Calibrating the Hull–White Mean-Reversion Function to the Initial Curve

Summary

The document gives the time-dependent drift function used in the Hull–White short-rate model to fit an initial discount curve. It expresses the drift in terms of the curve’s instantaneous forward rate, its time derivative, and the model’s mean-reversion and volatility parameters. The stated short-rate convention is mean reversion toward that time-varying function.

The discussion is limited to the formula for matching the initial term structure. Although the question also asks whether LIBOR and OIS curves must be simulated jointly for swaption pricing and whether parameters from a Hull–White tree can be reused in Monte Carlo, the accepted answer does not address those points. It also does not discuss numerical estimation of the forward-rate derivative, parameter calibration, or model limitations.

Key ideas

• The Hull–White drift function is chosen to reproduce the initial discount curve.
• Its expression depends on the instantaneous forward rate and its time derivative.
• The formula also depends on mean reversion and volatility under the stated short-rate convention.
• The answer does not resolve multi-curve simulation or parameter transfer from a tree model.

Tags

# Hull-White Monte Carlo simulation - mean reversion function


# Hull-White Monte Carlo simulation - mean reversion function












Quite new to implementing Hull white model in Monte Carlo simulation, hope to get help for 1. how to get the function $\theta$ in the following formula (the function used to match initial term structure)? 2.In pricing swaptions, floating is libor fwd and discounting is OIS, does this mean two curves need to be simulated jointly? 3. is it ok to get the $\theta, \alpha$ and $\sigma$ from HWTree and use in MC simulation? thanks

## Answer by rvignolo (score 4, accepted)

https://quant.stackexchange.com/a/58692

Given a initial discount bond $P^M(0, T)$ curve, the expression for $\theta(t)$ in the Hull White Short Rate model is a know result given by:

$$ \theta(t) = \frac{1}{\kappa} \cdot f'(0, t) + f(0, t) + \frac{1}{2} \cdot \left( \frac{\sigma}{\kappa} \right)^2 \cdot \left( 1 - e^{-2 \kappa t} \right). $$

I have used a notation where the spot rate dynamics is given by:

$$ dr(t) = \kappa \cdot (\theta(t) - r(t)) \cdot dt + \sigma \cdot dW(t). $$

Note that $f(t)$ is the instantaneous forward rate, given by:

$$ f(t, T) = - \frac{\partial}{\partial T} \ln \left( P(t, T) \right). $$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.