How is the Hull-White model calibrated to match the observed yield curve?

Question

AcadiFi · Accepted Answer

Hull-White has dynamics dr = (theta(t) - a*r)dt + sigma*dW. The function theta(t) is chosen so that model zero-coupon bond prices match observed market prices exactly on the calibration date.

The calibration has two parts:

1) Fit theta(t) to today's curve. Given market forward rates f(0,t) and constants a, sigma, theta(t) is determined by theta(t) = df(0,t)/dt + a*f(0,t) + (sigma^2 / 2a)*(1 - exp(-2at)). This is a closed-form relationship — no optimization, just differentiation of the observed forward curve.

2) Calibrate a and sigma to option prices. With theta determined, the free parameters a (mean reversion speed) and sigma (volatility) are chosen to minimize squared pricing errors against liquid instruments — typically ATM swaptions or caps. This requires iterative optimization.

```mermaid
graph LR
    A[Market Curve] --> B[theta(t)]
    C[Swaption Vols] --> D[a, sigma]
    B --> E[Calibrated HW Model]
    D --> E
```

At a desk like Ironbridge Derivatives, calibration runs every morning before trading. The model then prices Bermudan swaptions and callable bonds using trees or Monte Carlo consistent with the benchmark surface.

Key insight: fitting the curve exactly eliminates static arbitrage; fitting vol surface controls dynamic model risk. Both are necessary for trading desks.

How is the Hull-White model calibrated to match the observed yield curve?

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