A long$-$term US investor holds $100$ shares of the SPY $($an S&P$500$ index ETF$),$ but is worried about a pull$-$back in the overall stock market during the summer. The following is the option chain quote for SPY for the August maturity. Use the nearest strike prices.

SPDR S&P $500$ ETF Trust $($SPY$)$

Closing Price

$\backslash$table$[[$Closing Price,$$414\mathrm{.}65],[$Last Price,Open Interest,Strike,Last Price,Puts$],[29\mathrm{.}54,9,194,{395}_{?},6\mathrm{.}81,15,817],[25\mathrm{.}6,6,163,{400}_{?},7\mathrm{.}85,27,112],[22\mathrm{.}35,4,870,{405}_{?},8\mathrm{.}88,9,296],[18\mathrm{.}21,7,114,{410}_{?},10\mathrm{.}3,17,282],[14\mathrm{.}52,4,642,{415}_{?},11\mathrm{.}82,8,168],[11\mathrm{.}3,12,555,{420}_{?},13\mathrm{.}84,9,029],[8\mathrm{.}86,6,714,{425}_{?},15\mathrm{.}98,3,220],[6\mathrm{.}27,35,026,{430}_{?},18\mathrm{.}3,2,290],[4\mathrm{.}08,13,997,435,23,6,532]]$

a$.$ Construct a hedge to protect the position against a market decrease of more than $5\%.$ Explain the strategy, thow it is constructed, and the total cost to the investor.

b$.$ The investor wants to offset the cost of the hedge in part $($a$)$ by giving up some upside if the market increases by $5\%$ or more. Explain the combined strategy, how it is constructed, and the total cost to the investor.

$5.$ Suppose that put options on a stock with strike prices $$30$ and $$35$ cost $$4$ and $$7,$ respectively. How can the options be used to create $($a$)$ a bull spread and $($b$)$ a bear spread? Construct a table that shows the profit and payoff for both spreads.