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 fourstep 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$.$