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    Forwards, Futures and Swaps

    Forward ContractsReadings: Hull, Section 1.3, Sections 5.1-5.7,5.10, Sections 4.6 and 4.7

    Contracts and Trading MechanicsA forward contract is a contract to buy or sell an underlying asset at a predetermined price K (delivery price ) on a specified future date T .

    Long party agrees to buy the underlying asset at the delivery price K at time T .Short party agrees to sell the underlying asset at the delivery price K at time T .

    At the contract inception no money changes hands. The contract is settled at maturityT :the short delivers the asset to the long in return for cash amount K . The forward contractis a contract to exchange the underlying asset for a pre-negotiated amount of cash at a

    pre-specified future date.

    Counterparties: bilateral over-the-counter (OTC) contracts negotiated between twocounterparties (between two financial institutions or between a financial institution andits customer).

    Payoff at Expiration:The long receives the asset worth S T ( spot price of the asset at maturity of the forward T )and pays the delivery price K . Thus the cash flow ( payoff ) from the long forward positionat maturity T is equal to the difference between the sport price and the delivery price,S T K . Some forward contracts are physically settled, while some are cash settled.

    The short receives the cash amount K and delivers the asset worth S T in exchange. Thus

    the cash flow from the short forward position at maturity T is K - S T .

    Notation:T : contract maturity;t :current time, 0 t T ;S t or simply S : current spot price at t ;

    K : delivery price specified at contract inception;( , , )V K t T or simply V : present value at time 0t of the previously initiated forward

    contract with delivery price K and maturity T ; F (t ,T ) or simply F : forward price at time t for settlement at time T ;

    ( , )r t T or r : Zero-coupon risk-free interest rate per annum, expressed with continuous

    compounding, for the period from t to T .

    Wesuppose that the following assumptions hold for some market participants:

    There are no frictions in the market - no transaction costs, bid/ask spreads, andno trading restrictions (such as restrictions on short sales).The market participants can lend and borrow at the same risk-free rate ofinterest.

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    Market participants take advantage of arbitrage opportunitiesinstantaneouslyas they occur.

    We do not need to require that these assumptions hold for all market participants (theycertainly do not hold for most individuals). It is sufficient to require that they hold for

    some market participants, such as large financial institutions.

    Fixing the Forward Price at Contract Inception: The Case of Assets with KnownCash Income

    Assume all future cash income amountsgenerated by the asset (e.g. dividends on a stockor coupons on a bond) between t and T are known at time t and let I t be their PV at t . Ifthe asset produces cash flows iC at times it t , then the PV is

    ( )ir t t t i

    i

    I e C ,

    where it is assumed for simplicity that the same risk-free interest rate applies for all timesit .

    Proposition. The arbitrage-free forward price at time t is given by:

    ( )( , ) ( )r T t t t F t T e S I .

    Proof. The proof is by arbitrage arguments.

    First Arbitrage Argument

    Suppose that at time t you are able to enter into a forward contract with somecounterparty with the delivery price (forward price) ( )( , ) ( )r T t t t F t T e S I .

    Consider the following strategy.

    At time t :Step 1.Take a short position in the forward contract with maturity T and with delivery

    price (forward price) F (t ,T ).Step 2. Borrow t S dollars until time T at the risk-free rate r .

    Step 3. Use the loan to buy the asset at the current spot price t S .

    This strategy requires no initial cash investment.

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    At time T :Step 1.Settle the forward contract by selling the asset for the delivery price F (t ,T );Step 2. Use the amount ( )r T t t e S of the proceeds to repay the loan with interest. The cash

    income generated by the asset plus interest amounts to ( )r T t t e I by time t .

    Profit and Loss (P&L): a riskless arbitrage profit of ( )( , ) ( ) 0r T t t t F t T e S I .

    Since we assumed no arbitrage opportunities, we have a contradiction.

    Second Arbitrage Argument

    Suppose that at time t you are able to enter into a forward contract with somecounterparty with the delivery price (forward price) ( )( , ) ( )r T t t t F t T e S I .

    Consider the reverse strategy.

    At time t :Step 1. Take a long position in the forward contract with delivery price ( , ) F t T .Step 2. Sell short the asset at spot price t S and receive cash amount t S .Step 3. Invest the cash received from the short sale at the risk-free rate r .

    At time T :Step 1. Your cash balance is now equal to ( ) ( )r T t t t e S I (The investment of the cash

    amount equal to t S at time t at rate r is now worth( )r T t

    t e S . The short seller also has to pay the income generated by the asset to the party the asset was borrowed from. The PVof that income at time t is t I . Its value at time T is ( )r T t t e I . Thus, this amount issubtracted).Step 2.Settle the forward contract by taking delivery of the asset in exchange for the

    payment of delivery price ( , ) F t T .Step 3. Close out the short position by delivering the asset to the party the asset was

    borrowed from when the short sale was initiated.

    Profit and Loss (P&L): a riskless arbitrage profit of ( )( ) ( , ) 0r T t t t e S I F t T .

    Since we assumed there are no arbitrage opportunities, we have a contradiction.

    Thus, both inequalities are ruled out, and the following relationship must hold in themarket that does not allow arbitrage opportunities:

    ( )( , ) ( )r T t t t F t T e S I . This completes the proof.

    Note: The second arbitrage argument requires that the asset can be sold short. This can berelaxed. It is enough to require that there is a significant number of investors in the

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    market who hold the asset purely for investment purposes (as opposed to forconsumption, such as for physical commodities). If the forward price is too low, they willfind it attractive to sell the asset and take a long position in a forward contract.

    Fixing the Forward Price at Contract Inception: The Case of Assets with KnownDividend Yield

    Suppose the underlying asset pays dividends continuously at the known constantdividend rate q (dividend yield , say, 5% per annum with continuous compounding), sothat over an infinitesimal time interval dt one unit of the asset pays qdt units in dividends(worth qS t dt at the current asset price).

    Proposition. The arbitrage-free forward price at time t for delivery at time T is:

    ( )( )( , ) r q T t t F t T e S .

    Proof.

    First Arbitrage Argument

    Suppose ( )( )( , ) r q T t t F t T e S .

    At time t :Step 1.Take a short position in the forward contract with delivery price F (t ,T ). Step 2. Borrow ( )q T t t e S dollars for the period from t toT at the risk-free rate r .

    Step 3.Buy( )q T t

    e units of the underlying asset at the spot price( )q T t

    t e S .

    Note that by time T the ( )q T t e initial units of the asset accumulate one unit due todividends received during the period from t to T . To see this, recall that one unit of theasset pays qdt units in dividends over an infinitesimal time period. Suppose at time t weown t X units of the asset. Over an infinitesimal time interval dt , X t units of the asset pay

    t X qdt new units in dividends. Thus, the change in the total number of units t X held attime t is

    t t dX qX dt ,and t X solves the differential equation

    t t

    dX qX

    dt .

    Thus, 0qt

    t X e X .

    In particular, if at some fixed time t we have ( )q T t t X e units, then at time T> t we will

    have one unit of the asset, ( ) ( ) ( ) 1q T t q T t q T t T t X e X e e .

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    At time T :Step 1.Settle the short forward position by selling the one unit of the assetfor the delivery

    price F (t ,T ) .Step 2. Use the amount ( )( )r q T t t e S of proceeds to repay the loan with interest.

    P&L: a riskless arbitrage profit of( )( )( , ) 0r q T t t F t T e S .

    Thus, we have a contradiction with our assumption of no arbitrage.

    Second Arbitrage Argument

    Suppose ( )( )( , ) r q T t t F t T e S .

    Do the reverse strategy.

    At time t :Step 1.Take a long position in the forward contract with delivery price F (t ,T ). Step 2. Sell short ( )q T t e units of the asset.Step 3. Invest the proceeds of the short sale ( ( )q T t t e S dollars) at the rate r .

    At time T :Step 1. Settle the long forward position by taking delivery of one unit of the asset for thedelivery price F (t ,T ).Step 2. Close out the short position by delivering the asset to the party it was borrowedfrom.Step 3. The remaining cash amount is equal to the initial proceeds of the short sale plusinterest minus the delivery price paid for the asset.

    P&L: a riskless arbitrage profit ( )( ) ( , ) 0r q T t t e S F t T .

    We have a contradiction with our assumption of no arbitrage.

    Thus, bot

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