Setting Boundary Conditions that You Don’t Know

Assume that you are somehow involved in the aerodynamic imbalance estimates for wind turbines which begin to vibrate if not properly balanced. For doing a wind simulation of the rotors (comparable to a wind tunnel simulation for a car), you must restrict your calculation domain to a region in the neighborhood of the wind turbine. The air pressure and the wind speed at the boundary of your calculation domain are then set more or less arbitrary, and dangerous cases have to be found out systematically or by educated guesses. (The dangerous cases have to be un-risked. Sorry for that one.) By the way: Read Chapter 5 of the Workout to learn about upwinding, flowlines and appropriate discretisations of convection-dominated flows (and also why this is relevant for your finance application).

We study the influence of boundary conditions for a one-factor Hull-White model, here, for the sake of simplicity, a Vasicek model. The short rate r satisfies the stochastic differential equation

with the usual notation. The parameter b, the so-called reversion speed, pulls back the short rate to the equilibrium. This means that short rates far away from the equilibrium are very unlikely and that possibly wrong boundary conditions should not play a major role as long as these artificial boundaries are far away from the equilibrium. Can we confirm this argument?

We valuate a zero coupon bond that matures in 10 years from today. The parameters of the Vasicek model are chosen as b = 0.1, sigma = 0.01, and a = 0 meaning that negative interest rates can and will occur. We set artificial (Dirichlet) boundary conditions for the value of the bond at r = - 0.2 and at r = 0.2  and assume these boundary values to be 0, 2, 4 or 6 times the notional value. To solve the Vasicek differential equation, we use implicit time-stepping and appropriate upwinding to treat the convection correctly. Here are the present values of the bond  (for the different boundary values) as function of today’s short rate.

We observe that the boundary layers (the region influenced by the boundary conditions) are very thin. Of course, the calculation domain has to be chosen carefully: It should not be too small for accuracy reasons nor too big for performance reasons.

Read more about boundary conditions in Chapter 6 of the Workout