Check the following combinations of puts and calls and determine whether they conform to the put$-$call parity rule for European options. If you see any violations, suggest a strategy.

a$.$ July $155$

b$.$ August $160$

c$.$ October $170$

Repeat problem $13$ using American put$-$call.

The following option prices were observed for a stock on July $6$ of a particular year. Use this information to solve problems $11$ through $16.$ Unless otherwise indicated, ignore dividends on the stock. The stock is priced at $165\mathrm{.}13.$ The expirations are July $17,$ August $21,$ and October $16.$ The risk$-$free rates are $0\mathrm{.}0516,0\mathrm{.}0550,$ and $0\mathrm{.}0588,$ respectively.

$\backslash$table$[[,$CALLS,,PUTS$],[$STRIKE$,$JUL,AUG,OCT,,JUL,AUG,OCT$],[155,10\mathrm{.}50,11\mathrm{.}80,14\mathrm{.}00,,0\mathrm{.}20,1\mathrm{.}25,2\mathrm{.}75],[160,6\mathrm{.}00,8\mathrm{.}10,11\mathrm{.}10,,0\mathrm{.}75,2\mathrm{.}75,4\mathrm{.}50],[165,2\mathrm{.}70,5\mathrm{.}20,8\mathrm{.}10,,2\mathrm{.}35,4\mathrm{.}70,6\mathrm{.}70],[170,0\mathrm{.}80,3\mathrm{.}20,6\mathrm{.}00,5\mathrm{.}80,7\mathrm{.}50,9\mathrm{.}00,]]$Repeat problem $13$ using American put$-$call