A 3-year convertible bond with a face value of $$\$ 100$$ has been issued by company ABC. It pays a coupon of $$\$ 5$$ at the end of each year. It can be converted into $A B C$ 's equity at the end of the first year or at the end of the second year. At the end of the first year, it can be exchanged for 3.6 shares immediately after the coupon date. At the end of the second year, it can be exchanged for 3.5 shares immediately after the coupon date. The current stock price is $$\$ 25$$ and the stock price volatility conditional on no default is $25 \%$. No dividends are paid on the stock. The risk-free interest rate is $5 \%$ with continuous compounding. The yield on bonds issued by $\mathrm{ABC}$ is $7 \%$ with continuous compounding and the recovery rate is $30 \%$.
(a) Use a three-step tree to calculate the value of the bond.
(b) How much is the conversion option worth?
(c) What difference does it make to the value of the bond if the bond is callable for $$\$ 115$$ immediately before the coupon payment at the end of years 1 and 2 ?
(d) Explain how your analysis would change if there were a dividend payment of $$\$ 1$$ on the equity at the 6-month, 18 -month, and 30 -month points. Detailed calculations are not required.