The carbon bonds issued by countries or enterprises can solve the problem of funds in low carbon economy growth. Now most of carbon bonds pay fixed interest rates, and a few pay floating rates. The diversity of carbon bonds can attract more investors to participate green energy projects. The London Accord project group proposed the index linked carbon bonds in the World Band Government Borrowers' Forum in May 2009, and pointed out that the interest paid regularly may be linked to carbon price, governments' carbon emission targets, in-country fossil fuel prices or tariff feed-in prices. In this paper, the interests are considered to be linked with carbon prices in the condition of stochastic risk-free interest rate, and a partial differential equation is established for carbon bond interests. Also, a fitted finite volume method is employed to solve the resulting partial differential equation numerically, and on the basis of the valuation for zero-coupon bonds, the price of carbon bonds is obtained. Finally, some data are utilized for the calibration of the parameters in the established pricing models, and some numerical examples are presented and the effects of parameters on solutions are also demonstrated, which can provide references for the issuers of carbon bonds.