The stock of GreatSteps Inc. is priced at $82 per share today. No dividend is expected from this company within a few years. Based on the historical data, analysts estimate that the volatility of the stock price is 60% per annum. The interest rate is 3%. A put option on the stock strikes at 80 and matures in exactly seven months from now.

(a) Suppose the put is European. What is the value of the put if you value it in a seven-step binomial model? Display the binomial tree in your answer.

(b) Now, suppose the put is American and value it again using a seven-step binomial model. By how much is the American put value higher than its European version? Indicate the nodes on which the American put will be exercised.

(c) If the volatility jumps instantaneously to 70% per annum today, will the European put value increase or decrease according to your seven-step binomial model? By how much? What if the put is American?

(d) If the European put is quoted for $18.42 in the exchange now, with what volatility will your seven-step binomial model give the option price quoted in the market? (This volatility is called the implied volatility of this put option; it is implied by the premium quoted or traded for in the market.) What if the put is American?