be specified per time-step, i.e. we have a "smile", and the tree is built by solving for the probabilities corresponding to specified values of the underlying at each node. Once calibrated, the interest rate lattice is then used in the valuation of various of the fixed income instruments and derivatives. The approach for bond options is described aside—note that this approach addresses the problem of pull to par experienced under closed form approaches; see . For swaptions the logic is almost identical, substituting swaps for bonds in step 1, and swaptions for bond options in step 2. For caps (and floors) step 1 and 2 are combined: at each node the value is based on the relevant nodes at the later step, plus, for any caplet (floorlet) maturing in the time-step, the difference between its reference-rate and the short-rate at the node (and reflecting the corresponding day count fraction and notional-value exchanged). For callable- and putable bonds a third step would be required: at each node in the time-step incorporate the effect of the embedded option on the bond price and / or the option price there before stepping-backwards one time-step. (And noting that these options are not mutually exclusive, and so a bond may have several options embedded; hybrid securities are treated below.) For other, more exotic interest rate derivatives, similar adjustments are made to steps 1 and onward. For the "Greeks", largely as for equity, see under next section. An alternative approach to modeling (American) bond options, particularly those struck on yield to maturity (YTM), employs modified equity-lattice methods. Here the analyst builds a CRR tree of YTM, applying a constant volatility assumption, and then calculates the bond price as a function of this yield at each node; prices here are thus pulling-to-par. The second step is to then incorporate any term structure of volatility by building a corresponding DKC tree (based on every second time-step in the CRR tree: as DKC is trinomial whereas CRR is binomial) and then using this for option valuation. Since the 2007–2012 global financial crisis, swap pricing is (generally) under a "multi-curve framework", whereas previously it was off a single, "self discounting", curve; see . Here, payoffs are set as a function of LIBOR specific to the tenor in question, while discounting is at the OIS rate. To accommodate this in the lattice framework, the OIS rate and the relevant LIBOR rate