Hull White Trinomial Trees Arvid Kjellberg Jakub Lawik

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

And we define the price at time t of zero-coupon bond that pays off $1 at 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 � Short rate And we define the price at time t of zero-coupon bond that pays off $1 at 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.

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) r = θ = b/a � 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.

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 � SR drift depends on time short-term rate makes arbitrage-free with respect to observed prices � Does not incorporate � Short rate dynamics: mean reversion � σ (instantaneous SD) constant � θ(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 θ(t) deterministic function of time which calibrated against 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 σ : volatility of the short rate �Not directly provided by the market (inferred from data of IR derivatives)

Trinomial tree example � Call option, two step, Δt=1, strike price =0. 40. Our account amount $100. Probabilities: 0. 25, 0. 5 & 0. 25 � Payoff 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% I 2. 31% 0. 00% at the end of second time step: � Rollback precedure as: (pro 1*valu 1+. . . +pro 3*valu 3)e-rΔt

Trinomial tree example �Alternative � The branching possibilities 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 Δt rate of Δt, (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: σ = 0. 01, a = 0. 1, Δt = 1 ΔR=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 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 Q 0, 0 is 1 α 0=3. 824% Maturity 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 Rate(%) 3. 43 3. 824 4. 183 4. 512 4. 812 5. 086 α 0 right price for a zero-coupon bond maturing at time Δt Q 1, 1=probability *e-rΔt=0. 1604 Q 1, 0=0. 6417 and Q 1, -1=0. 1604.

How to build a tree? � Bond price(initial structure) = e-0. 04512 x 2=0. 913 � Solving for alfa 1= 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: Q 2, 2=0. 0182, Q 2, 0=0. 4736, Q 2, -1=0. 2033 and Q 2, -2=0. 0189. � Calculate α 2, Q 3, j’s will be found as well. We can then find α 3 and so on…

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 % 3, 5 % 4, 0 % 3, 2 % 4, 6 % 4, 0 % 3, 0 %

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