$2.$The market price of common stock of the same company is $20$ yuan per share, the price of convertible bonds is $105$ yuan per share, which can be converted into $5$ shares, and the price of nonconvertible bonds with the same characteristics is $102$ yuan per share. Suppose there is an equal chance that the price of the common shares of the company will change to $$18$ or $$22$ one year from now. Suppose there is an opportunity for convertible arbitrage,

$(1)$ Please calculate the range of value multiples for the number of convertible bonds and common stocks traded.

$(2)$ If the probability of the common stock price rising to $22$ yuan one year later is obviously greater, how to maximize the value of arbitrage retum? Please calculate the arbitrage return at this time.$(3)$ Is convertible arbitrage risk$-$free or risk$-$free? Why? How do you think the average stock price moves to increase the arbitrage room?

$(4)$ Please calculate the common stock price at the critical point with or without arbitrage space.