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For swaptions the logic is almost identical, substituting swaps for bonds in step 1, rast swaptions for bond options in step 2. Thetasensitivity to time, is likewise estimated given the option price at the first node in the tree and the option price for the same spot in a later time step.
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More recent models, in fact, are designed around direct convergence to Black-Scholes. For multiple underlyersmultinomial lattices  can be built, although the number of nodes increases exponentially with the number of underlyers. Lattice models have been developed for equity analysis here,   particularly as relates to distressed firms. For DKC, the first step is to recover the state prices corresponding to each node in the tree, such that these are consistent with observed option prices i.
Once calibrated, the interest rate lattice is then used in the valuation of various of the fixed income instruments and derivatives. Here, calibration means that the interest-rate-tree reproduces the prices of the zero-coupon bonds —and any other interest-rate sensitive securities—used in constructing the yield curve ; note the parallel to implied trees above, and compare Bootstrapping finance. Some exotic optionssuch as barrier optionsare also easily modeled here; note though that for other Path-Dependent Optionssimulation would be preferred.
DKC is effectively a discretized local volatility model. For caps and floors step 1 and 2 are combined: For exotic options the trinomial model or adaptations is sometimes more stable and accurate, regardless of step-size. For vanilla optionsas the number of steps increases, the results rapidly converge, and the binomial model is then preferred due to its simpler implementation. Construct an interest-rate tree, which, as described in the text, will be consistent with the current term structure of interest rates.
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This distinction means that for equilibrium-based models the yield curve is an output from the model, while for arbitrage-free models the yield curve is an input to the model. Lricing valuation oc proceeds as standard, with these substituted for p. An alternative approach to modeling American bond options, particularly those struck on yield to maturity YTMemploys modified equity-lattice methods. For similar reasons, real options and employee stock options are often modeled using a lattice framework, though with modified assumptions.
In this case, the Lattice is sometimes referred to as a "bush", and the number of nodes grows exponentially as a function of number of time-steps. For equity and commodities the application is as follows.