Jane Lee and Elijah Norton run a currency trading business out of Chicago, IL$.$ While taking an international finance course in college together, Jane and Elijah discovered they had a knack for trading in currencies. They started trading using their savings and, along the way, joined forces as they realized that they had diverging, yet complimentary, views. After trading for several years and earning some impressive returns, they were able to attract investors into their business and quit their jobs to trade full$-$time.

As is the norm with their business, they are discussing their proposed trades this morn$$ing$.$ Jane proposes to purchase three$-$month call options on the Japanese yen. Currently, the Japanese yen is trading at the rate of $95\mathrm{.}28$ cents per $100$ yen. The market expects the Japanese yen to appreciate slightly over the next three months, evidenced by the $90-$day forward rate of $95\mathrm{.}71$ cents per $100$ yen. However, Jane is confident that the Japanese yen would appreciate even more based on her analysis of the current eco$$nomic situation and reach $$1\mathrm{.}00$ per $100$ yen over the next three months. Japanese yen call options with a strike price of $96$ cents per $100$ yen are available at a premium of $1\mathrm{.}35$ cents per $100$ yen. The contract size is $1,000,000$ yen.

Jane proposes to purchase $500$ call option contracts for $$67,500$ and tells Elijah that they could be in for an almost $200\%$ return on this trade. Elijah is$,$ however, unsure and feels that Jane may be too optimistic about the yen appreciation.

Show Jane$$s expected profit and rate of return on this trade if yen appreciates to $$1\mathrm{.}00$ per $100$ yen as she expected. Show what the profit and return on their investment will be if Elijah is correct and the yen appreciates only to the forward rate. Even more, what if the yen fails to appreciate at all? What will be the rate of return on this trade then? To better understand this proposed trade, determine the future spot rate at which Jane will break even.