are jointly modeled in a three-dimensional tree, constructed such that LIBOR swap rates are matched. With the zeroeth step thus accomplished, the valuation will proceed largely as previously, using steps 1 and onwards, but here with cashflows based on the LIBOR "dimension", and discounting using the corresponding nodes from the OIS "dimension". Hybrid securities Hybrid securities, incorporating both equity- and bond-like features are also valued using trees. For convertible bonds (CBs) the approach of Tsiveriotis and Fernandes (1998) is to divide the value of the bond at each node into an "equity" component, arising from situations where the CB will be converted, and a "debt" component, arising from situations where CB is redeemed. Correspondingly, twin trees are constructed where discounting is at the risk free and credit risk adjusted rate respectively, with the sum being the value of the CB. There are other methods, which similarly combine an equity-type tree with a short-rate tree. An alternate approach, originally published by Goldman Sachs (1994), does not decouple the components, rather, discounting is at a conversion-probability-weighted risk-free and risky interest rate within a single tree. See , Contingent convertible bond. More generally, equity can be viewed as a call option on the firm: where the value of the firm is less than the value of the outstanding debt shareholders would choose not to repay the firm's debt; they would choose to repay—and not to liquidate (i.e. exercise their option)—otherwise. Lattice models have been developed for equity analysis here, particularly as relates to distressed firms. Relatedly, as regards corporate debt pricing, the relationship between equity holders' limited liability and potential Chapter 11 proceedings has also been modelled via lattice. The calculation of "Greeks" for interest rate derivatives proceeds as for equity. There is however an additional requirement, particularly for hybrid securities: that is, to estimate sensitivities related to overall changes in interest rates. For a bond with an embedded option, the standard yield to maturity based calculations of duration and convexity do not consider how changes in interest rates will alter the cash flows due to option exercise. To address this, effective duration and -convexity are introduced. Here, similar to rho and vega above, the interest rate tree is rebuilt for an upward and then downward parallel shift in the yield curve and these measures are calculated numerically given the corresponding changes in bond value. References Bibliography Options (finance) Short-rate models Mathematical finance