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# Hull-White model parameters

Hull White One Factor Model Parameters

To back out the parameters in the one factor **Hull White Model** , we use the approach adopted by some practitioners when calibrating the model to market prices that are not correlation sensitive.

We estimate first the mean reversion parameter from a time series historical one-year maturity spot rates (the most recent one year), then prefix the mean reversion parameter and calibrate only the short rate volatility parameter to current market prices of Options on Bund Futures (the market prices are updated daily on the Fixed Income Derivatives page at www.eurexchange.com).

To estimate the mean reversion parameter, we discretize the one factor Hull White model as follows:

- $ dr(t)=[\theta(t)-ar(t)]dt+\sigma dW(t),\! $

- $ r(t+1)-r(t)=[\theta(t)-ar(t)]dt+\epsilon (t+1),\! $

- $ r(t+1)=\theta(t)+(1-a)r(t)+\epsilon (t+1),\! $

- $ 1-a=\hat{\beta}=\frac{\rho \sigma (r_{t+1}) \sigma (r_{t}) }{\sigma^2 (r_{t})}=\hat{\rho},\! $

- $ a=1-\hat{\rho},\! $

The mean reversion paramer is the $ a $ value estimated from historical data. With this $ a $ parameter fixed, the short rate volatility $ \sigma $ parameter is obtained using the nonlinear least squares optimization method to minimize the squared relative price errors between model and market prices.

- $ \sum_{i=1...n}(\frac{modelPrice_i - marketPrice_i}{marketPrice_i})^2.\! $

We used the approach described above to calibrate to At-the-Money market prices of Options on Bund Futures with maturity March 10, 2011, and obtained the following set of Hull White parameters for three different valuation dates (EUR market):

- 28.10.2010: a= 0.016426, b= 0.013594, relativePriceErrorSq = 0.000088
- 05.11.2010: a= 0.019046, b= 0.014312, relativePriceErrorSq = 0.000004
- 16.11.2010: a= 0.016130, b= 0.012812, relativePriceErrorSq = 0.000113

The initial zero coupon rates (EUR market) used as inputs in ThetaScript implementations are respectively:

- 28.10.2010: R=[0.0086,0.0107,0.0128,0.0150,0.0174,0.0198,0.0220,0.0239,0.0254,0.0266]
- 05.11.2010: R=[0.0082,0.0096,0.0114,0.0134,0.0157,0.0181,0.0203,0.0223,0.0240,0.0254]
- 16.11.2010: R=[0.0082,0.0103,0.0125,0.0149,0.0174,0.0199,0.0222,0.0242,0.0259,0.0272]