PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

MODELING FINANCIAL SCENARIOS 185Ho and Lee [25] discuss a discrete time model of the noarbitrageapproach and include a time-dependent drift so thatobserved market prices of all bonds can be replicated. Thecontinuous-time equivalent of the Ho-Lee model isdr t = (t)dt + ¾dB t : (2.4)The time-dependent drift ((t)) of the Ho and Lee model isselected so that expected future interest rates agree with marketexpectations as reflected in the existing term structure. This driftis closely related to implied forward rates. Hull and White [28]use Ho and Lee’s [25] time-dependent drift to extend the equilibriummodels of Vasicek and CIR. The one-factor Hull-Whitemodel isdr t = ·((t) ¡ r t )dt + ¾dB t : (2.5)Heath, Jarrow, and Morton [23] (hereafter HJM) generalizethe arbitrage-free approach by allowing movements across theentire term structure rather than a single process for the shortrate. HJM posit a family of forward rate processes, f(t,T). Inthis familydf(t,T)=¹(t,T,f(t,T))dt + ¾(t,T,f(t,T))dB t , (2.6)wheref(t,T)=¡ @ lnP(t,T) : (2.7)@TChoosing between an arbitrage-free term structure model andan equilibrium model often depends on the specific application.Despite their initial appeal, arbitrage-free approaches often havedisadvantages. 3 Some of these include the following:² Arbitrage-free models are most useful for pricing purposes,especially interest rate derivatives. Since derivatives are pricedagainst the underlying assets, a model that explicitly capturesthe market prices of those underlying assets is superior to modelsthat more or less ignore market values. Hull [27] comments3 In addition to the references in this section, Tuckman [42] provides an excellentoverview of the advantages and disadvantages of equilibrium models vs. arbitrage-freemodels.