I am currently studying the N$-$factor Merton model $(1973)$ within the Heath$-$Jarrow$-$Morton $($HJM$)$ framework and need to understand the full derivation process, not just the final result. I've noticed in past inquiries, I've received the end solution without the accompanying steps, which are crucial for my learning.

Specifically, I am looking for a comprehensive explanation of the following components:

The solution for the short rate

$$

$($

$$

$)$

r$($t$),$ where

$$

$($

$$

$)$

$=$

$$

$$

$=$

$1$

$$

$$

$$

$($

$$

$)$

r$($t$)=$

i$=1$

N

$$

x

i

$$

$($t$),$ and

$$

$$

$$

$($

$$

$)$

$=$

$$

$$

$$

$$

$+$

$$

$$

$$

$$

$$

$($

$$

$)$

dx

i

$$

$($t$)=\backslash$theta

i

$$

dt$+\backslash$sigma

i

$$

dW

i

$$

$($t$)$ for

$$

i in

$\{$

$1$

$,$

$2$

$,$

$...$

$,$

$$

$\}$

$\{1,2,...,$N$\},$ and an explanation of how the covariance matrix

$\backslash$Gamma

$\backslash$Gamma affects this solution.

How the zero$-$coupon bond price

$$

$($

$$

$,$

$$

$)$

P$($t$,$T$)$ is derived from the risk$-$neutral expectation of the discount factor.

The derivation of the instantaneous forward rate

$$

$($

$$

$,$

$$

$)$

f$($t$,$T$).$

The definition and roles of the volatility terms

$$

$$

$($

$$

$)$

$\backslash$xi

i

$$

$($t$)$ and

$$

$$

$($

$$

$)$

$\backslash$psi

i

$$

$($T$)$ and how

$$

$$

$($

$$

$,$

$$

$)$

$=$

$$

$$

$($

$$

$)$

$$

$$

$($

$$

$)$

$\backslash$sigma

i

$$

$($t$,$T$)=\backslash$xi

i

$$

$($t$)\backslash$psi

i

$$

$($T$)$ for

$$

i in

$\{$

$1$

$,$

$2$

$,$

$...$

$,$

$$

$\}$

$\{1,2,...,$N$\}$ factors into the model.

Could you please provide a detailed, step$-$by$-$step derivation, highlighting each part of the process? This is crucial for me to grasp the underlying principles and the mathematical rigor involved. It would be greatly beneficial if the explanation can address the intuition behind each step as well as the technical details..

Derive $N-$ factor Merton $(1973)$ model in the HJM framework. Therefore, derive:

the solution for $r(t)={\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i(t),$ where $d{x}_i(t)={}_{i}dt+{}_{i}d{W}_i(t),$ for iin $,$ dots, $N,$ and where $$ is

the $(NN)$ covariance matrix for the $N$ factor innovations with ${}_{i,j}dt=E(d{W}_i(t)d{W}_j(t))={}_i{}_j{}_{i,j}dt$

being the instantaneous covariance between innovations in factors i and $j$ and ${}_{i,j}$ is the correlation coefficient

between the $i\text{'}$th and $j\text{'}$th factors,

the zero $-$ coupon bond price $P(t,T),$

the instantaneous forward rate $f(t,T),$

the ${}_i(t)$ and ${}_i(T)$ terms such that ${}_i(t,T)={}_i(t){}_i(T)$ for the iin $,$ dots, $N$ factors. Derive the $N-$ factor Merton $(1973)$ model in the HJM framework. Therefore, derive: hhbx

the solution for $r(t)={\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i(t),$ where $d{x}_i(t)={}_{i}dt+{}_{i}d{W}_i(t),$ for iin$\{1,2,$dots,$N\},$ and where $$ is

the $(NN)$ covariance matrix for the $N$ factor innovations with ${}_{i,j}dt=E(d{W}_i(t)d{W}_j(t))={}_i{}_j{}_{i,j}dt$

being the instantaneous covariance between innovations in factors i and $j$ and ${}_{i,j}$ is the correlation coefficient

between the $i\text{'}$th and $j\text{'}$th factors,

the zero$-$coupon bond price $P(t,T),$

the instantaneous forward rate $f(t,T),$

the ${}_i(t)$ and ${}_i(T)$ terms such that ${}_i(t,T)={}_i(t){}_i(T)$ for the iin$\{1,2,$dots,$N\}$ factors.