The following exchange rates are quoted simultaneously in Johannesburg, Frankfurt, Singapore and Paris:

Johannesburg: USD$/$ZAR $18\mathrm{.}8300$

Frankfurt: GBP$/$USD $1\mathrm{.}2550$

Singapore: ZAR$/$SGD $0\mathrm{.}0725$

Paris: SGD$/$USD $0\mathrm{.}7255$

a$-$ Calculate all the possible cross rates in the above and then indicate if there is a possibility for three$-$point arbitrage.

b$-$Suppose Jay have R$1$ million to perform an arbitrage trade. Also, suppose the transaction cost for executing all the arbitrage trades is R$1550.$ If the transaction costs are included, is there still a profitable arbitrage sequence available? Prove the answer by calculating the profit $($or loss$)$ earned from arbitrage if Jay has ZAR $1$ million to use.