Financial Simulation Modeling in Excel

HULL-WHITE TRINOMIAL TREE

While the Hull-White interest rate model has a nice analytical solution in the form of equation 4.7, we still need an arbitrage-free method of pricing an option. Equation 4.7 can provide us with a price, but that requires knowledge about the instantaneous rate at maturity, something that we could never know a priori. The best that could be done in an arbitrage-free market is to guess at the most probable rate, given the current observation of the yield curve. To accomplish this, Hull and White developed a trinomial tree method to track the migration of the interest rate up to maturity and from there work backward to compute the expected price of the option. A trinomial tree is much like the binomial tree we discussed earlier with the exception that there is now a third option where there exists a possibility that the rate remains the same at the next time step.

Building a Hull-White tree involves two primary stages. The first is to determine the optimal step size and structure of the tree depending on the desired time interval. The second is to realign the mean rate throughout the tree to match the term structure observed from the bond market. To understand why step 2 is necessary we must first understand how step one is accomplished. The first step involves simplifying equation 4.6 into a form that is easier to solve. By setting the drift term, θ, and the initial rate to zero, we can rewrite equation 4.6 with a new variable, r*, as equation 4.13

The ...