Deriving the Hull–White Calibration Condition from Bond Prices

Summary

The document asks how to derive a maturity derivative used to show when a Hull–White short-rate model is calibrated to the market instantaneous forward rate. It starts from the model’s zero-coupon bond price, which contains an integral involving the function B(u,T), and questions a step that differentiates an expression whose upper limit and integrand both depend on maturity T.

The proposed resolution is to expand B(u,T) into its constant and exponential components, then isolate the maturity-dependent exponential factor before differentiating the integral. The author says this makes the disputed identity straightforward. A further second-derivative expression for the log bond price is also stated, involving the endpoint value b_T and an integral term. The post gives no full derivation, parameter definitions, or numerical example, and its explanation points to an external answer rather than reproducing the algebra. Readers should therefore treat it as a hint about how to rearrange the integral, not a complete proof.

Key ideas

• The question concerns differentiating a Hull–White bond-price integral with respect to maturity.
• The integration limit and B(u,T) both depend on maturity, so the derivative requires care.
• The suggested step is to separate the constant and exponential parts of B(u,T) before differentiating.
• The post states a second maturity derivative of the log bond price but does not show its full derivation.

Tags

# Proof of the Hull & White Model calibration


# Proof of the Hull & White Model calibration












I have a question about the demonstration of the formula which states that: If we have an Hull & White Model for the short rate diffusion such that

Then the model is fully calibrated if and only if:

Where

f^M is the market instantaious forward rate.

Now to demonstrate this formula I managed to arrive at the fact that the Zero Coupon bond price under the Hull and White Model for maturity T is given by

Where:

And :

Now we only need to derive the bond prices in order to get the instantanious forward rate, so we get to the following:

My question is about the derivation of

Which is somehow equal to

How can we proove this? It seems to me that this derivation is incorrect because of the fact that the T variable is present both in the integral limits and in B(u,T).

Thanks in advance

## Answer by Xman (score 1, accepted)

https://quant.stackexchange.com/a/45950

I found the answer to my question!

It consists of separating the terms $1$ and $e^{-a(T-u)}$ from $B(u,T)$ and isolating the $e^{aT}$ term from the integral and it's straight forward then!

All in all it's the form of $B(u,T)$ that makes it possible to state such a formula.

PS: In order to continue the demonstration, we need to derive $-\frac{\partial^2}{\partial^2 T}\ln(P(t,T)) = \frac{\partial^2}{\partial^2 T}(\int_t^T b_u.B(u,T) du ) = b_T - a.\int_t^T b_u.e^{-a(T-u)} du $ ...

This is also a straight forward formula...

Thanks all!

Regads

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.