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Valuation of fixed and variable rate mortgages: binomial tree versus analytical approximations

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  • Werner Hürlimann

Abstract

The recombining binomial tree approach, which has been initiated by Cox et al. (J Financ Econ 7: 229–263, 1979 ) and extended to arbitrary diffusion models by Nelson and Ramaswamy (Rev Financ Stud 3(3): 393–430, 1990 ) and Hull and White (J Financ Quant Anal 25: 87–100, 1990a ), is applied to the simultaneous evaluation of price and Greeks for the amortized fixed and variable rate mortgage prepayment option. We consider the simplified binomial tree approximation to arbitrary diffusion processes by Costabile and Massabo (J Deriv 17(3): 65–85, 2010 ) and analyze its numerical applicability to the mortgage valuation problem for some Vasicek and CIR-like interest rate models. For fixed rates and binomial trees with about thousand steps, we obtain very good results. For the Vasicek model, we also compare the closed-form analytical approximation of the callable fixed rate mortgage price by Xie (IAENG Int J Appl Math 39(1): 9, 2009 ) with its binomial tree counterpart. With respect to the binomial tree values one observes a systematic underestimation (overestimation) of the callable mortgage price (prepayment option price) analytical approximation. This numerical discrepancy increases at longer maturities and becomes impractical for a valuable estimation of the prepayment option price. Copyright Springer-Verlag 2012

Suggested Citation

  • Werner Hürlimann, 2012. "Valuation of fixed and variable rate mortgages: binomial tree versus analytical approximations," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 35(2), pages 171-202, November.
    • Handle: RePEc:spr:decfin:v:35:y:2012:i:2:p:171-202
    DOI: 10.1007/s10203-011-0117-z
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    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • G21 - Financial Economics - - Financial Institutions and Services - - - Banks; Other Depository Institutions; Micro Finance Institutions; Mortgages

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