Pricing of adjustable rate mortgage subject to prepayment and default risk
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In this paper, a reduced-form approach is adopted to price Adjustable Rate Mortgages subject to prepayment and default. The reduced-form approach to valuation facilitates the utilization of real life data in asset pricing. A given ARM is priced using the risk-neutral pricing principle, incorporating both default and prepayment risks. In the process, the default and prepayment risks are assumed to be exogenous and empirically estimated. The key feature of the model in this paper is the way the default and prepayment risks are modelled. Going beyond the conventional way of treating prepayment and default risks as deterministic, we assume a particular type of stochastic data generating process which will better capture the dynamic nature of the default and prepayment behaviors. Pricing of a given ARM proceeds in several stages. At stage one, under a Cox proportional hazard framework, the effects on prepayment and default risks of an individual borrower underlying a mortgage are captured in a multiplicative way, while the baseline prepayment and default hazards are allowed to be independent of the idiosyncrasies of the individual borrowers. The effects of the individuals are thus estimated using observed histories on a large number of comparable mortgages. At stage two, the baseline hazards are assumed to come from stochastic data generating processes that can be modelled as CIR-type diffusion processes. The diffusion processes are estimated using a particle filtering technique in a state space framework where the underlying state variables can only be observed indirectly through measurement equations. At stage three, after a twofactor CIR model for the term structure of interest rate is assumed and estimated using a particle ltering technique, we calibrate the default and prepayment hazard processes so that the model would produce a price of a given mortgage that was consistent with what was observed in the marketplace. Finally, we conduct tests on the out-of-sample pricing performance and demonstrate the application of the model in mortgage pricing strategy.