Hull & White Trinomial Trees

1. ArvidKjellberg- JakubLawik - Juan Mojica - Xiaodong Xu Hull & White Trinomial Trees

2. The outline of our Project • by Journal of Derivatives in Fall 1994 • Whereto use it?  ifthereis 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

3. 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+Δt investor earn on average r(t)Δt • Payoff:

4. And we define the price at time t of zero-coupon bond that pays off $1at time T by: Theoretical background 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. • Short rate • And we define the price at time t of zero-coupon bond that pays off $1at time T by: • If R (t,T) is the continuously compounded interest rate at time t for a term of T-t: • Combine these formulas above: • 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.

5. 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 • 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 σ are chosen. r = θ = b/a

6. 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: • σ (instantaneous SD)  constant • θ(t) defines the average direction that r moves at time t

7. 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

8. 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 • θ(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

9. Volatility (estimation and structure) • Input parameters for HWM • a : relative volatility of LR and SR • σ : volatility of the short rate • Not directly provided by the market (inferred from data of IR derivatives)

10. Trinomialtreeexample • Call option, two step, Δt=1, strike price =0.40. Our account amount $100. Probabilities: 0.25,0.5 & 0.25 • Payoff at the end of second time step: • Rollback precedure as: (pro1*valu1+...+pro3*valu3)e-rΔt

11. Trinomialtreeexample • 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.

12. Howtobuild a tree? • HWM for instantaneous rate r: • First Step assumptions: • all time steps are equal in size Δt • rate of Δt,(R) follows the same procedure: • New variable called R* (initial value 0)

13. 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

14. How to build a tree? • With initial parameters: σ = 0.01, a = 0.1,Δt = 1 ΔR=0.0173,jmax=2,we get:

15. How to build a tree? • Second step  convert R* into R tree by displacing the nodes on the R*-tree • Define αi as α(iΔt),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 stage • Q0,0 is 1 • α0=3.824% • α0 right price for a zero-coupon bond maturing at time Δt • Q1,1=probability *e-rΔt=0.1604 • Q1,0=0.6417 and Q1,-1=0.1604.

16. How to build a tree? • Bond price(initial structure) = e-0.04512x2=0.913 • Solving for alfa1= 0.05205 • It means that the central node at time Δt 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 α2,Q3,j’s will be found as well. We can then find α3 and so on…

17. How to build a tree? • Finally we get:

18. Modelpresentation (excel)

19. Option valuation issues • Underlying interest rate • Payoff date • American options

20. Modelpresentation (excel) 5,4% 5,0% 4,6% 4,0% 4,0% 3,7% 4,0% 3,5% 3,2% 3,0%

21. ThankYou!!!