Question

Swap Valuation

1) Derive a single valuation formular for a receiver swap valid all over the contract lifetime, that is from time t to time Tn, both include

2) Modify formula derived at the previous step to allow for disentangled resetting and payment dates.

Answer 1

1). A receiver swap is a type of plain vanilla interest rate swap where the swap trader is paying floating interest and receiving fixed interest on a certain notional amount. In other words, we can say that he has short position in floating coupon paying bond and long position in fixed coupon paying bond of same par value. And we assume that the notioanl principal is both received and paind at the end of the swap. by doing so, we find that value of a receiver swap is just the difference between values of fixed and floating bonds.

So, in this case, the value of the swap would be;

V_swap = B_fix - B_fl ..................................................................................................(1)

Where B_fix is derived by discounting back all the future cash flows to the present value. And, to value the floating rate bond, we note that the bond is worth the notional principal immediately after a coupon payment is made becaue at this time the bond is a "fair deal" where the borrower pays LIBOR for each subsequent period.

Suppose notional principal is L, floating rate (LIBOR) is r. Let's say the agreement to enter the swap deal was signed on time t and the first payment was exchanged at t₁ where the floating coupon of amount k is paid and fixed coupon c₁ is received. As discussed earlier, at time t₁ , floating rate bond will be at par, means the value of the floating rate bond at t₁ will be (L+k). And its present value is

...................................................................................(2)

(where r₁ is the LIBOR for time t₁ prevailing in the market at time t)

Value of fixed rate bond is calculated by discounting back all the future coupon payments as well as principal amount.

.......................................................................(3)

Where R is fixed interest rate

Putting the values from (2) and (3) in equation (1), we get, Value of a receiver swap =