$15$ Using a one$-$step binomial model, calculate the risk neutral probability of an up move $,$ given a risk$-$free rate of $5\%,$ and up factor of $1\mathrm{.}15$ and a down factor of $0\mathrm{.}85,$ for an option expiring in six months.

$16.$ Using the information from the previous question calculate the value of a six moth call struck at $$80,$ assuming the stock is currently trading at $$75$

$23.$Assume NFLX is currently trading at $$465.$ A nine month call option, stuck at $$460,$ is available for $$9\mathrm{.}65.$ What is the time value of this call option?

$24.$Assume prices on the S&P $500$ are normally distributed with a mean of $4,000$ index points and an standard deviation of $600$ index points. What is the probability that the S&P $500$ fall below $3,200$ over next year?

$25.$ Assume the probability of the stock market rising in any given year can be modelled as a binomial distribution, and is $65\%.$ What is the probability that the stock market rises $12$ or times during the next twenty years?