The stock of Company ABC is currently trading at a price of $50. According to analyst forecasts, the share price will be either $45 or $55 at the end of six months. Suppose that the risk-free interest rate is 10% per annum with continuous compounding. What is the value of a six-month European put option with a strike price of $50 using the Delta-hedging method as discussed in the lecture notes? Verify your result using the Risk-neutral method.

Acompany manufactures x units of product a and y units of product b, on two machines, i and ii. it has been determined that the company will realize a profit of $3 on each unit of product a and $4 on each unit of product b. to manufacture a unit of product a requires 7 min on machine i and 5 min on machine ii. to manufacture a unit of product b requires 8 min on mchine i and 5 min on machine ii. there are 175 min available on machine i and 125 min available on machine ii in each work shift. how many units of a product should be produced in each shift to maximize the company's profit p?