# Valuing Prepayment and Default in a Fixed-Rate Mortgage: A Bivariate Binomial Options Pricing Technique

## Article excerpt

Kau et al. (1992, 1994), hereafter KKME, value a fixed-rate mortgage and its embedded options to default and prepay by adapting a two-state explicit finite-difference technique. Their continuous-time model has been used successfully to reveal a number of theoretical insights into mortgage pricing. Yet, the complexity of applying the finite-difference technique to mortgage pricing suggests the need to discover a new, perhaps simpler technique for use in a mortgage pricing model. In the spirit of Occam's razor,(1) we present a new, simpler model which uses the bivariate binomial options pricing technique developed by Nelson and Ramaswamy (1990) and Hilllard, Schwartz and Tucker (1995) to simultaneously value the prepayment and default options in a fixed-rate mortgage.(2) In addition, we explore the conditions under which prices generated by the bivariate binomial converge toward or diverge from the prices generated by the KKME model.

Our required ingredients for a new mortgage model include the following: (1) a two-state model, (2) that underlying processes are not limited to lognormal distributions, (3) allowance of correlation between state variables, (4) a backwards pricing model and (5) use of risk-neutral pricing and correctly calculated probabilities. We require a two-state model with a stochastic house price and interest rate in order to simultaneously price default and prepayment. For greater generality and empirical fidelity, we do not wish to be limited to lognormal diffusions. Given the possibility of correlation between our two underlying state variables, our model must allow the underlying variables to be correlated. In order to calculate the value of delay, a backwards pricing model is required. Finally, rather than choosing probabilities for our lattice in an ad hoc fashion, the model must correctly use risk-neutral valuation with calculated risk-neutral probabilities.

In recent years, the field of finance has investigated the pricing of options with multiple-state variables.(3) Boyle (1988) extends the one-state binomial model of Cox, Ross and Rubinstein (1979) to one that involves two state variables and uses a trinomial (three-jump) approach rather than a binomial approach. Leung and Sirmans (1990) and Ho, Stapleton and Subrahmanyam (1993) apply the Boyle model to the pricing of fixed-rate mortgages. However, the Boyle approach is applicable only to lognormal diffusions, violating our second requirement.

He (1990) and Barraquand and Martineau (1995) present simplified multistate option pricing models that we examined as potential basic models that could be adapted to our purpose.(4) However, He's model uses a much simpler interest-rate process, requires uncorrelated state variables, and assumes equally probable states, thus violating our second, third and fifth requirements. Barraquand and Martineau's simulation approach uses the forward-pricing Monte Carlo technique and assumes constant interest rates, thus violating requirements (4) and (1) respectively.

Our search ultimately led to Nelson and Ramaswamy (1990), hereafter NR. NR demonstrate how a binomial model is used to approximate nearly all diffusions once the heteroskedasticity of each process is removed. The transformation that is performed to remove the heteroskedasticity (transforming the original stochastic processes into constant variance processes) results in the simplified approach of Cox, Ross and Rubinstein (1979).(5) The paths that the new processes follow are used to generate a recombining (computationally simple) binomial lattice. Hilliard, Schwartz and Tucker (1995), hereafter HST, developed a bivariate binomial options pricing technique by extending the NR procedures for removing heteroskedasticity, and by implementing Hull and White's (1990)(6) procedure for removing the correlation between state variables. The NR-HST technique allows us to meet our five requirements.

The results of the KKME mortgage pricing model are the most obvious for comparison with our new technique. …