悉尼大学固定收益证券课件Lecture 10.ppt

interest rate derivatives,lecture 10,dr. andrew ainsworth,finc3019 fixed income securities,last week,securitisation in australia collateralised debt obligations sub-prime crisis,introduction,forwards futures u.s. treasury futures delivery options basis cost of carry pricing hedging eurodollar futures australian interest rate futures reading: sundaresan ch. 17,forwards versus futures,forward contracts,a buyer of a forward contract agrees to purchase one unit of the underlying asset at a specified future time the maturity or settlement or delivery date the price agreed to in the contract is the forward price the forward price is determined when the contract is written and does not change over the life of the contract a buyer of a forward contract is long in the forward market a seller of a forward contract is short in the forward market the forward price agreed to by the buyer and the seller is such that the buyer pays and receives nothing when the contract is entered into at maturity the short position provides the underlying asset to the long position and receives the forward price in return,forward contract cash flows,a trader buys a forward contract on coal on day 1 that will mature on day 4 the forward and spot (cash) prices of coal are given below cash flows occur only on the settlement date as the cash price rises, the long forward position becomes more valuable a loss would have eventuated if the cash price on maturity was less than 203 the forward price must equal the spot price on the maturity date why?,futures contracts,a buyer of a futures contract agrees to purchase one unit of the underlying asset at a specified future time the maturity date the price agreed to in the contract is the futures price the futures price is determined when the contract is written the futures price agreed to by the buyer and the seller is such that the buyer pays and receives nothing when the contract is entered into the parties to a futures contract will need to make or receive daily instalment payments toward the eventual purchase of the underlying asset the sum of these daily payments and those made at maturity will equal the futures price when the contract was entered into,futures contracts: marking to market,the daily change in the futures price determines the daily instalments long position makes payment equal to change in daily futures price if futures price falls receives payment equal to change in daily futures price if futures price rises short position receives payment equal to change in daily futures price if futures price falls makes payment equal to change in daily futures price if futures price rises this process is called marking to market in effect the futures contract is rewritten each day value of contract after daily settlement equals zero,futures contract cash flows,the cash flows from a futures contract will differ from a forward contract at maturity the balance due will be the futures price agreed to at the outset the futures price must equal the spot price on the maturity date why? why is reinvestment risk relevant for futures contracts? may need to liquidate investments to make payments or reinvest cash flows received,design of futures contracts,futures contracts are traded on exchanges and therefore the contracts need to be standardised physical delivery versus cash settlement? are the deliverable assets are in competitive supply? traders can corner the market u.s. treasury futures allows the choice of which treasury bond to deliver margins investor must deposit collateral as initial margin position in order to open a futures position if initial margin position falls below the maintenance margin, a margin call occurs and the investor will need to deposit cash to restore the initial margin options for the short position choice of physical delivery location choice of delivery date,payoffs under a forward/futures contract,price at maturity,payoff,futures vs. forwards,forwards differ from futures contracts in many ways: futures contracts are marked to market this causes differences in the price due to reinvestment risk and interest rate uncertainty forwards trade in over-the-counter (otc) markets futures trade on organised exchanges sydney futures exchange (sfe), chicago board of trade (cbot) forwards have credit risk because of the contract between counterparties futures involve clearinghouses that monitor margins and non-performance future contracts are standardised forward contracts are customised,uses of interest rates futures,hedge remove risk associated with a portfolio of bonds we can undertake a duration and/or convexity hedge speculate if an increase in interest rates is expected we will sell futures contracts there is substantial leverage because margin requirements are small short sales are much easier in the futures market than the spot market lower transaction costs arbitrage the generation of riskless profits cash and carry arbitrage reverse cash and carry arbitrage,u.s. treasury futures,u.s. treasury futures,u.s. treasury futures trade on the chicago board of trade futures traded on u.s. treasury instruments bills: 13-week notes: 2, 3, 5 and 10-year bonds: 30-year we will focus on 30-year t-bond futures an important feature of these futures is that they require physical delivery there are many options embedded in t-bond futures that make them difficult to accurately price these options are available to the seller during the delivery month,u.s. treasury bond futures contract specifications,what happens at expiry?,a trader can take an offsetting position before expiry to close out their trade if you are long futures then you take a short position prior to the expiry of the contract and you will not need to make physical delivery or a trader can make physical delivery to close out the futures position this will result in a loss because of the valuable delivery options that the short position in futures has physical delivery can occur on any day in the trading month for t-bond futures,delivery options,quality option short futures position has choice as to which bond to deliver can deliver on any day of the delivery month wildcard option futures trading closes at 2pm chicago time yet delivery can occur up until 8pm if deliverable securities were to experience a large price decline between 2pm and 8pm then the investor who is short can profit each day of the delivery month has an option that lasts for six hours end-of-the-month (timing) option the futures contract stops trading seven business days before month-end but delivery can occur up until the last business day the short investor has flexibility as to when to deliver the underlying during this period,conversion factor,the quality option gives traders with the short position the choice as to which bond to deliver they will want to deliver the cheapest bond the cbot uses a conversion factor to adjust the futures price either upwards if the coupon rate on the underlying bond is greater than 6% or downwards if the coupon rate is less than 6%: x is the number of months by which maturity exceeds n, so x = 0, 3, 6 or 9 the conversion factor seeks to make each of the bonds available to be delivered similar, so that they are all yielding 6% a high coupon bond is expensive so the trader is rewarded,conversion factor,to determine the compensation paid by the investor with the long position the invoice price we use the futures price (h) multiplied by the conversion factor (cf) then add the accrued interest (ai): if the futures price = 102.25 and the conversion factor for a 5.25% t-bond maturing in 28 years and 3 months = 0.8984 and accrued interest = 0.6420, what will be the invoice price? the trader with the short position can undertake this analysis with all eligible bonds to determine the cheapest to deliver (ctd),basis,the basis is the difference between the quoted price of the deliverable t-bond less the futures price for delivery at some future date s, multiplied by the conversion factor: long the basis: long spot and short futures size of futures position is the conversion factor long delivery options (quality, timing and wildcard options) short the basis: short spot and long futures short delivery options (quality, timing and wildcard options) note that accrued interest does not appear as it is included in both the invoice and spot prices,basis,no arbitrage ensures that the basis in the delivery month is greater than zero for u.s. treasury futures if it werent then a trader could sell futures and immediately deliver the bond to make a riskless profit note that this is only the case for futures where short traders have delivery options in the case of a forward contract, the basis can be positive or negative depending on the cost of carry outside of the delivery month, the basis can be positive or negative,valuing a u.s. treasury futures contract,how do we value a u.s. t-bond futures contract? this isnt a trivial task because of the delivery options but, we can estimate a theoretical forward price this does not include any of the delivery options we also disregard marking to market and associated cash flows the process by which we can value a forward contract is through a cash and carry arbitrage this gives rise to the forward-spot parity relationship,valuing a u.s. treasury futures contract,let us first consider how a trader can acquire a bond if at date 0 a trader wants to hold at future date 1 one unit of a bond, they have two options they can buy a forward contract at date from a seller who will deliver one unit of the bond at a price f0 or they can borrow money to buy the bond at date 0 let us consider the fraction of a year between date 0 and date 1 as t,valuing a u.s. treasury futures contract,the cash flows from these two operations are: p = spot price of bond f = forward price r = repo rate (financing rate) c = coupon rate t = time in years to futures delivery date,valuing a u.s. treasury futures contract,the cost of both of these operations is equal to zero at date 0 as there are no cash flows so, if there are not opportunities for arbitrage then the cash flows at date 1 should also be equal if we define cy as the current yield of the bond: we arrive at the price of a forward that precludes arbitrage and is the forward-spot parity relationship: note: it is not the price of a futures contract! why?,example,what is the theoretical forward price if the repo rate is 6% p.a., the spot price of a deliverable 10% t-bond is 97.54 and there are 6 months until delivery? the current yield is 10/97.54 = 10.25%,arbitrage opportunities,we can also think of the forward-spot parity relationship as a way to identify arbitrage opportunities cash and carry arbitrage if the price at which you can sell a bond in the forward market (at maturity of forward contract) is greater than the cost of financing the bond, then the investor should sell forward and finance the bond if not, then the investor should buy forward and sell the bond in a repurchase transaction note that there are no cash flows selling the futures has no cash flow borrowing and buying the bond has no current cash flow as it is fully financed,example: cash and carry trade,a 12% coupon 20-year bond with a par value of 100 is selling at par and is deliverable against a futures contract that settles in 3 months the futures price is 107 the current repo rate is 8% p.a. we can sell the futures contract at 107 borrow 100 for three months at 8% p.a. purchase the bond for 100,example: cash and carry trade,no initial cash outlay for this strategy at settlement of the futures contract: quoted price of bond 107 accrued interest 3 total proceeds 110 from the loan: repay principle 100 interest 2 total outlay 102 profit 8,example: reverse cash and carry trade,if the futures price had been 92 instead of 107 we could execute a different trade we can buy the futures contract at 92 lend 100 for three months at 8% per year short sell the bond for 100 at settlement of the futures contract we outlay 92 + 3 = 95 from the loan we receive 100 + 2 =102 we make a profit of 7 in reality arbitrageurs would act to eliminate such profitable opportunities in the previous two examples, if the futures price had been 99 there would not be arbitrage profits as inflows = outlays = 102,financing bonds,futures traders,repo dealer,post bond as collateral,receive cash,buy bond,pay cash,sell futures,unwinding the trade,take delivery of t-bond,pays cash and interest (f),receive cf x h (invoice price),deliver bond,arbitrage opportunities,a long position in a t-bond forward contract is equivalent to borrowing and buying the underlying bond a short position in a t-bond forward is equivalent to shorting the t-bond and investing at the risk free rate,cost of carry,the difference between the repo rate and the current yield is the cost of carry it is the cost of financing the position less the cash yield on the underlying security income earned by the long basis holder (short futures) on spot treasury position = accrued interest (from spot settlement to futures delivery) less financing cost (from spot settlement to futures delivery) theoretical price of a forward contract = spot price + the cost of carry the shape of yield curve affects cost of carry,cost of carry,if the repo rate is higher than the current yield then there is negative carry if there is negative carry (r cy) the forward price will be higher than the spot price (f p) if the current yield is greater than the repo rate then there is positive carry if there is positive carry (cy r) the forward price will be lower than the spot price (f p) if there is zero carry, the forward price will equal the spot price remember we have ignored interim cash flows from marking to market and delivery options if the borrowing and lending rates differ the futures price will have an upper and lower boundary,the value of futures,the theoretical price refers to the price of a forward contract which does not provide the short investor with delivery options these options will reduce the price of a futures contract relative to a forward contract why? theoretical price of a futures contract = spot price + the cost of carry value of quality option value of timing option value of wildcard option the exact value of these options is an empirical question,basis after carry,forward price represents the price at which the bond can be sold forward to break even invoice price represents the actual revenue from delivering the bond in the futures market the difference between the two is the profit or loss of selling futures and borrowing and buying the bond and is the basis after carry: forward price is the cost of carrying a deliverable bond through repo financing until delivery the conversion factor multiplied by the conversion factor is the revenue associated with delivery the difference is the profit or loss from a cash and carry position in the futures market,basis after carry,if the basis after carry is negative then arbitrage profits can be obtained in futures sell futures and borrow and buy the cheapest to deliver bond both are usually positive, however basis after carry can be separated into: basis less reinvested cash inflows plus financing costs when the carry is positive, the bac is less than the basis the bond with the lowest bac is the cheapest to deliver,hedging using t-bond futures,in order to hedge a position in an underlying bond using t-bond futures we need to examine the pvbp of each security: where f = the hedge ratio we need to take account of the cheapest to deliver bond: the number of contracts to purchase depends on the par value of the futures contract and the bond to be hedged,example,a $100m par 5.4% july 2027 t-bond has a pvbp of 486 a 5.8% may 2025 t-bond is the cheapest to deliver with a conversion factor of 0.9670 and pvbp of 387 each futures contract is on $100,000 par amount what is the number of futures contracts required to hedge this position?,eurodollar futures,eurodollar market,eurodollars are bank deposits denominated in usd but are located outside the u.s. it is an interbank market where participating banks can borrow and lend from each other the rate at which banks offer loans top each other is the london interbank offer rates (libor) it is determined by the british bankers association (bba) the spot market extends out to 10 years in maturity but the depth and liquidity is concentrated at the shorter end of the market libor is seen as the benchmark rate for many markets swaps, commercial paper and futures,eurodollar market,how s libor determined? a panel of large banks provide the rates that they will lend to each other at the panel includes banks such as barclays, credit suisse, deutsche, hbos, jp morgan, ubs, etc the four largest and the four lowest rates are eliminated and the remaining 8 rates are averaged to get libor libor should reflect the credit quality of the banks of the panel if there is a financial crisis that effects the bank, then we should see libor increase to reflect this increased credit risk interest on eurodollar time deposits is calculated on an actual/360 basis,eurodollar futures,eurodollar futures trade on may different exchange around the world they are cash settled to the 90-day libor that is derived from the 90-day eurodollar time deposit with a principal value of $1m the contract matures at 11am london time on the second london business day immediately preceding the third wednesday of the contract month one basis point change in the interest rate is equal to $25 gain/loss for each futures contract 90/360 days there is no delivery or timing options to complicate the pricing settlement price of the eurodollars futures contract will converge to libor 100 x (1 libor) so if libor is 5% then the futures settlement price is 95,intermarket spreads,eurodollar futures contracts can be used to implement strategies designed to profit from changes in the spread with treasury se