The global options market is facing heightened volatility due to prevailing economic uncertainties, including geopolitical tensions and fluctuating interest rates. Investors are increasingly using options to hedge against potential market downturns, leading to a surge in both call and put option trading volumes. The mixed economic data from major economies like the U$.$S$.,$ China, and the Eurozone is contributing to this uncertainty, making options a preferred tool for managing risk. Additionally, the varying central bank policies across different regions are adding to the unpredictability, influencing traders to adopt more sophisticated strategies to navigate the current market landscape.

The risk$-$free interest rate of one of the European countries is $4\%,$ given the term structure is flat. MIODEN is a high$-$tech company operating mainly in Europe. A non$-$dividend paying stock of MIODEN is selling at $$36$ in the country$$s stock market. In its derivatives market, an American put on this stock costs $$5$ with the strike price of $$32$ and the expiration date is in nine months.

$1.$ What is the upper and lower bounds for the price of an American call on the same stock with the same strike price and expiration date using the given information. $(25$ marks$)$

$2.$ Traders are exploring arbitrage opportunities in the options market to earn arbitrage profits. If the market price of the American call in Part $1$ is $$11.$ Explain carefully the arbitrage opportunities, if exist. Will the at$-$least profit case occur? Explain.