Hull-White Model

1、Hull and White model copyright THC 2003. All rights reserved by Thomas Ho Company, New York, NY. USA www. reference number23classification #50405InstructionsVersion1.0levelintermediatepublication dateMay-03authorHanyang Financial Engineering LabaffiliationHanyang Universityinputsinitial spot rates,

2、spot volatilitiesemail addressleesbhanyang.ac.krlast revised dateDec-03outputsarbitrage free spot ratesreferencesHull, J. and A. White, 1994, Numerical Procedures for Implementing Term Structure Models II: Two-Factor Models, Journal of Derivatives, 37-48, Winter.Hull, J. and A. White, 1996, Using Hu

3、ll-White Interest Rate Tree, Journal of Derivatives, 26-36, Spring.Ch. 5 of the The Oxford Guide to Financial Modeling by Thomas S.Y. Ho and Sang Bin Lee, 2003, Oxford University PressDescriptionsArbitrage free interest rate model that automatically fit the Black volatilitiesfinancial model classInt

4、erest rate modelissuerFinancial engineermodel typearbitrage free rate modelrisk sourcesinterest rate risksrisk distributionnormaleconomic assumptionspartial equilibrium modeltechnical assumptionsdiscrete time modelkey wordsLinksdatafinancial modelsInputsperiod123spot rate0.038240.045120.05086spot vo

5、l0.010.010.01parameter a0.1parameter dt1Outputs0.0971610.0693710.079841node B0.0382400.0520500.062520node Anode C0.0347300.045199node D0.027879period012Interim CalculationsThe Hull-White model for the instantateous short rate r is Its discrete version is First stage Step 1 Calculate the spacing betw

6、een interest rates using 0.01732051 ( )drtar dtdZqs=-+ ( )dRtaR dtdZqs=-+* is 0 at time0dRaR dtdZ Rs= -+3 RtsD =DR DStep 2 Determine Jmax and Jmin where the interest process switches from branching (a) to (c) and from branching (a) to (b), respectively(a)(b)(c)Jmax is the smallest integer greater th

7、an 0.184/(adt)Jmax 2Jmin -2Step 3Calculate the probabilities of the trinomial process at each node depending on where the node is located and the rate R at each nodenode ABCDEFj010-121Pu0.16666667 0.12166667 0.16666667 0.22166667 0.88666667 0.12166667Pm0.66666667 0.65666667 0.66666667 0.65666667 0.0

8、2666667 0.65666667Pd0.16666667 0.22166667 0.16666667 0.12166667 0.08666667 0.22166667R00.017320510-0.01732050.03464102 0.01732051Second stageA conversion of the tree for R* into a tree for RStep 1 Determine the prices of the Arrow-Debreau securities. Q(0,0),Q(1,1),Q(1,0),Q(1,-1)Q(i,j)is the price of

9、 a security which pays $1 if node (I,j) is reached and zero otherwise.Q(1,1)0.16041365Q(1,0)0.64165461Q(0,0)Q(1,-1)10.16041365 Step 2Determine alpha at period 1 to match the tree for R* to the initial term structure.objective function2.6174E-16aplha at period 10.05205002node ABCDEFj010-121Pu0.166666

10、67 0.12166667 0.16666667 0.22166667 0.88666667 0.12166667Pm0.66666667 0.65666667 0.66666667 0.65666667 0.02666667 0.65666667Pd0.16666667 0.22166667 0.16666667 0.12166667 0.08666667 0.22166667R0.038240.06937053 0.05205002 0.03472951Step 3Determine the prices the Arrow-Debreau securities Q(2,2),Q(2,1)

11、,Q(2,0),Q(2,-1),Q(2,-2)Q(2,2)0.01820898Q(2,1)0.19979709Q(1,1)Q(2,0)0.16041365 0.47359376Q(1,0)Q(2,-1)0.64165461 0.20326121Q(0,0)Q(1,-1)Q(2,-2)10.16041365 0.01885081Step 4Determine alpha at period 2 to match the tree for R* to the initial term structure.objective function2.3747E-08aplha at period 20.

12、06252node ABCDEFj010-121Pu0.16666667 0.12166667 0.16666667 0.22166667 0.88666667 0.12166667Pm0.66666667 0.65666667 0.66666667 0.65666667 0.02666667 0.65666667Pd0.16666667 0.22166667 0.16666667 0.12166667 0.08666667 0.22166667R0.038240.06937053 0.05205002 0.03472951 0.09716102 0.07984051node Enode Fn

13、ode Gnode Hnode I copyright THC 2003. All rights reserved by Thomas Ho Company, New York, NY. USA www. Hull, J. and A. White, 1994, Numerical Procedures for Implementing Term Structure Models II: Two-Factor Models, Journal of Derivatives, 37-48, Winter.Hull, J. and A. White, 1996, Using Hull-White I

14、nterest Rate Tree, Journal of Derivatives, 26-36, Spring.Ch. 5 of the The Oxford Guide to Financial Modeling by Thomas S.Y. Ho and Sang Bin Lee, 2003, Oxford University Press ( )drtar dtdZqs=-+GHI0-1-20.16666667 0.22166667 0.086666670.66666667 0.65666667 0.026666670.16666667 0.12166667 0.886666670-0

15、.0173205-0.034641GHI0-1-20.16666667 0.22166667 0.086666670.66666667 0.65666667 0.026666670.16666667 0.12166667 0.88666667Determine Jmax and Jmin where the interest process switches from branching (a) to (c) and from branching (a) to (b), respectivelyCalculate the probabilities of the trinomial process at each node depending on where the node is located and the rate R at each nodeQ(0,0),Q(1,1),Q(1,0),Q(1,-1)is the pri