Division of Applied Mathematics School of Education, Culture and Communication Mälardalen University Box 883, SE-721 23 Västerås, Sweden

Hull & White trinomial trees-An excel application

MMA 708, Analytical Finance IIJan Röman, Consulting senior lecturer

2008-12-06

Jakub Lawik Arvid Kjellberg

Xiaodong XuJuan Mojica

AbstractThe Hull & White tree-building procedure was first issued by Journal of Derivatives in Fall 1994. It is widely used by practitioners in the financial world. This procedure is appropriate for model which there is a function x = f(r) of the short rate r that follows a mean reverting arithmetic process. It also can be used to implement the Ho-Lee model, the Hull-White model, and so on. It is also a tool that can be used for developing a wide range of new models.

Our seminar report is about the Hull and White trinomial tree building procedure, we provide more details on the way in which Hull-White trinomial trees can be used in future interest rates model. Example of the implementation of the model using market data is also presented. We show how the model can be implemented by using excel.

Background

The short rate

The short rate, r, ate time t is the rate that applies to an infinitesimally short period of time at time t.It is sometimes referred to as the instantaneous short rate.Bond prices, option prices and other derivative prices depend only on the process followed by r in the risk-neutral world. We consider that the risk-neutral world in a very short time period between t and t+Δt, investors earn on average r(t) Δt. The payoff of fT at time T is:

Ê [e−r̂ (T−t ) f T ]

r̂ is the average value of r between t and T, and Ê denotes expected value for risk-neutral.And we define the price at time t of zero-coupon bond that pays off $1at time T by:

P ( t , T )=Ê [e−r̂ (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 )= −1T−t

ln Ê [e−r̂(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

dr=(b−ar ) dt+σ dV

Vasicek model assumes that the short rate is normal distributed and has a so-called “mean reverting process” (under Q). The drift in interest rate will disappear If we put r = θ = b/a. So this value represents the mean value of the short rate. And a is a measure of how fast the short rate will reach the long-term mean value. The Vasicek model was further extended in the Hull-White model.For the price at time t of a zero-coupon bond that pays $1 at time T:

P ( t , T )=A (t ,T )e−B (t ,T ) r(t )

In this equation, r(t) is the value of r at time t,

B (t , T )=1−e−a(T −t )

a

A ( t , T )=exp [ ( B (t ,T )−T+t )(a2b−σ2

2)

a2 −σ2 B (t ,T )2

4 a ]When a = 0, B (t,T) = T-t and A(t,T) = exp[σ2 (T-t)3/6]Finally, we get:

R (t , T )= −1T−t

lnA (t ,T )+ 1T−t

B (t , T )r ( t)

It shows that the entire term structure can be determined as a function of r(t) once a,b,and σ are chosen.

The Ho-Lee modelAs the first, one works on the arbitrage-free yield-based model. Ho-Lee assumes a normally distributed short-term rate. The short rate’s drift depends on time, so it makes arbitrage-free with respect to observed prices. It does not incorporate mean reversion and the short rate dynamics are represented by:

lnA ( t , T )=ln P (0 ,T )P(0 , t)

+(T−t ) F (0 , t )−12

σ2t (T−t )2

Where σ, the instantaneous standard deviation of the short rate,is constant and θ(t) is a function of time chosen to ensure that model fits the initial term structure. θ(t) defines the average direction that r moves at time t. This is independent for the level of r. Ho and Lee’s parameter that concerns the market price of risk proves to be irrelevant when the model is used to price interest rate derivatives.The variable θ(t) can be calculated by:

θ ( t )=F t(0 , t )+σ 2t

Which F(0,t) is the instantaneous forward rate for a maturity t.As an approximation, θ(t) equals Ft(0,t).It means that the average direction of the short rate will be moving in the future is almost equal to the slope of instantaneous forward curve

In the Ho-Lee model, the price of zero-coupon bond at time t in terms of short rate is:

P ( t , T )=A (t , T )e−r ( t )(T−t )

Which:

lnA ( t , T )=ln P (0 ,T )P(0 , t)

+(T−t ) F (0 , t )−12

σ2t (T−t )2

In this equation, time zero is today. Time t and T are general times in the future with T>=t. It defines the price of a zero-coupon bond at a future time t in terms of the short rate at time t and the prices of bonds today. The latter can be calculated from today’s term structure.

Hull-White One-factor model

The Hull & White model generalized by Vasicek model with time dependent parameter.

dr=(θ (t )−a (t )r ) dt+σ ( t ) dV (t)

In this formula, θ(t) is a deterministic function of time which calibrated against the theoretical bond price. The constants a and σ is calibrated against the volatility. Hull & White model can be characterized as the Ho-Lee model with mean reversion at rate and also as the Vasicek model with a time dependent reversion level. At time t, the short rate reverts to θ(t)/a at rate a, the Ho-Lee model is a particular case of Hull & White model with a=0.

V(t) is a standard Brownian motion under the risk-neutral measure, a>= 0 is the speed of mean-reversion, and σ > 0 is the volatility. The short rate is given by

The θ(t) function can be calculated by:

θ (t )=F t(0 , t)+ σ2 (1−e−2at )2 a

The last term in this equation is usually fairly small. If we ignore it, the equation implies that the drift of the process for r at time t is Ft(0,t)+a[F(0,t)-r].

Bond prices at time t in the Hull&White model are given by

P (t , T )=A (t ,T )e−B (t ,T ) r(t )

α 1=ln [0.1604 e−0.01732+0.6417+0.1604 e0.01732

0.9137 ]=0.5205

Where

B (t , T )=1−e−a(T −t )

a

And

lnA ( t , T )=ln P (0 ,T )P(0 , t )

+B(t ,T )F (0 ,t )− 14 a3 σ 2 ( e−aT−e−at )2(e2at−1)

These equations define the price of a zero-coupon bond at a future time t in terms of the short rate at time t and the price of bonds today.The latter can be calculated from today’s term structure.

Volatility parameter estimation and structureTwo of the input data necessary for the Hull-White model are volatility parameters a and σ where a gives the relative volatility of the long and short rates and σ is the absolute volatility of the short rate. Unlike the initial term structure these volatility parameters are not directly provided by the market.Therefore they have to be inferred from market data of interest rate derivatives.

The procedure of calibrating the Hull-White model to market prices is done by choosing a and σ according to:

Min∑i

¿¿

Where Pi is the market price of the ith interest rate derivative and Vi is the corresponding model price.

Trinomial trees

We are going to show one call option example for a two step trinomial interest tree with each time step equal to 1year in length that Δt=1.Our account amount for $100. Upper number at each node is rate and lower number is value of instrument. We assume that the up, middle and down probabilities are 0.25,0.5 and 0.25,respectively at each node.

0.00% 0.00% E 4.40%

0.00% B 3.81% F 3.88%

A 3.23% 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%

The tree is used to value a derivative that provides a payoff at the end of the second time step of

Where 0.40 is the strike price, R is the Δt-period rate and at the final nodes, the value of the derivative equals the payoff.

Node E :100(o.44-0.40)=4, Node (F,G,H,I)=0

Node B:[0.25x4+o.5x0+0.25x0]e -0.0381x1=0.963

Finally, Node A [0.25x0.963+0+0]e -0.0323=0.233

Sometimes, it is convenient to modify the standard branching pattern, which is used at all nodes above. Three alternative branching possibilities are shown below:

How to build a treeFollowing the method of Hull and White you will have a two step procedure to create your trinomial tree. We will show how this procedure can be applied to the Hull and White model of the instaneous short rate:

dr=[θ ( t )−ar ] dt+σdz

First step

In this first step we have to make a few assumptions. We firstly assume that all time steps are equal in size (∆ t ) and we also assume that the rate of ∆ t (R) follows the same equation as r above which would make it look like this:

dR=[θ ( t )−aR ]dt +σdz

To create the tree we start by defining a new variable called R* which has an initial value of 0 and is defined by:

dR ¿=−aR¿dt +σdz

We define ∆ R as the spacing between interest rates on the tree and we set:

∆ R=σ √3∆ t

We have to create a trinomial tree. And we do this by first of all defining which of the three branching techniques to use at each node,.(upwards, downwards and straight). We do this by firstly define (i,j) as the node where t = i∆ t and R* = j∆ R. The variable needs to be positvie whilst j can be either a positive or a negative integer. To determine what branching method to use at each node we need to make sure that all probabilities from that node is positive.

When the constant a is positive we use the downward branching for a sufficiently large j. And at the other extreme, for a sufficiently negative j we use the upward branching. In between we use the normal branching method. In the excelsheet we have jmax wich we define as the value of j when we change branching method from using the normal branching method to the downward branching method. As you could probably understand then jmin is defined as the value of j when we change from the normal branching method to the upward method.

Hull and White has proved that the probabilities are positive as long as:

jmax=min∫¿(¿0.184/ (a ∆ t))¿

And

jmin=− jmax

We need to define the probabilities for the different branches as well. The probabilities for the normal branching need to fulfil these equations

pu∗∆ R−pd ∆ R=−aj ∆ R ∆ t

pu ∆ R2+ pd ∆ R2=σ2 ∆ t+a2 j2 ∆ R2 ∆ t2

pu+ pm+ pd=1

By using

∆ R=σ √3 ∆ t

we can derive pu,pm and pd so it looks like this:

pd=16+ 1

2(a2 j2 ∆ t 2+aj ∆t )

pm=23−a2 j2 ∆ t2

pu=16+ 1

2(a2 j2 ∆ t 2−aj ∆ t )

If the branching is upward:

pd=76+ 1

2(a2 j2 ∆ t 2+3 aj ∆ t )

pm=−13

−a2 j2 ∆ t2−2aj ∆ t

pu=16+ 1

2(a2 j2 ∆ t 2+aj ∆ t )

and finally if branching downward:

pd=16+ 1

2(a2 j2 ∆ t 2−aj ∆ t)

pm=−13

−a2 j2 ∆ t2+2 aj ∆ t

pu=76+ 1

2(a2 j2 ∆ t 2−3 aj ∆ t )

As an illustration for the first part we can set σ = 0.01, a = 0.1 and Δt = 1. And we get:

Second step:

The second stage in the tree construction procedure is to convert the tree for R* into tree for R. In this case, we displacing the node on the R*-tree that the initial term structure of interest rates can be matched.

α (t )=R ( t )−R¿(t)

For example:

Suppose that the continuously compounded zero rates in the first stage,we get our table below:

Maturity

Rate(%)

0.5 3.431.0 3.8241.5 4.1832.0 4.5122.5 4.8123.0 5.086

The value of Q0,0 is 1 and the value of α0 is the right price for a zero-coupon bond maturing at time Δt. α0 is set equal to initial Δt-period interest rate. So α0=3.824% at Δt=1 definesthe position of the initial node on our R-tree. Then we can calculate Q 1,1,Q1,0,and Q1,-1 as well by using our table.

Q1,1=probability *e-rΔt=0.1604

Q1,0=0.6417 and Q1,-1=0.1604.

Since then, we are going to decide α1 at maturing time 2Δt.The price for this bond as seen at node B is e-( α1+0.01732) as ΔR=0.01732 when Δt=1. Similarly, the price at node C is e - α1and price at D is e-( α1+0.01732) as well, finally we get:

From the initial structure, the bond price is e-0.04512x2=0.913. Substituting for Q’s in equation above:

It means that the central node at time Δt in the tree for R corresponds to an interest rate of 5.205% shows in our figure.

We calculate Q2,2, Q2,1,Q2,0,Q2,-1 and Q2,-2,by using Q values determined previously. As an example for Q2,1,it is the value of a security which pays off $1 if node F is reached and zero, otherwise node F can be reached only from B and C which their interest rate are 6.937% and 5.205%. The value at node B of a security that pays $1 at node Fas 0.6566e -0.06937,the value of node C is 0.1667e -0.05205.So:

Using the same method, we get Q2,2=0.0182,Q2,0=0.4736,Q2,-1=0.2033 and Q2,-2=0.0189.

The next step is to calculate α2,Q3,j’s will be found as well. We can then find α3 and so on.

ConclusionThe Hull-White tree building procedure is a flexible approach to constructing trees for a wide range of different one-factor models of the term structure. The tree building is exactly consistent with the initial term structure. In our study, we have shown how the basic procedure presented and extended.

At the same time, it shows that negative interest rates can appear in our Hull & white model which is a shortcoming of the model.

List of references Lecture notes www.rotman.utoronto.ca/~amackay/fin/USE TREE 1a.pdf Options, Futures and other derivatives. 6th ed. Wikipedia