Use the information in the following table to determine the lower bound of the October $170$ put option assuming European options. The option prices were observed for a stock on July $6$ of a particular year. Ignore dividends on the stock. The stock is priced at $$165\mathrm{.}13.$ The expirations are August $21$ and October $16.$ The risk free rates are $5\mathrm{.}50\%$ for August and $5\mathrm{.}88\%$ for October.

Calls Puts

Strike Aug Oct Aug Oct

$$155$ $$11\mathrm{.}80$ $$14\mathrm{.}00$ $$1\mathrm{.}25$ $$2\mathrm{.}75$

$$160$ $$8\mathrm{.}10$ $$11\mathrm{.}10$ $$2\mathrm{.}75$ $$4\mathrm{.}50$

$$165$ $$5\mathrm{.}20$ $$8\mathrm{.}10$ $$4\mathrm{.}70$ $$6\mathrm{.}70$

$$170$ $$3\mathrm{.}20$ $$6\mathrm{.}00$ $$7\mathrm{.}50$ $$9\mathrm{.}00$

Question $21$ options:

$$3\mathrm{.}87$

$$0$

$$2\mathrm{.}18$

$$5\mathrm{.}37$