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1 Derivatives Notes on Futures/Forwards Notes on Futures/Forwards Robert Wood Robert Wood Distinguished Professor of Finance Distinguished Professor of Finance
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Nov 28, 2014

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Derivatives

Notes on Futures/ForwardsNotes on Futures/Forwards

Robert WoodRobert Wood

Distinguished Professor of FinanceDistinguished Professor of Finance

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Structure of Futures Market

ClearinghouseClearinghouse CustomerCustomer

Futures CommissionMerchant (FCM)EXCHANGE MEMBERSEXCHANGE MEMBERS

ClearingClearing

MembersMembersNonclearNonclear

MembersMembers

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Structure of Futures Market

Futures Commission Merchant Exchange Members

Floor Broker (Commission broker)Floor Trader (Local)

Day tradersScalpersPosition traders

Clearinghouse

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FCMFCM SELLERSELLERFCMFCMBUYERBUYER

Floor BrokerFloor Broker Floor BrokerFloor Broker

TRADING PITTRADING PIT

Floor BrokerFloor Broker Floor BrokerFloor Broker

FCMFCM FCMFCMClearinghouseClearinghouse

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Liquidating a Futures position

Physical Delivery Offsetting Exchange of Futures for Physicals

Flexibility

Cash Delivery

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Margin Requirement Initial Margin Maintenance Margin

variation margin

Open account with $5000 on Feb 20Feb 25: Buy 2 May contracts, Margin needed 2x3000=$6000Feb 26: Add $1000 to margin to meet $6000 requirementFeb 26: Marking to market gain 1400; Account value 7400Feb 27: Marking to market loss 2500; Account value 4900;

Above maintenance margin of 4200Feb 28: Marking to market loss 1000; Account value 3200;

margin call of 2800

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Futures Trading Open Outcry Board Trading

insufficient liquidity for open outcry

Electronic Trading Opening and Closing Call Settlement Price Price Limits Position limits

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Changing Commodity Trading Changing Commodity Trading VolumeVolume

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Key Players

HedgersWheat farmerCereal manufacturer

Speculators -- selling insurance Arbitrageurs -- correct mispricing Speculators and arbs cannot survive without

hedgers

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Open InterestOpen Interest

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Notation T: time until delivery date (in years) S: price of the asset underlying the forward

contract today K: delivery price in the forward contract f: value of a long position in the forward

contract today F: forward price today r: risk-free rate of interest per annum today,

with continuous compounding, for an investment maturing at the delivery date (i.e., in T years)

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Forwards vs. Futures Forward contracts:

Customized size and amountNegotiated rate -- more costlyMore difficult to cancel/reverseDo not exist for all markets

Futures contractsDaily mark to marketStandardized sizes and expirationsDo not exist for all markets

Prices may vary slightly 90 percent of forward contracts are delivered. Only

five percent of futures contracts are delivered.

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Pricing Forwards: No Income on Underlying 90-day Forward contract on stock Stock price is $20, 90-day bond yields 4% What is the forward price? Strategy 1

Buy forward contract with forward price F Strategy 2

Buy stock for $20 Payoff the same for both strategies Thus price today should be the same

F = S(1+r)T = (20)(1+0.04)(0.25) = $20.20

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Arbitrage: An Example Assume that the forward price is $21 Forward overpriced and/or stock underpriced Strategy

Buy stock, sell forward, borrow $20 @ 4% Cash flow at execution = 0 Cash flow at delivery

Sell stock through forward for $21Pay back borrowing and interestProfit = 21 - (20)(1+0.04)(0.25) = $0.80

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Pricing Forwards: Known Cash Income on Underlying

1 year forward on a bond Bond

matures in 5 years, pays $50 in interest every six months, price today is $1200

Interest rates6-month: 8%, 1-year: 10%

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Pricing Forwards: Known Cash Income on Underlying

Strategy 1Buy bond

Strategy 2buy forwardbuy 6-month bond that pays $50buy 1 year bond that pays $50

Strategies have same payoff Cost of strategies should be same

S=F(1+r)-T+I1(1+r1)-T1 + I2(1+r2)-T2

F = (1200 – 50(1+0.08)-0.5 – 50(1+0.1)-1)(1+0.1) = 1217.08

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Arbitrage: An Example Assume forward price is $1230 Forward is overpriced / bond is underpriced Strategy

Buy bond at a cost of $1200Sell forwardBorrow $1200

Pay back $50 in 6 months (PV = $48)Pay back remaining in 1 year (PV = $1152)

On deliverySell bond through forward for $1230Pay back borrowing (FV = 1273 - 50 = 1223)

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Pricing Forwards with Storage Costs

Assume that the PV of storage costs is U If underlying has income with PV of I

F = (S - I)(1+r)T

Since costs are negative income, replace -I with U to getF = (S + U)(1+r)T

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Pricing Forwards: Storage Costs 6-month forward on Gold Spot price of gold = $300/oz. Storage costs for gold = $1/six months/oz. paid up front 6-month interest rate = 5% What is the forward price? F = (S + U)(1+r)T

S = 300, U = 1 F = (300 + 1)(1+0.05)(0.5) = 308.43

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Arbitrage: An Example Assume that the forward price is 317.20 Forward is overpriced / gold is underpriced Sell forward

Plan to sell/deliver gold in 6 months for $317.20/oz. Buy gold for $300/oz. and pay storage of $1 for 6 months Borrow $301 for 6 months at 5% Profit = 317.20 - (301)(1+0.05)(0.5) = 317.20 - 308.43 =

8.77

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Basis and Cost of Carry Basis = Spot Price - Forward Price Contango Market

Forward Price > Spot PriceHedgers are net short; speculators longCost of carry determines forward priceF = S(1+c)T or c = (F/S)(1/T)-1Spot price of gold = 300, 6-month forward price is

308.62c = (308.62/300)(1/0.5)-1 = 5.83%

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Basis ConvergenceBasis Convergence

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Patterns of Futures PricesPatterns of Futures Prices

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Basis and Convenience Yield Assume that spot price of gold is $300 and the cost of carry is

5.67% per annum, storage costs are $1 per six months, interest rate is 5%

Expected 6- month forward price is $308.62 You observe the forward price is $305 Strategy

Buy forward Sell 1 oz. of gold for $300 and save on storage costs of $1 Invest $301 at 5% Profit would be $3.62

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Convenience Yield

Strategy requires sale of gold Forward price may indicate that holders of

gold do not want to sell their holdingsConvenience related to holding gold

F = S(1+c-y)T, y is convenience yield 305 = (300)(1+0.0567-y)(0.5) or y = 2.31% Attach a value of 2.31% to owning gold

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Using Models in Practice Futures on Stock Indices and Exchange Rates

Underlying security has continuous incomeIndex: Income is dividend yield on indexFX: Income is the interest rate in “foreign” currency

Futures on bonds with no couponsUnderlying security has no income

Futures on bonds with interestUnderlying security has discrete income

Futures on commoditiesUnderlying security has storage costs

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Use of Futures: Arbitrage Use pricing model to determine theoretical price Compare the theoretical price with market price

Strategy if futures is underpricedBuy futures, Sell underlying, Invest proceeds of sale till

delivery

Strategy if futures is overpricedSell futures, Buy underlying, Borrow money required for

purchase

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Futures Index Arbitrage: An Example

Consider a futures contract on a stock market indexCurrent index value = 8300Delivery date = 6 months6-month interest rate = 8%Dividend yield on index is 5%Futures price is 8494Theoretical futures price is (8300)(1+0.08-0.05)

(0.50) or 8423.58

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Futures-Index Arbitrage Forward is overpriced

Sell forward, Buy (1+q)-T units of index, Finance purchase of index with borrowing at r% till deliverySell forward (forward price = 8494)Buy 0.9759 (=(1+0.05)-(0.5)) units of the index for 8099.97 Borrow 8099.97 at 8% for 6 months

Reinvest all dividends back in indexOn delivery date

Sell index through forward for 8494Pay back borrowing, payback = 8099.97(1+0.08)(0.5) =

8417.74Profit = 8494 - 8417.74 = 76.26

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Risk Management: Hedging Activity that controls the price risk of a position Combine the derivative with the underlying Position in derivative is opposite to that in

underlying Hedge ratio is the number of units of the

derivative to the underlying Hedge ratio minimizes the overall exposure to

price risk

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Problems with Simple Strategy The “commodity” you are interested in does not

have a futures contractUse a futures contract with an underlying that is

closely related to the “commodity”

The date you need the “commodity” does not match the delivery dateUse a futures contract that has delivery as close to

(and greater than) the date you are interested in

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Managing the Risk of a PortfolioAssume that you are managing a $2m portfolio of stocksAssume that you are managing a $2m portfolio of stocks

You want to hedge the portfolio using a futures contractYou want to hedge the portfolio using a futures contract

Futures contract on the S&P 500Futures contract on the S&P 500

Current futures price is 1024.20Current futures price is 1024.20

Contract is for 250 units of the indexContract is for 250 units of the index

Beta of portfolio relative to the S&P 500 is 3.68Beta of portfolio relative to the S&P 500 is 3.68

Number of contracts = (3.68)(2,000,000/(250*1024.2) = 29Number of contracts = (3.68)(2,000,000/(250*1024.2) = 29

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Hedging

0

1

2

3

Avg Std Dev

Unhedged

Hedged

-202468

Jan Feb Mar Apr May Jun

Unhedged

Hedged

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Forward Rate Agreement(illustrating forward rates)

Used by large, international banks Buy a security from a bank Receive 15% for a year from the bank The interest will be received in the period

starting 6 months and ending 18 months from now

What is the 15% based on?1-year Forward rate in 6 months ( 0.5 r1.5)

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Forward Rate Agreements (cont.)

Consider an FRA for a future period T to T* What should the FRA interest rate be? Spot rates for T and T* are r and r* (1+r*)T* = (1+r)T(1+k)(T*-T)

or k = [(1+r*)T*/(1+r)T][1/(T*-T)] - 1 2 year rate is 12% and the 4 year rate is 14% k = [(1+0.14)4/(1+0.12)2][1/(4-2)] – 1 = 16.04% =

2r4

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Forward Rate Agreements (cont.)

Assume you entered into an FRA one year ago for 16% from year 2 to year 4Invest $100 in 2 years, receive 137.71 in 4

years (2 years later) Today, 1 year later, the 1-year and 3-year spot

rates are 14% and 15% What is the value of the FRA today? 137.71(1+0.15)-3 – 100(1+0.14)-1 = 2.83

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Futures on Long-Term Debt

Underlying pays known cash income T= time to delivery S = Spot cash price PVI = Present value of income until

delivery date of futures r = T-year zero-coupon yield Futures price F = (S-PVI)(1+r)T

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Problem with using formula

Cash Spot Price vs. Quoted Spot PriceAccrued Interest

Choice of deliverable bondAny bond that satisfies requirementsConversion factor--standardizes deliverables

Wild card option2pm futures (settlement price), 4pm underlying, 8pm intent

notice

Cash Futures price vs. Quoted Futures PriceAccrued interest and conversion factor

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Applications of Financial Futures Changing maturity of T-bill investments Converting floating-fixed rates Immunization

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Forwards with Short-Term Rates Underlying is a short-term instrument

90 day maturity; No coupon Value of underlying at time 0 is S = 100(1+r*)-T*

Forward price F at time T is S(1+r)T = [100(1+r)T] / (1+r*)T*

Forward rate from T to T* is

rf = [(1+r*)T* /(1+r)T] [1/(T*-T)] - 1 Forward Price F at time T = 100(1+rf)-(T*-T)

240 days240 days

Transfer t-bill at T for FTransfer t-bill at T for FInitiate futures contractInitiate futures contract

150150 240240

T-bill T-bill Matures at Matures at T*T*

00

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Short-term Rate Forward Example 150-day forward

Obtain a 90-day zero-coupon bond at delivery

150-day spot rate is 10% 240-day spot rate is 12% Forward price?

Forward rate = [(1+r*)T* /(1+r)T] [1/(T*-T)] - 1

= [1+0.12) (2/3) /(1+0.10) (5/12) ] [1/(0.25)] -1 = 15.41%Forward price = 100(1+0.1541)-(0.25) = 96.48

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Repo Arbitrage The 100-day forward price on a 90-day underlying is 98 The 190-day spot rate is 10% F = 98 = 100(1+r)T /(1+r*) -T* = 100(1+r)0.2739 /(1+0.1)-0.5205

Implied 100-day spot rate is 11.33% Compare implied with actual If implied > actual

Lend at implied, borrow at actual (Type 1) If implied < actual

Borrow at implied, lend at actual (Type 2)

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Repo Arbitrage

100-day spot rate is 11% < 11.62% Borrow at the spot rate for 100 days Lend at the implied spot rate

Sell the futures contract Invest borrowed money at spot rate for 190 days such that

payoff is $100 Amount borrowed = 100(1+0.1)-(190/365) = 95.16 Amount owed in 100 days = 95.16(1+0.11)(0.274) = 97.92 Profit = 98-97.92 = $0.08

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Repo ArbitrageRepo Arbitrage

100-day spot rate is 12% < 11.62%100-day spot rate is 12% < 11.62% Lend at the spot rate for 100 daysLend at the spot rate for 100 days Borrow at the implied spot rateBorrow at the implied spot rate

Buy the futures contractBuy the futures contractBorrow money at spot rate for 190 days such that payoff Borrow money at spot rate for 190 days such that payoff

is $100is $100

Amount borrowed = 100(1+0.1)Amount borrowed = 100(1+0.1)-(0.5205)-(0.5205) = 95.16 = 95.16 Amount obtained in 100 days=95.16(1+0.12)Amount obtained in 100 days=95.16(1+0.12)(0.274)(0.274) = =

98.1698.16 Profit = 98.16-98 = $0.16Profit = 98.16-98 = $0.16

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T-Bills and T-Bill Futures

T-Bill quotes and cash priceCash price = 100 - (n/360)(Quoted price)

T-Bill futures quotes and cash pricesFutures quote = 100 - 4(100 - Cash price)

60-day forward on 90-day T-bill150-day T-bill quoted at 960-day interest rate is 6% per annumCash price on 150-day T-bill = 100 - (150/360)(9) = 96.25Cash futures price = (96.25)(1+0.06)(60/360) = 97.19Quoted futures price = 100 - (4)(100-97.19) = 88.76

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Hedging Interest Rate Risk

DurationSensitivity of value to interest rate

Equalize durations of futures and portfolioEqual sensitivity for futures and portfolio

(NF)(F)(DF)=(S)(DS)

Number of futures NF = (SDS)/(FDF)

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Stock Index Futures

S&P500--the dominate stock index contract Cost of carry:

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Stock Index Futures Index arb -- program trading Portfolio insurance

Dynamic hedging--replicate option w/cash and stock

Motivation – 1984 ERISA act Market crash

Barings Bank -- risk management