The common stock of Company XYZ is currently trading at a price of $$45.$ Both a put and a call option are available for XYZ stock, each having an exercise price of $$43$ and an expiration date in exactly six months. The current market prices for the put and call are $$1\mathrm{.}40$ and $$3\mathrm{.}70,$ respectively. The risk$-$free holding period return for the next six months is $4\%,$ which corresponds to an $8\%$ annual rate.

a$.$ For each stock price in the following sequence, calculate the expiration date payoffs and profits $($net of the initial purchase price$)$ for the following positions: $(1)$ buy one XYZ call option, and $(2)$ short one XYZ call option

$($assume the call is uncovered$)$:

Buy $1$ call:

Expiration Date

XYZ Stock Price Expiration Date

Derivative Payoff Initial Derivative

Premium

Net Profit

$$20$

$$25$

$$30$

$$35$

$$40$

$$45$

$$50$

$$55$

$$60$

Short $1$ call $1$ call:

Expiration Date

XYZ Stock Price Expiration Date

Derivative Payoff Initial Derivative

Premium

Net Profit

$$20$

$$25$

$$30$

$$35$

$$40$

$$45$

$$50$

$$55$

$$60$

b$.$ Calculate the price at which the position will break even $($i$.$e$.,$ produce a net profit of zero$)$ for both scenarios above.