Simulating Hull–White Rates and Zero-Coupon Bonds

Summary

The post concerns Monte Carlo pricing of a zero-coupon bond under the Hull–White short-rate model. Its central clarification is that the questioned volatility-related term is deterministic, because it is derived from the initial yield curve, so it does not need to be simulated as a random process. The answer also recommends working with an integrated form of that deterministic term, which requires only the first derivative of forward rates.

For bond pricing, the answer advises simulating the short rate jointly with its time integral as a joint Gaussian process, rather than simulating rates first and numerically integrating them afterward. It points readers to a standard reference for a fuller explanation. The post gives conceptual guidance but no equations, implementation details, numerical results, or discussion of discretization error; practical accuracy still depends on the simulation setup and yield-curve inputs.

Key ideas

• The Hull–White adjustment term described in the post is deterministic when the initial yield curve is fixed.
• Using its integrated form requires only the first derivative of forward rates.
• The short rate and its time integral can be modeled together as a joint Gaussian process for bond pricing.
• The answer points to a reference for detailed treatment but gives no implementation or error analysis.

Tags

# Hull white model Monte Carlo simulation Zero Coupon Bond


# Hull white model Monte Carlo simulation Zero Coupon Bond












I am trying to use Hull White Model to price a zero coupon bond by Monte Carlo Simulation. The basic idea is under this equation:

Under Hull White Model, I want to generate every short rate (r) and integrate them to get the price. Based on the HW model, the dr(t) process includes a v(t) term as follow. I don't quite understand how to simulate thee v(t) process under Monte Carlo, can anyone help?

## Answer by g g (score 3)

https://quant.stackexchange.com/a/44971

You do not model $v(t)$ by Monte-Carlo! As your excerpt explains $\phi(t)$ is a deterministic function of the initial yield curve and accordingly $v(t)$ is deterministic as well. Two further remarks: (i) You should not base the model on $v(t)$ but on an integrated $v(t)$, since this only involves the first derivative of the forward rates. (ii) You should not model $r(t)$ and then integrate to find $\int{r(t)}$ but model both at the same time as a joint Gaussian process.

All these issues are clearly (and in detail) explained in the standard reference Glasserman (Chapter 3.3 Gaussian short rate models).

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