25. The current stock price of Alcoa is $65, and the stock does not pay dividends. The instantaneous risk-free rate of return is 7%. The instantaneous standard deviation of Alcoco's stock is 30%. You want to purchase a put option on this stock with an exercise price of $70 and an expiration date 30 days from now. According to the Black-Scholes OPM, you should hold __________ shares of stock per 100 put options to hedge your risk. Please show written out, how to do it step by step, not with excel.

Suppose there is also a 1-year European put option on the same stock as in Question 3 with exercise price $30. The current stock price is also $25 and the stock price, in 1 year, will be either $35 (up by 40%) or $20 (down by 20%). The interest rate is 8%. This stock does not pay dividend. What is the value of the put option? Please use risk neutral probability method and assume discrete discounting.  (2) What is put-call parity in option pricing? What needs to be true in order for put-call parity to hold?

24. You reconsider the ESPR trade. Now you decide to buy the 55 call and sell the 60 call. This is called a 25. You are looking again at the ESPR trade. The chances of the stock being above breakeven are 24.16%, and of being above $60 per share is 14.19%. What is your expected maximum return against risk?

A stock priced at $50 has two possible outcomes, either increases to $60 or decreases to $42. If the stock goes up, the payoff of the stock's call option = S5. But, if the stock price declines to $42, the calls will expire worthless. Given these two possible outcomes, what is the Hedge Ratio of the Call option? Round your answer to two decimals. The listed correct answer was 0.28. I could not find a way to get this.