Black–Derman–Toy model

In mathematical finance, the Black–Derman–Toy model (BDT) is a popular short-rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see Lattice model (finance) § Interest rate derivatives. It is a one-factor model; that is, a single stochastic factor—the short rate—determines the future evolution of all interest rates. It was the first model to combine the mean-reverting behaviour of the short rate with the log-normal distribution, and is still widely used.

Short-rate tree calibration under BDT: Step 0. Set the risk-neutral probability of an up move, p, to 50% Step 1. For each input spot rate, iteratively: adjust the rate at the top-most node at the current time-step, i; find all other rates in the time-step, where these are linked to the node immediately above (r_u; r_d being the node in question) via $\ln(r_{u}/r_{d})/2=\sigma _{i}{\sqrt {\Delta t}}$ (this node-spacing being consistent with p = 50%; Δt being the length of the time-step); discount recursively through the tree using the rate at each node, i.e. via "backwards induction", from the time-step in question to the first node in the tree (i.e. i=0); repeat until the discounted value at the first node in the tree equals the zero-price corresponding to the given spot interest rate for the i-th time-step. Step 2. Once solved, retain these known short rates, and proceed to the next time-step (i.e. input spot-rate), "growing" the tree until it incorporates the full input yield-curve.

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