Word gets out that you are doing such a great job in your new role and so it is not long before you are asked back to Uni to give a guest lecture. Specifically, you have been asked to go through the calculation of various option prices using both binomial trees and the Black$-$Scholes model.

The example to be used is as follows:

The S&P $500,$ which covers the $500$ largest companies on the U$.$S$.$ stock market, is currently trading at $5,240\mathrm{.}65($index points$).$ The dividend yield of the index is $4\mathrm{.}2\%$ per annum and the risk$-$free interest rate in U$.$S$.$ is $5\mathrm{.}1\%$ per annum $($both with continuous compounding for all maturities$).$ The volatility of the index is also estimated to be $29\mathrm{.}5\%$ per annum. Given your expertise in option valuation you decide to use a four step binomial tree to calculate the following derivative values $($in units of index points, to two decimal places$)$:

$($a$)$ A $10-$month European call option on the index with a strike of $5,100.$ Calculate also the value of the option by using the Black$-$Scholes formula. Compare the values and comment.

$($b$)$ A $10-$month American call option with a strike of $5,100.$ Is the answer different to the answer from $($a$)?$ Explain why if so$.$

$($c$)$ A long position in a forward contract on the index for delivery in $10$ months at a price of $5,100.$ Calculate also the theoretical value of this forward position. Compare these values and comment. $($Hint: the fair forward price that would be agreed for new contracts today is different to $5,100$ and hence the forward contract in question has a non$-$zero value.$)$

$($d$)$ An American down$-$and$-$out barrier call option with a strike of $5,100$ and knockout barrier of $4,300$ maturing in $10$ months. An American down$-$and$-$out call option gives the holder the right to buy the underlying asset at the strike price at any time on or prior to the expiration date so long asthe price of that asset did not go below a pre$-$determined barrier during the option$$s lifetime. When the price of the underlying asset is below the barrier, the option is "knocked$-$out" and no longer carries any value. Comment on the value of this option relative to the option in $($b$)$ and explain any differences. Would the value of this option change if the knockout barrier was increased from $4,300$ to $4,600?$ Comment on the reason why.