A "payment$-$option ARM" is a type of residential mortgage loan introduced in the $1980$s$.$ It is a

type of adjustable$-$rate mortgage in which the borrower has the option of making one of several

different payments. In the years leading up to the $2008$ financial crisis these types of mortgages

grew in popularity due to the ability of borrowers to make a low minimum monthly payment.

For the purpose of this problem, there are three payment options:

Option $1$: fully$-$amortizing

Option $2$: interest$-$only $($non$-$amortizing$)$

Option $3$: payment option. The borrower can make a minimum monthly payment. This payment

may not necessarily amortize the balance, nor cover interest.

Assume a borrower is interested specifically in a loan with the following characteristics:

Loan type: $\frac{3}{27}$ hybrid ARM

Initial rate: $3\mathrm{.}5\%$ annual

Reference rate: $1-$month SOFR

Fully$-$indexed rate: $1-$month SOFR $+3\mathrm{.}5\%$

SOFR at origination: $2\mathrm{.}5\%$

Adjustment period: once per year $($every $12$ months$)$

Rate cap: adjusted rate can be no higher than fully$-$indexed rate at origination $+1\%$

Rate floor: adjusted rate can be no lower than fully$-$indexed rate at origination $-1\%$

Minimum monthly payment allowed under Option $3$: $$500$ per month

The borrower wants a loan for a property worth $$250,000.$ They will make a $4\%$ down payment.

You expect SOFR on the first interest rate adjustment $($reset$)$ date to be $4\%.$

Answer the following:

Compute the monthly payment during the first year in which the rate adjusts under

Option $1.$

Compute the monthly payment during the first year in which the rate adjusts under

Option $2.$

Compute the difference in UPB between Options $1$ and $2$ at the end of the first year

during which the rate adjusts.

Compute the UPB for Option $3$ at the end of the first year during which the rate adjusts.