Using Short-Rate Trees to Infer Forward Term Structures

Summary

The document asks how to obtain longer-term interest rates from a calibrated short-rate model when valuing a mortgage-backed bond. Its motivating example links borrower prepayment decisions and penalty calculations to longer-maturity rates, while the model generates short rates over time. The specific question is whether simulated paths from a short-rate tree, such as BDT or Ho-Lee, can provide a term structure conditional on the short rate at a given simulation month.

This is a conceptual question rather than a worked method: the source gives no answer, derivation, calibration details, or evidence about how to construct those conditional rates. It highlights a modeling issue relevant to mortgage valuation and option-adjusted spread analysis, but does not establish whether the proposed inference is valid or how it should be implemented. Any solution would depend on the model’s state representation and the intended definition of the conditional term structure.

Key ideas

• Mortgage prepayment assumptions may depend on longer-maturity rates.
• A short-rate tree generates model states that raise the question of how to obtain conditional term rates.
• The document frames the issue in the context of mortgage-backed bond valuation and OAS.
• It provides no method or answer, so the proposed inference remains unresolved.

Tags

# Inferring a term structure when using a short-rate model


# Inferring a term structure when using a short-rate model












I'm relatively new to working with interest rate models and I am having some conceptual difficultly considering how a short-rate model can be utilized when calculating OAS of a mortgage-based bond, whose prepayment behavior will be based on longer term rates in any given month ($t$). For example, as a simple model assumption, a mortgage borrower might elect to prepay when 10 yr rates are 100 bps lower than their current coupon rate on the loan. A prepayment penalty calculation within the prepay model might also be based on current long term rates (such as the US10yr or US7yr).

In the context of a short-rate model this is my question specifically: if I calibrate a short-rate tree (such as BDT or Ho-Lee) then generate sample paths from this tree, can I infer a term structure of interest rates given a short-rate $r$ in a month $t$ at any point in the model/simulation?

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