shares of stock XYZ at T15 exactly six months from today sells for T1000. You can also raise additional funds which can be immediater invested, if desired, by selling call options with the above characteristics. In addition, a 6-month riskless zero-coupon bond with T100 face value (the amount you will make at the end of 6 months) sells for T90. You have decided to limit the number of call options that you buy or sell to at most 50. You consider three scenarios for the price of stock XYZ six months from today: the price will be the same as today, the price will go up to T40, or drop to T12. Your best estimate is that each of these scenarios is eguallg likely. 1. Formulate and solve (in Solver) a linear program to determine the portfolio of stocks, bonds, and options that maximizes expected prot. (a) What happens to protability if the price of XYZ goes to T40 per stock? 2. Suppose you want a prot of at least T2000 in an},r of the three scenarios. Write and solve (in Solver] a linear program that will maximize your expected prot under this additional constraint. (a) How does the solution compare to the earlier case described in [1). 3. Hiskless prot is dened as the largest possible prot that a portfolio is guaranteed to earn, no mat- ter which scenario occurs. Formulate and solve (in Solver) the model to determine the portfolio that maximizes riskless prot for the above three scenarios
