A firm owes to creditors a single amount of $$100$ million in $3$ years. It is estimated that the value of the firm's assets is today $$110$ million, and its volatility is $20\%.$ The risk$-$free rate is $5\%.$

A$)$ Using DerivaGem, calculate the value of equity and debt for this firm.

B$)$ Now assume that the creditors have imposed a barrier of $$95$ million, which leads to automatic default if the asset value crosses this barrier any time before debt maturity. Using DerivaGem, show that a European call option on the value of the firm with exercise price $$100$ million in $3$ years is worth the sum of the values of a European down$-$and$-$out call and a European down$-$and$-$in call option with the same parameters. Then, re$-$evaluate equity and debt and compare your results with those of question A$).$ How do your answers change, if the barrier is $$90$ million?