HullWhite Model
Contents

HullWhite Model

HullWhite Tree

Example: HullWhite Tree Calibration

Appendix: Interest Rate Derivative PDE
HullWhite Model
This section is adapted from Brigo and Mercurio (2006). As an extension of the Vasicek model, HullWhite model assumes that the short rate follows the meanreverting SDE:
where and are positive constants; and is timedependent function that will be used to fit the current zero curve.
To solve this SDE, first apply Ito lemma to
Then integrate both sides over [s,t],
In order to fit the term structure of interest rates, the timedependent must satisfy
where is the market observed instantaneous forward rate at time 0 for the maturity T.
Then can be rewritten as
where
Therefore, conditional on is normally distributed with mean and variance given respectively by
HW model is an affine term structure model where the continuouslycompounded spot rate is an affine function in the short rate, i.e.,
The zero coupon bond price is given by
where
We can also find the closeform formulas for zerocoupon bond options, caps/floors, and swaptions. See Brigo and Mercurio (2006) for detail.
HullWhite Trinomial Tree
To construct HW tree, it is helpful to decompose the short rate into the following format:
Where


(1) 


(2) 


(3) 
With this decomposition in hand, the tree construction task can be achieved in two steps. In the first step one constructs the trinomial tree for . Then in the second step, one shifts the tree by to bring it in line with the initial term structure.
Step One: Construct the Symmetric Trinomial Tree
From equations (2) and (3), it is known that
Denote the tree nodes by where the time index i ranges from 0 to N and the space index j ranges from some to some . Using the results in A.F (Brigo and Mercurio 2006), we have


(4) 
where .
Now, given the node , we need to locate its subsequent nodes and with the respective transition probabilities . This is done as follows. First find the space in the y direction as


(5) 
Then locate the level k by


(6) 
where round(x) function indicates the closest integer to the real number x. We then set


(7) 
At last the transition probabilities are chosen in a way to match the conditional mean and variance,


(8) 
where


(9) 
Step Two: Displace the Tree
The second step consists of displacing the tree nodes to obtain the corresponding tree for r. An easy way to do so is through equation (1), as in class HullWhite::FittingParameter, where the instantaneous forward rate is approximated by
This approach has to approximate continuouslycompounded rate with short rate r(0), therefore doesn’t fit exactly the zero curve.
The other way uses helps from ArrowDebrew prices. Denote be the displacement at time , and be the ArrowDebrew price of node . A state price security or an ArrowDebreu security is defined as a contract which pays off $1 in a particular state at a particular time and pays zero in all other states. Its price (net present value) is referred to as ArrowDebreu price.
The values of and are calculated recursively as follows.
1. Initialize
2. Find
3. With in hand, calculate
where is the probability of moving from node to node .
4. With in hand, find by solving
that leads to
5. Loop step 3 and 4 to discover and for further steps (i++).
6. Short rate on each node
Example: HullWhite Calibration
This section illustrates the HullWhite model calibration process with a real example. It calibrates the HullWhite tree to the LIBOR market on Monday, May 16, 2011. The settlement date is Wednesday, May 18, 2011.
HullWhite model has three parameters: , , and . In this example, the first two will be calibrated to LIBOR swaptions; and the third one will be calibrated to LIBOR spot curve. To begin with, and are initialized as
Another input is LIBOR spot curve, which is given by
The rates are continuously compounded. Conventions are Act/360 and Modified Following.
The calibration process is carried out in four steps:
Step 1. construct trinomial tree for process ;
Step 2. displace to obtain the tree for ;
Step 3. price swaptions on this tree;
Step 4. calibrate the tree to swaptions market.
Step One: Construct Trinomial Tree for Process x(t)
The outcome of this step is shown in the following figure. This trinomial tree of is symmetric.
Figure 1 – Trinomial Tree of x(t)
Now let’s walk through the steps to create this figure.
1. From equation (2), , or node (0,0) is 0.
2. Consider node (0,0). From equation (4)
3. Go through equations (3) – (5) to locate its descendants: nodes (1,1), (1,0) and (1,1).
4. Using (8) and (9) to get transition probabilities from to its descendants.
By now node (0,0) is finished.
5. Now consider node (1,1). Follow step 2 to 4,
and
as well as
6. Process similarly node (1,0) and node (1,1), then move on to the next tenor (6m). Then the figure will be created.
When the volatility is timedependent, this trinomial tree is recombining.
Step Two: Displace x(t) to Obtain the Tree for r(t)
We can iteratively find state prices and , then shift the symmetric tree from step one by to obtain the HullWhite tree. This has been explained in Section 2. Let’s follow the procedure introduced in that section.
1. Initialize
2. Find displacement
3. Move on to time step 1, compute the state prices for the three nodes:
4. The displacement for time step 2
5. Move on to time step 2, compute the state prices for the five nodes. For example, for the middle node (2,0), it has three incoming nodes: (1,1), (1,0), and (1,1), then
6. Given the five state prices on time step 2, the displacement of this time step is
The results are shown in the following table.
By displacing the symmetric trinomial tree in step one with corresponding , the HullWhite tree is constructed as in the following figure.
Note that the rates on node (1,1) and node (2,2) are negative. Hull White model can produce negative rates due to normal distribution.
Step Three: Price Swaptions on the Tree
Before we can proceed to swaption pricing, it needs to point out the way in calculating the discounting factor D(t,T) on the short rate tree. To calculate the discount factor, for example , we need to retrieve the rate at the beginning of the period, in this case 0.0026 on time . This contrasts with the general case when the term structure curve is used, where we usually retrieve the rate from the end of the period, in this case 0.0026 on time . This rule applies to other time steps as well.
Now we are ready to price on this tree a 3mx6m ATM European payer Swaption with notional $1,000 and ATM rate 0.007427. The swaption can be exercised only on , giving the owner the right to enter into the long position (pay fixed) of a 3mx6m forward starting swap. Therefore the value of this swaption on each of the three nodes on , or nodes (1,j), j= 1, 0, 1, is simply
So it comes down to price the underlying swap, which is priced by discounting its fixed leg cash flows and floating leg cash flows respectively, via the following formula,
First consider the fixed leg. It pays on the amount
where the year fraction according to 30/360 day count. The results are shown in the following table.
Now we explain how to discount one step backward, from time step 3 to from step 2. Later we will use the same logic to discount backward the floating leg.
Consider the node (2,2) for instance. It has three descendants: node (3,3), (3, 2), and (3, 1). Denote the value of node (3, j). In this particular case
To roll one step back,
We continue to deal with other four nodes in step 2 and then move on to step 1. This leads to Table 3.
For the floating leg, we can do it similarly by first identifying the cash flows and then discounting them. An alternative and quicker way is through equivalent cash flows.
Unlike fixed leg, floating leg has tenor of 3 months (see chapter LIBOR Rates). Therefore, it contains two cash flows. The first one resets at time and pays at time ; the second one resets at time and pays at time .
Look at the second cash flow. On node (2,2), the rate is fixed at
So the payment made at time is
which is equivalent to
on time after discounting. Thus by starting with via equivalent cash flows, it saves us one step of backward induction. The results are shown in the following table.
In Table 4, the two floating cash flows are treated independently, and then added up together. Cash flows start with equivalent cash flows. For example, second cash flow pays at time step 3. Its equivalent cash flow on node (2, 2) at time step 2 is
After figuring out the other four nodes at time step 2, they are discounted back to time step 1 via the same procedure as has been seen in the fixed leg part. Total value at time step 1 is the sum of the first and second cash flows.
Now we have treated both floating and fixed legs, it is ready to price the swap and the 3mx6m swaption. The pricing procedure is shown in the following table.
In Table 5, the fixed leg column and floating leg column are inherited from Table 3 and 4, respectively. Then a long swap position receives floating leg while pays fixed leg. The Swaption is only exercised when it is in the money, or underlying swap has positive value. The last column, NPV, is the product of the Swaption value column and state price column. Finally the NPV of swaption price is the sum of NPV column, or 0.739527.
This example is done in Excel. In comparison, the accompanying C++ code gives 0.739461.
Step Four: Calibrate the Tree to Swaptions Market
Step three calculates the 3mx6m swaption on the HW tree. It is known as the model price, which depends on the model parameters. In this case, it depends on the (initial) value of and .
Market calibrates the HullWhite model to swaption volatility cube by minimizing
where model price is calculated by following step three; and is the market price, obtained by plugging the volatility quotes into Black model. Pay attention to whether the volatilities are quoted as log vol or normal vol (see Chapter LIBOR Volatility).
The optimization can be achieved by iterations. In that case, a new HullWhite tree will be constructed for each iteration (when α and σ change).
Bloomberg Commands: SWPM, VCUB.
Appendix
This appendix derives PDE for Interest Rate Derivatives (IRDs) in short rate model. Let the short rate SDE be
An interest rate derivative (IRD) has payoff V at time T. Its value at time t is
Using Ito lemma
by defining
We construct a hedging portfolio with two instruments with two different maturities, and
In order to be riskfree
and noarbitrage
which shows that the market price of risk is independent of maturity T
Then
Substitute it into the drift equation we obtain the PDE
A faster way to get the PDE under riskneutral measure uses martingale property. Let the bank account numeraire be
We know that an instrument with payoff , is a martingale under riskneutral measure Q. Therefore,
whose drift term should be 0. It leads to,
Reference
[1] Brigo, D. and Mercurio, F (2006). Interest rate models: theory and practice: with smile, inflation, and credit. Springer Verlag.
[2] Daglish, T. Lattice methods for noarbitrage pricing of interest rate securities. The Journal of Derivatives. 2(18), pp. 7—19, 2010.