Intensity-based valuation of residential mortgages: An analytically tractable model

Vyacheslav Gorovoy, Vadim Linetsky*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

This paper presents an analytically tractable valuation model for residential mortgages. The random mortgage prepayment time is assumed to have an intensity process of the form ht = h0(t) + γ (k - rt)+, where h0(t) is a deterministic function of time, rt is the short rate, and γ and k are scalar parameters. The first term models exogenous prepayment independent of interest rates (e.g., a multiple of the PSA prepayment function). The second term models refinancing due to declining interest rates and is proportional to the positive part of the distance between a constant threshold level and the current short rate. When the short rate follows a CIR diffusion, we are able to solve the model analytically and find explicit expressions for the present value of the mortgage contract, its principal-only and interest-only parts, as well as their deltas. Mortgage rates at origination are found by solving a non-linear equation. Our solution method is based on explicitly constructing an eigenfunction expansion of the pricing semigroup, a Feynman-Kac semigroup of the CIR diffusion killed at an additive functional that is a linear combination of the integral of the CIR process and an area below a constant threshold and above the process sample path (the so-called area functional). A sensitivity analysis of the term structure of mortgage rates and calibration of the model to market data are presented.

Original languageEnglish (US)
Pages (from-to)541-573
Number of pages33
JournalMathematical Finance
Volume17
Issue number4
DOIs
StatePublished - Oct 2007

Keywords

  • Area functional
  • CIR diffusion
  • Eigenfunction expansion
  • Hazard process
  • Mortgage
  • Prepayment intensity
  • Pricing semigroup

ASJC Scopus subject areas

  • Accounting
  • Finance
  • Social Sciences (miscellaneous)
  • Economics and Econometrics
  • Applied Mathematics

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