Consider a stock with initial value $S_0=100\mathrm{.}00.$ Every day, this stock has a $50\%$ chance of a

$+1\%$ return, and a $50\%$ chance of a $-1\%$ return, i$.$e$.$ the stock either goes up by a factor of

$u=1\mathrm{.}01$ or down by a factor of $d-0\mathrm{.}99.$ You have purchased a call option with a strike price

of $K=100,$ and a maturity of $T=2$ year, which is $n=500$ trading days. The random payoff

of this option is

$P=$max$\{S_T-K,0\},$

where $S_T$ is the random terminal value of the stock price.

$($a$)$ In order to help understand the distribution of the the terminal stock price $S_T,$ we use kernel

density estimation. Write a function $y=$KDE$(x,x,b)$ which implements kernel density

estimation using a Gaussian kernel.

Here:

$x$ is a vector of sample points $x=[x_1,$dots.$x_n]$ drawn from some distribution $F$ whose

probability density function we are trying to approximate.

$b$ is a bandwidth parameter.

The output is defined to be

KDE$(x,x,b)$:$=\frac{1}{nb}{\displaystyle \underset{i=1}{\overset{n}{}}}K(\frac{x-x_i}{b})$

where $K(x)$ is the standard normal PDF$.$

$($b$)($i$)$ In a script file Q$1.$m$,$ write code that will output a plot of a random path of the stock

price $S_0,S_1,$dots,$S_n$ vs $0$:$n.$

$($ii$)$ We now investigate the distribution of the terminal stock price $S_T$ using simulation.

In your script Q$1.$m write code that constructs a vector $SS$ with $M=10000$ pos$-$

sible terminal stock price values $S_T.$ Plot a histogram of your vector $SS($using

'Normalization', $\text{'}$pdf$\text{'}).$ On the same graph, plot a kernel density estimate of

the PDF$,$ using your function KDE, with bandwidth determined by the Silverman Rule:

$b$:$=1\mathrm{.}06std\frac{x}{n^{\frac{1}{5}}},$ where $n$ is the number of data points.

$($iii$)$ Next, we compute the probability that this option expires in$-$the$-$money $($i$.$e$.$ that the

option is worth something at expiry$)$ using simulation: In your script Q$1.$m$,$ compute

and display the proportion of the values of $SS$ that are $>K.$

$($iv$)$ Still in Q$1.$m$,$ use an appropriate built$-$in matlab function that will compute and display

the exact value of $P(S_T>K).($Your answer should be close to that in $($iii$).)$

$($v$)$ Finally, still in Q$1.$m$,$ write code that uses simulation to compute and display an ap$-$

proximation of the expected payoff of this call option.