The initial stock price $($S$0)$ is $35,$ the stock vola$-$tility is $0\mathrm{.}40(40\%$ per annum$),$ and the risk$-$free rate is $0\mathrm{.}01(1\%$ per annum$).$ Consider a European put option whose strike price is equal to $35,$ with a time$-$to$-$maturity of one year. The dividend yield is $0\mathrm{.}03(3\%$ per annum$).$ Use a simple two$-$step binomial tree to value the above European put option. Set u equal to the expo$-$nential of sigma times the square root of the maturity time divided by the number of time steps $($i$.$e$.,$ exp$(($T$/$n$)1/2)$ and d equal to the exponential of minus one times sigma times the square root of the maturity time divided by the number of time steps $($i$.$e$.,$ exp$(-($T$/$n$)1/2).$

After that, increase the number of time steps $($n$)$ and repeat the binomial tree value calculation. So$,$ for example, use a binomial tree with three, then four, then five, then six, etc. time steps to value the option. I$$ll leave it up to you how many binomial trees you use in your calculations. I would expect that the average group does binomial trees with two, three, four, and five time steps. But the sky is the limit$,$ and the more binomial trees you calculate the better for your mark.

Don$$t show each and every binomial tree you do in the coursework assignment. Showing one tree is certainly enough for illustrative purposes. Keep the page limit in mind!

Plot the value of the European put option $($y$-$axis$)$ against the number of time steps used in the binomial tree $($x$-$axis$).$ Also plot the Black$-$Scholes value $($e$.$g$.,$ as horizontal line$)$ in the graph. Do more time steps render the binomial tree value closer to the Black$-$Scholes $(1973)$ value? Explain in intuitive language why this is the case $($you would want to look at your textbook$).$