Checking Hull–White Simulated Discount Factors Against Market Curves

Summary

The document addresses an apparent mismatch between simulated discount factors in a Hull–White one-factor Monte Carlo model and the market discount curve at the initial time. Its central point is that the model’s no-arbitrage setup is intended to reproduce the market discount factor in expectation. A finite simulation may show small differences because of numerical and sampling error, so individual paths need not each match the market curve.

The suggested diagnostic is to compare the average of simulated discount factors with the market discount factor across maturities or times. The answer reports that a plotted comparison shows the two curves nearly coinciding, with a near-zero difference. It also supplies the model’s bond-pricing relationship, which adjusts the initial market curve using the evolving state variable and variance terms. The discussion is brief and offers no details about simulation size, discretization, calibration, or the size of the observed errors; the comparison is a practical check, not a complete implementation guide.

Key ideas

• Hull–White discount factors are designed to match the market curve in expectation under the no-arbitrage model.
• Finite Monte Carlo samples can differ slightly from theoretical values because of numerical and sampling error.
• Compare the mean simulated discount factor with the corresponding market discount factor as a diagnostic.
• The bond-pricing expression incorporates the initial curve, the model state, and variance adjustments.

Tags

# Current discount rate of Hull White One-Factor Monte Carlo Simulation


# Current discount rate of Hull White One-Factor Monte Carlo Simulation












I have a question about the Hull-White One-Factor Monte Carlo Simulation. As we know under the Hull-White One-Factor Model, the short rate follows a random process. So basically, every simulation path is different than each other, which can generate different discount factor value for today. As a result, when I generate the evolution of discount factors(i.e. 1 year bond) over time, it cannot converge at time 0.

How can I solve this problem to make each path' today discount rate converge. Thank you so much!

## Answer by sh lee (score 3)

https://quant.stackexchange.com/a/54078

The average of simulated discount factors from the Hull-White model and market discount factor are the same in theory but very similar in the simulation due to numerical error.

I draw one figure which compares two discount factors and shows their difference.

- red line : mean of simulated discount factors

- blue line : market discount factor

- green line : difference of two discount factor

You can find two discount factors are nearly same and their difference is nearly zero.

I think it is useful to check the process for applying the no-arbitrage condition as follows.

$$P(t,T) = \frac{P(0,T)}{P(0,t)} \exp \left( -x(t)B(t,T) + \frac{1}{2}\{V(t,T)-V(0,T)+V(0,t)\} \right)$$

I hope this helps.

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