$.$ On January $2,2021,$ for past services, Rosen Corp. granted Lynne Boles, its president, $30,000$ stock appreciation rights that are exercisable immediately and expire on January $2,2022.$ On exercise, Boles is entitled to receive cash for the excess of the market price of the stock on the exercise date over the market price on the grant date. Boles did not exercise any of the rights during $2021.$ The market price of Rosen's stock was $$30$ on January $2,2021,$ and $$45$ on December $31,2021.$ As a result of the stock appreciation rights, Rosen should recognize compensation expense for $2021$ of:

$$

a$.$ $$0.$

b$.$ $$180,000.$

c$.$ $$450,000.$

d$.$ $$900,000.$