Chapter 14

1. Chapter 14 Swap Pricing

2. Outline • Intuition into swap pricing • Solving for the swap price • Valuing an off-market swap • Hedging the swap • Pricing a currency swap

3. Intuition Into Swap Pricing • Swaps as a pair of bonds • Swaps as a series of forward contracts • Swaps as a pair of option contracts

4. Swaps as A Pair of Bonds • If you buy a bond, you receive interest • If you issue a bond you pay interest • In a plain vanilla swap, you do both • You pay a fixed rate • You receive a floating rate • Or vice versa

5. Swaps as A Pair of Bonds (cont’d) • A bond with a fixed rate of 7% will sell at a premium if this is above the current market rate • A bond with a fixed rate of 7% will sell at a discount if this is below the current market rate

6. Swaps as A Pair of Bonds (cont’d) • If a firm is involved in a swap and pays a fixed rate of 7% at a time when it would otherwise have to pay a higher rate, the swap is saving the firm money • If because of the swap you are obliged to pay more than the current rate, the swap is beneficial to the other party

7. Swaps as A Series of Forward Contracts • A forward contract is an agreement to exchange assets at a particular date in the future, without marking-to-market • An interest rate swap has known payment dates evenly spaced throughout the tenor of the swap

8. Swaps as A Series of Forward Contracts (cont’d) • A swap with a single payment date six months hence is no different than an ordinary six-month forward contract • At that date, the party owing the greater amount remits a difference check

9. Swaps as A Pair of Option Contracts • Assume a firm buys a cap and writes a floor, both with a 5% striking price • At the next payment date, the firm will • Receive a check if the benchmark rate is above 5% • Remit a check if the benchmark rate is below 5%

10. Swaps as A Pair of Option Contracts (cont’d) • The cash flows of the two options are identical to the cash flows associated with a 5% fixed rate swap • If the floating rate is above the fixed rate, the party paying the fixed rate receives a check • If the floating rate is below the fixed rate, the party paying the floating rate receives a check

11. Swaps as A Pair of Option Contracts (cont’d) • Cap-floor-swap parity Write floor Long swap Buy cap + = 5% 5% 5%

12. Solving for the Swap Price • Introduction • The role of the forward curve for LIBOR • Implied forward rates • Initial condition pricing • Quoting the swap price • Counterparty risk implications

13. Introduction • The swap price is determined by fundamental arbitrage arguments • All swap dealers are in close agreement on what this rate should be

14. The Role of the Forward Curve for LIBOR • LIBOR depends on when you want to begin a loan and how long it will last • Similar to forward rates: • A 3 x 6 Forward Rate Agreement (FRA) begins in three months and lasts three months (denoted by ) • A 6 x 12 FRA begins in six months and lasts six months (denoted by )

15. The Role of the Forward Curve for LIBOR (cont’d) • Assume the following LIBOR interest rates:

16. The Role of the Forward Curve for LIBOR (cont’d) LIBOR yield curve % 5.62 0 x 12 5.57 0 x 9 5.50 0 x 6 5.42 spot 0 6 9 12 Months

17. Implied Forward Rates • We can use these LIBOR rates to solve for the implied forward rates • The rate expected to prevail in three months, 3f6 • The rate expected to prevail in six months, 6f9 • The rate expected to prevail in nine months, 9f12 • The technique to obtain the implied forward rates is called bootstrapping

18. Implied Forward Rates (cont’d) • An investor can • Invest in six-month LIBOR and earn 5.50% • Invest in spot, three-month LIBOR at 5.42% and re-invest for another three months at maturity • If the market expects both choices to provide the same return, then we can solve for the implied forward rate on the 3 x 6 FRA

19. Implied Forward Rates (cont’d) • The following relationship is true if both alternatives are expected to provide the same return:

20. Implied Forward Rates (cont’d) • Using the available data:

21. Implied Forward Rates (cont’d) • Applying bootstrapping to obtain the other implied forward rates: • 6f9 = 5.71% • 9f12 = 5.77%

22. Implied Forward Rates (cont’d) LIBOR forward rate curve % 5.77 9 x 12 5.71 6 x 9 5.58 3 x 6 5.42 spot 0 3 6 9 12 Months

23. Initial Condition Pricing • An at-the-market swap is one in which the swap price is set such that the present value of the floating rate side of the swap equals the present value of the fixed rate side • The floating rate payments are uncertain • Use the spot rate yield curve and the implied forward rate curve

24. Initial Condition Pricing (cont’d) At-the-Market Swap Example A one-year, quarterly payment swap exists based on actual days in the quarter and a 360-day year on both the fixed and floating sides. Days in the next 4 quarters are 91, 90, 92, and 92, respectively. The notional principal of the swap is $1. Convert the future values of the swap into present values by discounting at the appropriate zero coupon rate contained in the forward rate curve.

25. Initial Condition Pricing (cont’d) At-the-Market Swap Example (cont’d) First obtain the discount factors:

26. Initial Condition Pricing (cont’d) At-the-Market Swap Example (cont’d) First obtain the discount factors:

27. Initial Condition Pricing (cont’d) At-the-Market Swap Example (cont’d) Next, apply the discount factors to both the fixed and floating rate sides of the swap to solve for the swap fixed rate that will equate the two sides:

28. Initial Condition Pricing (cont’d) At-the-Market Swap Example (cont’d) Apply the discount factors to both the fixed and floating rate sides of the swap to solve for the swap fixed rate that will equate the two sides:

29. Initial Condition Pricing (cont’d) At-the-Market Swap Example (cont’d) Solving the two equations simultaneously for X gives X = 5.62%. This is the equilibrium swap fixed rate, or swap price.

30. Quoting the Swap Price • Common practice to quote the swap price relative to the U.S. Treasury yield curve • Maturity should match the tenor of the swap • There is both a bid and an ask associated with the swap price • The dealer adds a swap spread to the appropriate Treasury yield

31. Counterparty Risk Implications • From the perspective of the party paying the fixed rate • Higher when the floating rate is above the fixed rate • From the perspective of the party paying the floating rate • Higher when the fixed rate is above the floating rate

32. Valuing an Off-Market Swap • The swap value reflects the difference between the swap price and the interest rate that would make the swap have zero value • As soon as market interest rates change after a swap is entered, the swap has value

33. Valuing an Off-Market Swap (cont’d) • An off-market swap is one in which the fixed rate is such that the fixed rate and floating rate sides of the swap do not have equal value • Thus, the swap has value to one of the counterparties

34. Valuing an Off-Market Swap (cont’d) • If the fixed rate in our at-the-market swap example was 5.75% instead of 5.62% • The value of the floating rate side would not change • The value of the fixed rate side would be lower than the floating rate side • The swap has value to the floating rate payer

35. Hedging the Swap • Introduction • Hedging against a parallel shift in the yield curve • Hedging against any shift in the yield curve • Tailing the hedge

36. Introduction • If interest is predominantly in one direction (e.g., everyone wants to pay a fixed rate), then the dealer stands to suffer a considerable loss • E.g., the dealer is a counterparty to a one-year, $10 million swap with quarterly payments and pays floating • The dealer is hurt by rising interest rates

37. Introduction (cont’d) • The dealer can hedge this risk in the eurodollar futures market • Based on LIBOR • If the dealer faces the risk of rising rates, he could sell eurodollar futures and benefit from the decline in value associated with rising interest rates

38. Hedging Against A Parallel Shift in the Yield Curve • Assume the yield curve shifts upward by one basis point • The present value of the fixed payments decreases • The present value of the floating payments also decreases, but by a smaller amount • The net effect hurts the floating rate payer

39. Hedging Against A Parallel Shift in the Yield Curve (cont’d) • The dealer could sell eurodollar (ED) futures to hedge • Need one ED futures contract for every $25 change in value of the swap • Need to choose between the various ED futures contracts available

40. Hedging Against A Parallel Shift in the Yield Curve (cont’d) • How to choose between the ED futures contracts available? • With a stack hedge, the hedger places all the futures contracts at a single point on the yield curve, usually using a nearby delivery date • With a strip hedge, the hedger distributes the futures contracts along the relevant portion of the yield curve depending on the tenor of the swap

41. Hedging Against Any Shift in the Yield Curve • The yield curve seldom undergoes a parallel shift • To hedge against any change, determine how the swap value changes with changes at each point along the yield curve

42. Hedging Against Any Shift in the Yield Curve (cont’d) • Steps involved in hedging: • Convert the annual LIBOR rate into effective rates :

43. Hedging Against Any Shift in the Yield Curve (cont’d) • Steps involved in hedging (cont’d): • Next, determine the number of futures needed at each payment date:

44. Tailing the Hedge • Futures contracts are marked to market daily • Forward contracts are not marked to market • This introduces a time value of money differential for long-tenor swaps • Hedging equations would overhedge

45. Tailing the Hedge (cont’d) • To remedy the situation, simply reduce the size of the hedge by the appropriate time value of money adjustment (tail the hedge):

46. Tailing the Hedge (cont’d) Tailing the Hedge Example Assume we have determined that we need 100 ED futures contracts for delivery two years from now. The two-year interest rate is 6.00%. How many ED futures do you need if you tail the hedge?

47. Tailing the Hedge (cont’d) Tailing the Hedge Example (cont’d) You need 89 ED futures contracts:

48. Pricing A Currency Swap • To value a currency swap: • Solve for the equilibrium fixed rate on a plain vanilla interest rate swap for each of the two countries • Determine the relevant spot rates over the tenor of the swap • Determine the relevant implied forward rates • Find the equilibrium swap price for an interest rate swap in both countries