I am trying to understand the relationship between two methods of pricing callable bonds in the risk-neutral pricing framework.

Problem statement
-----------------

Let's consider zero-coupon bond with face value 1, expiring in 1 year and callable at 6 months with strike $K =0.4$. I am going to work in the Ho-Lee model in which the spot rate follows the generalized Brownian motion
$$ dr(t) = \theta(t) dt + \sigma d\hat{w}(t). $$
For the purposes of obtaining a numerical result, I will assume $\sigma = 0.16$ and $r(0) = 0.3$ and choose the fitting function to be $\theta(t) = 0$ (unrealistic values in practice).

I can show numerically that the price obtained from the finite-difference method (Method 2) converges to the analytical price (Method 1). **I would like to obtain the analytical result directly from the PDE in Method 2**.

Method 1
--------

The zero-coupon discount curve is given by
$ P(t,T) = e^{-r(t) (T-t) + \frac{\sigma^2}{6}(T-t)^3} $.
Applying Ito's lemma we obtain
$$ dP(t,T) = P(t,T) r(t) dt - P(t,T)(T-t)\sigma d\hat{w}(t). $$
It follows that a European call option with expiration $T\_i$ on the ZCB with maturity $T\_j>T\_i$ can be priced using the Black pricing formula using a time-depedent volatility,
\begin{align}
 \mathbf{CZCB}(t,T\_i,T\_j,K)
 & = P(t,T\_i)\mathbb{E}^{Q\_i}\_t\left[\max(G(T\_i)-K,0)\right] \\
 & = P(t,T\_i)\left[G(t)N(y+\bar{\sigma}\sqrt{T\_i-t})-KN(y)\right]
\end{align}
where
\begin{equation}
G(t) = \frac{P(t,T\_j)}{P(t,T\_i)}, \quad \quad y = \frac{\log\left(\frac{G(0)}{K}\right)-\frac{1}{2}\bar{\sigma}^2(T\_i-t)}{\bar{\sigma}\sqrt{T\_i-t}}, \quad \quad
 \bar{\sigma} =\sigma(T\_j-T\_i).
\end{equation}
Thinking of the callable bond as a straight bond minus the call option we obtain
\begin{equation}
 P(0,T\_e)-\mathbf{CZCB}(0,T\_c,T\_e,K) = \$ \, 0.344467,
\end{equation}
where I have set $T\_e = 1$, $T\_c = 0.5$, and $K = 0.4$

Method 2
--------

Thinking of the callable bond as a spot-rate contingent claim we see that the bond price satisfies the PDE,
\begin{equation}
 \frac{\partial g}{\partial t} + \frac{1}{2}\sigma^2 \frac{\partial^2 g}{\partial r^2} - rg(r,t) = 0.
\end{equation}
In a fully explicitly discretization with $t\_n = n \Delta t$ ($0 \leq n \leq N$) we have
\begin{equation}
 g\_n^j = h\_n^j , \quad \quad h\_n^j = \frac{1}{1+r\_j\Delta t} \bigl[p g\_{n+1}^{j+1}+(1-2p) g\_{n+1}^{j} + p g\_{n+1}^{j-1}\bigr], \quad \quad p = \frac{\sigma^2\Delta t}{2\Delta r^2}.
\end{equation}
with the following boundary conditions in the $t$-direction (assuming $N$ is even)
\begin{align}
g^j\_N & = 1 \\
g^j\_{N/2+1} & = \max(K,h^j\_{N/2+1})
\end{align}
where the second condition accounts for the possibility of exercise at the 6 month mark.