Financial Derivatives Section 1

Financial Derivatives Section 1 Forwards & Futures Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 1 / 49

Outline 1 Introduction Definitions-Terminology Standardization Marking-to-Market Futures Prices 2 Theoretical Pricing of Futures Futures on Investment Assets with no Income Futures on Investment Assets with Income Futures on Currency Rates Futures on Commodities 3 The Use of the Futures Contracts Hedging with Futures Speculation Using Futures Futures and arbitrage opportunities M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 2 / 49

Definitions What is a Forward/Future contract? Forward/Futute contract is a binding agreement to buy or sell an asset at a specified time in the future for a specified price. The Parties The party that has agreed to buy has the long position. The party that has agreed to sell has the short position. Example Company A contacts Company B and agrees to buy 1,000 barrels of crude oil at 58.7$ each, on 15/10/2018 (todays price is 60.75$). Nothing is exchanged when they sign the contract. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 4 / 49

Terminology Underlying Asset: The asset that underlies the deal. Examples: crude oil, stocks, stock index, currency rate, corn etc. Its spot price will be denoted by S(t). Maturity: The future time when the transaction will take place. It will be denoted by T. Delivery Price: The fixed price at which the transaction will be executed. It will be denoted by K. Forward/Future Price: It is the market price that would be agreed today for delivery of the asset at a specified maturity date. As the futures/forward contract is signed, the futures/forward price becomes delivery price. the futures/forward price is defined as the price that will make the contract have a zero value on its inauguration date. It will be denoted by F (t, T ). M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 5 / 49

Futures/Forwards Payoffs Long Position Consider an investor who takes a long position (will buy the underlying asset) on a future/forward contract and he keeps his position open until the maturity of the contract (at time T ). The payoff of his position will be: Payoff of long position = S(T ) K Short Position Consider an investor who takes a short position (will sell the underlying asset) on a future/forward contract and he keeps his position open until the maturity of the contract (at time T ). The payoff of his position will be: Payoff of short position = K S(T ) M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 6 / 49

Payoff Diagrams Figure : Long Position Payoff Figure : Short Position Payoff M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 7 / 49

Futures vs Forwards Futures Traded in organized exchange Contracts are standardized Daily settlement (margin account) Prices posted daily on news Delivery rarely occurs Not specified delivery date Forwards Over-the-counter transaction Contracts are negotiable Settlement at the maturity Private agreement Delivery usually occurs Fixed delivery date What is the additional risk that a position of a forward contract carries? M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 8 / 49

Standardization of a future contract Underlying asset: The asset has to be well defined and specified (e.g. in the case of commodity assets, the quality has to be fully determined). Contract size: The amount of the asset to be delivered. Delivery month: The period during the month at which the delivery can be made. Delivery arrangements: Details on the delivery location and transportation of the commodity assets. Settlement price: The exact way at which the future price is calculated at the end of each day. Position/Price movement limits: The specified limits imposed by the exchanged in order to avoid trading irregularities. Regulation: The regulating authority. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 9 / 49

Standardization of a future contract M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 10 / 49

The Clearinghouse The role of the clearinghouse The exchange clearinghouse is an adjunct of the exchange and acts as an intermediary in the futures contracts. It keeps track of all the transactions and calculates the net position of each of its member. It demands: A margin account. The deposit of an initial margin (usually 2%-10%). Payments of the margin according to daily calculation of the investor s net position. This procedure is called marking-to-market. The balance of the margin account should not be lower that a given amount called maintenance margin. Marking-to-Market It is the accounting act of recording the changes of the futures price. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 11 / 49

Forward Prices vs Futures Prices What makes them differ? Usually, we assume that the prices (and hence the pricing models) are the same. What determines the difference of the forwards and the futures prices is the uncertain interest rate and in particular its relationship with the underlying asset. Case 1: Corr(S, r) = 0 (ceteris paribus) = Forwards Price = Futures Price. Case 2: Corr(S, r) > 0 (ceteris paribus) = Futures Price > Forwards Price. Case 3: Corr(S, r) < 0 (ceteris paribus) = Futures Price < Forwards Price. Important Notice: A forward contract is an OTC transaction and as such it carries credit risk. In other words, each party encounters the risk that the other party is not going to fulfil its obligations determined by the contract. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 12 / 49

Futures Prices vs Spot Prices Why do the prices coincide at maturity? If F (T, T ) > S(T ) an arbitrage opportunity emerges: Buy at the asset at S(T ) and go short at the futures. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 13 / 49

Futures Prices vs Spot Prices, an example Spot crude oil price vs future (Oct18) crude oil price Basis = spot price - futures price It can be positive or negative.what does determine its sign? M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 14 / 49

General Notes The theoretical futures pricing is based on the non-arbitrage assumption. The nature of the underlying asset determines whether pricing by arbitrage is feasible. We focus only on the tradeable underlying assets. The futures pricing model depends mainly on the nature of the underlying asset. There are two main categories of underlying assets: Investment assets Consumption assets such as stocks, bonds, gold, silver such as copper, oil M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 16 / 49

Standard Assumptions Investors are subjected to no transaction costs. The underlying asset is tradeable. Investors can borrow and lend money at the same risk-free interest rate. Investors exploit arbitrage opportunities immediately (in other words, we assume that there is no arbitrage opportunity). There is no constrains on the short-selling on the underlying asset. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 17 / 49

Price of Futures on Assets with No-Income Finding the non-arbitrage price The price of the futures on an asset that pays no dividend until the maturity is given by: r(t t) F (t, T ) = S(t)e where: T t is time to maturity and r is the constant interest rate. Why is that? Suppose that F (t, T ) > S(t)e r(t t). An arbitrage opportunity arises: At time t: Go short on the futures and borrow S(t) and buy the asset. At time T : Sell the asset at F (t, T ) and return S(t)e r(t t) to the bank. r(t t) Risk free profit = F (t, T ) S(t)e M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 18 / 49

Price of the Futures on Assets with Known Income Finding the non-arbitrage price The price of the futures on an asset that pays known income until maturity is given by: where: r(t t) F (t, T ) = (S(t) I (t, T ))e I (t, T ) is the present value (at time t) of the income (dividend) provided within the period [t, T ]. Examples of such assets are: stocks, coupon bonds etc. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 19 / 49

Price of the Futures on Assets with Known Income cont d Similarly... Suppose that F (t, T ) > (S(t) I (t, T ))e r(t t). An arbitrage opportunity arises: At time t: Go short on the futures and borrow S(t) and buy the asset. At time of the Income: Invest the Income at the risk-free rate until time T. At time T : Sell the asset at F (t, T ), get the income I (t, T )e r(t t) from the asset and return S(t)e r(t t) to the bank. r(t t) Risk free profit = F (t, T ) (S(t) I (t, T ))e M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 20 / 49

The Effect of a Known Dividend Yield Dividend yield The dividend yield is the financial ratio that shows how much an asset pays out as a percentage of the asset price. q(t) = D(t) S(t) where D(t) is the dividend paid at time t. Continuous dividend yield The continuous dividend yield is the approximate continuous return of an asset if each dividend is reinvested in the asset. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 21 / 49

From Discrete to Continuous Dividend Yield Suppose that the (underlying) asset pays discrete dividend yields q(t 1 ), q(t 2 ),..., q(t N ) over the period [t, T ]. Then if all dividends are reinvested on the asset, we have S(t) S(T )(1 + q(t 1 ))(1 + q(t 2 ))...(1 + q(t N )). The continuous dividend yield q(t, T ) is the solution of equation: That is e q(t,t )(T t) = (1 + q(t 1 ))(1 + q(t 2 ))...(1 + q(t N )) q(t, T ) = 1 ln(y (t, T )), T t where Y (t, T ) = (1 + q(t 1 ))(1 + q(t 2 ))...(1 + q(t N )). Price of futures on asset with dividend yield (r q(t,t ))(T t) F (t, T ) = S(t)e M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 22 / 49

Stock Index Futures A stock index follows the value of a hypothetical portfolio, examples: S&P500, NASDAQ, the FTSE100 and the Greek FTSE20. The futures contracts on stock indices are settled in cash and not by delivering the whole portfolio. Since all the investors can theoretically create the index portfolio, the index is considered as a tradeable asset. The price of futures on stock index Index is considered as an asset that pays continuous dividend yield, hence its futures price is given by: (r q(t,t ))(T t) F (t, T ) = S(t)e M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 23 / 49

The Price of the Futures on Foreign Currency Rates The underlying asset of these contracts is a certain number of units of a foreign currency. The futures price of such futures is given by: F (t, T ) = S(t)e (r r f )(T t) where: S(t) is the spot exchange rate at time t. r is the domestic risk-free interest rate. r f is the foreign risk-free interest rate. Its proof is left as an exercise. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 24 / 49

The Price of the Futures on Investment Commodities Storage Cost The main thing about commodities is that they incur storage cost. It can be considered as negative income. Futures price on investment assets Standard examples of commodities that are investments assets are gold and silver. Futures on such assets have the following price: r(t t) F (t, T ) = (S(t) + U(t, T ))e F where: (r+u(t,t ))(T t) (t, T ) = S(t)e U(t, T ) is the present value (at time t) of all the storage costs during the period [t, T ] and u(t, T ) is the storage costs per annum as a proportion of the spot price. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 25 / 49

The Price of the Futures on Consumption Commodities Futures price on consumption assets For assets that are held only for consumption the above pricing with arbitrage arguments can not be used. Individuals and companies keep the commodity because of its consumption value (futures contracts can not be consumed). The arbitrage arguments can only provide an upper bound for the price: r(t t) F (t, T ) (S(t) + U(t, T ))e F (r+u(t,t ))(T t) (t, T ) S(t)e M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 26 / 49

Convenience Yield Definition The convenience yield is the solution y of the equation F (t, T )e y(t t) = S(t)e (r+u(t,t ))(T t) F (t, T ) = S(t)e (r y+u(t,t ))(T t). What does it mean? Convenience yield can be considered as the return of an asset obtained by the ability to profit from temporary local shortages or the ability to keep a production running. For the investment asset is equal to zero. It reflects the market s expectations concerning the future availability of the commodity. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 27 / 49

Cost of Carry Definition The cost of carry incorporates the relationship of the futures prices and the price of the underlying asset. It is defined as: c = interest rate + storage cost - income earned Cases Underlying asset No dividend payment Dividend payment Investment commodity Consumption commodity Cost of carry r r q r + u r + u y M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 28 / 49

Why Do We Use Futures? 1 Hedging (reducing the risk exposure). 2 Speculation (increasing the risk exposure for making profit). 3 Arbitrage (An investment that provides certain profit without any risk). M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 30 / 49

Hedging Definition Hedging is an action that reduces the risk exposure of a financial position by making the final outcome more certain (locking the price). Basic positions Short hedge: When an investor knows that he will sell an asset in the future and wants to lock-in the price Go short in futures. Long hedge: When an investor knows that he needs to purchase an asset in the future and wants to lock-in the price Go long in futures. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 31 / 49

Hedging cont d Perfect hedge (example) Suppose that a company wants to buy oil In Oct 2018. The price of one barrel brent crude oil is now (Mar 2018) $60.75. It can lock-in the price by going long on oil futures contract with maturity Oct 2018, with delivery price $58.7 per barrel. No risk about the price No worries. Reasons why perfect hedging is not always possible Difference on maturity of the available futures and the desired one. Difference on the underlying asset of the available futures and the desired one. A combination of the above. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 32 / 49

Hedging cont d Choices of the hedging on futures The choice of the underlying asset. The choice of the delivery month. The number of the futures contracts. Basis (recalled) b(t) = S(t) F (t, T ) Facts The basis risk arises from the uncertainty about the future interest rates and asset yield. For commodity assets basis risk may be substantial. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 33 / 49

Basis Risk Different maturity Suppose for now that r = 0. Consider an investor who has to sell the asset at some time t 2 in the future. He wants to lock-in the price by going short in futures contracts on the same asset, but there is no available contract with maturity equal to t 2. He chooses a delivery month that is as close as possible (but later than) t 2 (why later?). Suppose that the closer available maturity is T > t 2. Now: He goes short on the futures contract with maturity T. At time t 2 : He closes the short position on the futures and sells the asset at S(t 2 ). M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 34 / 49

Basis Risk cont d The effective price The effective price of the asset is given by: S(t 2 ) + F (0, T ) F (t 2, T ) }{{} = F (0, T ) + b(t 2) Profit/Loss from futures Two possible outcomes 1 S(0) S(t 2 ), which pushes the inequality F (0, T ) F (t 2, T ). He loses some gain from the price increase but he was more certain. 2 S(0) S(t 2 ), which pushes the inequality F (0, T ) F (t 2, T ). The effect of the good hedging. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 35 / 49

Basis risk cont d Different maturity and asset Suppose for now that r = 0. Consider an investor has to deliver the asset A at some time t 2 in the future. He wants to lock-in the price by going short in futures contracts on asset A, but there is no available contract on A. He chooses to go short in futures contract on an asset B, which is closely correlated to asset A. Now: He goes short on the futures contract on asset B with maturity T. At time t 2 : He closes the short position on the futures and sells the asset A at S(t 2 ). Effective price = S(t 2 ) + F B (0, T ) F B (t 2, T ) M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 36 / 49

Optimal Hedging How much futures should I long/short? Consider an investor that owns an asset with price S(t) and he wants to hedge the risk of this price by entering into futures contracts written on a different asset. How much futures he should long/short? Let assume that he shorts h contracts. The portfolio value change is given by: Π(t) = S(t) h F (t, T ). M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 37 / 49

Optimal Hedging cont d Minimizing of variance The investor goal is to minimize the variance of the portfolio value changes by choosing the right hedge ratio. Var( Π) = σ 2 S + h 2 σ 2 F 2hρσ S σ F where: σ S is the standard deviation of S, σ F is the standard deviation of F and ρ is the correlation between S and F. In other words, his hedging problem is the following: arg min{var( Π)} h R M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 38 / 49

Optimal Hedging cont d Finding the optimal h Facts Var( Π) h Var( Π) h ĥ = ρσ S σ F = = 2hσ 2 F 2ρσ F σ S, hence = 0 Cov( S, F ) σf 2. In case where Cov( S, F ) > 0 the investor goes short on ĥ futures contracts, whereas when Cov( S, F ) < 0he goes long ĥ futures contracts. A special case is when the futures is on the same underlying asset, where ĥ = 1. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 39 / 49

Hedging Using Index Futures Another hedging example Stock index futures can be used to hedge the risk on a well-diversified portfolio of stocks. The stock portfolio risk can be hedged by buying futures written on each stock held. This may be to expensive and even not feasible (non available futures, low liquidity). Alternatively, we can short h = β Π(t) S(t) M where, M is the multiplier of the future and β is the beta of the stock portfolio. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 40 / 49

An Example of Hedging Using Index Futures An example of perfectly hedging Consider firm A that has a well-diversified portfolio of stocks traded which worths 500,000 and has (estimated) beta equal to 1.5. Suppose that A wants to perfectly hedge this stock portfolio for two months by using the benchmark index futures with maturity in four months. If the current price of the index is 1,200 units A should short: where 5 is the appropriate multiplier. 500, 000 1.5 = 125 contracts 1, 200 5 M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 41 / 49

Changing the Beta Arrange the risk exposure Consider an investor that wishes to change the beta of his (well-diversified) portfolio from β to β. It has to options to do so: 1 Change the portfolio weights. 2 Use futures contracts on stock index. Using index futures for changing beta If β < β long (β β) Π(t) S(t) M If β > β short (β β ) Π(t) S(t) M It is clear that by shorting β Π(t) zero. S(t) M future contracts. future contracts. contracts the beta of the portfolio becomes M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 42 / 49

An Example of Hedging Using Index Futures Changing the beta Consider our previous example of firm A. Suppose now that A expects that the market will fall in the two following months and wants to reduce the beta of its stock portfolio. It chooses to change beta to 0.5 by using futures contracts. 500, 000 He should short: (1.5 0.5) = 83 contracts. 1, 200 5 Suppose that r = 2%, q = 4% p.a. and in the following two months market total return is 3%. The CAPM indicates that the return of the portfolio will be: r p = r/6 + β(r M r/6) = 0.33% + 1.5( 3% 0.33%) = 4.96%. After two months the company is going to close its futures position by going long on 83 contracts of the index futures. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 43 / 49

Speculation With Futures It is a high leverage speculation (it requires the margin account). Speculators never participate the delivery (their positions closed before the maturity). They generally increase the liquidity of the futures markets and allow hedgers to easily transfer risk. The general idea is: Short futures when you believe price is going to decline. Long futures when you believe price is going to increase. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 44 / 49

The Classification of the Speculators Some classes of speculators that use futures are the following: Scalpers: They keep their position open over a very small time interval exploiting the bid-ask spread and the small fluctuation on the prices (a lot of small profits end up substantial). Day traders: They close their positions at the end of each session (no matter the outcomes). Position traders: They keep their position open during longer time intervals (polar opposite of day traders). M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 45 / 49

Speculation Strategies with Futures Spread with futures: A strategy that is formed by taking position in two or more futures simultaneously. Interdelivery spread: A strategy according to which the speculator goes long and short on the same futures contract which have different maturities. Intercommodity spread: A strategy according to which the speculator goes long and short on futures contracts with the same maturity but on different commodities. Intermarket spread: A strategy according to which the speculator goes long and short on futures contracts on the same or related assets and maturity but in different exchange markets. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 46 / 49

What are the Arbitrageurs Looking for? A futures market arbitrageur mainly looks for price discrepancies between: (a) The same futures contract but different markets. Standard examples are the currency and the metals (gold, silver) markets. There are high transaction costs involved. Such arbitrage opportunities are rare and they can be exploited only by fast moving (program trading). (b) The futures price that is traded and the one that they can replicate. The arbitrageur compares the actual futures price with the theoretical one (obtained by the interest rate he can use). Profits are obtained only by large positions. In both cases there is no risk undertaken since one position is against the other. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 47 / 49

An Arbitrage Example On 08/01/10 the S&P500 index is 1,1141.69 units and its futures price for maturity Dec. 2010 is 1,125.10. Suppose that an arbitrageur can use the 1-year LIBOR which is 0.98%. The dividend the S&P500 index (is estimated to) give until the maturity is 2.33%. We have the following data: T t = 245 252, F (t, T ) = 1, 125.10, S(t) = 1, 1141.69, r = 0.98%, q = 2.33% and M = 250. The theoretical futures price of the arbitrageur is: F Th (t, T ) = S(t)e (r q)(t t) = 1, 126.8. Since, F (t, T ) < F Th (t, T ) the arbitrageur: On 08/01/10: Goes long the futures contract and short selling the index. On 16/12/2010: He returns 2.33% and receives 1,126.8 and buys the index for 1,125.10. The arbitrage profit per contract is (1,126.8-1,125.10)*250=$425,80. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 48 / 49

Summary of Section 1 Definition and terminology of forwards and futures. Futures vs forwards. Pricing methodology (cases). Hedging with futures. Speculation with futures. Arbitrage with futures. M. Anthropelos (Un. of Piraeus) Forwards and Futures Spring 2018 49 / 49