Chapter 8 Swaps. Introduction to Swaps. A swap is a contract calling for an exchange of payments, on one or more dates, determined by the difference in two prices. A swap provides a means to hedge a stream of risky payments.
Spot price – Swap price – Spot price = – Swap price
Swap Payment Spot Purchase of Oil
0.517 / 0.483 – 1 = 7%.
If the deal is priced fairly, the interest rate on this loan should be the implied forward interest rate.
On net, XYZ pays 6.9548%:
XYZ net payment = – LIBOR + LIBOR – 6.9548% = –6.9548%
Floating Payment Swap Payment
where ni=1 P(0, ti) r(ti-1, ti) is the PV of interest payments implied by the strip of forward rates, and ni=1 P(0, ti) is the PV of a $1 annuity when interest rates vary over time.
Thus, the fixed swap rate is as a weighted average of the implied forward rates, where zero-coupon bond prices are used to determine the weights.
This equation is equivalent to the formula for the coupon on a par coupon bond.
Thus, the swap rate is the coupon rate on a par coupon bond.
For example, the December swap rate can be computed using equation (8.3): (1 – 0.9485)/ (0.9830 + 0.9658 + 0.9485) = 1.778%. Multiplying 1.778% by 4 to annualize the rate gives the December swap rate of 7.109%.
Equation (8.4) is equal to equation (8.2), when k = 1.
The PV of the market-maker’s net cash flows is
($2.174 / 1.06) + ($2.096 / 1.062) – ($4.664 / 1.063) = 0
This equation is equivalent to equation (8.2), with the implied forward rate, r0(ti-1, ti), replaced by the foreign-currency-denominated annuity payment translated into dollars, R*F0,ti .
The commodity swap price is a weighted average of commodity forward prices.
where Qti is the quantity of gas purchased at time ti .
When Qt = 1, the formula is the same as equation (8.11), when the quantity is not varying.
Let f0(ti) denote the forward price for the floating payment in the swap. Then the fixed swap payment is