Step $1$:

At $t=0,$ you sell $($short$)$ a European put to a buyer.

As the seller, how can you do to make this short position one$-$period $($from $t=0$ to $\frac{3}{12})$ neutral to the change in the underlying stock price?

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Step $2$: At $t=\frac{3}{12}$ and $S_{\frac{3}{12}}=22.$

$(1)$ How much do you have in your bank account?

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$(2)$ When the price of the underlying stock goes up to $$22,$ the delta of the put option increase from ${}_0^p=-0\mathrm{.}4936$ to ${}_u^p.$ Show ${}_{u^p}^p.$

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$(3)$ What's delta of the portfolio constructed at time $t=0$ if you keep the delta unchanged $($i$.$e$.,{}_0^p=-0\mathrm{.}4936)$

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$(4)$ To make the portfolio delta neutral $($i$.$e$.,{}_u^{pf}=0)$ when $S_{\frac{3}{12}}=22,$ how should you do$?$

Step $3$: $,$ At $t=\frac{3}{12}$ and $S_{\frac{3}{12}}=18$

$(1)$ How much do you have in your bank account?

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$(2)$When the price of the underlying stock goes down to $$18,$ the delta of the put option decrease from ${}_0^p=-0\mathrm{.}4936$ to ${}_d^p.$ Show ${}_d^p.$

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$(2)$ To make the portfolio delta neutral $($i$.$e$.,{}_d^{pf}=0)$ when $S_{\frac{3}{12}}=18,$ how should you do$?$

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Step $4$: Show your profits or losses when the stock price on the option maturity date is $S_{\frac{6}{12}}=24\mathrm{.}2($from $S_{\frac{3}{12}}=22).$

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Show your profits or losses when the stock price on the option maturity date is $S_{\frac{6}{12}}=19\mathrm{.}8$ from $S_{\frac{3}{12}}=22).$

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Step $5$: Show your profits or losses when the stock price on the option maturity date is $S_{\frac{6}{12}}=19\mathrm{.}8($from $S_{\frac{3}{12}}=18).$

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Show your profits or losses when the stock price on the option maturity date is $S_{\frac{6}{12}}=16\mathrm{.}2($from $S_{\frac{3}{12}}=18).$

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