Consider the following option trading strategies:$$a$)$ The options with strike prices R$80,$ R$85,$ and R$90$ cost R$10,$ R$13,$ and R$18,$ respectively.$$Suppose a trader creates a butterfly spread from the options by trading a$$total of $500$ options, determine the maximum net loss to the butterfly spread after$$considering the cost of the options. $[4]$b$)$ The table consists of call and put premiums for $3-$month European options with$$different strike prices for a non$-$dividend paying stock with S$0=$ R$40$ and r $=0\mathrm{.}08$p$.$a$.$ continuously compounded. A trader is considering two strategies using the$$Strike Price Call Option Premium Put Option Premium$$R$35$ R$6\mathrm{.}13$ R$0\mathrm{.}44$R$40$ R$2\mathrm{.}78$ R$1\mathrm{.}99$R$45$ R$0\mathrm{.}97$ R$5\mathrm{.}08$information on the options provided in the table. The first strategy is a strangle$$consisting of a R$35-$strike put and a R$45-$strike call. The second strategy is a R$40-$strike straddle.

i$.$ Explain how the options can be used to create the first strategy.

ii$.$ Explain how the options can be used to create the second strategy.

iii. Let S be the stock price in three months. Construct a table that shows the$$profit for the first strategy.

iv$.$ Let S be the stock price in three months. Construct a table that shows the$$profit for the second strategy.

v$.$ Let S be the stock price in three months. Determine the range of stock prices$$in $3$ months for which the first strategy outperforms the second$$strategy$.$