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Intensity‐Based Valuation Of Residential Mortgages: An Analytically Tractable Model

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  • Vyacheslav Gorovoy
  • Vadim Linetsky

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.

Suggested Citation

  • Vyacheslav Gorovoy & Vadim Linetsky, 2007. "Intensity‐Based Valuation Of Residential Mortgages: An Analytically Tractable Model," Mathematical Finance, Wiley Blackwell, vol. 17(4), pages 541-573, October.
    • Handle: RePEc:bla:mathfi:v:17:y:2007:i:4:p:541-573
    DOI: 10.1111/j.1467-9965.2007.00315.x
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    References listed on IDEAS

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    1. Pavel Cizek & Wolfgang Karl Härdle & Rafal Weron, 2005. "Statistical Tools for Finance and Insurance," HSC Books, Hugo Steinhaus Center, Wroclaw University of Technology, number hsbook0501.
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    Cited by:

    1. Ebrahim, M. Shahid & Shackleton, Mark B. & Wojakowski, Rafal M., 2011. "Participating mortgages and the efficiency of financial intermediation," Journal of Banking & Finance, Elsevier, vol. 35(11), pages 3042-3054, November.
    2. Zhe Cheng & Scott Robertson, 2017. "Endogenous current coupons," Finance and Stochastics, Springer, vol. 21(4), pages 1027-1071, October.
    3. Cheikh Mbaye & Frédéric Vrins, 2022. "Affine term structure models: A time‐change approach with perfect fit to market curves," Mathematical Finance, Wiley Blackwell, vol. 32(2), pages 678-724, April.
    4. Robert J. Shiller & Rafal M. Wojakowski & M. Shahid Ebrahim & Mark B. Shackleton, 2017. "Continuous Workout Mortgages: Efficient Pricing and Systemic Implications," Cowles Foundation Discussion Papers 3016, Cowles Foundation for Research in Economics, Yale University.
    5. Dongjae Lim & Lingfei Li & Vadim Linetsky, 2012. "Evaluating Callable and Putable Bonds: An Eigenfunction Expansion Approach," Papers 1206.5046, arXiv.org.
    6. Mikhail Chernov & Brett R. Dunn & Francis A. Longstaff, 2018. "Macroeconomic-Driven Prepayment Risk and the Valuation of Mortgage-Backed Securities," Review of Financial Studies, Society for Financial Studies, vol. 31(3), pages 1132-1183.
    7. Shiller, Robert J. & Wojakowski, Rafal M. & Ebrahim, M. Shahid & Shackleton, Mark B., 2019. "Continuous Workout Mortgages: Efficient pricing and systemic implications," Journal of Economic Behavior & Organization, Elsevier, vol. 157(C), pages 244-274.
    8. Almas Naseem & R. Reesor, 2015. "Risk and reward of home equity borrowing for investment in Canada, a stochastic analysis," Computational Management Science, Springer, vol. 12(1), pages 45-79, January.
    9. Ahmad, F. & Hambly, B.M. & Ledger, S., 2018. "A stochastic partial differential equation model for the pricing of mortgage-backed securities," Stochastic Processes and their Applications, Elsevier, vol. 128(11), pages 3778-3806.
    10. Scott Robertson & Zhe Cheng, 2015. "Endogenous Current Coupons," Papers 1510.02010, arXiv.org.
    11. Matthew Lorig, 2011. "Pricing Derivatives on Multiscale Diffusions: an Eigenfunction Expansion Approach," Papers 1109.0738, arXiv.org, revised Apr 2012.
    12. Tetsuya Ishikawa & Scott Robertson, 2017. "Optimal Investment and Pricing in the Presence of Defaults," Papers 1703.00062, arXiv.org.
    13. Lim, Dongjae & Li, Lingfei & Linetsky, Vadim, 2012. "Evaluating callable and putable bonds: An eigenfunction expansion approach," Journal of Economic Dynamics and Control, Elsevier, vol. 36(12), pages 1888-1908.

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