$15\mathrm{.}1.$ What does the Black$-$Scholes$-$Merton stock option pricing model assume about the probability distribution of the stock price in one year? What does it assume about the probability distribution of the continuously compounded rate of return on the stock during the year?

$15\mathrm{.}2.$ The volatility of a stock price is $30\%$ per annum. What is the standard deviation of the percentage price change in one trading day?

$15\mathrm{.}3.$ Explain the principle of risk$-$neutral valuation.

$15\mathrm{.}4.$ Calculate the price of a $3-$month European put option on a non$-$dividend$-$paying stock with a strike price of $$50$ when the current stock price is $$50,$ the risk$-$free interest rate is $10\%$ per annum, and the volatility is $30\%$ per annum.

$15\mathrm{.}5.$ What difference does it make to your calculations in Problem $15\mathrm{.}4$ if a dividend of $$1\mathrm{.}50$ is expected in $2$ months?

$15\mathrm{.}6.$ What is implied volatility? How can it be calculated?

$15\mathrm{.}7.$ A stock price is currently $$40.$ Assume that the expected return from the stock is $15\%$ and that its volatility is $25\%.$ What is the probability distribution for the rate of return $($with continuous compounding$)$ earned over a $2-$year period?

$15\mathrm{.}8.$ A stock price follows geometric Brownian motion with an expected return of $16\%$ and a volatility of $35\%.$ The current price is $$38.$

$($a$)$ What is the probability that a European call option on the stock with an exercise price of $$40$ and a maturity date in $6$ months will be exercised?

$($b$)$ What is the probability that a European put option on the stock with the same exercise price and maturity will be exercised?

$15\mathrm{.}9.$ Using the notation in this chapter, prove that a $95\%$ confidence interval for $S_T$ is between $S_0{e}^{(-\frac{{}^2}{2})T-1\mathrm{.}96\sqrt[\phantom{2}]{T}}$ and $S_0{e}^{(-\frac{{}^2}{2})T+1\mathrm{.}96\sqrt[\phantom{2}]{T}}.$

$15\mathrm{.}10.$ A portfolio manager announces that the average of the returns realized in each year of the last $10$ years is $20\%$ per annum. In what respect is this statement misleading?

$15\mathrm{.}11.$ Assume that a non$-$dividend$-$paying stock has an expected return of $$ and a volatility of $.$ An innovative financial institution has just announced that it will trade a security that pays off a dollar amount equal to $ln{S}_T$ at time $T,$ where $S_T$ denotes the value of the stock price at time $T.$

$($a$)$ Use risk$-$neutral valuation to calculate the price of the security at time $t$ in terms of the stock price, $S,$ at time $t.$

$($b$)$ Confirm that your price satisfies the differential equation $(15\mathrm{.}16).$

$15\mathrm{.}12.$ Consider a derivative that pays off $S_T^n$ at time $T,$ where $S_T$ is the stock price at that time. When the stock pays no dividends and its price follows geometric Brownian motion, it can be shown that its price at time $)(T$ has the form $h(t,T)S^n,$ where $S$ is the stock price at time $t$ and $h$ is a function only of $t$ and $T.$

$($a$)$ By substituting into the Black$-$Scholes$-$Merton partial differential equation, derive an ordinary differential equation satisfied by $h(t,T).$

$($b$)$ What is the boundary condition for the differential equation for $h(t,T)?$

$($c$)$ Show that $h(t,T)=e^{[0\mathrm{.}5{}^{2}n(n-1)+r(n-1)](T-t)},$ where $r$ is the risk$-$free interest rate and $$ is the stock price volatility.