The valuation and optimal policies of puttable convertible bonds

American-style convertible bonds commonly contain the put provision that allows the investors to put or sell their holdings to the issuer at preset prices and dates. The embedded put option includes a free boundary in addition to the conversion boundary. Because of the correlation of two moving boundaries with the convertible price, the valuation of puttable convertible bonds remains a classical problem in quantitative finance. This paper presents the valuation model of puttable convertible bonds under the Black-Scholes framework. We distinguish between the conventional pricing model and the current work by the realization of a jump in the put price across the hitting time. The jump condition permits the derivation of two recombining differential systems and we explore the impact of jump effect on the pricing dynamic of this innovative financial derivative