Tài chính doanh nghiệp - Chapter 14: Swap pricing

© 2004 South-Western Publishing1Chapter 14Swap Pricing2OutlineIntuition into swap pricingSolving for the swap priceValuing an off-market swapHedging the swapPricing a currency swap3Intuition Into Swap PricingSwaps as a pair of bondsSwaps as a series of forward contractsSwaps as a pair of option contracts4Swaps as A Pair of BondsIf you buy a bond, you receive interestIf you issue a bond you pay interestIn a plain vanilla swap, you do bothYou pay a fixed rateYou receive a floating rateOr vice versa5Swaps 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 rateA bond with a fixed rate of 7% will sell at a discount if this is below the current market rate6Swaps 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 moneyIf because of the swap you are obliged to pay more than the current rate, the swap is beneficial to the other party7Swaps as A Series of Forward ContractsA forward contract is an agreement to exchange assets at a particular date in the future, without marking-to-marketAn interest rate swap has known payment dates evenly spaced throughout the tenor of the swap8Swaps 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 contractAt that date, the party owing the greater amount remits a difference check9Swaps as A Pair of Option ContractsAssume a firm buys a cap and writes a floor, both with a 5% striking priceAt 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%10Swaps 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 swapIf the floating rate is above the fixed rate, the party paying the fixed rate receives a checkIf the floating rate is below the fixed rate, the party paying the floating rate receives a check11Swaps as A Pair of Option Contracts (cont’d)Cap-floor-swap parity 5%+=5%5%Write floorBuy capLong swap12Solving for the Swap PriceIntroductionThe role of the forward curve for LIBORImplied forward ratesInitial condition pricingQuoting the swap priceCounterparty risk implications13IntroductionThe swap price is determined by fundamental arbitrage argumentsAll swap dealers are in close agreement on what this rate should be14The Role of the Forward Curve for LIBORLIBOR depends on when you want to begin a loan and how long it will lastSimilar 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 )15The Role of the Forward Curve for LIBOR (cont’d)Assume the following LIBOR interest rates:Spot (0f3)5.42%Six Month (0f6)5.50%Nine Month (0f9)5.57%Twelve Month (0f12)5.62%16The Role of the Forward Curve for LIBOR (cont’d) LIBOR yield curveMonths069125.425.505.575.62spot0 x 60 x 90 x 12%17Implied Forward RatesWe can use these LIBOR rates to solve for the implied forward ratesThe rate expected to prevail in three months, 3f6The rate expected to prevail in six months, 6f9The rate expected to prevail in nine months, 9f12The technique to obtain the implied forward rates is called bootstrapping18Implied Forward Rates (cont’d)An investor canInvest 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 maturityIf the market expects both choices to provide the same return, then we can solve for the implied forward rate on the 3 x 6 FRA19Implied Forward Rates (cont’d)The following relationship is true if both alternatives are expected to provide the same return:20Implied Forward Rates (cont’d)Using the available data:21Implied Forward Rates (cont’d)Applying bootstrapping to obtain the other implied forward rates:6f9 = 5.71%9f12 = 5.77%22Implied Forward Rates (cont’d) LIBOR forward rate curveMonths03695.425.585.715.77spot3 x 66 x 99 x 12%1223Initial Condition PricingAn 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 sideThe floating rate payments are uncertainUse the spot rate yield curve and the implied forward rate curve24Initial 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. 25Initial Condition Pricing (cont’d)At-the-Market Swap Example (cont’d) First obtain the discount factors:26Initial Condition Pricing (cont’d)At-the-Market Swap Example (cont’d) First obtain the discount factors:27Initial 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:28Initial 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:29Initial 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. 30Quoting the Swap PriceCommon practice to quote the swap price relative to the U.S. Treasury yield curveMaturity should match the tenor of the swapThere is both a bid and an ask associated with the swap priceThe dealer adds a swap spread to the appropriate Treasury yield31Counterparty Risk ImplicationsFrom the perspective of the party paying the fixed rateHigher when the floating rate is above the fixed rateFrom the perspective of the party paying the floating rateHigher when the fixed rate is above the floating rate32Valuing an Off-Market SwapThe swap value reflects the difference between the swap price and the interest rate that would make the swap have zero valueAs soon as market interest rates change after a swap is entered, the swap has value33Valuing 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 valueThus, the swap has value to one of the counterparties34Valuing 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 changeThe value of the fixed rate side would be lower than the floating rate sideThe swap has value to the floating rate payer35Hedging the SwapIntroductionHedging against a parallel shift in the yield curveHedging against any shift in the yield curveTailing the hedge36IntroductionIf interest is predominantly in one direction (e.g., everyone wants to pay a fixed rate), then the dealer stands to suffer a considerable lossE.g., the dealer is a counterparty to a one-year, $10 million swap with quarterly payments and pays floatingThe dealer is hurt by rising interest rates37Introduction (cont’d)The dealer can hedge this risk in the eurodollar futures marketBased on LIBORIf the dealer faces the risk of rising rates, he could sell eurodollar futures and benefit from the decline in value associated with rising interest rates38Hedging Against A Parallel Shift in the Yield CurveAssume the yield curve shifts upward by one basis pointThe present value of the fixed payments decreasesThe present value of the floating payments also decreases, but by a smaller amountThe net effect hurts the floating rate payer39Hedging Against A Parallel Shift in the Yield Curve (cont’d)The dealer could sell eurodollar (ED) futures to hedgeNeed one ED futures contract for every $25 change in value of the swapNeed to choose between the various ED futures contracts available40Hedging 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 dateWith a strip hedge, the hedger distributes the futures contracts along the relevant portion of the yield curve depending on the tenor of the swap41Hedging Against Any Shift in the Yield CurveThe yield curve seldom undergoes a parallel shiftTo hedge against any change, determine how the swap value changes with changes at each point along the yield curve42Hedging Against Any Shift in the Yield Curve (cont’d)Steps involved in hedging:Convert the annual LIBOR rate into effective rates :43Hedging 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: 44Tailing the HedgeFutures contracts are marked to market dailyForward contracts are not marked to marketThis introduces a time value of money differential for long-tenor swapsHedging equations would overhedge45Tailing 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):46Tailing 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?47Tailing the Hedge (cont’d)Tailing the Hedge Example (cont’d) You need 89 ED futures contracts:48Pricing A Currency SwapTo value a currency swap:Solve for the equilibrium fixed rate on a plain vanilla interest rate swap for each of the two countriesDetermine the relevant spot rates over the tenor of the swapDetermine the relevant implied forward ratesFind the equilibrium swap price for an interest rate swap in both countries