Pricing callable–puttable convertible bonds with an integral equation approach

In this paper, the pricing problem of callable–puttable convertible bonds written on a single underlying asset is studied with an integral equation (IE) approach. The complication of the pricing exercise results from the tangled presence of callability, puttability, as well as conversion, which have led to possible coexistence of two moving boundaries at the same time, depending on the call price, the put price, and the conversion ratio. If a callable–puttable convertible bond needs to be priced at a time sufficiently far away from the expiry, only the moving boundary associated with the puttability needs to be dealt with. When the pricing time is closer to expiry beyond a critical point, it is then possible to have two distinct cases. While the two moving boundaries associated with conversion and puttability coexist in one case, they may both disappear in another case; callability remains to be the only issue that needs to be dealt with. Furthermore, there exists another critical value, beyond which a callable–puttable convertible bond can be treated as its vanilla counterpart. Mathematically, various different scenarios demand different systems of IEs to be formulated and solved numerically.