Monte Carlo is the leading method used to price mortgages and equilibrium mortgage interest rates are determined by the use of Monte Carlo simulation models. The mortgage applications use interest rate models as the main random factor. One factor interest rate models under equilibrium were developed by Vasicek [#!kn:VasicekJFE1977!#] and Cox, Ingersoll and Ross [#!kn:CIREconometrica1985b!#]1.3. Richard [#!kn:RichardJFE1978!#] and Cox, Ingersoll and Ross [#!kn:CIREconometrica1985b!#] extended the one factor CIR model to two factors independently.
The key base of modern multi-factor interest rate models with closed form solutions are the multivariate normal models of Langetieg [#!kn:LangetiegJF1980!#] and the two factor square root model of Richard [#!kn:RichardJFE1978!#] and Cox, Ingersoll and Ross (CIR) [#!kn:CIREconometrica1985b!#] .
Special cases of Langetieg's model are Ho-Lee, , [#!kn:HoLeeJF1986!#], Jamshidian, [#!kn:JamshidianML1987!#] [#!kn:JamshidianJF1989!#] [#!kn:JamshidianML1989a!#] , and Hull and White and [#!kn:HullWhiteRFS1990!#] .
Later models built on Langetieg include the quadratic model Longstaff [#!kn:LongstaffJFE1989!#], Beaglehole et al., [#!kn:BeagleholeTenneyJFI1991!#] [#!kn:BeagleholeTenneyJFE1992!#] , Jamshidian [#!kn:JamshidianFuji1993!#] 1.4 Eterovic [#!kn:EterovicHarvard1994!#], Ahn, Dittmar , and Gallant [#!kn:AhnDittmarGallantRFS2002!#], Leippold and Wu [#!kn:LeippoldWuJFQA2002!#] and others, Lin Chen's [#!kn:LinChenHarvard1995!#] 3 factor model and the affine model of Duffie and Kan [#!kn:DuffieKanMF1996!#], and the Heath, Jarrow, and Morton (HJM) methodology [#!kn:HJMEconometrica1992!#].
Langetieg derived a Boness-like formula for options on stocks with his interest rate model applying Merton's argument on Black-Scholes. Hull and White showed that Langetieg's approach could be slightly modified to apply to their version of the Langetieg model. State prices or option prices for these models were developed by Cox, Ingersoll and Ross [#!kn:CIREconometrica1985b!#] Jamshidian [#!kn:JamshidianJF1989!#], Longstaff [#!kn:LongstaffJFE1990!#], Hull and White [#!kn:HullWhiteRFS1990!#], Beaglehole et al. [#!kn:BeagleholeTenneyJFI1991!#], Milne and Turnbull [#!kn:TurnbullMilneRFS1991!#], Chen and Scott [#!kn:ChenScottRFS1992!#], Longstaff and Schwartz [#!kn:LongstaffSchwartzJF1992!#], Constantinides [#!kn:ConstantinidesRFS1992!#], Lin Chen [#!kn:LinChenHarvard1995!#], Manistre [#!kn:ManistreSOA1997!#], Duffie, Pan and Singleton [#!kn:DPSEconometrica2000!#] and others. Earlier work on state prices or Green's functions in finance traces back to McKean [#!kn:McKeanIMR1965!#], Garman [#!kn:GarmanBerkeley1976!#], Cox, Ingersoll and Ross [#!kn:CIREconometrica1985a!#], Breeden and Litzenberger [#!kn:BreedenLitzenbergerJB1978!#], Banz and Miller [#!kn:BanzMillerJB1978!#], Ingersoll [#!kn:Ingersoll1987!#], and Merton [#!kn:Merton1990!#]. In addition, Lucas [#!kn:LucasEconometrica1978a!#] derived state prices building on the foundation of Hakansson's [#!kn:HakanssonUCLA1966!#], [#!kn:HakanssonEconometrica1970a!#] approach to optimal savings, consumption and portfolio choice. Lucas developed a state price using marginal rates of substitution that arise in the Hakansson type optimizations of consumption, savings and portfolio decisions in a multiple-period context. Cox, Ingersoll and Ross and Vasicek's equilibriums are also built on the Hakansson foundation.
Because of the prepayment models and the frequent use of interest
rate models without closed form solution, Monte Carlo and Low
Discrepancy Sequences are often used for mortgage pricing and risk
analysis. An example of a model without a closed form is the
DMRP
1.5 model, see for example Groover [#!kn:GrooverPTSNL1992!#], Groover et al. [#!kn:GrooverTenneyPTSNL1992a!#] and Craighead
et al. [#!kn:CraigheadTenneySOA1997!#].