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8/13/2019 2 Forwards Futures Swaps

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

( )( )

( , ) r q T t

t F t T e S .

Marking to market a seasonedfor ward posit ion at some time after the contractinception


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At contract inception, PV of a forward contract is zero. At some time during the life ofthe contract its PV is generally different from zero.

Proposition. PV of the (long) forward contract is equal to the difference of the current forward price F (t ,T ) at time t and the delivery price K written into the forward contract

discounted from the forward maturity to the present time:( )( , , ) ( ( , ) )r T t V K t T e F t T K .

Proof. Consider a long forward contract with delivery price K that was initiated at sometime prior to t . Establish a new short forward contract at the forward price ( , ) F t T currently prevailing at time t . Hold both contracts to maturity. The payoff at maturity is:

( ) ( ( , ) ) ( , )T T S K F t T S F t T K .

This is independent ofT

S (fixed amount of cash) with the present value at time t equalto:

( ) ( ( , ) )r T t e F t T K .

This completes the proof.

Note: This relationship is valid for all types of underlying assets, including consumptionassets such as commodities.

For assets with known cash income, substituting ( )( , ) ( )r T t t t F t T e S I we obtain:

( ) ( )( , , ) ( ( , ) )r T t r T t t t V K t T e F t T K S I e K

For assets with known dividend yield we substitute ( )( )( , ) r q T t t F t T e S and obtain:

( ) ( ) ( )( , , ) ( ( , ) )r T t q T t r T t t V K t T e F t T K e S e K

Forwards Summary:

1. Underlying provides no income:

( )( , ) r T t t F t T e S , ( )( , , ) r T t t V K t T S e K .

2. Known cash income:

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

3. Known yield q:


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

Example: Currency Forwards

If the underlying asset is a foreign currency, then the forward exchange rate is:

( )( )( , ) d f r r T t t F t T e S ,

where r d is the domestic risk-free interest rate (with continuous compounding) and r f is theforeign risk-free interest rate (with continuous compounding) and S is the price of oneunit of foreign currency expressed in domestic currency (for example, USD 1.30 per oneEuro). The foreign currency can be thought of as the asset that pays continuous dividendyield at the rate r f

Confusing point: currency spot, forward, and futures quotations.

FX futures are always quoted in the market as the number of US dollars needed to buyone unit of the foreign currency. In contrast, most FX spot and forward exchange ratesare quoted in the market as the number of units of foreign currency needed to purchaseone dollar. Euro, British pound, and Australian and New Zealand dollars are exceptionsto this rule: spot and forward rates for these currencies are quoted in the same way as inthe futures market (as the number of US dollars needed to purchase one unit of thecurrency).

Forward Interest RatesReading: Hull, Sections 4.6, 4.7

Forward rates are interest rates implied by the current spot rates for periods of time in thefuture.

Example

Given: R(0,1): spot rate for one year R(0,2): spot rate for two yearDetermine:

f (0;1,2): forward rate for an investment starting at the end of the first year and lasting forone year (through the end of the second year).

Investors can lock in the forward rate for future investing and borrowing. Consider twoalternatives:1) Invest $1 at the rate R(0,2) for two years.2) Invest $1 at the rate R(0,1) for one year and enter into a forward contract to invest the

proceeds at the end of the first year for one more year at the forward rate f (0;1,2).


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To prevent arbitrage, the future value of your investment should be equal under bothalternatives. This leads to the arbitrage relationship:

2 (0,2) (0;1,2)(0,1) (0;1,2) 2 (0,2) (0,1). R f Re e e f R R

If this relationship is not satisfied, there is an arbitrage opportunity.

If f is greater than the theoretical value implied by this relationship, borrow for twoyearsat the interest rate R(0,2), invest for one year at the interest rate R(0,1) and enter intothe forward contract to invest for the second year at the rate f .

If f is less than the theoretical value, do the opposite: borrow for one year at R(0,1), enterinto the forward contract to borrow for the second year at f, and invest for two years at

R(0,2).

General case:

(0, ) R T : spot rate of interest applying from today (time zero) to time T (0, *) R T : spot rate of interest applying from today to time T*, *T T (0; , *) f T T : forward rate for the period of time between T and T* as seen at time 0.

Then the arbitrage relationship is:

* (0, *) (0, ) ( * ) (0; , *)T R T TR T T T f T T e e e ,

( * ) (0; , *) * (0, *) (0, ),T T f T T T R T TR T

and * (0, *) (0, )(0; , *)

*T R T TR T

f T T T T

.

This equation can be re-written as

(0; , *) (0, *) ( (0, *) (0, ))*T

f T T R T R T R T T T

If the yield curve is upward slopping with (0, *) (0, ) R T R T , then the forward rate is greater than the spot rate (0, *). R T

A forward rate agreement (FRA ) is a contract initiated at time t that specifies aninterest rate to be applied (for either borrowing or lending) to the agreed upon principalfor a future period of time [ T , T *]. See Hull Section 4.7 for details.Note: In class we usually work with continuously compounded interest rates to simplify.In the markets each quoted interest rate has its own compounding frequence (e.g. annual,


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semiannual, etc) and day count conventions. When working with real data, be careful tomake sure you know exactly the market convention in that market.

Swaps

Readings: Hull, Chapter 7 (we will not have enough time to cover swaps in detail inclass; read this chapter on your own)

Swaps are essentially Multi-Period Forward Contracts that let you lock in the currentswap rate (or price) during the entire life of the swap (with multiple periods).

Example of a Commodity SwapAn airline needs to buy its supply of jet fuel of N barrels per quarter over the next threeyears. It wishes to hedge its exposure to changes in the spot price of fuel. It enters into aswap with a swap dealer. Each quarter over the next three years, the airline pays the swapdealer an amount equal to the price of N barrels of jet fuel assuming a pre-negotiated

fixed price K per barrel . In exchange, it receives from the swap dealer an amount equal tothe price of N barrels of fuel priced at the then-current spot price ( floating market price ),so it can buy its fuel on the spot market.

Question: How do we set this fixed price K ?

This commodity swap is equivalent to a portfolio of twelve forward contracts, expiring atthe end of the 1 st quarter, 2 nd quarter, etc., each having the same pre-negotiated price K .Thus, PV of the commodity swap at time 0 is equal to the sum of PVs of the twelveforward contracts ( n = 12 in this case):

1(0) (0, )[ (0, ) ]

n

i ii

V N P t F t K ,

where (0, )(0, ) i i R t t i P t e is the discount factor (price of a zero-coupon bond with onedollar face value maturing at time it ) and (0, )i F t is the it -maturity forward price of one

barrel.

At inception, the swap is set up so that it is neither an asset nor liability. Its set up at zeroPV, V(0) = 0. Solving for the price K , we find:

1

1

(0, ) (0, )

(0, )

n

i ii

n

ii

F t P t K

P t

Thus, the swap price is the weighted average of the twelve forward prices. Depending onthe term structure of interest rates P (0,t i), some of the individual forward contracts have

positive PV and some negative PV. But the whole portfolio of twelve forward contracts


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has zero PV (the swap has zero PV at inception). Positive and negative PVs of individualforwards in the swap cancel out.

At some time during the life of the swap (seasoned swap), the swap generally has a non-zero PV. Marking to market a seasoned commodity swap position at some time t after

inception, assuming t is between ( i-1)th and ith payment dates, t i-1 < t


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Substituting this relationship in the formula for K , we finally arrive at the expression forthe currency swap rate:

1

1

(0, )

(0) (0, )

n

f ii

n

d ii

P t

K S P t .

Compare this with the formula for currency forward price with one delivery date t :

( ) (0, )(0, ) (0) (0)(0, )

d f r r t f

d

P t F t S e S

P t .

Interest Rate Swap

Counterparty A pays Counterparty B interest at a fixed rate of R% per annumsemiannually. B pays A interest at six-month LIBOR rate. Such a swap is equivalent to a

portfolio of forward rate agreements, one for each six-month period. The swap rate issuch a rate that sets the present value of this portfolio of forward rate agreements equal tozero.

Applications of interest rate swaps:Managing interest rate riskAsset/liability managementLowering borrowing costs

Interest rate swaps are covered in Hull, Ch. 7.

Plain Vanilla Interest Rate Swap

A pays B interest at the fixed rate S % per annum semiannually. B pays A six-monthLIBOR rate.

Notation:

N : notional (principal) of the swapT : final maturity of the swap (also called swap tenor )t = 0: contract inceptiont : some time during the life of the swapt i: payment dates, t ii , i = 1, 2,, n, t n T n

: time between two payments L(t i , t i+1): LIBOR rate between t i and t i+1 quoted at t i (simple interest rate)S : fixed swap rate P t ( , )0 : price of a zero-coupon bond with the face value of one dollar paid at time t


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LIBOR Rates

LIBOR rates are quoted as simple interest rates. Suppose we invest one dollar at time t until time t . At time t we will get

1 ( , ) ( , ) L t t L t t ,

where ( , ) L t t is the LIBOR rate with tenor (time to maturity) quoted at time t and( , ) L t t is the day-count fraction (fraction of a year between t until t ). For

example, suppose a ninety-day LIBOR rate is quoted at 6% per annum. We invest onedollar for ninety days. How much will we have in three months? The day count fractionis 90 / 360 1/ 4 L , and we will have $1.015. In general

( , ) ( , ) / 360 L t t d t t ,

where

( , )d t t is the actual number of days between t until t .

LIBOR is a floating reference rate used in floating rate notes and swaps.

Floating rate notes (FRNs)

In contrast with the standard bonds and notes that have fixed coupons, floating rate noteshave coupons that change (float) as the reference interest rate (typically LIBOR) changes.

Consider a floating rate note with $1 principal and maturity at imte T. Coupons are C (t i) paid at times t i = i , where is the time between two coupon payments.

The coupon to be paid at time t i+1 is set at time t i equal to the current simple interest ratefor the time period :

C (t i+1) = L (t i ,t i+1).

It is easy to see that the FRN is always worth par at each coupon reset date t i. The proof is by induction starting from maturity:

1

1 1

( ) 1 11 ( , ) 1 ( , )

i

i i i i

C t L t t L t t

for 1,...,1,0.i n .

In practice, the interval between two coupons is often either or of a year (quarterlyor semiannual coupons) and the reference interest rate is LIBOR (or LIBOR + somespread).

Establishing a swap rate


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1

1

(0) (0) (0) (0, ) [1 (0, )]

(0, ) (0, )

n

fix float ii

n

i

i

V V V NS P t N P T

NS P t NP T N

A par swap has zero present value at inception V (0) = 0, and the swap rate is found to be:

1

1 (0, )

(0, )n

ii

P T S

P t

The swap curve is a swap rate S = S (T ) considered as a function of the swaps maturity (tenor) T .

Futures ContractsReadings. Hull, Chapter 2, Section 5.8Futures vs. Forwards

Forwards Futures- Private OTC contracts - Traded on exchange- Not standardized - Standardized- Usually one specific - Delivery options:delivery date and the timing option,asset are specified un- quality optionambiguously- Settled at - Settled dailycontract expiration- Counterparty credit risk - No credit risk

Futures contracts are specified by the exchange, while forward contracts are private OTCarrangements.

Futures glossary (you should know the meaning of these terms): Hedge tradeDay tradeSpread tradeOpen interestDaily volumeLifetime highs/lows

Concept check: when a futures contract is traded on the exchange floor, it may be thecase that the open interest increases by one, stays the same, or decreases by one. Whatdoes this mean?


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Futures Trading MechanicsDeposit initial margin with a futures broker (margins set by the exchanges dependingon the contracts volatility and changed often). Daily settlement: at the end of each day each futures account is marked to market and

actually adjusted to reflect the current end-of-day P&L: deposits are made and lossesare withdrawn. Futures contracts are actually re-settled each day: PV of a futures position is always zero at the end of the day: V is resettled each day to zero.In effect,the futures contract is closed out and re-written again at a new price F each day.

Many financial futures are settled in cash(e.g., S&P 500 futures).

Relationship between Forward and Futures Prices when Interest Rates areDeterministic

Question: futures contracts are settled every day, while forward contracts are settled at

maturity. Does this result in any differences between futures and forward prices?

Proposition. Futures and forward prices are equal when interest rates are deterministic(known in advance).

The proof is given in Technical note 24 to Section 5.9 under the assumption of flat termstructure of interest rates. The proof can be extended to deterministic term structure ofinterest rates that is not necessarily flat.

What happens when interest rates are stochastic?If asset price changes are positively correlated with interest rate changes, then the futures

price tends to be greater than the forward price. Gains in futures resulting from theincrease in the asset price will be realized when interest rates tend to be higher, thusearning greater interest. Vice versa, if asset price changes are negatively correlated withinterest rate changes, the futures price tends to be lower than the forward price. Thedifference is not significant for short-dated contracts, but may become significant asmaturity increases and the absolute value of the correlation between the asset and interestrates increases.

Commodity Futures

Commodity storage costs and commodity inventory value must be accounted for in

evaluating commodity futures. 1. Investment commodities are commodities held by a significant number of investorssolely for investment purposes (gold and silver).

Storage costs can be regarded as negative income. Let t U denote the present value of allstorage costs to be incurred to store the commodity from the current time t until maturityT . Then the futures price of an investment commodity (gold, silver) is:


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

If storage costs are proportional to the spot price of the commodity and expressed as the percentage rate u per annum, they can be regarded as negative dividend yield:

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

2. Consumption commodities

There are certain benefits associated with holding physical commodities in stock (keep production running, avoid stock-outs, take advantage of local short-term shortages, etc.)

Thus for consumption commodities we only have the inequality:

( )( , ) ( ) r T t t t

F t T S U e or

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

This inequality is due to the first arbitrage (an arbitrageur can borrow cash, buycommodity, open a short futures contract, store commodity until maturity of the futurescontract, deliver it against the short position, and realize a profit net of storage costs andinterest on financing the commodity purchase the so-called cash-and-carry arbitrage).

However, even if futures is lower than the right hand side of the inequality, the holders of

physical commodity may not wish to sell the physical commodity and buy the futures(i.e., they may not wish to execute the second arbitrage), since they need the physicalcommodity for current consumption.

Suppose y is an effective payment rate (yield) for the potential inconvenience of a stock-out for commodity, called convenience yield . It is defined as follows:

( )( )( , ) ( ) r y T t t t F t T S U e

or

( )( )( , ) r u y T t t F t T e S .

The convenience yield reflects the market expectations concerning the future availabilityof the physical commodity. The greater the possibility that shortages will occur, thehigher the convenience yield. On the other hand, if users of the commodity have highinventories, there is little chance of shortages in the near future and the convenience yield


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tends to be low. If inventories are low, shortages are more likely and the convenienceyield is usually higher.

For investment assets, convenience yield should be equal to zero. Otherwise, there is anarbitrage opportunity (second arbitrage).

Cost of carry = (interest paid to finance the asset) + (storage costs, if any) (incomeearned on the asset, if any)

Cost of carry rate for different assets:Assets with no dividends and storage costs: r Stock index: r - q Foreign currency: r d r f Commodity: r + u

Then, the futures price can be expressed as follows ( c cost of carry rate):

Investment asset: F = ecT

S Consumption asset: F = e (c

y)T S

Hedging with Futures and ForwardsReadings: Hull, Chapter 3 and Sections 5.9-5.14

Hedging Summary

Short Hedge

Current position: plan to sell the asset at a future date (e.g., investment portfolio manager

who owns a stock, commodity producer expects to sell the commodity in the future, amanufacturer anticipating revenue in some foreign currency that will need to beconverted to domestic currency).

Risk: Asset price may fall and lower price will be realized in the future asset sale.

Hedge: short hedge. Take a short position in forward or futures contracts on the asset tolock in the sale price.

Long Hedge

Current position: plan to buy the asset at a future date (e.g., commodity or energy useranticipates future purchases, a company anticipates making a purchase in foreigncurrency at a future date).

Risk: Asset price may rise and higher price will have to be paid in the future purchase.


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Hedge: long hedge. Take a long position in forward or futures contracts on the asset tolock in the purchase price.

Own asset Asset price may fall Short hedge

Hedging does not necessarily improve the overall financial outcome in a given situation.What it does is reduce risks by making the outcome more certain .

Why do firms hedge?

Hedging is a transaction designed to reduce or, in some cases, eliminate risk.

Some reasons why firms hedge include:

Reduce or eliminate risks that are not related to the firms core business and expertise

(such as currency, interest rate, commodity, and credit risks for a manufacturer).Achieve greater stability of earnings as a result.Hedging reduces the likelihood of bankruptcy (and thus reduces the likelihood ofincurring the cost of financial distress).Hedging improves the firms credit standing, thus facilitating more favorablefinancing terms, and reducing the costs of doing business.Thus, hedging is not a zero-sum game!In some cases hedging may or may not be done, depending on the prevailing practicesin a particular industry, and possible de sirability or undesirability for a company to

be different from its competitors.

Hedging with futures is often not perfect.

Problems: 1. The underlying risk exposure and the underlying asset in the futures contract may notmatch exactly.2. Futures contracts with the desired expiration may not be available. The hedge mayrequire the futures contract to be closed out before expiration. Alternatively, a rollinghedge may be required for long-term hedges.

Basis Risk

The asset to be hedged may not be exactly the same as the asset specified in the futurescontract basis risk .

Basis = (Spot price of the asset to be hedged - Futures price of the contract used)

Decision process to optimally choose a futures contract for hedging:


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1. Choice of the underlying asset that best matches the risk exposure you want tohedge

2. Choice of the delivery month3. Choice of the appropriate number of futures contracts to use for the hedge

Minimum Variance Hedge

The hedge ratio is the number of futures contracts one should use to hedge a particularrisk exposure.

Suppose a hedger is long the asset and short h futures contracts.

Notation:S : random change in the spot price, S , during the life of the hedge (random variable) F : random change in the futures price, F , during the life of the hedge (random

variable)

S : standard deviation of S (estimated from historical data) F : standard deviation of F (estimated from historical data): correlation between the random variables S and F (estimated from historical data;

assume 0 )h: hedge ratio (to be determined)

The change in value of the hedged position during the life of the hedge is:

S h F .

The variance of the change in value of the hedged position is:

2 2 2 2S F S F Var h h .

Generally, the objective of the hedge is to maximize the hedgers expected utility.Without the detailed knowledge of the hedgers utility, we make a simplifyingassumption that the objective of hedging is to minimize risk of the hedged position asmeasured by the variance of its change .

Thus, the goal is to minimize variance:

0Var h

.

The hedge ratio h that minimizes the variance is

S

F

h


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20

and is called the minimum variance hedge ratio .

How effective is the hedge compared to doing nothing?

Calculate the residual variance of the hedged position:

2 2 2 2S F S F Var h h .

Substituting the minimum variance hedge ratio h we have

2 2(1 ) S Var ,

and the residual standard deviation of the hedged position is

21 S StDev ,

i.e., the minimum variance hedge reduces the standard deviation by a factor of 21 as compared to the original unhedged position.

Example of Stock Index Futures

A stock index tracks the performance of a representative portfolio of stocks:,i i

i

I w S wi are weights of stocks in the index.

Types of indexes:Equally-weighted indexMarket capitalization weighted index.

S&P 500 index is a market-capitalized index ( wi reflect number of shares outstanding).

As an approximation, a stock index is often regarded as an asset that pays a continuous proportional dividend at the rate q, where q is the average annualized dividend yieldduring the life of the contract. Hence, the forward price of the index can be approximatedas follows (t=0 in the following):

( )(0, ) r q T F T e S

A more precise calculation requires estimating dividends of each particular stock to be paid during the life of the futures contract:

0(0, ) ( )rT F T e S I ,

where 0 I is the PV of all income (future dividends) to be received during the life of thefutures contract:


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21

( 0, )0

i i R t t i

i

I e D ,

where D i are dividends paid at time it and (0, )i R t are discount rate from it to time zero.

For relatively short maturities, the differences between forward and futures prices

resulting from the stochastic behavior of interest rates are insignificant. We will assumethat the interest rates are deterministic and stock index forward and futures prices areequal.

Example: S&P 500 index futures. The contract size (dollar value of the underlying index portfolio) is equal to 250 x (value of the index). The tick size (minimum price move) isequal to 0.10. The dollar value of one tick is $250 x 0.10 = $25.

Index Arbitrage

1. If ( )r q T F e S - buy stocks, short futures.

2. If ( )r q T F e S - sell stocks, buy futures.

Sell the overpriced and buy the underpriced asset.

Index arbitrage trades are executed through program trading - computer generates theorders that go directly into the electronic trading system of the NYSE and/or other tradingvenues.

Practical issues in index arbitrage and program trading:

In practice, rather than trading the entire index portfolio of 500 stocks (for S&P 500), asmaller portfolio of stocks is often traded to save on transaction costs. This portfolioshould be highly correlated with the index. One needs to solve an optimization problem to select this replicating portfolio. One needs to minimize a tracking error the trackingerror between this smaller portfolio and the target index. Due to the fact that there aresome transaction costs of trading (including market impact and possible executiondelays), there is some tolerance 0 such that index arbitrage is done when:

1. If ( )( )r q T F e S - buy stocks, short futures.

2. If ( )( )r q T F e S - sell stocks, buy futures.

Caveat: the normal relationship between index futures and the underlying spot index maysometimes be violated due to severe liquidity problems and delays in order execution.During the October 87 crash futures traded at up to 18% discount to the cash index!

Hedging Stock Portfolio Risk with Index Futures


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P : current value of the stock portfolio to be hedged, F : current value of one futures contract (futures price times the contract size, e.g., S&P500 futures: 250 x (index value)).

An optimal number of contracts to short when hedging the stock portfolio (optimal hedge

ratio) is P F

, where is the stock portfolio beta defined by: P M

,

where is the correlation between changes in the portfolio value and the stock market asa whole (index is a proxy for the market), and P and M are standard deviations of

percentage changes in the portfolio value and the market as a whole, respectively.

Changing the beta of a stock portfolio (levering and un-levering betas)

Case1: Reducing market exposure (un-levering beta)To reduce beta of your stock portfolio from to some target *, > *, sell short the

quantity ( *) P F of stock index futures contracts.

Case 2: Increasing market exposure (levering beta)To increase the beta of your stock portfolio from to some target *, < * , buy the

quantity ( * ) P F

of stock index futures contracts.

Download - 2 Forwards Futures Swaps

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