Explain how to value a convertible bond within an arbitrage-free framework

In an arbitrage-free framework a convertible is broken into components valued separately. A plain convertible is worth its straight value plus a call option on the shares covered by the conversion ratio. If the issuer can call the bond, subtract the issuer's call; if the holder can put it, add the put. So a callable, putable convertible = straight bond + call on shares − issuer call + holder put. Higher share price volatility raises the value of the conversion option; higher interest rate volatility raises both bond options. Because the components interact, practitioners use trees or simulation that model both rates and share prices. Watch the signs of the embedded options.