2. The stock of ABC, Inc. is trading at $10, and you have modeled a one-level binary tree, with the stock going up to $12 or down to $9. The interest rate for the relevant time period is 10%. ABC has decided to issue a new type of security, the Powerstock. Each share of the Powerstock pays an amount equal to the square of the stock price at the end of the same time period.

(a) You are planning to underwrite the issuance of the Powerstock that requires you to purchase a large number of shares of the Powerstock. As a trader, you plan to hedge the purchase immediately by buying or selling short the common stock of ABC. For each share of the Powerstock that you purchase, you have to:

(b) Now that your holdings include a large number of shares of the Powerstock due to the underwriting activity, you decide to offer options on the Powerstock, expiring one period from now, with a strike of $100. The price of this option is:

(c) You subsequently determine that the shares of ABC are not quite as volatile as you had expected. Therefore you revise the model such that the interest rate remains at 10% for the period and the shares, trading now at $10, go to $10.5 in the up state and down to $9.5 in the down state. The risk-neutral probability of the stock going to the up state is now: