53Ho-Lee Model V3

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Ho-Lee Model© copyright THC 2003. All rights reserved by Thomas reference number20classification50401Version1.0levelintermediateInstructionspublication dateAug-03This worksheet implements an aauthorHanyang Financial Engineering Lab.stochastic movement of the termaffiliationHanyang Universityemail addressleesb@hanyang.ac.krinputlast revised dateDec-03outputreferencesHo, T.S.Y., and S. Lee, 1986, Term Structure MovementHo, T.S.Y., 2000, A Closed-form Binomial Interest Rate MResearch PaCh.5 of "The Oxford Guide to Financial Modeling" by Thomas S.Y. Ho and SDescriptions

financial model class1 factor interest rate modelissuer/modelerN/Amodel typerelative valuation modelrisk sourcesinterest raterisk distributionnormaleconomic assumptionsarbitrage-free money market measuretechnical assumptionsm=1key wordsarbitrage-free interest rate model, time-varying volatility, mean reversion pro

Linksdatahttp://www.federalreserve.gov/financial models7. Bond Arithmetic, 8. Bond Model, 20. Ho-Lee Model

Inputsyear01234initial yield curve0.0600.0600.0650.0700.075initial discount function p(n)1.0000000.9417650.8780950.8105840.740818one period forward curve 0.0600.0600.0700.0800.090lognormal spot volatility (σS)00.07750.07750.07750.0775lognormal forward volatility (σf)00.07750.07750.07750.0775the number of partitions a year (m)1f(n)·σf(n+1)=(n+1)·r(n+1)·σS(n+1)-n·r(n)·σS(n)

time to maturity (T) at each node1forward price0.9417650.9323940.9231160.9139310.904837delta (δ)10.9907430.9892090.9876770.986147

convexity adjustment1.000001.990741.980051.967971.954561.989211.977021.963481.987681.973991.98615

numerator of convexity adjustment at time n1.0000001.9907433.9387367.733494denominator of convexity adjustment at time n1.9907433.9387367.73349415.04646

Outputs

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(1)(1)2()(1)ininPnPPn



################################Discount function lattice########################################year01234

0.1282350.1087540.1142850.0909050.0963540.1003350.0746610.0800550.0839540.086385Interest rate lattice0.060.0653610.0692050.0715540.072435year01234

0.1087540.1025540.0963540.0963540.0901540.083954expected one year rate over a 3-year horizon0.0901540.0839540.0777540.071554year01234

Interim Calculations for checking the consistency of the model0.07750.07750.07750.07750.07750.0775Time-varying forward volatility lattice0.07750.07750.07750.0775year01234

forward volatility*forward rate0.004650.0054250.00620.006975spot volatility by conversion formula0.07750.07750.07750.0775Prof. Ho, this is for the purpose of checking the volatility conversion formula. From this, we can see that the Ho-Lee mode

spot volatility by definition (the same definition as yield volatility by BDT)1.0000000.9280581.0000001st spot volatility0.8780950.9367291.000000year01234checking the arbitrage-free condition0.8780950.0746610.0653610.0775

1.0000000.9131051.0000000.8520370.9230661.0000003-year zero coupon bond0.8105840.8693790.9331361.000000year01234checking the arbitrage-free condition0.8105840.0800630.0699882nd spot volatility0.0775

ln()()nniiPTrTT

(,1,)(,,)()2()ninirniTrniTTfT

(1)(1)2()(1)ininPnPPn

1.0000000.8969511.0000000.8241200.9081431.0000000.7738270.8435050.9194741.0000004-year zero coupon bond0.7408180.7994290.8633460.9309461.000000year01234checking the arbitrage-free condition0.7408180.0854690.0746193rd spot volatility0.0775

0.0967200.09090.085095expected 2-year bond yield at the end of period 20.08510.07930.073470year01234

0.8796470.7945420.8920040.7351860.8157560.9045340.6952250.7630490.8375370.9172415-year zero coupon bond0.6703200.7283160.7919690.8599000.930126year01234checking the arbitrage-free condition0.6703200.090880.0792554th spot volatility0.0775003. All rights reserved by Thomas Ho Company, New York, NY. USA www. thomasho.com tom.ho@thomasho.comInstructionsThis worksheet implements an arbitrage-free interest rate movement model. This model takes the complete term structure as given astochastic movement of the term structure such that the movement is arbitrage free.

initial yield curve, constant lognormal spot volatility of interest ratediscount function lattice, Interest rate latticeements and Pricing of Interest Rate Contingent Claims, Journal of Finance, Vol.41, 1011-1029.Rate Model, Research Paper, Owen School of Business Administration, Vanderbilt University.Modeling" by Thomas S.Y. Ho and Sang Bin Lee, 2003, Oxford University Press