Futures and Forward Contracts PDF

CHARACTERISTICS AND EVALUATION OF FUTURES AND FORWARD CONTRACTS D. LAUTIER - UE 109 - 2021/22 1 1 Section 1. Spot - Forwards - Futures Section 2. The principle of arbitrage and the pricing of commodity futures S...

CHARACTERISTICS AND EVALUATION OF FUTURES AND FORWARD CONTRACTS D. LAUTIER - UE 109 - 2021/22 1 1 Section 1. Spot - Forwards - Futures Section 2. The principle of arbitrage and the pricing of commodity futures Section 3. Examples of commodity futures Section 4. The pricing of futures contracts: generalization Section 5. The value of a forward contract D. LAUTIER - UE 109 - 2021/22 2 2 Section 1. Spot – Forward - Futures 1.1. The spot market 1.2. The forward market 1.3. The characteristics of the transactions in the spot and the forward markets 1.4. The evolution from the forward to the futures market 1.5. The futures market 1.6. The conclusion of the futures transactions 1.7. The convergence of the basis 1.8. The differences between the forward and the futures transactions 1.9. Spot prices – forward prices – futures prices D. LAUTIER - UE 109 - 2021/22 3 3  1.1. The spot market (cash market) Underlying market Immediate exchange of the underlying asset Private agreement negotiated between the two counterparties D. LAUTIER - UE 109 - 2021/22 4 4 1.2. The forward market Private agreement negotiated between two counterparties Delivery of the underlying is deferred Firm instrument - Forward purchase (long position on the derivative) - Forward sale (short position on the derivative) The functions of the forward transactions - Hedging of the price risk - Supply and delivery D. LAUTIER - UE 109 - 2021/22 5 5 1.3. The characteristics of the transactions on the spot and the forward markets Physical markets Decentralization Lack of standardization But on early stage: - standardization - centralization D. LAUTIER - UE 109 - 2021/22 6 6  1.4. From forward to futures markets The limits of forward transactions Standardization trend in order to facilitate endorsement Counterparty risk remains Creation of the Clearing House D. LAUTIER - UE 109 - 2021/22 7 7 After a long process, creation of futures markets for some underlying assets The futures market does not replace the spot and forward markets: it is their complementary market Futures markets do not exist for all underlying assets D. LAUTIER - UE 109 - 2021/22 8 8 1.5. The futures market Negotiation of standardized forward contracts: futures contracts Details : - The quality of the underlying asset - The volume (nominal) - The place of delivery - The delivery date Contract rigidity D. LAUTIER - UE 109 - 2021/22 9 9  Role of the Clearing House Liquidity function: Centralization of the exchanges Function of counterparty risk management: - Daily valuation (settlement prices) - Daily margin calls to cover losses - Payment of profits corresponding to gains (Zero sum game) - Deposit payment - Required for all transactions (purchase or sale) - In case you are unable to absorb the losses of a day D. LAUTIER - UE 109 - 2021/22 10 10 Counterparty risk management in OTC markets Bilateral netting Collateralisation / Initial Margin Revaluation of collateral and margin call Going through a clearing house: Cleared OTC product D. LAUTIER - UE 109 - 2021/22 11 11 Bilateral Netting / CCP CC CCP CC CP D. LAUTIER - UE 109 - 2021/22 12 12  Some regulatory elements Three objectives of the G20 after the great crisis (2009, 2011) - Increasing transparency - Mitigating systemic risk - Fighting market abuse Reforms coordinated globally by the Financial Stability Board (FSB) Provisions transcribed in the US: Dodd-Frank / CFTC - CFTC : Commodity Futures and Trading Commission Provisions transcribed in Europe: EMIR / ESMA - EMIR : European Market Infrastructure Regulation - ESMA: European Securities and Markets Authority D. LAUTIER - UE 109 - 2021/22 13 13 Potential difficulties Clearing house "too big to fail”: moral hazard? Going through a Clearing house encourages greater risk-taking? Going through a Clearing house: - learning costs - treasury costs Unbalances in the markets of the assets used for collaterals ? D. LAUTIER - UE 109 - 2021/22 14 14 1.6. The conclusion of a futures contract Physical delivery Compensation (countertrade) Cash settlement D. LAUTIER - UE 109 - 2021/22 15 15  1.7. The convergence of the basis The basis is the difference between the futures price F and the spot price S : F(t,T) – S(t) At the expiry of the contracts T, the futures and spot prices must be identical : F(T,T) = S(T) Absence of equality results in arbitrage opportunities If, a few days prior to the expiry of the futures contract: - The price of the physical = 140 - The futures price = 145 D. LAUTIER - UE 109 - 2021/22 16 16 One will: - Sell a futures contract at 145 - Execute it by delivering the underlying, purchased at 140 - Make a profit of 5 Consequences: 1. Price of physical (underlying) rises 2. Price of contract falls 3. Equality between the two prices is restored D. LAUTIER - UE 109 - 2021/22 17 17 1.8. Differences between forwards and futures FUTURES FORWARD TRANSACTIONS TRANSACTIONS STANDARDIZATION EXTREME NON-EXISTENT EXPIRATION EXCEPTIONAL NORMAL OF CONTRACTS EFFECTIVE DELIVERY EFFECTIVE DELIVERY LIQUIDITY AMPLE ALMOST ZERO INITIAL OUTLAY DEPOSIT NO PAYMENT MARGIN CALLS IMPERFECT PERFECT HEDGING INEXPENSIVE COSTLY REVERSIBLE IRREVERSIBLE D. LAUTIER - UE 109 - 2021/22 18 18  1.9. A brief history of the creation of futures markets Cereals Non-ferrous metals Tropical products Oilseed Animals Currencies Financial fixed income Petroleum products Equity Indices ……. D. LAUTIER - UE 109 - 2021/22 19 19 Key Dates for derivative markets 1850 – 1930 Forward market on agricultural products 1930 – 1970 Futures markets on raw materials 1970-1985 Futures markets on financial assets 1985-2000 The era of OTC markets 2000-….. Towards a new organization of derivatives markets D. LAUTIER - UE 109 - 2021/22 20 20 1.9. Spot price - Forward price - Futures price Physical Market - Immediate delivery - Spot transaction - Spot price As many spot prices as transactions No standardization - Delayed delivery - Forward transaction - Forward price As many forward prices as transactions / maturities No standardization Contract market - Futures transaction - Futures price Standardization As many futures prices as delivery dates D. LAUTIER - UE 109 - 2021/22 21 21  D. LAUTIER - UE 109 - 2021/22 22 22 … D. LAUTIER - UE 109 - 2021/22 23 23 1.9. Spot price - Forward price - Futures price Physical market - with immediate delivery spot transaction spot price - with forward delivery forward transaction forward price - non standardization Contract market futures transaction futures price - standardization Parallelism in the evolution of prices Imperfect parallelism D. LAUTIER - UE 109 - 2021/22 24 24  Section 2. The principle of arbitrage and the pricing of commodity futures 2.1. Contango, backwardation and the theory of storage 2.2. The formulation of the evaluation of commodity futures 2.3. Application example 2.4. The discrepancy between the theoretical equilibrium price of a futures contract and the price observed in the market D. LAUTIER - UE 109 - 2021/22 25 25 2.1. Contango, backwardation and the theory of storage 2.1.1. The basis and its evolution 2.1.2. Contango and storage costs 2.1.3. The cash and carry arbitrage 2.1.4. Backwardation and the convenience yield 2.1.5. The reverse cash and carry arbitrage D. LAUTIER - UE 109 - 2021/22 26 26 2.1.1. The basis and its evolution At any time (except at maturity), there is a difference between the futures and the spot prices This difference (F(t,T) – S(t)) is called basis: If F(t,T) > S(t) : Premium (Contango) If F(t,T) < S(t) : Discount (Backwardation) D. LAUTIER - UE 109 - 2021/22 27 27  Evolution of the temporal basis Premium/ F(t,T) – S(t) Discount F(t + Δt, T) –S(t + Δt) F(T, T) –S(T) t t + Δt T D. LAUTIER - UE 109 - 2021/22 28 28 Evolution of the basis Premium/ Discount January March June D. LAUTIER - UE 109 - 2021/22 29 29 2.1.2. Contango (premium) and storage costs Contango F(t,T) = S(t) + CS(t,T) F(t,T) price of futures contract with maturity T, S (t): spot price at present date t CS (t, T): storage cost between t and T Storage costs: Costs of deterioration and obsolescence Costs and insurance premiums related to storage Financial costs (interest rates) Variations in the value of the product D. LAUTIER - UE 109 - 2021/22 30 30  2.1.3. Cash and carry arbitrage Spot price = 100 Storage costs = 6 Theoretical futures price = Fair price = 106 Observed futures price = 109 Cash and Carry : - Purchase Cash: - 100 - Future sale: +109 - Storage and delivery - 6 + 3 The level of contango is limited to the cost of storage D. LAUTIER - UE 109 - 2021/22 31 31 2.1.4. Backwardation (discount) and the convenience yield If storage costs are positive, how could the futures price be less than the spot price? -There is an income associated with the holding of the physical commodity: the convenience yield D. LAUTIER - UE 109 - 2021/22 32 32 What is the convenience yield ? It is an implicit income Stockholding enables: - To limit the costs and the delays in deliveries - To meet unexpected increases in demand - To ensure continuity of operations Holding goods is especially important when the stocks are rare The convenience yield is: - High when stocks are scarce - Low when stocks are abundant D. LAUTIER - UE 109 - 2021/22 33 33  - The convenience yield must be deducted from the costs of storage - When the convenience yield is greater than the costs of storage, the market is in backwardation D. LAUTIER - UE 109 - 2021/22 34 34 The relation between the spot and the futures prices becomes: F(t,T) = S(t) + CS(t,T) - CY(t,T) F(t,T): futures price at t, of the futures contract with maturity T, S (t): spot price at t, CS(t,T): storage costs between t and T, CY(t,T): convenience yield between t and T. D. LAUTIER - UE 109 - 2021/22 35 35 2.1.5. Reverse cash and carry arbitrage: Spot price = 105 Storage cost = 3 Convenience yield = 8 Theoretical futures price = Fair price = 100 Observed futures price = 97 Reverse Cash and Carry: - Cash selling : + 105 - Futures purchase : - 97 - Storage cost : + 3 - Convenience yield : - 8 Limits to arbitrage : available stock + 3 D. LAUTIER - UE 109 - 2021/22 36 36  Conclusion Inventories are abundant in contango (premium) - The arbitrage mechanism works well - Contango is stable and limited to storage costs Inventories are rare in backwardation (discount) - The arbitrage mechanism works less well - Backwardation is less stable; subjective limit D. LAUTIER - UE 109 - 2021/22 37 37 2.2. The pricing of futures in physical commodities: formulation S(t) = spot price of the underlying asset F(t,T) = price of the futures contract Cp (t,T) = net cost of carry expressed in absolute terms c(t,T) = net cost of carry expressed as an annual percentage t = present date T = expiration of the contract T - t = time to maturity of the contract (maturity) D. LAUTIER - UE 109 - 2021/22 38 38 F(t,T) =S(t) + CS(t,T) – CY(t,T) = S(t) + Cp(t,T) Cost of carry as a simple interest rate: F(t,T) = S(t) + S(t) [ ct (T - t) ] = S(t) [ 1+ ct (T - t) ] Cost of carry as a compounded interest rate: F(t,T) = S(t) (1 + ct)T-t Cost of carry as a continuously compounded rate F(t,T) = S(t) exp (ct(T-t)) D. LAUTIER - UE 109 - 2021/22 39 39  Reminder: from discrete time to continuous time Valuation in discrete time for a maturity (T-t), cost of carry c and spot price S Decomposition into m sub-periods: Valuation in continuous time for maturity (T-t): D. LAUTIER - UE 109 - 2021/22 40 40 2.3. Application example Evaluation of a futures contract with a maturity of three months - Spot price = 100 - 3-month interest rate (cost of funds) = 10% - Holding cost (pure storage cost) = 1% - Income from tangible assets (convenience yield) = 5% Net annual cost of carry (for a period of 3 months): c = 10 + 1 - 5 = 6% D. LAUTIER - UE 109 - 2021/22 41 41 The price of the futures contract will be: D. LAUTIER - UE 109 - 2021/22 42 42  2.4. The discrepancy between the theoretical equilibrium price of a futures contract and the market price The net cost of carry is not determined with certainty Presence of imperfections: - Transaction costs - Commissions - Deposit and margin calls Expectations and speculation D. LAUTIER - UE 109 - 2021/22 43 43 Real evolution of the basis Theoretical evolution of the basis t t+1 t+2 t+3 t+4 t+5 t+6 t+7 t+8 t +9 D. LAUTIER - UE 109 - 2021/22 44 44 Section 3. Some examples of commodity futures Oil contracts traded on Nymex Light sweet crude oil Other contracts Wheat contracts traded in the United States D. LAUTIER - UE 109 - 2021/22 45 45  Light sweet crude oil Trading Unit: 1,000 U.S. barrels (42,000 gallons). Price Quotation: U.S. dollars and cents per barrel. Trading Hours: (All times are New York times) - Electronic trading is conducted from 6:00 PM until 5:00 PM via the CME Globex® trading platform, Sunday through Friday. There is a 60-minute break each day between 5:00 PM (current trade date) and 6:00 PM (next trade date). D. LAUTIER - UE 109 - 2021/22 46 46 Trading Months: The current year and the next six years. A new calendar year will be added following the termination of trading in the December contract of the current year Minimum Price Fluctuation: $0.01 (1¢) per barrel ($10.00 per contract). Maximum Daily Price Fluctuation: $10.00 per barrel ($10,000 per contract) for all months. If any contract is traded, bid, or offered at the limit for five minutes, trading is halted for five minutes. D. LAUTIER - UE 109 - 2021/22 47 47 Delivery F.O.B. seller's facility, Cushing, Oklahoma, at any pipeline or storage facility with pipeline access to Enterprise, Cushing storage, or Enbridge, Cushing storage, by in-tank transfer, in-line transfer, or inter-facility transfer (pumpover). D. LAUTIER - UE 109 - 2021/22 48 48  Deliverable Grades Domestic Crudes (Deliverable at Par) 1.Deliverable Crude Streams : West Texas Intermediate, Low Sweet Mix, New Mexican Sweet, North Texas Sweet, Oklahoma Sweet, South Texas Sweet. - Sulfur: 0.42% or less by weight as determined by ASTM Standard D4294, or its latest revision; - Gravity: Not less than 37 degrees American Petroleum Institute (“API”), nor more than 42 degrees API as determined by ASTM Standard D 287, or its latest revision; - Viscosity: Maximum 60 Saybolt Universal Seconds at 100 degrees Fahrenheit as measured by ASTM Standard D445 and as calculated for Saybolt D.Seconds by ASTM Standard D LAUTIER - UE 109 - 2021/22 49 - …. 49 Foreign crudes - U.K.: Brent Blend (for which the seller shall be paid a 30 c. per b. discount below the last settlement price) - Nigeria: Bonny Light (for which the seller shall be paid a 15 cent per barrel premium above the last settlement price) - Nigeria: Qua Iboe (for which the seller shall be paid a 15 cent per barrel premium above the last settlement price) - Norway: Oseberg Blend (for which the seller shall be paid a 55 cent per barrel discount below the last settlement price) - Colombia: Cusiana (for which the seller shall be paid 15 cent per barrel premium above the last settlement price) - ….. D. LAUTIER - UE 109 - 2021/22 50 50 Inspection Inspection shall be conducted in accordance with pipeline practices. A buyer or seller may appoint an inspector to inspect the quality of oil delivered. However, the buyer or seller who requests the inspection will bear its costs and will notify the other party of the transaction that the inspection will occur. D. LAUTIER - UE 109 - 2021/22 51 51  Oil contracts traded on Nymex (CME Group) Heating Oil Gasoline Crude Oil Volume 1,000 barrels 1,000 barrels 1,000 barrels $ 0.0001 per $ 0.0001 per 1 cent per Price quotation gallon or $ 4.2 per gallon or $ 4.2 per barrel, or $10 contract contract per contract Maximum price $ 0.2 per gallon $ 0.2 per gallon $ 7.5 per barrel for fluctuation for the nearest 2 for the nearest 2 the nearest 2 months; months months; $ 0.06 for other ; $ 0.06 for other $ 3 for other maturities maturities maturities D. LAUTIER - UE 109 - 2021/22 52 52 Heating Oil Gasoline Crude Oil Last day of Last business day Last business day 3 working days transaction of the month of the month before the 25th of preceding the preceding the the month delivery month delivery month preceding the delivery month Point of Cushing, New York New York delivery Oklahoma Monthly Monthly Consecutive contracts listed contracts listed months for the Expiry dates for the current for the current current and the year and the year and the next five years, next 3 calendar next 3 calendar then maturities at years + 1 month years + 1 month 7, 8, 9 years D. LAUTIER - UE 109 - 2021/22 53 53 Wheat contracts traded in the United States CME Group Kansas City BOT Wheat N°2 Northern Spring N°2 Hard Red Winter quality N°2 Soft Red N°2 Hard Red Winter N°2 Dark Northern Spring Volume 5,000 bushels 5,000 bushels Price ¼ cent per bushel, or $12.5 ¼ cent per bushel, or $12.5 quotation per contract per contract D. LAUTIER - UE 109 - 2021/22 54 54  Wheat contracts traded in the United States CBOT Kansas City BOT Max limit of price 35 cents per bushel 35 cents per bushel fluctuation March, May, July, March, May, July, Expiry dates September, December September, December Last The business day prior to The business day transaction the 15th calendar day of the preceding the 15th calendar day contract month day of the month D. LAUTIER - UE 109 - 2021/22 55 55 Section 4. The pricing of futures contracts on financial assets 4.1. Futures on zero coupon bonds 4.2. Futures on coupon-bearing bonds 4.3. Futures on equities and equity indices D. LAUTIER - UE 109 - 2021/22 56 56 4.1. Futures on zero coupon bonds F = S + Cp Theoretical price using a simple interest rate: F = S[ 1 + r (T - t) ] Theoretical price using a compounded interest rate: F = S (1 + r)T-t Theoretical price using a continuously compounded rate: F = S e r(T-t) D. LAUTIER - UE 109 - 2021/22 57 57  Example r = 8% S = 90 T - t = 1 year F* = 90 (1+8%) = 97,2 F* = 90 exp (0,08) = 97,49 D. LAUTIER - UE 109 - 2021/22 58 58 4.2. Futures on coupon-bearing bonds Formulation : F*= theoretical equilibrium price of the futures contract F = futures price observed in the market S = spot price of the bond r = the interest rate applicable to lending and borrowing operations until maturity y = the coupon rate of the bond until the maturity date, calculated from the nominal rate D. LAUTIER - UE 109 - 2021/22 59 59 Arbitrage At t : Borrowing S on the money market rate Bond purchase at price S Sale of a futures contract with maturity T at a price F At T (1 year later) : Actual delivery of the underlying asset at price F Collection of accrued interest: yS Repayment of loan: S + rS D. LAUTIER - UE 109 - 2021/22 60 60  Result: Amount received : F + yS Amount paid : S + rS At market equilibrium, the arbitrage profit is zero: (F + yS ) - (S + rS) = 0 The theoretical price of the futures contract is: F* = S + S (r - y) More generally : F* = S (1+ (r – y)(T-t)) F* = S (1 + (r-y))T-t F*= S e(r-y)(T-t) D. LAUTIER - UE 109 - 2021/22 61 61 4.3. Futures on equities and equity indices: Formulation S = price of the equity or equity index on the spot market F* = theoretical price of futures contract r = risk-free interest rate d = dividend rate D. LAUTIER - UE 109 - 2021/22 62 62 Simple interest rate: F* = S [1 + (r - d)(T - t)] Compounded interest rate: F* = S [1 + (r - d)]T – t Continuously compounded interest rate: F* = S e(r-d)(T – t) D. LAUTIER - UE 109 - 2021/22 63 63 Example Price of index CAC 40 = 6,000 d = 4% r = 6% T - t = 6 months D. LAUTIER - UE 109 - 2021/22 64 64 The futures price and the income associated with holding the underlying asset Commodities: convenience yield Coupon bearing bonds: coupons Stock index and equity: dividends Currencies: remuneration at the money market rate D. LAUTIER - UE 109 - 2021/22 65 65 Section 5. The value of a forward contract 5.1. Value of a contract at maturity 5.2. Value of a contract at its initiation 5.3. Value of a contract before its expiry D. LAUTIER - UE 109 - 2021/22 66 66  Market value For the financial statements (mark-to-market), or for an internal report For a speculator who wants to close his position For a hedger whose need for hedging disappeared D. LAUTIER - UE 109 - 2021/22 67 67 5.1. Forward contract at maturity 5.1.1. Contract purchase In January Buy a forward maturing in June at a price F = 100 In June: H1. The spot price of the underlying is S = 110 The forward contract involves paying 100 for something that costs 110. Its value is: 110 - 100 = 10 H2. The spot price is S = 85 It would have been better to buy the underlying on the spot market. The forward contract involves paying 100 for something that costs 85. Its value is: 85 - 100 = - 15 68 D. LAUTIER - UE 109 - 2021/22 68 At the end of the contract, its value is either positive or negative Symmetrical nature of the instrument Let us note: - fT: the contract value at maturity T - S (T): the spot price of the underlying at maturity - F (0, T): the forward price at the purchase of the contract at time 0 At T, for a long position, the contract value is: fT = S(T) - F(0,T) D. LAUTIER - UE 109 - 2021/22 69 69  Profile of profits and losses at the expiry of a forward contract, long position fT = S(T) - F(0,T) Profit 45° F(0,T) Price of underlying asset S(T) Loss D. LAUTIER - UE 109 - 2021/22 70 70 5.1.2. Sale of a forward contract In March, sell a forward contract at price F(0,T) = 80, with September maturity In September: - H1. The spot price is S(T) = 120 The value of the forward contract is negative: 80-120 = - 40 - H2. The spot price is S(T) = 70 The value of the forward contract is positive: 80 - 70 = 10 At time T, for a short position, the contract value is: fT = F(0,T) – S(T) D. LAUTIER - UE 71 109 - 2021/22 71 Profile of profits and losses at the expiry of a forward contract, short position Profit F(0,T) Price of underlying asset S(T) Loss fT = F(0,T) – S(T) D. LAUTIER - UE 109 - 2021/22 72 72  Zero-sum game Buyer: Profit fT = S(T) - F(0,T) F(0,T) Price of underlying S(T) Loss Seller: D. LAUTIER - UE 109 - 2021/22 fT = F(0,T) – S(T) 73 73 5.2. Contract value at initiation At time 0, the value of the negotiated contract: f0 Forward price at the time of contract: F(0,T) What would the value of this contract be, if it was sold instantly? Forward price at the time of immediate resale: F(0,T) At the time of its purchase, at 0, the forward contract’s value for a long position: f0 = F(0,T) - F(0,T) = 0 The same is true for a short position D. LAUTIER - UE 109 - 2021/22 74 74 Fundamental principle: the value of a forward contract is zero, at its initiation This is also true for a futures contract. Essential difference with option contracts D. LAUTIER - UE 109 - 2021/22 75 75  5.3. Valuation of a contract before its expiry 5.3.1. Long position: contract purchase In January, buy a forward maturing in June at a price: F (0, T) = 100 In March: the agent wants to sell the contract In March, the price of the forward maturing in June is: F (t, T) = 80 What is the March value of the contract expiring in June that was purchased in January? This value depends on: - Changes in prices between January and March - The time left until June D. LAUTIER - UE 109 - 2021/22 76 76 By cancelling the contract today, the buyer avoids (gives up) a future loss (profit) The present value of the future loss (profit) depends on: - F(0,T) the delivery price associated with the forward contract that has been bought at time 0 - F(t,T) the forward price associated with this contract today The value ft of this contract, for the buyer who wants to sell it, is: - r is the interest rate D. LAUTIER - UE 109 - 2021/22 77 77 If in March, the price of the forward contract for June maturity is F = 80, the buyer may wish to cancel his position The interest rate for 3 months is 1% Value of the position: (80 – 100) exp (- 0.01 x 0.25) = - 19.95 The buyer pays 19.95 to his/her counterparty to exit his/her position Counterparty risk on this position The buyer is found in a speculative position D. LAUTIER - UE 109 - 2021/22 78 78  Valuation of a forward contract prior to maturity, short initial position Let us note: - F(0,T) the delivery price associated with a forward contract, sold at time 0 - F (t, T) the forward price associated with this contract today The value ft of the purchased contract is: D. LAUTIER - UE 109 - 2021/22 79 79 Example: Value of a long position in a forward contract - September 16, N, forward purchase of 300,000 barrels of oil, at a price F (0, T) = $ 110, due in June N + 1 H1. On December 16, the forward price for delivery in June N +1 is: F (t, T) = $ 80 / b - The buyer wishes to close his/her position - 6 months interest rate: r = 5% Value of the long position on the forward contract: ft = 300,000 x (80 - 110) x exp(-0.05 x 0.5) = - 8,777,789.21$ The forward contract is out of the money : ft < 0 D. LAUTIER - UE 109 - 2021/22 80 80 - H2: Forward price: F(t,T) = 135 $/b Value of the long position in the forward contract: ft = 300,000 (135 – 110) x exp(-0.05 x 0.5) = 7,314,824.34$ Forward contract is in the money: ft > 0 - H3: Forward price: F(t,T) = 110 Value of the long position on the forward contract: 300,000 (110 – 110) x exp(-0.05 x 0.5) = 0 Forward contract is at the money: ft = 0 D. LAUTIER - UE 109 - 2021/22 81 81