# Seller Swap Model
Seller Swap Valuation
One party sells mortgage pools on its balance sheet and pays the bond interest by entering into a pay-fixed swap with CHT, and receives the interest from MBS pool sold to CHT. This is a seller swap. The fixed leg is semi annual, and the float leg, MBS coupons, is monthly. In addition, the MBS sold to the trust generates principal cash flows. CHT buys new-pooled mortgages from the party with this principal flows every month until the maturity of the swap.
CHT sold bullet bonds of total notional *N* that pays *C*% semi annual coupon for 60 months to investors for *P* dollars. Using these proceedings, CHT bought a mortgage pool, whose dollar price was *P*, from the party and entered into a “seller swap” with the party. CHT will buy new mortgage pools from the principal payments generated from the mortgage pools that already bought from the party
The party agreed to pay CHT the coupon of the bond, dollars every six months for 5 years. On the floating side, CHT agreed to pay, monthly, all the coupons generated by the mortgage pools it bought from the party for 60 months.
Let  denotes the first day of month *i,* for  assuming  is the inception of the swap and  denotes the coupon the party receive at time  for  Furthermore, let  and denote the time  dollar price and unit price of one dollar of notional, respectively of the mortgage pool, sold to CHT at time  Moreover, assume the valuation date *t* belongs to , where *m* is an integer between 1 and 60. Then
The fair value of the Seller-Swap 
where  is time *t* price of a zero coupon bond that matures at time  and  is the price of one dollar notional of the bond at inception.
The time *t* price of the fixed leg, which includes the notional at maturity, is time *t* price of the bond less present value of the notional, *N*.
We show that the time *t* value of the float- leg is equal to the sum of time *t* dollar prices of the mortgages the party sold to CHT less expected discounted value of mortgages at the maturity of the swap.
Let  and  denotes the time  price and notional of all the mortgages the party sold to CHT up to and including time  for  respectively.
At inception, , the party sold one MBS,  to CHT.
Evolution of 
time  dollar Price of : measurable at time 
Coupon for the first month:  measurable at time 
Then  and 
Principal payment at time :  measurable at time 
time dollar Price of :  measurable at time 
Coupon for the second month:  measurable at time 
At time  the party received  and sold another MBS, to CHT. i.e.,
where is the time  price of  Additionally, we note that
 and 
where is the time price of  per one dollar of notional. Table 1 describes the evolution of  and  during the second month.
## Table 1
 
time  price  
Coupon for month 2  
Prepay at time   
time  price  
Here, we observe that the first two rows, *time*  *price* and *Coupon for month 2*, are measurable at time , and the last two rows, *Prepay at time* and *time*  *price* respectively, are measurable at time . The Coupon  the party received at time  in dollars is given by
At time , i.e., beginning of the third month, the party received  amount of cash and sold another MBS,  to CHT. We note that
 and 
where  is the time  price of  and  is the time  price of  per one dollar of notional.
Table 2 describes the evolution of   and  during the third month.
Table 2
   
Price at time    
Coupon for month 3   
Prepay at time    
Price at time    
Here, we note that the dollar price of at time , is equal to  and  the Coupon the party received at time  is given by
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