## 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 h_{t} = h_{0}(t) + γ (k - r_{t})^{+}, where h_{0}(t) is a deterministic function of time, r_{t} 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 language | English (US) |
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Pages (from-to) | 541-573 |

Number of pages | 33 |

Journal | Mathematical Finance |

Volume | 17 |

Issue number | 4 |

DOIs | |

State | Published - 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