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Nov 28, 2014

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DerivativesNotes on Futures/Forwards Robert Wood Distinguished Professor of Finance1

Structure of Futures MarketClearinghouse Customer

EXCHANGE MEMBERS Clearing Nonclear Members Members

Futures Commission Merchant (FCM)2

Structure of Futures MarketFutures Commission Merchant Exchange MembersFloor Broker (Commission broker) Floor Trader (Local)Day traders Scalpers Position traders

Clearinghouse3

BUYER

FCM Floor Broker

FCM Floor Broker

SELLER

TRADING PIT Floor Broker FCMClearinghouse

Floor Broker FCM4

Liquidating a Futures positionPhysical Delivery Offsetting Exchange of Futures for PhysicalsFlexibility

Cash Delivery

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Margin RequirementInitial Margin Maintenance Marginvariation margin

Open account with $5000 on Feb 20Feb 25: Buy 2 May contracts, Margin needed 2x3000=$6000 Feb 26: Add $1000 to margin to meet $6000 requirement Feb 26: Marking to market gain 1400; Account value 7400 Feb 27: Marking to market loss 2500; Account value 4900; Above maintenance margin of 4200 Feb 28: Marking to market loss 1000; Account value 3200; margin call of 2800 6

Futures TradingOpen Outcry Board Tradinginsufficient liquidity for open outcry

Electronic Trading Opening and Closing Call Settlement Price Price Limits Position limits7

Changing Commodity Trading Volume

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Key PlayersHedgersWheat farmer Cereal manufacturer

Speculators -- selling insurance Arbitrageurs -- correct mispricing Speculators and arbs cannot survive without hedgers9

Open Interest

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NotationT: 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) 11

Forwards vs. FuturesForward contracts:Customized size and amount Negotiated rate -- more costly More difficult to cancel/reverse Do not exist for all markets

Futures contractsDaily mark to market Standardized sizes and expirations Do not exist for all markets

Prices may vary slightly 90 percent of forward contracts are delivered. Only five percent of futures contracts are delivered.12

Pricing Forwards: No Income on Underlying90-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) = $20.2013

Arbitrage: An ExampleAssume 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 $21 Pay back borrowing and interest Profit = 21 - (20)(1+0.04)(0.25) = $0.80

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Pricing Forwards: Known Cash Income on Underlying1 year forward on a bond Bondmatures in 5 years, pays $50 in interest every six months, price today is $1200

Interest rates6-month: 8%, 1-year: 10%15

Pricing Forwards: Known Cash Income on UnderlyingStrategy 1 Buy bond Strategy 2 buy forward buy 6-month bond that pays $50 buy 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) = 1217.0816

Arbitrage: An ExampleAssume forward price is $1230 Forward is overpriced / bond is underpriced Strategy Buy bond at a cost of $1200 Sell forward Borrow $1200Pay back $50 in 6 months (PV = $48) Pay back remaining in 1 year (PV = $1152)

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

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Pricing Forwards with Storage CostsAssume 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 get F = (S + U)(1+r)T

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Pricing Forwards: Storage Costs6-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 ExampleAssume 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 CarryBasis = Spot Price - Forward Price Contango MarketForward Price > Spot Price Hedgers are net short; speculators long Cost of carry determines forward price F = S(1+c)T or c = (F/S)(1/T)-1 Spot price of gold = 300, 6-month forward price is 308.62 c = (308.62/300)(1/0.5)-1 = 5.83% 21

Basis Convergence

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Patterns of Futures Prices

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Basis and Convenience YieldAssume 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 StrategyBuy forward Sell 1 oz. of gold for $300 and save on storage costs of $1 Invest $301 at 5% Profit would be $3.6224

Convenience YieldStrategy 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 gold25

Using Models in PracticeFutures on Stock Indices and Exchange Rates Underlying security has continuous income Index: Income is dividend yield on index FX: Income is the interest rate in foreign currency Futures on bonds with no coupons Underlying security has no income Futures on bonds with interest Underlying security has discrete income Futures on commodities Underlying security has storage costs26

Use of Futures: ArbitrageUse pricing model to determine theoretical price Compare the theoretical price with market priceStrategy 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 ExampleConsider a futures contract on a stock market indexCurrent index value = 8300 Delivery date = 6 months 6-month interest rate = 8% Dividend yield on index is 5% Futures price is 8494 Theoretical futures price is (8300)(1+0.080.05)(0.50) or 8423.58

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Futures-Index ArbitrageForward is overpriced Sell forward, Buy (1+q)-T units of index, Finance purchase of index with borrowing at r% till delivery Sell 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 index On delivery date Sell index through forward for 8494 Pay back borrowing, payback = 8099.97(1+0.08)(0.5) = 8417.74 Profit = 8494 - 8417.74 = 76.26 29

Risk Management: HedgingActivity 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 risk30

Problems with Simple StrategyThe 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 in31

Managing the Risk of a PortfolioAssume that you are managing a $2m portfolio of stocks You want to hedge the portfolio using a futures contract Futures contract on the S&P 500 Current futures price is 1024.20 Contract is for 250 units of the index Beta of portfolio relative to the S&P 500 is 3.68 Number of contracts = (3.68)(2,000,000/(250*1024.2) = 2932

Hedging8 6 4 2 0 -2Jan Unhedged Hedged Feb Mar Apr May Jun

3 2 1 0 Avg Std Dev33

Unhedged Hedged

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)34

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% = 2 r435

Forward Rate Agreements (cont.)Assume you entered into an FRA one year ago for 16% from year 2 to year 4 Invest $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 DebtUnderlying 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)T37

Problem with using formulaCash Spot Price vs. Quoted Spot PriceAccrued Interest

Choice of deliverable bondAny bond that s

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