FUTURES MARKETS Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 1.1

The Nature of Derivatives A derivative is an instrument whose value depends (is derived from) on the values of other more basic underlying variables Futures Contracts Forward Contracts Swaps Options Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 1.2 2

Ways Derivatives are Used To hedge risks To speculate (take a view on the future direction of the market) To lock in an arbitrage profit To change the nature of a liability To change the nature of an investment without incurring the costs of selling one portfolio and buying another Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 1.3 4

Positions Every transaction involves two positions Long position (agreement to buy in the future) Short position (agreement to sell in the future) Zero – sum game 1.4 4

What is the benefit of derivative markets ? Efficiency Liquidity Ability to forecast trends Transparency Risk management Portfolio diversification Investment opportunities

Forward contracts A Forward contract is a contract where an investor A (buyer) agrees with another investor B (seller) That A will buy from B An underlying asset (underlying instrument) At a pre-specified price (forward price/rate) At a pre-specified future date (delivery date) E.g. agree to sell 10 million barrels of oil (underlying asset) at $100 per barrel (forward price) in 9 months (delivery date)

Over The Counter (OTC) → Investors determine the terms of the contract Forward contracts are between two parties (usually between financial institutions and/or their clients) and NOT via organized exchanges Over The Counter (OTC) → Investors determine the terms of the contract → Confidentiality

Disadvantages OTC Limited transparency Limited regulations No official body for approving new products Limited clearance procedures

Neutralize Position There is no secondary market As a result, the contract cannot be liquidated However, an investor can neutralize it by taking an exact opposite position (same duration, same size) Short in 10 million barrels of oil at $100 per in 9 months Long in 10 million barrels of oil at $100 per in 9 months

Profit and Losses (P&L) Long position on a forward contract Profit (loss) when the price of the underling asset at delivery is higher (lower) than the delivery price Short position on a forward contract Profit (loss) when the price of the underling asset at delivery is lower (higher) than the delivery price

Example On the 8th of March we agree with our bank to buy 1,000,000 Pound Sterling at the exchange rate $1.55 in 90 days On the 6th of August the rate is rate $/BP is at $1.60 Profit or loss?

On the 6th of August we pay $1,550,000 to the bank We take 1,000,000 BP We sell them in the market for $1,600,000 Profit : $50,000 The bank losses the same amount (zero-sum game) We ignore transaction costs

Returns Sτ = the spot price of the underlying at delivery date Κ = The delivery (forward) price Profit from a long position on a FC: (Sτ – Κ) Profit from a short position on a FC: (Κ – Στ)

In the example: Sτ = 1.60 Κ = 1.55 Long Position Profit (Sτ – Κ) = $1.60 – $1.55 = $0.05 For 1,000,000 BP = 1.000.000 x $0,05 = $50,000

Graphical representation

Introduction to hedging A simple example Consider a company that has 1000 units of a product that has a current price of 500 E/unit If the price goes up in 1 month e.g. at 540 E/unit Profit 40 Ε If the price goes down in 1 month e.g. at 460 E/unit Loss 40 Ε Mple

Graph

Uncertainty What can the Company do ? Assume a forward contract on the product exists short forward for 1000 units at 500 E

Graph

Forward contracts Forward contracts are similar to futures except that they trade in the over-the-counter market Forward contracts are popular on currencies and interest rates A US company will pay £10 million for imports from Britain in 3 months and decides to hedge using a long position in a forward contract

Futures Contracts A futures contract is an agreement to buy or sell an asset at a certain time in the future for a certain price By contrast in a spot contract there is an agreement to buy or sell the asset immediately (or within a very short period of time) Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 1.22 7

Organized markets In contrast to forward contracts, futures contracts are traded in organized exchanges The contracts are between the member and the exchange and have the “guarantee” of the exchange Every contract has the same characteristics and this standardization allows easy trading, liquidity, and easy clearing

Most important Derivative Exchanges in the US

Most important Derivative Exchanges except US

Futures Price The futures prices for a particular contract is the price at which you agree to buy or sell It is determined by supply and demand in the same way as a spot price Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 1.28 8

Electronic Trading Traditionally futures contracts have been traded using the open outcry system where traders physically meet on the floor of the exchange Increasingly this is being replaced by electronic trading where a computer matches buyers and sellers Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 1.29

Examples of Futures Contracts Agreement to: buy 100 oz. of gold @ US$600/oz. in December (NYMEX) sell £62,500 @ 1.9800 US$/£ in March (CME) sell 1,000 bbl. of oil @ US$65/bbl. in April (NYMEX) Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 1.30 9

Terminology The party that has agreed to buy has a long position The party that has agreed to sell has a short position Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 1.31 10

Example January: an investor enters into a long futures contract on COMEX to buy 100 oz of gold @ $600 in April April: the price of gold $615 per oz What is the investor’s profit? Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 1.32 11

Futures markets 5 parties involved (1) Derivatives Exchange (2) Clearing House (3) Regulator (4) Members (5) Investors (speculators, hedgers,arbitrageurs)

Derivatives Exchange → Usually belongs to company that not only offers the physical place where transactions take place but also provides security and sets the rules → In many countries it works with electronic order matching while in other it works with a pit Pit: the traders and brokers transact in the form of an auction-style open-outcry

The clearing house → clears all transactions → guarantees all trades → stands between the two parties Once a transaction takes place the clearer: (α) records the transaction (β) clears the transaction (γ) estimate the margins (δ) clear the obligations of the parties

Regulator → Securities and Exchange Commission (US) → Capital Market Committee (Greece) For example in Greece the CMC may take regulatory decisions that are equal to state laws It regulates, the Athens Exchange, the clearing organization, brokers, dealers, mutual funds, investment trusts, etc

Futures Contracts Available on a wide range of underlying Exchange traded Specifications need to be defined: What can be delivered, Where it can be delivered, & When it can be delivered Settled daily Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 2.38

Delivery If a futures contract is not closed out before maturity, it is usually settled by delivering the assets underlying the contract. When there are alternatives about what is delivered, where it is delivered, and when it is delivered, the party with the short position chooses. A few contracts (for example, those on stock indices and Eurodollars) are settled in cash Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 2.39

Some Terminology Open interest: the total number of contracts outstanding equal to number of long positions or number of short positions Settlement price: the price just before the final bell each day used for the daily settlement process Volume of trading: the number of trades in 1 day Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 2.40

Differences Futures - Forwards Futures are standardized (when, where, what) Standardization is necessary for the efficient running of the market (α) since everybody transacts in the same contract the market is liquid and the trade is easy (β) there is a possibility for large trades since the mechanics of buying/selling are easy to manage (γ) transaction costs are lower since the traders do not need to negotiate the terms in every trade

Other differences Futures are available for a large number of underlying assets: Weather derivatives Pork bellies Orange juice Coffee

Weather derivatives Farmers, theme parks, airliners, gas and power companies Heating degree days (HDD) or cooling degree days (CDD) contracts Heating degree days are one of the most common types of weather derivative.

daily settlement, mark to market Futures are settled daily i.e. at the end of the day the investors with losses pay the loss to the investors with gains

Margins Futures: when a contract is initiated no money change hands However, every party has to pay a “good faith” deposit or “Margin” to the clearing organization They are used as insurance A margin is cash or marketable securities deposited by an investor with his or her broker The balance in the margin account is adjusted to reflect daily settlement Margins minimize the possibility of a loss through a default on a contract Initial margin requirement and maintenance margin

Margins Futures exchanges determine and set futures margin rates. The margin is set based on the risk of market volatility. In a typical futures contract, the margin rate varies between 5 and 15% of the total contract value. For example, the buyer of a contract of wheat futures might only have to post $1,700 in margin. Assuming a total contract of $32,500 ($6.50 x 5,000 bushels) the futures margin would amount to around 5% of the contract value.

Margins & Leverage You buy one contract of a COMEX gold future at 1270 Each contract is for 100 ounces of gold; Initial margin= $4400 Sell one contract of COMEX gold future at 1275 Profit- $5 per ounce or $500 per contract If you bought the actual gold and made a $5 profit that would equate to a 0.3937% gain ($5/$1,270) However, since you bought the gold futures contract, the gain is calculated on the amount of margin posted for the trade or $4,400 and the profit would equate to an 11.36% gain ($500/$4,400)

Example of a Futures Trade An investor takes a long position in 4 December gold futures contracts on June 5 contract size is 100 oz. futures price is US$380 margin requirement is US$2,000/contract (US$8,000 in total) maintenance margin is US$1,500/contract (US$6,000 in total) Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 2.52

Example of a Futures Trade Position: 4 December Gold Futures Every Futures is for 100 oz. Price: $380 / oz Thus: 4 x 100 x $380 = $152,000 What dowe deposit: $2,000 x 4 = $8,000 Leverage

Example of a Futures Trade Assume that at the end of the day the price drops at $377 We loose $3 x 400 oz ($1,200) Our margin account will reduce by $1,200 (down to $6,800) Assume that at the end of next day the price drops a further $3 at $374 We loose another $1,200 and the margin account now stands at $5,600 Margin Call

Margins for stock futures in Greece

Τιμή ΣΜΕ Απριλίου στον FTSE20 Παράδειγμα Στις 13/4/2001 ένας κερδοσκόπος προσδοκά πως βραχυχρόνια ο δείκτης FTSE/ASE 20 θα κινηθεί ανοδικά. Για το λόγο αυτό αγοράζει 10 ΣΜΕ στο δείκτη με μήνα λήξης τον Απρίλη στην τιμή των 1.750. Υπολογίστε το κέρδος/ζημιά του γα τις υποθετικές τιμές κλεισίματος του ΣΜΕ που συνοψίζονται στις δυο πρώτες στήλες του παρακάτω πίνακα. Λύση Τα ΣΜΕ λήγουν την τρίτη Παρασκευή του Απρίλη, δηλαδή στις 20/4/2001. Τα ΣΜΕ απαιτούν αρχικό περιθώριο ασφάλισης 12% επί της αξίας των συναλλαγών δηλ 12% του 10*1750*5=87.5000 που είναι 10.500. Οπότε θα είναι: Ημέρα Τιμή ΣΜΕ Απριλίου στον FTSE20 Κέρδος/Ζημιά (σε μονάδες) (σε €) Λογαριασμός margin 13/4 1750 10500 14/4 1753 3 150 (=3x10x5€) 10650 (=10500+150) 15/4 1749 -4 -200 (=-4x10x5€) 10450* (+50) 16/4 1 50 10550 17/4 1748 -2 -100 18/4 1757 9 450 10950 19/4 1762 5 250 11200 20/4 1770 8 400 11600 Το σύμβολο * στην τελευταία στήλη δηλώνει ότι ο λογαριασμός πέφτει κάτω από το περιθώριο ασφάλισης και απαιτεί επιπλέον κατάθεση (margin call).

Example Trading with Margin

Margin Examples (17/12/2016)

Margin Examples (17/12/2016)

Trading Details, example, CME When selling or buying futures contracts, it is necessary to post a performance bond deposit with a futures broker. This is a small percentage of the value of each contract traded, representing the dollar value of the probable maximum price move in the next day’s market, and thus the likely maximum loss that could be incurred in that day’s trading. Because no one knows whether prices will move up or down by this amount, both the buy side and the sell side of all futures transactions post such a deposit. That way, the profiting side of the market can be immediately credited, out of the balances of the losing side of the market. This flow of payments is conducted by the CME Clearing House, in transactions with all clearing members, who in turn “settle up” with each of their own customers.

Trading Details, example, CME Suppose a hedger sells one December CME Lean Hog futures at $66/cwt. The total value of the contract is $26,400 (400 cwt. times $66/cwt.). The hedger started with a performance bond deposit of $800. By the end of the second day, the contract decreased in value by $320. The hedger would realize a gain if he bought it back for $320 less than the selling price. The $320 is credited to his account. Not until the fifth day does the futures price begin to rise again. This time the contract value has increased by 340, which is subtracted from his account balance.

Example Leverage

Convergence of Futures to Spot Price Spot Price Futures Price Spot Price Time Time (a) (b) Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 2.66

Forward Contracts A forward contract is an OTC agreement to buy or sell an asset at a certain time in the future for a certain price There is no daily settlement (unless a collateralization agreement requires it). At the end of the life of the contract one party buys the asset for the agreed price from the other party Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 2.67 10

Long & Short Hedges A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price A short futures hedge is appropriate when you know you will sell an asset in the future & want to lock in the price Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 3.70

Arguments in Favor of Hedging Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 3.71

Arguments against Hedging Shareholders are usually well diversified and can make their own hedging decisions It may increase risk to hedge when competitors do not Explaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 3.72

Basis Risk Basis is the difference between spot & futures Basis risk arises because of the uncertainty about the basis when the hedge is closed out Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 3.73

Long Hedge Suppose that F1 : Initial Futures Price F2 : Final Futures Price S2 : Final Asset Price You hedge the future purchase of an asset by entering into a long futures contract Cost of Asset=S2 – (F2 – F1) = F1 + Basis Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 3.74

Short Hedge Suppose that F1 : Initial Futures Price F2 : Final Futures Price S2 : Final Asset Price You hedge the future sale of an asset by entering into a short futures contract Price Realized=S2+ (F1 – F2) = F1 + Basis Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 3.75

Αντιστάθμιση με ΣΜΕ Τέσσερις κύριες αποφάσεις: (1) Αγορά ή πώληση ΠΣ & ΣΜΕ? (2) Χρόνος εφαρμογής αντιστάθμισης? (3) Ποιο συμβόλαιο θα επιλεγεί? (4) Ποιος είναι ο αριθμός των ΠΣ?

Αντιστάθμιση με ΣΜΕ (1) Αγορά ή πώληση ΠΣ & ΣΜΕ? (1) Αγορά ή πώληση ΠΣ & ΣΜΕ? Κατανόηση Θέσης μετρητοίς Ανάληψη αντίθετης στο παράγωγο Π.χ. αρνητική μετρητοίς – θετική παράγωγα (long hedge) Π.χ. θετική μετρητοίς – αρνητική παράγωγα (short hedge)

Αντιστάθμιση με ΣΜΕ (2) Χρόνος εφαρμογής αντιστάθμισης? (2) Χρόνος εφαρμογής αντιστάθμισης? Πρέπει να γίνει αντιστάθμιση ? Εάν ναι, τότε άμεση εφαρμογή Η ανάληψη θέσης στην αγορά ΣΜΕ πρέπει να γίνει αμέσως μετά τη λήψη της απόφασης για αντιστάθμιση.

Αντιστάθμιση με ΣΜΕ (3) Ποιο συμβόλαιο θα επιλεγεί? (3) Ποιο συμβόλαιο θα επιλεγεί? Σε κάθε χρονική στιγμή υπάρχουν πολλά συμβόλαια και πρέπει να επιλεγεί αυτό με την μικρότερη λήξη (που όμως είναι μετά την ημέρα λήξης της θέσης μετρητοίς) Στην αγορά διαπραγματεύονται πολλά ΣΜΕ για το ίδιο υποκείμενο αγαθό ή τίτλο, με διαφορετικούς χρόνους λήξης. Το ΣΜΕ με το μικρότερο χρόνο λήξης είναι εκείνο του οποίου η τιμή είναι περισσότερο συσχετισμένη με την τιμή του υποκείμενου αγαθού ή τίτλου. Αφού για να είναι επιτυχημένη η αντιστάθμιση θα πρέπει οι τιμές του ΣΜΕ και του υποκείμενου αγαθού να είναι όσο το δυνατόν περισσότερο συσχετισμένες, αυτό οδηγεί στην επιλογή του ΣΜΕ με τη μικρότερη λήξη.

Αντιστάθμιση με ΣΜΕ (4) Ποιος είναι ο αριθμός των Συμβολαίων (4) Ποιος είναι ο αριθμός των Συμβολαίων Ο αριθμός των ΣΜΕ που χρειάζονται για την αντιστάθμιση εξαρτάται από: Το μέγεθος (αξία) της θέσης στην spot αγορά. Το ποσοστό κάλυψης της θέσης στη spot αγορά που επιθυμεί ο επενδυτής να αντισταθμίσει. Το μέγεθος της αγοράς μετρητοίς και το ποσοστό κάλυψης της θέσης αυτής: ΠF = – ΠC (ΑΑ) ΠF = προθεσμιακή θέση ΠC = μέγεθος θέσης μετρητοίς ΑΑ = αναλογία αντιστάθμισης (πλήρη αντιστάθμιση, ΑΑ=1) Το αρνητικό πρόσημο υποδηλώνει την ανάληψη θέσης στη προθεσμιακή αγορά, αντίθετη από εκείνη της θέσης στη spot.

Optimal Hedge Ratio Ο επενδυτής που κάνει αντιστάθμιση, επιθυμεί εξ’ ορισμού να μειώσει όσο γίνεται τον κίνδυνο που αναλαμβάνει, δλδ να ελαχιστοποιήσει τον κίνδυνο μεταβολής του κέρδους του. Μπορεί να δειχθεί ότι το ποσοστό ΑΑ της θέσης του στη spot αγορά που πρέπει να αντισταθμίσει είναι: 3.81

Optimal Hedge Ratio Αριθμός συμβολαίων ΣΜΕ = Τρέχουσα αξία της θέσης στη spot αγορά που θα αντισταθμιστεί / Αξία ενός συμβολαίου ΣΜΕ 3.82

Παράδειγμα Μια εταιρεία γνωρίζει ότι θα χρειαστεί να αγοράσει 1 εκατομμύριο γαλόνια καυσίμου σε 6 μήνες. Η τυπική απόκλιση της μεταβολής της τιμής ανά γαλόνι του καυσίμου για το διάστημα 6 μηνών είναι 0,016. Η εταιρεία θέλει να αντισταθμίσει τον κίνδυνο με αγορά ΣΜΕ σε πετρέλαιο θέρμανσης. Η τυπική απόκλιση της μεταβολής της τιμής ΣΜΕ για το διάστημα 6 μηνών είναι 0,020 και ο συντελεστή συσχέτισης μεταξύ των μεταβολών των δύο τιμών είναι 0,8. Ένα συμβόλαιο ΣΜΕ σε πετρέλαιο θέρμανσης είναι για παράδοση 21.000 γαλονιών. 1) Τι θέση πρέπει να πάρει η εταιρεία για αντισταθμίσει τον κίνδυνό της; 2) Πόσα συμβόλαια ΣΜΕ πρέπει να χρησιμοποιήσει η εταιρεία;

Παράδειγμα 1) Αφού η εταιρεία θα χρειαστεί να αγοράσει 1 εκατομμύρια γαλόνια σε 6 μήνες, τότε θα πρέπει να αγοράσει ένα ΣΜΕ σε πετρέλαιο θέρμανσης (διαφορετικό υποκείμενο προϊόν), ώστε με τη λήξη του να βρεθεί να κατέχει ποσότητα καυσίμου. 2) Η εταιρεία σκοπεύει να αγοράσει 1.000.000 γαλόνια. Ο λόγος (αναλογία) αντιστάθμισης δίνεται από τη σχέση: Συνεπώς, ο αριθμός συμβολαίων ΣΜΕ θα είναι Επειδή όμως ο αριθμός συμβολαίων ΣΜΕ που μπορεί να αγοραστεί πρέπει να είναι ακέραιος, η εταιρεία θα αγοράσει 30 συμβόλαια, αναλαμβάνοντας η ίδια το μέρος του κινδύνου που δεν μπορεί να αντισταθμίσει. Στην περίπτωση που αγόραζε 31 συμβόλαια, θα είχε υπερβάλλουσα αντιστάθμιση, δλδ θα αναλάμβανε μεγαλύτερο μέρος του κινδύνου από το παράγωγο αντί για τον κίνδυνο του υποκείμενου προϊόντος.

The Futures/Forward Price In order top find the right future/forward price for an asset we employ the same methodology as for every investment Present Value approach Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 3.85

Notation S0: Spot price today F0: Futures or forward price today T: Time until delivery date r: Risk-free interest rate for maturity T Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.86

1. Gold: An Arbitrage Opportunity? Suppose that: The spot price of gold is US$600 The quoted 1-year futures price of gold is US$650 The 1-year US$ interest rate is 5% per annum No income or storage costs for gold Is there an arbitrage opportunity? Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.87 18

2. Gold: Another Arbitrage Opportunity? Suppose that: The spot price of gold is US$600 The quoted 1-year futures price of gold is US$590 The 1-year US$ interest rate is 5% per annum No income or storage costs for gold Is there an arbitrage opportunity? Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.88 19

The Futures Price of Gold If the spot price of gold is S & the futures price for a contract deliverable in T years is F, then F = S (1+r )T where r is the 1-year (domestic currency) risk-free rate of interest. In our examples, S=600, T=1, and r=0.05 so that F = 600(1+0.05) = 630 Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.89 20

When Interest Rates are Measured with Continuous Compounding F0 = S0erT This equation relates the forward price and the spot price for any investment asset that provides no income and has no storage costs Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.90

Another example 7-month forward on a zero coupon bind that matures in a year Spot price = $965, interest rate 5.5% βάση S0 = $965, e = 2.71828, r = 0.055, Τ = (7/12) F0 = S0 er Τ = 965 e 0.055 (7/12) = $996.46

When an Investment Asset Provides a Known Dollar Income F0 = (S0 – I )erT where I is the present value of the income during life of forward contract Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.92

Example What is the forward price if the dicopunt rate is 6%? 10-month forward on a stock that has a current price of $20 and will pay a dividend per share of $0.25 in 3, 6, 9 months from today What is the forward price if the dicopunt rate is 6%?

Ι = 0.25 e -0.06 (3/12) + 0.25 e -0.06 (6/12) + 0.25 e -0.06 (9/12) = 0.72791 F0 = (S0 – Ι) er Τ = (20-0.72791) e 0.06 (10/12) = $20.2

When an Investment Asset Provides a Known Yield F0 = S0 e(r–q )T where q is the average yield during the life of the contract (expressed with continuous compounding) Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.95

Valuing a Forward Contract Suppose that K is delivery price in a forward contract & F0 is forward price that would apply to the contract today The value of a long forward contract, ƒ, is ƒ = (F0 – K )e–rT Similarly, the value of a short forward contract is (K – F0 )e–rT Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.96

Forward vs Futures Prices Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different: A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price A strong negative correlation implies the reverse Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.97

Παράδειγμα Δίνεται προθεσμιακό συμβόλαιο διάρκειας 1 έτους σε μετοχή με τιμή σήμερα 50€ που πληρώνει μέρισμα 0,5€ σε 3 μήνες και 1€ σε 9 μήνες από σήμερα. Η τιμή παράδοσης του συμβολαίου είναι 55€. Το επιτόκιο είναι 4% σταθερό με συνεχή ανατοκισμό. 1) Υπολογίστε την αξία του προθεσμιακού συμβολαίου σήμερα. 2) Εξετάστε αν υπάρχουν ευκαιρίες εξισορροπητικής κερδοσκοπίας (arbitrage). Αν ναι, δείξτε πως μπορεί να τις εκμεταλλευτεί ένας επενδυτής. Αν δεν υπάρχουν τέτοιες ευκαιρίες εξηγείστε γιατί συμβαίνει αυτό.

Λύση 1) Αρχικά, θα πρέπει να υπολογίσουμε την αξία του προθεσμιακού συμβολαίου τη χρονική στιγμή t=0. Σύμφωνα με τα δεδομένα θα είναι: Το I0 είναι παρούσα αξία των μερισμάτων που καταβάλλονται στη διάρκεια ζωής του προθεσμιακού συμβολαίου: Συνεπώς: 2) Αφού το f0 δεν είναι μηδέν, υπάρχουν ευκαιρίες εξισορροπητικής κερδοσκοπίας. Επειδή είναι f0 <0 , παίρνουμε αρνητική θέση (short) στο προθεσμιακό συμβόλαιο και θετική θέση (long) στη μετοχή.

Λύση (συνέχεια) Τη στιγμή t=0 Λαμβάνουμε αρνητική θέση (short) στο προθεσμιακό συμβόλαιο Δανειζόμαστε 50€ με επιτόκιο 4% για 1 έτος Αγοράζουμε τη μετοχή (long) στη Spot αγορά για 50€ Τη στιγμή τα=1 Παραδίδουμε τη μετοχή για 55€ από το προθεσμιακό συμβόλαιο Έχουμε αξία μερισμάτων (μελλοντική αξία) που εισπράχθηκαν κατά τη διάρκεια κατοχής της μετοχής: Αποπληρώνουμε το δάνειο αξίας: Μένει συνολικό κέρδος χωρίς κίνδυνο ίσο με: 55+1,5252-52,04 = 4,4852€

Stock Index F0 = S0 e(r–q )T Can be viewed as an investment asset paying a dividend yield The futures price and spot price relationship is therefore F0 = S0 e(r–q )T where q is the dividend yield on the portfolio represented by the index during life of contract Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.101

Stock Index For the formula to be true it is important that the index represent an investment asset In other words, changes in the index must correspond to changes in the value of a tradable portfolio The Nikkei index viewed as a dollar number does not represent an investment asset Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.102

Index Arbitrage When F0 > S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures When F0 < S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.103

Index Arbitrage Index arbitrage involves simultaneous trades in futures and many different stocks Very often a computer is used to generate the trades Occasionally (e.g., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.104

Futures and Forwards on Currencies A foreign currency is analogous to a security providing a dividend yield The continuous dividend yield is the foreign risk-free interest rate It follows that if rf is the foreign risk-free interest rate Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.105

Futures on Consumption Assets F0  S0 e(r+u )T where u is the storage cost per unit time as a percent of the asset value. Alternatively, F0  (S0+U )erT where U is the present value of the storage costs. Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.106