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

188 MODELING FINANCIAL SCENARIOShave closed-form solutions for common interest rate dependentsecurities.Single- vs. Multifactor ModelsThe models presented above are all one-factor term structuremodels since there is only a single variable generating stochasticmovements in interest rates. One problem with one-factor modelsis that the single source of uncertainty drives all term structuremovements. As a result, yields of all maturities are perfectlycorrelated to the one stochastic factor and the range of potentialyield curves is limited. The effects of multi-dimensional movesin the term structure can have serious consequences on a portfolio’svalue. Reitano [35] demonstrates that even small nonparallelshifts in the yield curve can cause extreme changes inasset values.Introducing additional sources of uncertainty (such as allowingthe long end of the curve to fluctuate or introducing stochasticvolatility or both) provide for a fuller range of yield curvemovements and shapes. The downside is that introducing multipledimensions of yield curve movements quickly increases thecomplexity and tractability of the model. Choosing the numberof stochastic factors for a term structure model represents animportant balance between accuracy and simplicity.To illustrate an example of a multifactor term structure model,Hull and White [29] extend the one-factor Hull-White model [28]to include a stochastic mean reversion level:dr t =((t)+u t ¡ ar t )dt + ¾ 1 dB 1tdu t = ¡bu t dt + ¾ 2 dB 2t :(2.8)Similar to the one-factor Hull-White model, the instantaneousshort-term rate (r t ) reverts to some time-dependent reversion level((t)+u t ). The introduction of a stochastic process for u t in thesecond equation in Formula (2.8) shows that the mean reversionlevel is also variable. The effect of introducing this second