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Lecture 7. Pricing Forward and Future Contracts & Interest Rate Contracts Relating forward and future prices to the spot price & Hedging exposure to interest rate movements Options, Futures, Derivatives 10/15/07 back to start 1 Pricing Futures Contracts vs. Pricing Forward Contracts Forward contract pricing: F0 = S0erT . If interest rates vary then forward and future prices may no longer be the same. However, if the risk-free interest rate is constant and the same for all maturities, then the forward price for a contract is the same as the futures price. We sketch a proof. • Suppose futures contract last n days • Fi is the futures price at the end of day i with daily settlement • Deﬁne δ as the risk-free rate per day. • Strategy: – Take a long futures position of eδ at the start – Increase long position to ekδ at end of day k • At end of day i the position is (Fi − F i − 1)eδi. Assume proﬁt is compounded at risk-free rate until end of day n iδ (n−i)δ nδ (Fi − Fi−1) e e = (Fi − Fi−1) e Pn nδ • Value at end of n days is i=1 (Fi − Fi−1 ) e = (Fn − F0) enδ . • Fn = ST , the terminal spot price, so value of the strategy is (ST − F0) enδ . Options, Futures, Derivatives 10/15/07 back to start 2 Pricing Futures Contracts vs. Pricing Forward Contracts • An investment of F0 in a risk-free bond, combined with above strategy yields nδ nδ nδ F0e + (ST − F0) e = ST e • On the other hand, a forward contract with price G0 at day 0 yields a return of ST enδ . • In absence of arbitrage opportunities implies F0 = G0 Options, Futures, Derivatives 10/15/07 back to start 3 Futures on Commodities We consider pricing futures contracts on commodities that are primarily investment assets, such as precious metals. • Gold owners can earn income from leasing the gold. • The interest charged from leasing gold is the gold lease rate • Also holds for silver. In the absence of storage and income, the forward price of the commodity that is an investment asset is F0 = S0erT We treat storage costs as negative income, then set U to be the present value of all storage costs, net of income, during the life of a forward contract, then F0 = (S0 + U ) erT Options, Futures, Derivatives 10/15/07 back to start 4 Futures on Commodities, cont. Example: Consider a 1 year futures contract on an investment asset that provides no income. It costs $2 per unit to store the asset, with the payment being made at the end of the year. Assume that the spot price $450 per unit and the risk-free rate is 7% per annum for all maturities. This corresponds to r = 0.07, S0 = 450, T = 1, and U = 2e−0.07×1 = 1.865 The futures price, F0 is given by F0 = (450 + 1.865) e0.07×1 = $484.63 If the actual futures price is greater than 484.63, an arbitrageur can buy the asset and short 1-year futures contracts to lock in a proﬁt. If the actual futures price is less than 484.63, an investor who owns the asset can improve the return by selling the asset and buying futures contracts. If storage costs are proportional to the price of the commodity then it can be treated as negative yield: F0 = S0e(r+u)T Options, Futures, Derivatives 10/15/07 back to start 5 Consumption Commodities Commodities that are consumption assets (as opposed to investment assets) usually provide no income, but can be subject to signiﬁcant storage costs. We consider if our pricing model holds: rT F0 > (S0 + U ) e Arbitrageur should follow the following strategy: • Borrow an amount S0 + U at the risk-free rate and use it to purchase one unit of the commodity and to pay for storage. • Short a forward contract on one unit of the commodity. This would ideally lead to a proﬁt of F0 − (S0 + U )erT at time T . However, there is a tendency for S0 to increase and F0 to decrease until F0 > (S0 + U ) erT no longer holds. Therefore, F0 > (S0 + U ) erT cannot hold for long periods of time. Options, Futures, Derivatives 10/15/07 back to start 6 Consumption Commodities On the other hand, consider rT F0 < (S0 + U ) e Then an investor should do the following: • Sell the commodity, save the storage costs, and invest the proceeds at the risk-free interest rate. • Take a long position in a forward contract. The result is a riskless rate of (S0 + U )erT − F0 relative to the position they would’ve been had they held the commodity. Therefore, we seem to get F0 = (S0 + U ) erT for the pricing of the commodity. However, companies who keep consumption commodities in storage, do so because of its consumption value, not because of investment value. Such owners are reluctant to sell the commodity and buy forward contracts, because forward contracts cannot be consumed. There is nothing to stop F0 < (S0 + U ) erT from holding. All we can say is rT F0 ≤ (S0 + U ) e If storage is proportional to the spot price then (r+u)T F0 ≤ S0e Options, Futures, Derivatives 10/15/07 back to start 7 Convenience Yields Users of the consumption asset place diﬀerent value to the commodity than investors. • Users of corn futures keep production running • Beneﬁts from holding the physical asset are referred to as the convenience yield provided by the community • If the cost of storage is known and has present value U then the convenience yield y is deﬁned as yT rT F0e = (S0 + U ) e • If cost of storage is a proportional u to the spot price, then y is (r+u−y)T F0 = S0e Convenience yield measures the extent to which the left-hand side is less than the right-hand side in F0 ≤ (S0 + U ) erT or F0 ≤ S0e(r+u)T . Reﬂects the market’s expectations concerning the future availability of the commodity. • Greater the possibility of shortages, greater the convenience yield • Larger the inventories, the smaller the convenience yield Options, Futures, Derivatives 10/15/07 back to start 8 Cost of Carry Cost of carry is storage costs and interest that is paid to ﬁnance the asset less the income earned on the asset. • Nondividend-paying stocks, the cost of carry is r (no storage & no income) • Stock index, the cost of carry is r − q (income earned at rate q ) • Currency futures, the cost of carry is r − rf (diﬀerence in the foreign risk-free rate) • Commodity with income q and storage costs at rate u, the cost of carry is r − q + u Cost of carry is cT F0 = Soe for investment assets and (c−y)T F0 = S0e for consumption assets with y the convenience yield. Options, Futures, Derivatives 10/15/07 back to start 9 Futures prices and Expected futures prices • The market’s average opinion about what the spot price of an asset will be at a certain time in the future is the expected spot price • Speculators tend to hold long positions, since require greater compensation for the greater risk. Only trade if they expect to make money. • Hedgers tend to hold short positions and more willing to take losses, since their primary interest is reducing risk. • Together this implies futures price of an asset will be below the expected spot price - Keynes & Hicks. We see that • Relationship between futures prices and expected short is based on the relationship between risk and expected return • More later on Options, Futures, Derivatives 10/15/07 back to start 10 Summary • Futures and forward prices is convenient to divide futures contracts into two categories • Pricing occurs via arbitrage situation • Three cases, assume – Assets with no income: F0 = S0erT and Value of long foward contract with delivery price K is S0 − Ke−rT – Assets with known income: F0 = (S0 − I)erT and Value of long foward contract with delivery price K is S0 − I − Ke−rT – Assets with known yield: F0 = S0e(r−q)T and Value of long foward contract with delivery price K is S0e−qT − Ke−rT • Futures prices for stock indices, currencies, gold, and silver can be priced in the same way. • Pricing consumption asset futures need to take into account the beneﬁts of owning the asset that are not obtained from owning the futures contract. This is measured by the convenience yield. • Can obtain only upper bounds for futures price of consumption assets using arbitrage arguments. • Cost of carry is the storage cost plus ﬁnancing minus the income. For investment assets futures price is greater than the spot price by the amount reﬂecting the cost of carry. For consumption assets futures price is greater than the spot price by an amount reﬂecting the cost of carry net of the convenience yield. Options, Futures, Derivatives 10/15/07 back to start 11 Interest Rate Futures • Use Treasury bonds and Eurodollar futures contracts to hedge exposure to interest rate movements. • Interest rate futures allow companies to hedge themselves from risks associated to changes in the risk-free rates. Options, Futures, Derivatives 10/15/07 back to start 12 Day Count Conventions Deﬁnes the way in which interest is accrued over time. Day count is expressed in the form X/Y when computing interest between two dates. Number of days between dates × Interest earned in reference period Number of days in reference period Day count conventions provide for a way to determine the value of partial coupon payments. Options, Futures, Derivatives 10/15/07 back to start 13 Day Counts, cont. • US Treasury Bonds - Day counted as actual/actual reference period The interest earned between dates is the ratio of the number of days to the true length of the period of time. Example: Suppose a bond principal is $100, coupon payment dates are March 1 and September 1, the coupon rate is 8%. What is the interest earned between March 1 and July 3? The length of the time between March 1 and September 1 is 184 days, in which $4 of interest is earned. during the period. There are 124 days between March 1 and July 3. Therefore, 124 × 4 = $2.6957 184 Options, Futures, Derivatives 10/15/07 back to start 14 Day Counts, cont. • US Corporate and Municipal Bonds - Day counted as 30/360 Here, we assume 30 days per month and 360 days per year when carrying out calculations. Example: Consider the same example. Suppose a bond principle is $100, coupon payment dates are March 1 and September 1, the coupon rate is 8%. What is the interest earned between March 1 and July 3 using the 30/360 rule? We count the length of the time between March 1 and September 1 as 180 days, in which $4 of interest is earned. during the period. The total number of days counted between March 1 and July 3 is (4 × 30) + 2 = 122 so the interest earned is 122 × 4 = $2.7111 180 Options, Futures, Derivatives 10/15/07 back to start 15 Day Counts, cont. • US Money Market Instruments - Day counted as actual/360 Used for money market instruments in the US. Here, we calculate the actual number of days that have elapsed and divide by 360 days per year when carrying out calculations. Note - interest earned on an entire year is 365/360 times the quoted rate. Sometimes the money market rates are quoted using discount rates. Example: Consider the same example. Suppose a bond principle is $100, coupon payment dates are March 1 and September 1, the coupon rate is 8%. What is the interest earned between March 1 and July 3 using the actual/360 rule? We count the length of the time between March 1 and September 1 as 180 days, in which $4 of interest is earned. during the period. The total number of days counted between March 1 and July 3 is 124 so the interest earned is 124 × 8 = $2.7555 360 Options, Futures, Derivatives 10/15/07 back to start 16 US Money Market Instruments, cont. Consider a 91-day Treasure Bill with annualized rate of interest earned at 8% of face value. (91-day T-Bill quoted as 8) Suppose face value $100 then interest earned over the life of the Treasury Bill is 91 $100 × 0.08 × = $2.0222 360 The true interest rate is 2.0222 = 2.064% 100 − 2.0222 over the 91-day period. In general we have 360 P = (100 − Y ) n where P is the quoted price, Y is the cash price, n is the remaining life of the Treasury bill in days. Options, Futures, Derivatives 10/15/07 back to start 17 Quotes for Treasury Bonds • US Treasury bonds are quote in dollars and thirty-seconds of a dollar. The quote price is for a bond with face value of $100. • Example: A quote of 92-07 indicates a quote price for a bond with face value $100 is $92.21875 . • Quoted price is referred to as the clean price • Cash price is referred to as the dirty price Cash price = Quoted price + Accrued interest since last coupon date Options, Futures, Derivatives 10/15/07 back to start 18 Quotes for Treasury Bonds Options, Futures, Derivatives 10/15/07 back to start 19 Quotes for T-Bonds • Example: Suppose it is March 5, 2007, and the bond is 11% coupon bearing bond maturing on July 10, 2010 with quoted price of 95-16 ≡ $95.50. • Coupons for T-bonds are paid semiannually, so most recent coupon was January 10, 2007 and next coupon date is July 10, 2007. • 54 days between January 10, 2007 and March 5, 2007. • 181 days between January 1. 2007 and July 10, 2007. • On a bond with face value $100, coupon payments are $5.50. The accrued interest on March 5, 2007 is the share of the July 10 coupon accruing to the bondholder on March 5, 2007 54 × $5.5 = $1.64 181 • Case price for the $100 face value bond is $95.5 + $1.64 = $97.14 Options, Futures, Derivatives 10/15/07 back to start 20 Treasure Bond Futures • Treasury bond futures prices are quoted • Delivery can occur anytime during the delivery month • Each contract is for delivery of $100,000 face value of bonds. • Treasury bond futures are traded on CBOT. • Any bond with 15 years to maturity on the ﬁrst day of the delivery month and is not callable within 15 years from that day can be delivered • A callable bond contains provisions in which it can be bought back at a certain price at certain times during its life • Treasury Note Futures Contracts with a maturity between 6 1/2 and 10 years can be delivered. • 5 year Treasury Note Futures, any bond delivered has a life of about 4 or 5 years Options, Futures, Derivatives 10/15/07 back to start 21 Conversion Factors • Treasury bond futures contract allows the short position to choose to deliver any bond with maturity of more than 15 years and is not callable within 15 years. • When a bond is delivered, the conversion factor deﬁnes the price received by the party with the short position. • The quoted price applying to the delivery is conversion factor times most recent settlement price. If we include accrued interest then the cash received for a $100 face value of bond delivered is (Settlement price × Conversion factor) + Accrued Interest Example: Consider a bond with settlement price of 90-00 with conversion factor for the bond delivered of 1.3800. Assume the accrued interest at the time of delivery is $3 per $100 face value. The cash received by the short investor is (1.3800 × 90.00) + 3.00 = $127.20 Options, Futures, Derivatives 10/15/07 back to start 22 Calculating Conversion Factors • Conversion factor for a bond is equal to the quoted price the bond would have per dollar of principal on the ﬁrst day of the delivery month on the assumption that the interest rate for all maturities would equal 6% per annum with semiannual compounding • The bond maturity and the times to the coupon payments are rounded down to the nearest 3 months for calculation purposes. • After rounding, if the bond does not last for an exact number of 6-month periods (i.e. and extra 3 month period), the ﬁrst coupon is assume to be paid after 3 months and accrued interest is subtracted. Example: Consider a 10% coupon bond with 20 years and 2 months to maturity. • Coupon payments are assumed to be made at 6-month intervals until the end of the 20 years when the principal is paid. • Assume the face value is $100 • Given the discount rate of 6% per annum with semiannual compounding, then the face value is 40 −i −40 X 5 (1.03) + 100 (1.03) = $146.23 i=1 Dividing by the face value gives a conversion factor of 1.4623. Options, Futures, Derivatives 10/15/07 back to start 23 Calculating Conversion Factors, Example 2 Example: Consider a 8% coupon bond with 18 years and 4 months to maturity. • For the purpose of calculating the conversion factor, the bond is assumed to have exactly 18 years and 3 months to maturity. • Discounting at 6% per annum compounded semiannually yields 36 X 4 100 4+ i + 36 = $125.83 i=1 1.03 1.03 √ • Interest rate for a 3 month period is 1.03 − 1 = 0.014889. • Discounting back to the present gives a bond value as 125 = $123.99 1.014889 • Subtracting oﬀ the accrued interest of 2.0 yields $121.99 . Conversion factor is 1.2199. Options, Futures, Derivatives 10/15/07 back to start 24 Cheapest-to-Deliver Date • Any time during the delivery month, there are many bonds that can be delivered in CBOT Treasury bond futures contract. • They can vary widely. • Party with the short position can choose which of the available bonds is ”cheapest” to deliver. Short party receives: (Settlement price × Conversion factor) + Accrued interest Cost of purchasing is: Quoted bond price × Accrued interest Cheapest-to-deliver bond is the one which Quoted bond price − (Settlement price × Conversion factor) is the smallest. Options, Futures, Derivatives 10/15/07 back to start 25 Cheapest-to-Deliver Date Consider a short portfolio of three bonds. Assume the most recent settlement price is 93-08. Bond Quoted bond price Conversion Factor 1 99.50 1.0382 2 143.50 1.5188 3 119.75 1.2615 The cost of delivery of the bonds are Bond 1 :99.50 − (93.25 × 1.0382) = $2.69 Bond 2 :143.50 − (93.25 × 1.5188) = $1.87 Bond 3 :119.75 − (93.25 × 1.2615) = $2.12 Cheapest-to-deliver is Bond 2. Options, Futures, Derivatives 10/15/07 back to start 26 Cheapest-to-Deliver, cont. Recall the smallest Quoted bond price − (Settlement price × Conversion factor) deﬁnes the cheapest-to-deliver bond. Consider a short portfolio of four bonds. Assume the most recent settlement price is 101-12. Bond Quoted bond price Conversion Factor 1 125-05 1.2131 2 142-15 1.3792 3 115-31 1.1149 4 144-02 1.4.026 The cost of delivery of the bonds are Bond 1 :125.15625 − (101.375 × 1.2131) = $2.178 Bond 2 :142.46875 − (101.375 × 1.3792) = $2.652 Bond 3 :115.96875 − (101.375 × 1.1149) = $2.946 Bond 4 :144.06250 − (101.375 × 1.4026) = $1.874 Cheapest-to-deliver is Bond 4. Options, Futures, Derivatives 10/15/07 back to start 27 Determining Futures Price Assume that the cheapest-to-deliver and delivery dates are known, the T-Bond futures contract is a futures contract on a security providing the holder with known income. Therefore, rT F0 = (S0 − I) e where I is the present value of the coupons during the life of the futures contract, T is the time until the futures contract matures, and r is the risk-free interest rate applicable to a time period of length T . Example: • Consider a Treasury bond futures contract. It is known that the cheapest-to-deliver bond will be a 12% coupon bond with a conversion factor of 1.4000. • Suppose also that it is known that delivery will take place in 270 days. • Coupons are payable semiannually on the bond • The last coupon date was 60 days ago • Next coupon date is 122 days from now and another in 305 days. • The rate of interest is 10% per annum with continuously compounding Options, Futures, Derivatives 10/15/07 back to start 28 Example, cont. • The current quoted bond price is $120. • Cash price is obtained by adding the proportion of the next coupon payment that accrues to 60 the hold. The cash price becomes: 120 + 60+122 = 121.978 • A coupon of $6 will be received after 122 = 0.3342 years. The present value becomes 365 −0.1×0.3342 6e = 5.803 • Futures contract lasts for 270 = 0.7397 years. 365 • The cash futures price, if the contract were written on the 12% bond, is computed by subtracting oﬀ the accrued interest 148 125.094 − 6 × = 120.242 148 + 35 • From the deﬁnition of the conversion factor, 1.4000 standard bonds are considered equivalent to each 12%. The quoted futures price should be 120.242 = 85.887 1.4000 Options, Futures, Derivatives 10/15/07 back to start 29 Example 2 We do another example of pricing. It is July 30, 2005. The cheapest-to-deliver bond in a September 2005 Treasury bond futures contract is a 13% coupon bond, and delivery is expected to be made on September 30, 2005. Coupon payments on the bond are made on Feb. 4 and Aug. 4 each year. The term structure is ﬂat, and the rate of interest with semi-annual compounding is 12% per annum. The conversion factor for the bond is 1.5. The current quoted bond price is $110. What is the quoted futures price for the contract? Recall F0 = (S0 − I) erT . • There are 177 days between Feb. 4 and July 30 and 182 days between Feb. 4 and Aug. 4. The cash price of the bond is, therefore; 177 110 + × 6.5 = 116.32 182 • The rate of interest with continuous compounding is rC = 2 ln(1 + .12 ) = 0.1165 per year. 2 • A coupon of 6.5 will be received in 5 days = 001366 years time. The present value of the coupon is −0.01366×0.1165 6.5e = 6.490 Options, Futures, Derivatives 10/15/07 back to start 30 Example 2, cont. • The futures contract lasts for 62 days = 0.1694 years. The cash futures price if the contract were written on the 13% bond would be 0.1694×0.1165 (116.32 − 6.490) e = 112.02 • At deliver there are 57 days of accrued interest. The quoted futures price if the contract were written on the 13% bond would be 57 112.02 − 6.5 × = 110.01 184 • Using the conversion factor we get the quoted price of 110.01 = 73.34 1.5 Options, Futures, Derivatives 10/15/07 back to start 31 Eurodollar Futures • The most popular interest rate futures contract in the US is the 3-month Eurodollar futures contract traded on CME. • A eurodollar is a dollar deposited in a US or foreign bank outside the US. • The Eurodollar interest rate is the rate of interest earned on Eurodollars deposited by one bank with another bank. • Essentially the same as the LIBOR (London Interbank Oﬀer Rate) • 3-month Eurodollar futures contracts are contracts on the 3-month Eurodollar interest rate • Allow for investors to lock in an interest rate on $1 million for a future 3-month period. • Delivery months of March, June, September, December for up to 10 years into the future In other words, an investor in 2007 can use Eurodollar futures to lock in an interest rate for 3-month periods that are as far into the future as 2017. Options, Futures, Derivatives 10/15/07 back to start 32 Eurodollar Futures, cont. Options, Futures, Derivatives 10/15/07 back to start 33 Eurodollar Futures, cont. • Consider a March 2005 contract. • Settlement price is 97.63 and the contract ends on the third Wednesday of the delivery month = March 16, 2005. • The contract is marked to market in the usual way until this date. • On March 16, 2005 the settlement price is set equal to 100 − R, where R is the actual 3-month Eurodollar interest rate on that day, expressed with quarterly compounding and an actual/360 day count convention • In particular if the 3-month Eurodollar interest rate on March 16, 2005, turned out to be 2%, the ﬁnal settlement price would be 98. • Final marking to market to reﬂect this settlement price and all contracts are declared closed Options, Futures, Derivatives 10/15/07 back to start 34 Eurodollar Futures, cont. Remarks: • Contract designed so that each basis point (0.01%) results in a gain/loss of $25 for each long/short contract. One basis point up results in a gain of $25 for the long and a loss of $25 for the short. One basis point down results in loss of $25 for the long and a gain of $25 for the short since 1 1, 000, 000 × 0.0001 × = 25 4 • Since the futures quote is 100 minus the futures interest rate, an investor who is long gains when the interest rates fall and an investor who is short gains when interest rates rise. Options, Futures, Derivatives 10/15/07 back to start 35 Eurodollar Futures, cont. Example: On Feb. 4, 2004 an investor wants to lock in the interest rate that will be earned on $5 million for 3 months starting on March 16, 2005. • The investor goes long ﬁve March05 Eurodollar futures contract at 97.63 • On March 16, 2005 the 3=month LIBOR interest rate is 2%, so the ﬁnal settlement price proves to be 98.00. • The investor gains 5 × 25 × (9800 − 9763) = $4625 on the long futures position. • The interest earned on the $5 million for 3 months at 2% is 5, 000, 000 × 0.25 × 0.02 = 25, 000 or $25,000. The gain on the futures contract brings this up to $29,625. This is the interest that would have been earned if the interest rate had been 2.37%, i.e. (5, 000, 000 × 0.25 × 0.0237 = 29, 625). • Thus the interest rate is locked in at 2.37% = 100 - 97.63. Options, Futures, Derivatives 10/15/07 back to start 36 Eurodollar, cont. The exchange deﬁnes the contract price as 10, 000 × (100 − 0.25 × (100 − Q)) where Q is the quote. The settlement price of 97.63 for the March 2005 contract (from our table) corresponds to a contract price of 10, 000 × (100 − 0.25 × (100 − 97.63)) = $994, 075 In the above example, the ﬁnal contract price is 10, 000 × (100 − 0.25 × (100 − 98)) = $995, 000 Diﬀerence between the initial and ﬁnal contract price is $925, so the investor with the long position in 5 contracts gains 5 × 925 = $4, 625. Options, Futures, Derivatives 10/15/07 back to start 37 Homework Due Oct. 3, 5PM. • 5.6, 5.9, 5.15 • Graded: 4.27, 5.24, 5.25, 5.27 Options, Futures, Derivatives 10/15/07 back to start 38

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