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

The recombining binomial tree approach, which has been initiated by Cox et al. (J Financ Econ 7: 229–263, <CitationRef CitationID="CR16">1979</CitationRef>) and extended to arbitrary diffusion models by Nelson and Ramaswamy (Rev Financ Stud 3(3): 393–430, <CitationRef CitationID="CR43">1990</CitationRef>) and Hull and White (J Financ Quant Anal 25: 87–100, <CitationRef CitationID="CR30">1990a</CitationRef>), 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, <CitationRef CitationID="CR15">2010</CitationRef>) 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, <CitationRef CitationID="CR61">2009</CitationRef>) 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