Hull & White Trinomial Trees Arvid Kjellberg- Jakub Lawik - Juan Mojica - Xiaodong Xu The outline of our Project by Journal of Derivatives in Fall 1994 Where to use it? if there is a function x = f(r) of the short rate r that follows a mean reverting arithmetic process Our project: Hull and White trinomial tree building procedure Excel Implementation
Theoretical background Short Rate (or instantaneous rate) The interest rate charged (usually in some particular market) for short term loans. Bonds, option & derivative prices can depend only on the process followed by r (in risk neutral world) t - t+t t t investor earn on average r(t) t t Payoff: E e r (T t ) f t
And we define the price at time t of zero-coupon bond that pays off $1at time T by: Theoretical background his equation enables the term structure of interest rates at any given time to be obtained from the value of r at that time and risk-neutral process for Short rate And we define the price at time t of zero-coupon bond that pays off $1at time T by: P t , T E e r (T t )
If R (t,T) is the continuously compounded interest rate at time t for a term of T-t: P t , T e R (t ,T )(T t ) Combine these formulas above: R t , T 1 ln E e r ( T t ) T t This equation enables the term structure of interest rates at any given time to be obtained from the value of r at that time and risk-neutral process for r.
Vasicek model How is related to the Hull/White model? was further extended in the Hull-White model Asumes short rate is normal distributed Mean reverting process (under Q) Drift in interest rate will disappear if r = = b/a
a : how fast the short rate will reach the long-term mean value b: the long run equilibrium value towards which the interest rate reverts Term structure can be determined as a function of r(t) once a, b and are chosen. Ho-Lee model How is related to the Hull/White model? Ho-Lee model is a particular case of Hull & White model with a=0 Assumes a normally distributed short-term rate
SR drift depends on time makes arbitrage-free with respect to observed prices Does not incorporate mean reversion Short rate dynamics: (instantaneous SD) constant (t) defines the average direction that r moves at time t Ho-Lee model Market price of risk proves to be irrelevant
when pricing IR derivatives Average direction of the short rate will be moving in the future is almost equal to the slope of instantaneous forward curve Hull-White One-factor model No-arbitrage yield curve model Parameters are consistent with bond prices implied in the zero coupon yield curve In absence of default risk, bond price must pull towards par as it approaches maturity. Assumes SR is normally distributed & subject to mean reversion
MR ensures consistency with empirical observation: long rates are less volatile than short rates. HWM generalized by Vasicek (t) deterministic function of time which calibrated against the theoretical bond price V(t) Brownian motion under the risk-neutral measure a speed of mean-reversion Volatility (estimation and structure) Input parameters for HWM a : relative volatility of LR and SR
: volatility of the short rate Not directly provided by the market (inferred from data of IR derivatives) Trinomial tree example Call option, two step, t=1, strike price =0.40. Our t=1, strike price =0.40. Our account amount $100. Probabilities: 0.25,0.5 & 0.25 0.00% 0.00% E 4.40%(4)
0.00% B 3.81%(0.963) F 3.88% A 3.23%(0.233) C 3.29% G 3.36% 0.00% D 2.76% H 2.83%
0.00% 0.00% I 2.31% 0.00% 0.00% 0.00% Payoff at the end of second time step: max[100( R 0.40), 0] Rollback precedure as: (pro1*valu1+...+pro3*valu3)e
-rt=1, strike price =0.40. Our t Trinomial tree example Alternative branching possibilities The pattern upward is useful for incorporating mean reversion when interest rates are very low and Downward is for interest rates are very high. How to build a tree? HWM First for instantaneous rate r:
Step assumptions: all time steps are equal in size t=1, strike price =0.40. Our t rate of t=1, strike price =0.40. Our t,(R) follows the same procedure: New variable called R* (initial value 0) How to build a tree? : spacing between interest rates on the tree for error minimization.
Define branching techniques Upwards a << 0 Downwards a >> 0 Normal a=0 Define probabilities(depends on branching) probabilities are positive as long as: Straight / Normal Branching How to build a tree? With initial parameters: = 0.01, a = 0.1,t=1, strike price =0.40. Our t = 1 t=1, strike price =0.40. Our R=0.0173,jmax=2,we get:
How to build a tree? Second step convert R* into R tree by displacing the nodes on the R*-tree t R t R * t Define i as (it=1, strike price =0.40. Our t),Qi,j as the present value of a security that payoff $1 if node (I,j) is reached and 0.Otherwise,forward induction With continuously compounded zero rates in the first Maturity Rate(%) stage 0.5 3.43
Q0,0 is 1 0=3.824% 1.0 1.5 2.0 2.5 3.0 3.824 4.183 4.512 4.812 5.086 0 right price for a zero-coupon bond maturing at time t=1, strike price =0.40. Our t Q1,1=probability *e-rt=1, strike price =0.40. Our t=0.1604 Q1,0=0.6417 and Q1,-1=0.1604.
How to build a tree? Q1,1e ( 10.1732) Q1,0e 1 Q1. 1e ( 10.1732) Bond price(initial structure) = e-0.04512x2=0.913 0.1604e ( 10.1732) 0.6417 e 1 0.1604e ( 10.1732) 0.9137 Solving for alfa1= 0.05205 It means that the central node at time t=1, strike price =0.40. Our t in the tree for R corresponds to an interest rate of 5.205% Using the same method, we get:
Q2,2=0.0182,Q2,0=0.4736,Q2,-1=0.2033 and Q2,-2=0.0189. Calculate on 2,Q3,js will be found as well. We can then find 3 and so How to build a tree? Finally we get: Model presentation (excel) Option valuation issues
Underlying interest rate Payoff date American options Model presentation (excel) 5,4 % 5,0 % 4,0 % 3,7
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