As a valued member of the UTS alumni, you have been asked to be a guest lecturer for Lecture 8 and 9 for Derivative Securities (25620). You have been asked to go through calculating option prices using the binomial tree and Black-Scholes model. 

The S&P 5000 is currently standing at 2630, has a volatility of 20% per annum and a dividend yield of 2.5% per annum with continuous compounding. The risk-free interest rate is 3.10% per annum with continuous compounding. Given your expertise in index options you decide to use a four-step binomial tree to calculate the following option prices (to four decimal places): 

1. (a) a six-month short forward contract on the index, with delivery price of 2650. Calculate also the theoretical value of the forward contract. Compare and comment. 

1. (b) a European six-month put option with a strike of 2650. Calculate also the value of the option by using the Black-Scholes formula. Compare and comment. 

1. (c) an American six-month put option with a strike of 2650. 

1. (d) a European down-and-out barrier put option with a strike of 2650 and knockout barrier of 2400 maturing in 6 months. A down-and-out put option gives the holder the right to sell the underlying asset at the strike price on the expiration date so long as the price of that asset did not go below a pre-determined barrier during the option’s lifetime. When the price of the underlying asset falls below the barrier, the option is “knocked-out” and no longer carries any value. 

Can you actually show me clearly how to solve for 1D?