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Fixed Income and Interest-Rate Models / Short-Rate Models

The Hull-White Extended Vasicek Model: Calibrating to the Initial Curve

A derivatives desk pricing a 10-year Bermudan swaption needs a model that simultaneously (a) prices all vanilla bonds consistently with the observed swap curve and (b) generates realistic rate dynamics for the exercise boundary simulation. The Vasicek model fails (a): three parameters cannot fit 30+ market bond prices. The Ho-Lee model fails (b): zero mean reversion means the long end is as volatile as the short end, which distorts the Bermudan's exercise boundary. Hull and White (1990) resolve both problems by making the Vasicek long-run mean θ\theta a deterministic function of time: ϑ(t)\vartheta(t) is calibrated date by date to exactly fit the initial forward curve, while the mean reversion speed aa and volatility σ\sigma retain their Vasicek role of governing the dynamics.
Learning objectives. By the end of this note you should be able to:
  • write down the Hull-White SDE and identify the role of each parameter;
  • state the calibration formula for ϑ(t)\vartheta(t) in terms of the initial forward curve and model parameters;
  • decompose r(t)=x(t)+α(t)r(t) = x(t) + \alpha(t) and identify the deterministic shift α(t)\alpha(t);
  • state the Hull-White bond price formula P(t,T)=A(t,T)e−B(t,T)r(t)P(t,T) = A(t,T)e^{-B(t,T)r(t)} with B(t,T)B(t,T) expressed in closed form and A(t,T)A(t,T) expressed in terms of market prices;
  • price a European bond option using the Jamshidian-type formula;
  • articulate the key differences among Ho-Lee, Vasicek, and Hull-White.

In this lesson

    1. The Hull-White SDE
    1. Bond price formula
    1. European bond option pricing
    1. Hull-White in the trio: Ho-Lee, Vasicek, Hull-White
    1. Trinomial tree implementation

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