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FIXED-INCOME SECURITIES Chapter 11 Forwards and Futures FIXED-INCOME SECURITIES Chapter 11 Forwards and Futures

Outline • • • Futures and Forwards Types of Contracts Trading Mechanics Trading Strategies Outline • • • Futures and Forwards Types of Contracts Trading Mechanics Trading Strategies Futures Pricing Uses of Futures

Futures and Forwards • Forward – An agreement calling for a future delivery of Futures and Forwards • Forward – An agreement calling for a future delivery of an asset at an agreedupon price • Futures – Similar to forward but feature formalized and standardized characteristics • Key differences in futures – – Secondary market - liquidity Marked to market Standardized contract units Clearinghouse warrants performance

Key Terms for Futures Contracts • Futures price: agreed-upon price (similar to strike price Key Terms for Futures Contracts • Futures price: agreed-upon price (similar to strike price in option markets) • Positions – Long position - agree to buy – Short position - agree to sell • Interpretation – Long : believe price will rise (or want to hedge price decline) – Short : believe price will fall (or want to hedge price increase) • Profits on positions at maturity (zero-sum game) – Long = spot price ST minus futures price F 0 – Short = futures price F 0 minus spot price ST

Markets for Interest Rates Futures • The International Money Market of the Chicago Mercantile Markets for Interest Rates Futures • The International Money Market of the Chicago Mercantile Exchange (www. cme. com) • The Chicago Board of Trade (www. cbot. com) • The Sydney Futures Exchange • The Toronto Futures Exchange • The Montréal Stock Exchange • The London International Financial Futures Exchange (www. liffe. com) • The Tokyo International Financial Futures Exchange (TIFFE) • Le Marché à Terme International de France (www. matif. fr) • Eurex (www. eurexchange. com)

Instruments Instruments

Characteristics of Future Contracts • A future contract is an agreement between two parties Characteristics of Future Contracts • A future contract is an agreement between two parties • The characteristics of this contract are – – – The underlying asset The contract size The delivery month The futures price The initial regular margin

Underlying Asset and Contract Size • The underlying asset that the seller delivers to Underlying Asset and Contract Size • The underlying asset that the seller delivers to the buyer at the end of the contract may exist (interest rate) or may not exist (bond) – The underlying asset of the CBOT 30 -Year US Treasury bond future is a fictive 30 -year maturity US Treasury bond with 6% coupon rate • The contract size specifies the notional principal or principal value of the asset that has to be delivered – The notional principal of the CBOT 30 -Year US Treasury bond future is $100, 000 – The principal value of the Matif 3 -month Euribor Future to be delivered is euros 1, 000

Price • The futures price is quoted differently depending on the nature of the Price • The futures price is quoted differently depending on the nature of the underlying asset – When the underlying asset is an interest rate, the future price is quoted to the third decimal point as 100 minus this interest rate – When the underlying asset is a bond, it is quoted in the same way as a bond, i. e. , as a percentage of the nominal value of the underlying • The tick is the minimum price fluctuation that can occur in trading • Sometimes daily price movement limits as well as position limits are specified by the exchange

Trading Arrangements • Clearinghouse acts as a party to all buyers and sellers – Trading Arrangements • Clearinghouse acts as a party to all buyers and sellers – Obligated to deliver or supply delivery • Initial margin – Funds deposited to provide capital to absorb losses • Marking to market – Each day profits or losses from new prices are reflected in the account • Maintenance or variation margin – An established margin below which a trader’s margin may not fall • Margin call – When the maintenance margin is reached, broker will ask for additional margin funds

Conversion Factor • When the underlying asset of a future contract does not exist, Conversion Factor • When the underlying asset of a future contract does not exist, the seller of the contract has to deliver a real asset – May differ from the fictitious asset in terms of coupon rate – May differ from the fictitious asset in terms of maturity • Conversion factor tells you how many units of the actual asset are worth as much as one unit of the fictive underlying asset • Given a future contract and an actual asset to deliver, it is a constant factor which is known in advance • Conversion factors for next contracts to mature available on web sites of futures markets

Conversion Factor (Cont’) • Consider – A future contract whose fictitious underlying asset is Conversion Factor (Cont’) • Consider – A future contract whose fictitious underlying asset is a m year maturity bond with a coupon rate equal to r – Suppose that the actual asset delivered by the seller of the future contract is a x-year maturity bond with a coupon rate equal to c • Expressed as a percentage of the nominal value, the conversion factor denoted CF is the present value at maturity date of the future contract of the actual asset discounted at rate r • Example – Consider a 1 year future contract whose underlying asset is a fictitious 10 year maturity bond with a 6% annual coupon rate – Suppose that the asset to be delivered is at date 1 a 10 -year maturity bond with a 5% annual coupon rate

Invoice Price • The conversion factor is used to calculate the invoice price – Invoice Price • The conversion factor is used to calculate the invoice price – Price the buyer of the future contract must pay the seller when a bond is delivered – IP = size of the contrat x [futures price x CF] • Example – Suppose a future contract whose contract size is $100, 000, the future price is 98. The conversion factor is equal to 106. 459 and the accrued interest is 3. 5. – The invoice price is equal to IP = $100, 000 x [ 98% x 106. 459% + 3. 5% ] = $104, 329. 82

Cheapest-to-Deliver • At the repartition date, there are in general many bonds that may Cheapest-to-Deliver • At the repartition date, there are in general many bonds that may be delivered by the seller of the future contract • These bonds vary in terms of maturity and coupon rate • The seller may choose which of the available bonds is the cheapest to deliver • Seller of the contract delivers a bond with price CP and receives the invoice price IP from the buyer • Objective of the seller is to find the bond that achieves Max (IP - CP) = Max (futures price x CF – quoted price)

Cheapest-to-Deliver (Example) • Suppose a future contract – Contract size = $100, 000 – Cheapest-to-Deliver (Example) • Suppose a future contract – Contract size = $100, 000 – Price= 97 • Three bonds denoted A, B and C Quoted Price Bond A Bond B Bond C Conversion Factor IP-CP 103. 90 118. 90 131. 25 107. 145% 122. 512% 135. 355% 3, 065$ -6, 336$ 4, 435$ • Search for the bond which maximises the quantity IP-CP • Cheapest to deliver is bond C

Forward Pricing • Consider at date t an investor who wants to hold at Forward Pricing • Consider at date t an investor who wants to hold at a future date T one unit of a bond with coupon rate c and time t price Pt • He faces the following alternative cash flows – Either he buys at date t a forward contract from a seller who will deliver at date T one unit of this bond at a fixed price Ft – Or he borrows money at a rate r to buy this bond at date t -

Forward Pricing • Note that both trades have a cost equal to zero at Forward Pricing • Note that both trades have a cost equal to zero at date t. • Note also that the position at the end is the same (one unit of bond). • Then in the absence of arbitrage opportunities, the value of the two strategies at date t must be equal • From this we obtain or with R = 365 r/360 and C = 100 c/Pt

Forward Pricing - Example • On 05/01/01, we consider a forward contract maturing in Forward Pricing - Example • On 05/01/01, we consider a forward contract maturing in 6 months, written on a bond whose coupon rate and price are respectively 10% and $115 • Assuming a 7% interest rate, the forward price F 05/01/01 is equal to

Forward Pricing – Underlying is a Rate • Simply determine the forward rate that Forward Pricing – Underlying is a Rate • Simply determine the forward rate that can be guaranteed now on a transaction occurring in the future • Example – An investor wants now to guarantee the one-year zero-coupon rate for a $10, 000 loan starting in 1 year • Either he buys a forward contract with $10, 000 principal value maturing in 1 year written on the one-year zero-coupon rate R(0, 1) at a determined rate F(0, 1, 1), which is the forward rate calculated at date t=0, beginning in 1 year and maturing 1 year after • Or he simultaneously borrows and lends $10, 000 repayable at the end of year 2 and year 1, respectively • This is equivalent to borrowing $10, 000 x[1+R(0, 1)] in one year, repayable in two years as $10, 000 x(1+R(0, 2))2. – The implied rate on the loan given by the following equation is the forward rate F(0, 1, 1)

Futures Pricing • Price futures contracts by using replication argument, just like forward contracts Futures Pricing • Price futures contracts by using replication argument, just like forward contracts • Let’s consider two otherwise identical forward and futures contracts – Cash-flows are not identical because gains and losses in futures trading are paid out at the end of the day – Denoted as G 0 and F 0, respectively, current forward and futures prices • When interest rates are changing randomly – Cannot create a replicating portfolio – Cannot price futures contracts by arbitrage • However, short term bond prices are very insensitive to interest rate movements – Replication argument is almost exact

Futures versus Forward Pricing Futures versus Forward Pricing

Uses of Futures • Fixing today the financial conditions of a loan or investment Uses of Futures • Fixing today the financial conditions of a loan or investment in the future • Hedging interest rate risk – Because of high liquidity and low cost due to low margin requirements, futures contracts are actually very often used in practice for hedging purposes – Can be used for duration hedging or more complex hedging strategies (see Chapters 5 and 6) • Pure speculation with leverage effect – Like bonds, futures contracts move in the opposite direction to interest rates – This is why a speculator expecting a fall (rise) in interest rates will buy (sell) futures contracts – Advantages : leverage, low cost, easy to sell short

Uses of Futures – Con’t • Detecting riskless arbitrage opportunities using futures • Cash-and-carry Uses of Futures – Con’t • Detecting riskless arbitrage opportunities using futures • Cash-and-carry arbitrage – Consists in buying the underlying asset and selling the forward or futures contract – Amounts to lending cash at a certain interest rate X – There is an arbitrage opportunity when the financing cost on the market is inferior to the lending rate X • Reverse cash-and-carry – Consists in selling (short) the underlying asset and buying the forward or futures contract – Amounts to borrowing cash at a certain interest rate Y – There is an arbitrage opportunity when the investment rate on the market is superior to the borrowing rate Y