2 - Futures.docx

CHAPTER 2: THE CHARACTERISTICS AND EVALUATION OF FUTURES CONTRACTS Spot – forwards – futures The spot market (cash market) Underlying market (marché sous-jacent) Immediate exchange of the underlying asset Private agreement negotiated between the two counterparties (buyer and seller) Ex : stock m...

CHAPTER 2: THE CHARACTERISTICS AND EVALUATION OF FUTURES CONTRACTS Spot – forwards – futures The spot market (cash market) Underlying market (marché sous-jacent) Immediate exchange of the underlying asset Private agreement negotiated between the two counterparties (buyer and seller) Ex : stock market is a spot market The forward market • Private agreement negotiated between two counterparties • Delivery of the underlying is deferred • Firm instrument : ie it’s a fixed derivative. We agree on a future purchase / a future sale. • The functions of the forward transactions Hedging of the price risk Driving by supply and the facilitation of delivery Since delivery is going in the future, what comes with the forward market is uncertainty. The forward market was previous to the future markets. Terminology: Long position = buy Short position = sell The characteristics of transactions in the spot and the forward markets • Physical markets at the beginning, but as time went, the forward market has gone decentralized (computers, phone calls…) • Lack of standardization • But on early stage: standardization centralization The evolution from the forward market to the futures market • The limits of forward transactions : uncertainty (counterparty risk, although 67% counterparts are now backed by CCP) , opacity, sometimes not very liquid • Standardization trend in order to facilitate endorsement • Creation of the Clearing House since more regulations • 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. They serve different needs. Assets are not all exchanged in both markets. Futures markets do not exist for all underlying assets The futures market • Negotiation of standardized forward contracts: futures contracts. Some contracts are very specified, with a lot of details. • Details : The quality of the underlying asset The volume (nominal) The place of delivery The delivery date Especially commodity features need these specificities. Contract rigidity: once you enter the deal you cannot exit it until delivery. 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. You need to pay a deposit amount to be able to make the transaction (no matter buy or sell). Then the trading stocks move on the market, then your positions makes gains or losses. If you make gains, your deposit account grows. If you’re making a loss, you have to put money in your account. The losses are accumulating in the deposit account gradually until maturity. Purpose: monitor the exposure, so that the Clearing House is covered. 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 A margin call happens when a broker demands that an investor deposits additional money or securities so that the margin account is brought up to the minimum maintenance margin. A margin call occurs when the account value falls below the broker's required minimum value. Counterparty risk management in OTC markets • Bilateral netting • Collateralization (put an asset to guarantee your transaction) / Initial Margin (65% of transactions in 2007 and 71% in 2011) • Revaluation of collateral and margin call • Going through a clearing house: Cleared OTC product (67% in 2016) The CCP simplifies the process in case of several agents making transactions. Some regulatory elements: • 3 objectives of the G20 (2009, 2011) Increasing transparency Mitigating systemic risk (general market risk to which all operators are exposed, linked to the economy and not to a particular asset) for the sake of all market participants. Fighting market abuse, especially in high speed trading : limit unfair operations (ex: insider trading / buy a lot of stocks in order to make the price increase) • Reforms coordinated globally by the financial stability board (FSB) Provisions transcribed in the US: Dodd-Frank / CFTC (Trump wants to abolish it) Provisions transcribed in EU : EMIR / ESMA EMIR: EU market infrastructure regulation ESMA: EU securities and markets authority Potential difficulties • Clearing house is extremely important: “too big to fail”: moral hazard. The exposure in extreme days can be very important, and it is needed to be strict in the monitory of the clearing house. • Going through a Clearing House encourages greater risk-taking ? • Cost of going through a clearing house: Learning Treasury • Imbalances in the markets for collaterals It has become more expensive over the years to trade with a clearing house The conclusion of the transaction • Physical delivery of the asset Or • Compensation (countertrade) : prior to the delivery, you sell the contract to another counterparty and your position goes to 0. Compensation trade is a form of barter in which one of the flows is partly in goods and partly in hard currency Or • Cash settlement: you exchange the difference of money between the order and the real price Over time, the % of deliveries collapsed. Because huge impractibility of physical deliveries. The convergence of the basis The basis is the difference between the futures price F and the spot price S : F − S On expiry of the contracts, the futures and spot prices must be identical Absence of equality results in arbitrage opportunities : Ex: If, a few days prior to the expiry of the contract: The price of physical = 140 The futures price = 145 There is an arbitrage opportunity: you sell the expensive one and buy the cheap one. Sell a contract at 145 Execute by delivering 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 Future price is very close to the spot price during the delivery period. The differences between the forward and the futures transactions Initial outlay = how much money to pay in order to enter the instruments Liquidity in the futures market because of standardized exchanges. No liquidity on forwards market because every participants decides what they want to trade, you need to find somebody who wants to enter a particular position. Livraison physique des contrats futures fait dans seulement 1% des cas. Alors que c’est la norme pour le marché forward. Initial outlay: mise de fond initial: pour futures deposit et margin calls (qu’on verse tous les jours), alors que pour forward rien ne nous force à le faire même si ca se démocratise. Hedging : couverture : Futures : imperfect pour hedging parce que extrêmement standardisée mais du coup elle est inexpensive parce que elle est reversible (on peut facilement annuler sa position) Forward : perfect pour la couverture mais costly parce que elle est irreversible. Futures: Given the standardized instruments and particular business needs, the hedging of futures transactions is imperfect. You approximate volume and quality of the asset. Inexpensive because liquid market. Reversible: you can close the deal prior to maturity. Perfect hedging of forward transactions: No mismatch in volumes, delivery dates and underlying asset A brief history of the development of futures market Started with agricultural products, then currencies, financial fixed income, petroleum products, and then equity indices. The primary function of futures market was then hedging agricultural products. Key dates for derivatives markets: • 1850 – 1930 : forward markets 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, big increase in volumes, extreme popularity until the crisis. Since the crisis, more regulations in order to increase transparency and safeguard the agents. Spot prices – forward prices – futures prices • Physical market : with immediate delivery so we call it spot transaction with a spot price No standardization so as many spot prices as transactions no standardization with delayed delivery we call it forward transaction with a forward price as many forward prices as transactions and also maturities no standardization • Contract market: futures transaction with futures price standardization as many futures prices as delivery dates but not transaction because oil is oil no different types of oil • Parallelism in the evolution of prices : entre le marché physique et dérivé les prix varient de manière parallèle : si prix blé augmente, le futures ou forward blé augmente aussi. • Imperfect parallelism : un prix à 1 mois n’évolue pas de la même manière qu’un prix à 10 ans : donc la tendance va être la même (hausse ou baisse) mais pas un parfait parallélisme. Tableau prix: prior settle: prix du jour d’avant open : opening price volume : nombre de contrat négociés depuis l’ouverture The principle of arbitrage and pricing of commodity futures Contango, backwardation and the theory of storage The basis and its evolution Contango: descendant VS backwardation: ascendant Très important de regarder ce qui est surligné dans le chapitre 5. • 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 (appelée la base): If F > S : Premium (for commodities: contango market): cad la base est positive on parle de premium ou de contango (pour les commodities). If F < S : Discount (for commodities: backwardation market) Pour le graph : Evolution of the temporary basis The premium / discount is the basis. Ce que l’on voit c’est qu’en t (date d’achat), la différence entre F et S est maximale, ensuite on avance jusqu’à T la date de maturité où S=F. C’est expliqué par le fait qu’on a plus d’information et donc on est plus précis et par le fait que les couts de stockages sont plus faibles vu qu’on va les stocker moins longtemps. Evolution of the basis : The longer the maturity is, the bigger the basis is. Contango and storage costs Contango or premium is a situation where the futures price (or forward price) of a commodity is higher than the expected spot price: cad la base est positive cad F>S. Contango/premium : F(t, T) = S(t) + CS(t, T)  • F(t,T) price of futures contract with maturity T, • S (t): spot price in at present date t • CS (t, T): storage cost between t and T Storage costs: Costs of deterioration and obsolescence: perte de qualité, péremption par exemple Costs and insurance premiums related to storage: pour le pétrole c’est inflammable ca coûte en assurance Financial costs (directement associés aux taux d’intérêts): si j’achète du pétrole, l’argent que j’ai utilisé je ne peux plus l’utiliser pour d’autres investissements donc j’ai un cout financier d’immobilisation. Variations in the value of the product itself (product is not homogeneous by nature). The cash and carry arbitrage Cash-and-carry-arbitrage = market neutral strategy combining the purchase of a long position in an asset such as a stock or commodity, and the sale (short) of a position in a futures contract on that same underlying asset. J’achète cash et ensuite je porte (je stock) l’actif pour le livrer. Spot price = 100 Storage costs = 6 Theoretical futures price = Fair price = F= 106 Observed futures price = 109 qu’est ce qu’on a intérêt à faire ? Pour tous les arbitrages : on vend ce qui est chère on achète ce qui est pas chère On va faire un Cash (achat au comptant) and Carry (je vais stocker et livrer l’actif) arbitrage : on va acheter cash au comptant 100 : Purchase Cash: - 100 Futures sale: +109 Storage and delivery - 6 NET PROFIT = 3 The arbitrage opportunity comes from the difference between the fair price and the observed price. The level of contango is limited to the cost of storage Discrepancy = écart Backwardation 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 holding the physical commodity which is called the convenience yield. Il y a un rendement associé à la détention d’un actif, c’est le convenience yield. You want to keep the commodities even if storage costs When convenience yield is higher than storage costs : spot price is greater than the futures price. Discount (backwardation). What is the convenience yield ? On considère le sucre: on a du sucre chez sois, pourquoi ca nous rapport de l’argent de l’avoir chez sois ? Ca nous réduit nos couts de livraison (se rendre au supermarché) Convenience yield is the benefit or premium associated with holding an underlying product or physical good, rather than the associated derivative security or contract. • It is an implicit income • Stockholding (avoir du stock) enables: To limit the costs and the delays in deliveries: pour le sucre se rendre au supermarché To meet unexpected increases in demand To ensure continuity of operations: mon café je l’aime chaud, donc il me faut du stock pour pouvoir toujours avoir un sucre et un café chaud Holding goods is especially important when the stocks are rare : • The convenience yield is: High when stocks are scarce (rare) Low when stocks are abundant • The convenience yield must be deducted of the costs of storage • When the convenience yield is greater than the costs of storage (ce qui se passe si l’actif est rare pas du tout abundant), the market is in backwardation because the futures price becomes lower than the spot price. The relation between the spot and the futures prices becomes: F(t, T) = S(t) + CS(t, T) − CY(t, T)  • F(t,T) : price in t of 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. The reverse cash and carry arbitrage Les opérations d’arbitrage pour un marché en backwardation est reverse cash and carry. 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 (on vend le produit au comptant) Futures purchase : -97 Storage cost : +3 (je n’ai plus les couts de stockage) Convenience yield : -8 (I’m losing the ability to hold on to the commodity, I’m giving it to the upcoming holder, donc je perds le convenience yield) INCOME = 3 Limits to arbitrage : available stock (cash selling). We might end up in a situation where we don’t have enough commodity for a short position. In the exercise, it could be an issue (high convenience yield donc ca veut dire que la resource était déjà scarce). In a backwardation environment, stocks are not abundant and we’re limited to the amount of commodity to be shorted. 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 (par exemple on a une crainte sur l’approvisionnement d’une ressource) The pricing of futures in physical commodities: formulation Cost of carry = costs incurred as a result of an investment position. These costs can include financial costs, (ex: interest costs on bonds, interest expenses on margin accounts, interest on loans used to purchase a security).  S(t) = spot price of the underlying asset at t F(t,T) = price of the futures contract at t for T Cp (t,T) = net cost of carry expressed in absolute terms: le cout de portage nette exprimé en valeur absolue, c’est le cout de stockage moins convenience yield c(t,T) ou ct = net cost of carry expressed as an annual percentage t = the present T = expiration of the contract T - t = time to maturity of the contract (maturity) F(t, T)  = S(t) + CS(t, T) – CY(t, T) = S(t) + Cp(t, T)  • Cost of carry as a simple interest rate (taux d’interet simple) : F(t,T)= S(t)+ S(t)[ ct (T − t)] = S(t) * [1 + ct(T − t)] C’est donc un pourcentage du prix spot • Cost of carry as a compounded interest rate (intérêts composés) : F(t,T) = S(t) * (1+ct)T − t  • Cost of carry: continuously compounded F(t,T) = S(t) * exp[ct (T−t)] On va souvent utiliser le temps continue en finance To transform rate r with compounding m times per annum to continuous compounding: $k = m*ln(1 + \frac{r}{m})$ Reminder: from discrete-time to continuous time : on va utiliser les limites • Valuation in discrete time for a maturity (T-t), cost of carry c and spot price S: S(t) * (1+ ct)T  − t • Decomposition in m sub-periods : $S\left( t \right)*\left( 1 + \frac{\text{ct}}{m} \right)^{m*\left( T - t \right)}$ On va faire le passage à la limite : on passe du temps discret au temps continue ${lim\ (m\ \rightarrow infini)}\left( 1 + \frac{\text{ct}}{m} \right)^{m*\left( T - t \right)} = e^{\text{ct}*\left( T - t \right)}$ • Valuation in continuous time for maturity (T-t): S(t) * ect * (T−t) 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) (cout de portage) : c = 10 + 1 − 5 = 6% MAIS le 6% c’est un taux annuel donc ensuite il faut multiplier par 3/12 même si on dit que c’est le taux pour le portage 3 mois. The price of the futures contract will be: Simple interests: F = 100 * [1+ (0, 06 * 3/12)] = 101, 50  Compounded interests: $F = 100*\left( 1 + 0,06 \right)^{\frac{3}{12}} = 101,467\ $ Continuous interests : F  = 100 * e (0, 06 * 3/12)  = 101, 51  If it accumulated interest for 6 months: Simple interests : $F = 100*(1 + \left( 0,06*\frac{6}{12} \right) = 103$ Compounded interests: F =$100*\left( 1 + 0,06 \right)^{\frac{6}{12}} =$102,956 Continuous interests : F = $100*e^{0,06*\frac{6}{12}}$= 103,04 The discrepancy (écart) between the theoretical equilibrium price of a futures contract and the price observed in the market The discrepancy is explained by : • The net cost of carry is not determined with certainty: le cout de portage nette n’est pas determine avec certitude c’est subjectif. • Presence of imperfections in the market: transaction costs, commissions, deposit and margin calls: il y a des choses dans le marché qu’il n’y a pas dans la formule (commission par exemple : quelle est la commission de la chambre de compensation ?). • Expectations and speculation: we cannot be sure about particular values for a particular asset The basis (la base: F-S) va enfaite énormément bouger. Et plus on est loin de l’échéance plus la différence entre la base réelle et la base théorique va être grande. Avec les opérations d’arbitrages on va osciller entre des périodes de surévaluations et des périodes de sous évaluations ; et plus on se rapproche de la maturité plus les opérations d’arbitrages vont être faciles à saisir : moins couteuses. Examples of commodity futures • Oil contracts traded on Nymex: standardization because futures Light sweet crude oil (pétrole léger) Trading Unit : 1,000 U.S. barrels (42,000 gallons). Price Quotation : U.S. dollars and cents per barrel. Maximum Daily Price Fluctuation $10.00 per barrel ($10,000 per contract) for all months. If any contract is traded, bid, or offered at the limit for five minutes, trading is halted for five minutes. Why need of max price fluctuation ? In case of extreme volatility (ex attaque de drone en Arabie Saoudite), to defer market squeeze. When big price fluctuations we stop trading activities. Par exemple, avec attaque de drone, le prix fluctue trop, donc la chambre de compensation met en place la limite de 10$ et quand on atteind ces 10$ on coupe le marché. La chambre de compensation met une limite parce que elle veut s’assurer que tout le monde va payer ses pertes, pour couvrir ces pertes elle a les deposit, mais ces deposit (dans le cas du pétrole) ne couvre que 10$ de fluctuation max sur 1 jour. Donc quand on atteind ces 10$ soit elle fait sortir les acteurs de leur contrat soit les acteurs doivent augmenter leur déposit. Delivery : Free.Onwards.B. seller's facility (=the seller is not paying for the transfer but the buyer), For foreign crudes, the seller has to pay more or less. In case of less, the seller compensates. If higher, the price goes down. Need to all agree on the price. Si on ne peut livrer que du brent UK alors que le contrat était sur brent US, la chambre de compensation à fixer des prix de compensation : par exemple ici c’est 30 cents le barrel (slides). Disagreement can still occur. That is why there is 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. Tout ca on le dit pour comprendre le role des chambres de compensation et pour montrer à quel point les futures sont standardisés : tous les cas sont prévus. Other contracts • Wheat contracts traded in the United States The pricing of futures contracts: generalization Futures on zero coupon bonds : on rembourse tout à échéance F = S + Cp  Cp: pour le zéro il n’y a pas de convenience yield mais il y a cout de stockage. On considère que cout de portage est donc toujours positif. • 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 * er * (T−t) Important variable: interest rate Example over the course of 1 year: r = 8% S = 90 T - t = 1 year F *  = 90 * (1+8%) = 97, 2 F *  = 90 * exp (0,08) = 97, 49 F*= valeur théorique Futures on coupon-bearing bonds: tu recois interet tous les ans 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 On part d’une operation d’arbitrage cash and carry at t : • Borrowing S on the money market at rate r • 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 : donc on livre le bond • Collection of accrued interest: yS • Repayment of loan: S + rS Result: • Amount received : F + yS • Amount paid : S + rS At market equilibrium (s’il n’y a pas d’opportunité d’arbitrage), the arbitrage profit is zero: (F+yS) − (S+rS) = 0 D’où, the theoretical price of the futures contract is: F* = S + S(r − y) More generally : F* = S (1+ (r – y)(T-t))  = S(1+(r−y))T − t = S*e(r − y)(T − t) 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 • 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) Example • Price of index CAC 40 = 6000 • d = 4% • r = 6% • T - t = 6 mois cout portage ici est r-d=2 donc on est en report (contango) F = 6 000×[1+ (0,06− 0,04)*6/12]= 6160 F = $6000*\left\lbrack 1 + \left( 0,06 - 0,04 \right)^{\frac{6}{12}} \right\rbrack = 6059,7$ F = 6000 * e^((0,06-0,04)*6/12)= 6060,3 The value of forwards contract Il y a une difference entre la valeur du contrat et son prix. La valeur de marché est importante pour : • For the financial statements (bilan, compte résuultat…), or for an internal report • For a speculator who wants to close his position • For a hedger qui veut se retirer de sa position. Ce qui est dit pour forward est vrai pour futures. The price of a futures contract 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 Value of a contract at maturity Contract purchase On se place dans la position d’achat d’un contrat. In January Buy a forward maturing in June at a price F = 100 (c’est le prix, mais quelle est la valeur pour moi du contrat ?) : donc je m’engage à acheter en juin au prix 100. In June: 1er option: The spot price of the underlying is S = 110 (good news because we have made an agreement to purchase that is less than its current market price, we are going to buy cheaper). The forward contract involves paying 100 for something that costs 110. Its value is: 110 – 100 = 10 Donc la valeur de mon contra test 10 ici. 2ème option: 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 Donc la valeur du contrat est négative. Hence: • At the end of the contract, its value is either positive or negative. Symmetrical nature of the instrument : soit c’est positif soit c’est négatif: cf graphic • An agent who wants to hedge against the risk of price increases purchases a contract (long position) - If, at maturity, price increases occur, the contract has a positive value - If, at maturity, price increases do not occur, the contract has a negative value • Let us note: fT : the contract value at maturity T S (T) : the value 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) On a angle à 45° parce que pour 1€ de prix en plus, tu gagnes 1€ en plus, c’est symmetrique. For any price above the strike price, you make money. As the price of the underlying goes down, you end up losing money. Main gains = infinity. Maximum loss = the strike price F Sale of a forward contract (short position) In March, sell a forward contract at price F (0,T) = 80, with September maturity In September: 1er option: The spot price is S = 120 (bad news) It would have been better to sell at 120 rather than at 80 The value of the forward contract is negative: 80-120 = - 40 2ème option: The spot price is S = 70 The contract allows to sale at 80 which is higher than the market price of 70 The value of the forward contract is : 80 – 70 = 10 • The profile of the contract seller is perfectly symmetrical to that of the buyer • Let us note : fT the contract value at maturity T S (T) the value of the underlying at maturity F (0, T) the forward price at the contract sale, at time 0 At time T, for a short position, the contract value is :f(T) = F(0,T) − S(T) As the price moves around the strike price, the total gains to the system reach 0. Strike price = prix de l’underlying au moment du closing / Spot price = prix de l’underlying ajd sur le marché Value of a contract at initiation The moment we find the counterparty and close the deal. • 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? Si on veut se retirer et donc revendre le contrat • 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 • Fundamental principle: at initiation/closing, the value of a forward contract is zero. At initiation there is no monetary exchange, it is only a commitment contracted. • This is also true for a futures and swap contracts. • If this is not the case, there is an arbitrage opportunity • It is an essential difference with option contracts: to take a position on a futures contract, the agent does not pay a premium (he/she only pays a deposit, which is used to manage counterparty risk). The option contract gives you the right but not the obligation so you pay a premium. Value of a contract before its expiry ie at any intermediate point On a 2 possibilités : soit on revend son contrat, soit on essaye de se retirer du contrat (souvent pour forward parce que pas de liquidité sur le marché) pour cela on va demander d’annuler le contrat ou de le renégocier. Si on veut annuler, la contrepartie peut proposer un prix, sinon on est obligé d’exécuter le contrat. 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 • By canceling the contract today, the buyer avoids (gives up) a futur loss (profit), donc on va mesurer la perte (actualization) • The present value of the future loss (profit) is: 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 f(t) of this contract, for the buyer who wants to sell it, is: f(t) = (F(t;T)−F(0,T)) * e − r * (T−t) r is the interest rate. e (exponentiel) donc on est en taux continue et le - indique actualisation. • If in March, the price of the forward contract for June maturity is F = 80, the buyer may wish to cancel its position • The interest rate for 3 months is 1% • Value of the position : (80−100) * exp (−0,01*0,25) =  − 19.95 • The buyer has to pay 19.95 to his/her counterparty to exit his/her position if the counterparty accepts. • Counterparty risk on this position : la contrepartie peut se dire que le mec va pas pouvoir le payer donc peut accepter ce 19,95 au lieu d’attendre la maturité de juin. • The buyer is found in a speculative position: il prefère se dire les prix vont encore baisser (speculative) et donc annule son contrat. 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 f(t) of the purchased contract is: f(t)=(F(0, T) − F(T, t))*e − r(T − t) 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 We measure between December to June N+1 (exit position and maturity) H1. On December 16, the forward price for delivery in June N +1 is: F (t, T) = $ 80 / barrel - The buyer wishes to close his/her position - Interest rate for 6 months: r = 5% Value of the long position on the forward contract: ft = 300 000 * (80-110)*exp(-0,05*0,5) = - 8 777 789.21$ The forward contract is out of the money : f(t) < 0 H2 : Forward price: F(t,T) = 135 $/barrel Value of the long position in the forward contract : ft = 300 000 *(135-110)*exp(-0,05*0,5) = 7 314 824$ The forward contract is in the money: f(t) > 0 H3 : Forward price : F(t, T) = 110 Value of the long position in the forward contract : f(t)= 300 000*(110 – 110)*exp(-0,05*0,5) = 0 The forward contract is at the money ft = 0 Book : chapter 5: Determination of forward and futures prices Closing out a position = entering into the opposite trade to the original one. The vast majority of futures contracts do not lead to delivery. The reason is that most traders choose to close out their positions prior to the delivery period specified in the contract. Closing out a position means entering into the opposite trade to the original one. For example, the New York investor who bought a July corn futures contract on March 5 can close out the position by selling (i.e., shorting) one July corn futures contract on, say, April 20. The Kansas investor who sold (i.e., shorted) a July contract on March 5 can close out the position by buying one July contract on, say, May 25. In each case, the investor’s total gain or loss is determined by the change in the futures price between March 5 and the day when the contract is closed out. For delivery: Suppose that the party on the other side of investor A’s futures contract when it was entered into was investor B. It is important to realize that there is no reason to expect that it will be investor B who takes delivery. Investor B may well have closed out his or her position by trading with investor C, investor C may have closed out his or her position by trading with investor D, and so on. The usual rule chosen by the exchange is to pass the notice of intention to deliver on to the party with the oldest outstanding long position. Parties with long positions must accept delivery notices. However, if the notices are transferable, long investors have a short period of time, usually half an hour, to find another party with a long position that is prepared to accept the notice from them. Un hedger paye des impôts sur le profit total de son hedging (si ça dure 3 ans, il paiera des impôts 1 fois à la fin des 3 ans), alors qu’un spéculateur paye des impôts tous les ans sur les profits de chaque année. Forward contracts are easier to analyse than futures contracts because there is no daily settlement : only a single payment at maturity. Investment assets vs Consumption assets Investment asset = asset that is held for investment purposes by significant numbers of investors (ex: stocks, bonds, gold) Consumption asset = held primarily for consumption Short selling Short selling = selling an asset that is not owned. Purchase of share = purchase the shares in (t) + receive dividend + sell shares in (t+1) Short sale of shares = borrow shares and sell them in (t) + pay dividend + buy share in (t+1) and replace borrowed shares to close short position The investor is required to maintain a margin account with the broker to guarantee that the investor won’t walk away from the short position is share prices increase. Assumptions and notations Forward price for an investment asset Known income Pages 108-109: pas compris pourquoi des (-) dans les calculs de puissance : négatif quand on calcule la VA, positif quand on calcule la valeur capitalisée Known yield Valuing forward contracts Value of a forward contract on an investment asset that provides no income : f = (F0−K)e − r * T = S0 − K * er * T K : delivery price (part of the contract) Are forward prices and futures prices equal ? Theory: When the ST risk-free rate is constant, forward price with x delivery date = future price with x delivery date In real life, factors influence the prices: tax, transaction costs, treatment of margins Futures prices of stock indices F0 = SO * e(r−y) * T (même formule que dans cours) Pas à lire Futures on commodities Etapes de calcul avec storage costs : Calculer la NPV des storage costs que l’on paiera en date t Ajouter ces storages costs actualisés au prix du futures Cost of carry Delivery options Pas à lire A limit order specifies a particular price. The order can be executed only at this price or at one more favorable to the investor. Thus, if the limit price is $30 for an investor wanting to buy, the order will be executed only at a price of $30 or less. There is, of course, no guarantee that the order will be executed at all, because the limit price may never be reached. A stop order or stop-loss order also specifies a particular price. The order is executed at the best available price once a bid or offer is made at that particular price or a less- favorable price. Suppose a stop order to sell at $30 is issued when the market price is $35. It becomes an order to sell when and if the price falls to $30. In effect, a stop order becomes a market order as soon as the specified price has been hit. The purpose of a stop order is usually to close out a position if unfavorable price movements take place. It limits the loss that can be incurred. A stop–limit order is a combination of a stop order and a limit order. The order becomes a limit order as soon as a bid or offer is made at a price equal to or less favorable than the stop price. Two prices must be specified in a stop–limit order: the stop price and the limit price. Suppose that at the time the market price is $35, a stop–limit order to buy is issued with a stop price of $40 and a limit price of $41. As soon as there is a bid or offer at $40, the stop–limit becomes a limit order at $41. If the stop price and the limit price are the same, the order is sometimes called a stop-and-limit order. A market-if-touched (MIT) order is executed at the best available price after a trade occurs at a specified price or at a price more favorable than the specified price. In effect, an MIT becomes a market order once the specified price has been hit. An MIT is also known as a board order. Consider an investor who has a long position in a futures contract and is issuing instructions that would lead to closing out the contract. A stop order is designed to place a limit on the loss that can occur in the event of unfavorable Questions and exercises : ; What is the difference between a long forward position and a short forward position? Long position = buy Short position = sell 1.5 ; An investor enters into a short forward contract to sell 100,000 British pounds for US dollars at an exchange rate of 1.4000 US dollars per pound. How much does the investor gain or lose if the exchange rate at the end of the contract is (a) 1.3900 : gain = 1000 (b) 1.4200: loss = 2000 1.12 ; Explain why a futures contract can be used for either speculation or hedging. 1.19 ; A trader enters into a short forward contract on 100 million yen. The forward exchange rate is $0.0080 per yen. How much does the trader gain or lose if the exchange rate at the end of the contract is (a) $0.0074 per yen: gain = 60 000 (b) $0.0091 per yen: loss = 110 000 1.21 (forward only) ; "Options and futures are zero-sum games." What do you think is meant by this? The loss of A is the gain of B. 1.24 ; On July 1, 2011, a company enters into a forward contract to buy 10 million Japanese yen on January 1, 2012. On September 1, 2011, it enters into a forward contract to sell 10 million Japanese yen on January 1, 2012. Describe the payoff from this strategy. 2.1 ; Distinguish between the terms open interest and trading volume. - open interest: number of contracts outstanding - trading volume: volume of contracts exchanged 2.3 ; Suppose that you enter into a short futures contract to sell July silver for $17.20 per ounce. The size of the contract is 5,000 ounces. The initial margin is $4,000, and the maintenance margin is $3,000. What change in the futures price will lead to a margin call? What happens if you do not meet the margin call? 2.4. Suppose that in September 2012 a company takes a long position in a contract on May 2013 crude oil futures. It closes out its position in March 2013. The futures price (per barrel) is $68.30 when it enters into the contract, $70.50 when it closes out its position, and $69.10 at the end of December 2012. One contract is for the delivery of 1,000 barrels. Assume that the company has a December 31 year-end. What is the company's total profit? When is it realized? How is it taxed if it is : (a) a hedger (b) a speculator 2.5. What does a stop order to sell at $2 mean? The order gets activated when the market price is equal or inferior to $2 When might it be used? In order to limit the losses from an existing long position. What does a limit order to sell at $2 mean? You want to sell until the price reaches $2 When might it be used? Could be used to instruct a broker that a short position should be taken, providing it can be done at a price more favorable than $2. 2.7: What differences exist in the way prices are quoted in the foreign exchange futures market, the foreign exchange spot market, and the foreign exchange forward market? Futures : number of USD per unit of foreign currency Spot and forward : British pound, €, Australian dollar, New Zealand dollar. Other currencies are quoted as the number of units of foreign currency per USD. 2.8. The party with a short position in a futures contract sometimes has options as to the precise asset that will be delivered, where delivery will take place, when delivery will take place, and so on. Do these options increase or decrease the futures price? These options make the contract less attractive to the long position party and more attractive to the short position party. It thus tends to reduce the futures price. 2.9. What are the most important aspects of the design of a new futures contract? Quality, volume, place of delivery, time of delivery 2.10. Explain how margins protect investors against the possibility of default. 2.11. A trader buys two July futures contracts on orange juice. Each contract is for the delivery of 15,000 pounds. The current futures price is 160 cents per pound, the initial margin is $6,000 per contract, and the maintenance margin is $4,500 per contract. What price change would lead to a margin call? Under what circumstances could $2,000 be withdrawn from the margin account? 2.12. Show that, if the futures price of a commodity is greater than the spot price during the delivery period, then there is an arbitrage opportunity. Does an arbitrage opportunity exist if the futures price is less than the spot price? 2.15 ; At the end of one day a clearing house member is long 100 contracts, and the settlement price is $50,000 per contract. The original margin is $2,000 per contract. On the following day the member becomes responsible for clearing an additional 20 long contracts, entered into at a price of $51,000 per contract. The settlement price at the end of this day is $50,200. How much does the member have to add to its margin account with the exchange clearing house? 2.17 ; The forward price of the Swiss franc for delivery in 45 days is quoted as 1.1000. The futures price for a contract that will be delivered in 45 days is 0.9000. Explain these two quotes. Which is more favorable for an investor wanting to sell Swiss francs? 2.20 ; Live cattle futures trade with June, August, October, December, February, and April maturities. Why do you think the open interest for the June contract is less than that for the August contract in Table 2.2? 2.21. What do you think would happen if an exchange started trading a contract in which the quality of the underlying asset was incompletely specified? 2.22. "When a futures contract is traded on the floor of the exchange, it may be the case that the open interest increases by one, stays the same, or decreases by one." Explain this statement. 5.2: What is the difference between the forward price and the value of a forward contract? Forward price = the price at which you would agree to buy / sell an asset at a future time. Value of a forward contract = 0 when you first enter into it. As time passes, the underlying asset price changes and the value of the contract may become positive / negative. 5.3. Suppose that you enter into a 6-month forward contract on a non-dividend-paying stock when the stock price is $30 and the risk-free interest rate (with continuous compounding) is 12% per annum. What is the forward price? Forward price = 30 * e0, 12 * 0, 5  = 31, 86 5.4. A stock index currently stands at 350. The risk-free interest rate is 8% per annum (with continuous compounding) and the dividend yield on the index is 4% per annum. What should the futures price for a 4-month contract be? $F = 350*e^{0.08 - 0.04*\frac{4}{12}} = 354.7$ 5.5. Explain carefully why the futures price of gold can be calculated from its spot price and other observable variables whereas the futures price of copper cannot. Gold = investment asset If futures prices of gold go high: buy at spot price + sell contracts If futures prices decrease: sell at spot price + buy contracts Copper = consumption asset. If futures prices go high: buy at spot prices + sell contracts If futures prices decrease: does not make sens to buy futures 5.6. Explain carefully the meaning of the terms convenience yield and cost of carry. What is the relationship between futures price, spot price, convenience yield, and cost of carry? Convenience yield = benefits obtained from ownership of the physical asset Cost of carry = interest cost + storage cost – income earned. F0 = S0 * e(cost of carry−convenience yield) * T  5.7. Explain why a foreign currency can be treated as an asset providing a known yield. 5.9. A 1-year long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding. (a) What are the forward price and the initial value of the forward contract? Forward price = 40 * e0.1 * 1 = 44.21 Initial value of the forward contract = 0 (b) Six months later, the price of the stock is $45 and the risk-free interest rate is still 10%. What are the forward price and the value of the forward contract? Forward price = 45 * e0, 1 * 0, 5 = 47, 31 Value of the forward contract = 5.10. The risk-free rate of interest is 7% per annum with continuous compounding, and the dividend yield on a stock index is 3.2% per annum. The current value of the index is 150. What is the 6-month futures price? Futures price = 150 * e(7%−3.2%) * 0, 5 = 152.88 5.12. Suppose that the risk-free interest rate is 10% per annum with continuous compounding and that the dividend yield on a stock index is 4% per annum. The index is standing at 400, and the futures price for a contract deliverable in four months is 405. What arbitrage opportunities does this create? Theoretical futures prices = $400*e^{0.1 - 0.4*\frac{4}{12}} = 408,08$ The futures prices for contract is only 405. This shows that the price of the index is more expensive than what it should be. So you sell the index + buy futures contracts. 5.15. The spot price of silver is $15 per ounce. The storage costs are $0.24 per ounce per year payable quarterly in advance. Assuming that interest rates are 10% per annum for all maturities, calculate the futures price of silver for delivery in 9 months. 5.16. Suppose that F1 and F2 are two futures contracts on the same commodity with times to maturity, t1 and t2, where t2 > t1. Prove thatwhere r is the interest rate (assumed constant) and there are no storage costs. For the purposes of this problem, assume that a futures contract is the same as a forward contract. Demonstration par l’absurde: