The current price of a particular stock is $80. Each week the stock price goes up 15% or down 10% with some probability. a.) Find probabilities that the stock goes up or down which give that the expected return from investing in this stock is $0. These probabilities are called the “risk neutral” probabilities. b.) An American call option is a contract giving the buyer of the contract the right, but not the obligation to buy the stock at a future date (called the expiration date) from the seller for a predetermined price (called the strike price). Suppose you are interested in buying a call option which expires in three weeks with a strike price of $110. If the stock price is above $110 at the end of three weeks, then you can buy the stock for a lower price than it is worth, and you effectively earn the difference between the stock price and the strike price. If the stock price is below $110, the option expires worthless. What would a fair price for this option be (that is, how much is the expected earnings from this contract) if the probabilities that the stock goes up or down are given by your answer to part (a)? c.) Rather than using the probabilities in part (a), if banks offer an interest rate on cash holdings, it is common practice in financial modeling to find “risk neutral” probabilities so that the expected return on the stock is the same as the interest rate. Find these probabilities if the current interest rate is 5% per week. d.) What is the expected earnings from an American call option which expires in three weeks with a strike price of $110 under the probabilities from part (c)? (Note that the “present value” of $1 in three weeks is $1/(1.05)3= 0.86 since if we invest 86 cents at the interest rate, it will be worth $1 in three weeks). What is the present value of the American call option?