A call option has an exercise price of $$10\mathrm{.}00$ and an expiry date of six months. The underlying stock is currently trading for $$12\mathrm{.}00.$

$($a$)(3$ points$)$ What is the minimum price an investor would pay for this option? Could an investor profit if the call option was actually trading at $$1\mathrm{.}00?$

$($b$)(3$ points$)$ What is the maximum price an investor would pay for this option? If the option was trading for $$13\mathrm{.}00,$ how could an investor earn a risk$-$free profit?

This is a more analytical question than a quantitative one.

Question $2(10$ points$)$

A call option has a striking price of $$8\mathrm{.}10$ and an expiry date of six months. The underlying stock is currently trading at $$7\mathrm{.}60$ and has historically exhibited a standard deviation of $10\mathrm{.}6$ percent. The standard deviation is not expected to change. The interest rate is $5\mathrm{.}4$ percent. Using the Black$-$Scholes equation, estimate the value of the call option described above. What are some of the limitations of using the Black$-$Scholes equation?

Question $3(10$ points$)$

Suppose you have just purchased a share of stock in WYZ Corp for $$50.$ The stock is not expected to pay a dividend during the time you plan to hold it$.$ You have forecasted that the stock will be worth in one year either $ $65$ or $$40.$ Suppose further that you can sell a call option on WYZ stock with an exercise price of $$52\mathrm{.}50.$ It is a European$-$style contract that expires in exactly one year. What the option should sell for today if the one$-$year T$-$bill yield is $8\%?$

Question $4(7$ points$)$

Suppose that WYZ stock is currently valued at $$53$ and that call and put options on WYZ stock with an exercise price of $$50$ sell for $$6\mathrm{.}74$ and $$2\mathrm{.}51$ respectively. Both options can only be exercised in $6$ months. You can also buy a $6-$month T$-$bill with a $$50$ face value with an annualized rate$/$return of $6\mathrm{.}25\%.$ Does Put$-$Call parity hold? If not, what is the arbitrage trade you should make and what is the arbitrage profit?