# Hull-White 2-factor Model: 2) Zero Coupon Bond

This post derives the expression of zero coupon bond price of Hull-White 2-factor model.

Earlier posts on Hull-White 2-factor model

Let P(t,T) denotes the price of zero-coupon bond with maturity T at time t. Assuming Ft as the information generated by x(t) and y(t) up to time tP(t,T) have the following form.

To solve for P(t,T), we need to know the implementable expression for  except for φ(u) because φ(u) is not stochastic but deterministic process. Integrating x(u)+y(u) from time t to T, we can get the following result.

Here,  and . The derivation above is of the same logic for the case of Hull-White 1-factor model.

V(t,T) is also given using Itô isometry like Hull-White 1-factor model.

Similar to HW 1-factor model, we can find that  follows the normal distribution of a mean x(t)B1(t,T)+y(t)B2(t,T) and a variance V(t,T).

Using the fact that  with a normally distributed random variable Y which has a mean μ and a variance σ2, the price of zero-coupon bond becomes

It is argued that Hull-White model is consistent to the no-arbitrage assumption (perfect fit) if market discount factor P(0,T) satisfies the following condition.

Using the above relationship,

As the above expression for function φ(.) holds, the price of zero-coupon bond has the following form.

Substituting V(t,T) into the equation, the price of zero coupon bond P(t,T) is reformulated as

From this post, we know that most of derivation regarding HW 2-factor model is similar to the HW 1-factor model.

For additional insight on this topic visit the SH Fintech Modeling Blog.

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