Problem $1$:

Consider the following policy for hedging. Let $R_0$ be its return $($net present value$).$ If the price of the stock increases more than $3$ percent in the first six months, buy a European call option for the next six months. Otherwise, buy a European put option for the next six months. Set the exercise price of the put option at $$97$ and assume the price of the put option is $$2\mathrm{.}35.$ Assume the cost of the call option is $$3\mathrm{.}40,$ and sample size $N=1000.$ Use the following parameters for the geometric random walk model for the stock price movement.

S$0=100$;$\%$ Current Price So

$m=0\mathrm{.}08$;$\%$ Drift $m$

$s=0\mathrm{.}1$; $\%$Volatility $s$

$T=0\mathrm{.}5$; $\%$xpiration Time $T$

Dt $=0\mathrm{.}5$; $\%$Time interval for observing stock price

$r=0\mathrm{.}059$;$\%$ Risk$-$free Rate $r$

Kp$=97$; $\%$xercise Price K for put option

ic $=0\mathrm{.}03$;$\%$ Percentage increase for call option

Kc$=103$ ; $\%$Strike price for call option

$cc=3\mathrm{.}4$; Price of a call option

$cp=2\mathrm{.}35$; $\%$Price of a put option

$N=10000$;$\%$ Sample size

What is the distribution $($histogram$)$ of $R_0?$ Compute the relative frequency of $R_0$ based on the simulated sample, and graph the histogram.

What is the expect return of the hedging strategy? And its $90\%$ confidence interval?

What is the probability that What is the probability that $1?$

What $is$ the probability $of$ purchasing a put option $in$ a year?