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Question: Is there a generalized procedure for building a discrete (e.g. binomial) term structure model with risk-neutral branching probabilities that ensure positive probabilities under alternative measures (e.g. real-world, forward)?

Motivation: I'm interested in building a short rate binomial short-rate tree that can support both risk-neutral (for pricing) and real world (for risk analysis) measures. The two sources I have for constructing risk-neutral trees are Shreve: Volume I and Haugh's lecture notes. In both, they use risk-neutral probabilities of 0.5 (calibrating the rate via model parameters to match market prices of ZCB).

By introducing a market price of interest rate risk, one could change the measure from risk-neutral to real-world while using the same lattice of rates. There's a possibility that the market price of risk is large enough to push one of the two branching probabilities to be negative. In such a case, it would make sense to revise the risk-neutral probabilities to be something other than 0.5 such that the calibrated rates permit positive probabilities in the real-world measure as well. This seems needlessly iterative.

Hull & White [2016] introduce something similar to what I am proposing above on their trinomial tree approach. I'd like a simpler binomial model first before I attempt to implement their trinomial model.

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