Hull–White Calibration and Bond Pricing

Article Quant Q&A · Author: Shivam Anand

Summary

The document collects questions about calibrating one- and two-factor Hull–White models to cap prices, calculating instantaneous forward rates for the drift function, and obtaining short-rate paths after calibration. The replies distinguish the model’s short rate from an initial market rate used to construct discount factors. They also point to solving the model’s stochastic differential equation to generate short-rate paths.

For zero-coupon bond valuation, the answer explains that taking an expectation of the discounted short-rate integral is less practical than using the Hull–White model’s analytic bond-price representation in terms of its A and B functions. It further notes that, in a discrete simulation setup, the analytic coefficients should match the simulation scheme; an exact Euler step is cited as a way to avoid time-discretization error. The replies provide references rather than a full calibration recipe, and they do not work through the original questions about forward-rate derivatives or parameter estimation.

Key ideas

Hull–White short rates are generated by solving the model’s stochastic differential equation.
The model’s analytic zero-coupon bond formula uses A and B functions.
Monte Carlo bond pricing through the short-rate integral involves an expectation over simulated paths.
The analytic bond coefficients should be consistent with the simulation scheme being used.
The answers reference an exact Euler scheme but do not provide a complete calibration walkthrough.

Tags

Fixed income Derivatives pricing Risk management Backtesting

Full text

# Hull White help needed


# Hull White help needed












I've been trying to calibrate Hull-White 1 Factor & 2 Factor model using Caps but I've some major doubts about my methodology, and would really appreciate some help.

I am using these formulas

For getting the instantaneous forward rates needed in theta formula i used the central difference method for getting the derivatives at the discrete time intervals t

(from excel data.....assumed P(t,T)= math.exp(r*t))

1) In the A formula for P(t,T) which formula should we use? e^-rt or the Ae^-B*r(t)...If I have to use the latter one, what should I take r? the same r value i used initially for calculating DFs? or the r(t) analytical formula after the Hull White calibration? (I calibrated a,sigma value using sum squared min error with implied caplet values from formula given above and market values that i had)

2) After the calibration step, how do i get the r(t) values? i have to use monte carlo simulation

Thank you once again for your time, I am completely new to this field & would appreciate a helping hand. or the

## Answer by Christian Fries (score 3)

https://quant.stackexchange.com/a/42276

On the Monte-Carlo Simulation of the Hull-White Model: You can find the specification of the Euler Scheme simulation in https://ssrn.com/abstract=2737091 . The paper gives the exact Euler step, i.e. the simulation step does not have a simulation time discretisation error. An implementation of this in Java is available as part of http://finmath.net/finmath-lib/, see also https://github.com/finmath/finmath-lib

The code is (currently) here: https://github.com/finmath/finmath-lib/blob/master/src/main/java/net/finmath/montecarlo/interestrate/models/HullWhiteModel.java

API Documentation here: http://finmath.net/finmath-lib/apidocs/net/finmath/montecarlo/interestrate/models/HullWhiteModel.html

On the formula of the zero bond: The representation of the zero bond with r involves an integral and even worse, an expectation ($P(t,T) = E( exp(-\int_t^T r(\tau) d\tau )$). The core advantage of the HW Model is that you have an analytic formula for P in terms of A and B. So you would use that formula. Note that the paper mentioned above also gives the correct values for A and B under the simulation scheme. Its an advantage to have the correct analytic formula under the simulated model instead of using the formula derived from the continuous time SDE, but being in a discretised setup.

## Answer by numerairX (score 2)

https://quant.stackexchange.com/a/42217

1) the latter one. r is modeled short rate (specific to hull white 1 & 2).

2) r is calculated by solving the SDE of the above mentioned model. You can refer to this document for detailed analytical solutions.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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

Hull–White Calibration, Bond Pricing, and Short-Rate Simulation

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