Futures & SWAPSFinancial Derivatives

Shanghai Fall 2014

Week 6-7 FINC 5880

Futures Contract• A commitment today to transact in the future with the purpose

to settle the price for future delivery of the underlying asset

• Highly standardized and exchange traded securities (www.cme.com)

• Started in agriculture but developed towards all kinds of financial futures (stock indices, interest rates, currencies…)

• Traded electronically (globex)

• Traders in options post a good faith deposit at a Broker account (MARGIN) in order to guarantee contract performance

Future contracts contain

• Delivery date of a commodity at maturity• The agreed price (to be paid at maturity)• The spec/grade of the commodity:

– No 2 hard winter wheat or No 1 soft red wheat…

• Means of delivery• The trader who takes the long position of the

contract is said to buy a contract• The trader who takes the short position of the

contract is said to sell the contract.

Futures (buy and sell)

• Buying or selling a contract is only figurative no money changes hands at the start of the contract

• The future price will only be paid at maturity of the contract

• A price increase at maturity favors the buyer of the contract

• A price decrease at maturity favors the seller of the contract

• For every buyer there is a seller and the profit of the buyer is equal to the loss of the seller and vice versa (zero sum game)

Profit and Loss on contract

• Profit to Long= Spot price at maturity-original future price

• Profit to Short= Original future price-Spot price at maturity

• Every long position is offset by a short one

Buy call option versus buying futures contract…

• The buyer of the call can walk away (no obligation to exercise the option)

• If the asset price drops the futures buyer is exposed to considerable losses! The call option buyer can loose no more than the premium of the option…

• If Pt= price of future at maturity and Fo=Future price today

• Long futures profit= Pt-Fo• Short futures profit= Fo-Pt

Pay off Futures…

Futures contracts…

Currencies Commodities Equity Indices

GBP Corn Dow

YEN Wheat S&P Midcap

EURO Soybeans Nasdaq

SFR Milk NYSE

AUD Coffee Russell 2000

Mex Peso Cotton FTSE

Brazilian Real Sugar CAC

Canadian $ Rice DAX

SKR Orange juice HEX

Clearing House (CH)

• Trading done more and more over electronic platforms (EUREX)

• The clearinghouse stands in between buyer and seller; the seller of the contract hold a short position with the CH the buyer a long position

• The clearinghouse net position is always zero…since it takes a short and long position at the same time and for the same size and spec and maturity…

• Almost all traders liquidate their positions before the end of maturity of the contract (just use the contract to hedge the price risk)

• Before maturity open interest (contract outstanding) normally dries up quickly since all traders are reversing their positions…

Marking to Market…Margin

• Futures contracts rarely result in actual delivery…• Margins required are normally 5-15% of the contract

value so example:• If the current future price for wheat is $2.06/bushel (1

bushel= 35.24 Liters) then the value of the contract (5000 bushels standard) $ 10,300 if the margin required is 10% the Margin is $1,030 (per contract of 5000 bushels)

• Futures on more volatile assets require a higher Margin %...

• If the margin reaches a level below the Maintenance Margin a Margin call will follow (to protect the clearing house from any potential losses in the future)

Class assignment 1: Margins

• Suppose the Maintenance Margin on a Wheat Futures contract is 5% (Initial Margin 10%) Current Future Price $2.06

• A standard wheat contract is 5000 bushels

• How many $ cents does the wheat price need to fall to trigger a Margin call on the long position (buyer) ?

Answer…

• Value contract $2.06*5000 bushels=$10,300 so initial margin 10%*$10,300=$1,030

• Maintenance margin thus at $515

• Every 1$ct price drop triggers $0.01*5000 bushels=$50 margin

• $515/$50= about 11 $ cents price drop is needed to trigger the margin call…

Convergence Property

• A commodity available from two markets (spot or future) should have the same value (priced identically)

• Thus the Future price at maturity=spot price at maturity

• The future price and spot price converge at maturity…

Daily Mark to Market….(example)

Day (price today $17.10) delivery Day 5

Profit/Loss per ounce of Silver of long position

Daily Proceeds for 5000 ounces standard contract Silver

1 Future Price: $ 17.20 $0.10 $ 500

2 Future Price: $ 17.25 $0.05 $ 250

3 Future Price: $ 17.18 -$0.07 -$ 350

4 Future Price: $ 17.18 $0 $ 0

5 Future Price: $ 17.21 $0.03 $ 150

Total $ 550

Commodity versus financial

• Delivery of Futures on commodities is in that commodity

• Delivery of Futures on the Stock indices is in cash equal to the value of that index at maturity…

• The S&P500 index calls for delivery of $250 times the index. So if S&P500=1200 that is a cash settlement of $300,000 in return for $250*futures price.

Regulating Body

• CFTC=Commodities Futures Trading Commission maintains trading and authorizes trading in new contracts…

Example

• You think crude oil prices will increase:• You buy oil futures• Each contract requires buying 1000 barrels (159 liters)• Current Future price (delivery Nov) $52.67• For every $1 price increase of crude oil the long position

(buyer) gains $1000• For every $1 price drop the short position (seller) gains

$1000• If the spot price is $54.67 at maturity the long side of the

contract will benefit $2000• The short side will loose an identical amount…

Why buy futures?

• You can make the same profit as on buying the underlying asset but your investment is only a fraction of the value of that asset (the margin 5-15%)

• The lower the margin (investment) the higher the potential returns…

Class assignment 2: Calculate Return%

• You buy oil at $52.67 and sell it in the spot market later for $54.67; your % return is?

• You buy a future contract with initial margin 10% (1000 barrels) at $52.67 (spot today=future price today) at maturity the price has gone up $2 to $54.67;your % return is?

Class Assignment 3: Futures Oil Distributor

• An oil distributor plans to sell 100,000 barrels of oil in Nov and hedges his price risk with selling 100 (1000 barrels each) futures contracts in November

• The current price $52.67• Assume the price of oil at maturity can only be

$50.67, $52.67 or $54.67 per barrel ; • show that the futures contract offers a perfect

hedge against the price risk!

Answer…no matter the price of oil at maturity the

proceeds are the same …

P=$50.67 P=$52.67 P=54.67

Revenues from oil sale

100,000*P=

$5,067,000

100,000*P=

$5,267,000

100,000*P=

$5,467,000

Profit on Futures contract

$2*100,000=

(Fo-Pt)*100,000

$200,000

$ 0

(Fo-Pt)*100,000

(no difference with buy price)

-$2*100,000=

(Fo-Pt)*100,000

-$200,000

Total Proceeds $5,267,000 $5,267,000 $5,267,000

Generalizing…

• Total revenues from the sales of oil by the distributor (class assignment)=

• Pt (per barrel) being the price at maturity • Plus the difference between the future price at

moment of buying futures (Fo) and the price at maturity (Pt)

• Thus Proceeds=Pt+(Fo-Pt)=Fo the position is perfectly hedged against any price movement in the future at the current future price….

Class Assignment 4: Futures Electricity Power Company

• An Electricity Power Company plans to buy 100,000 barrels of oil in Nov; show that if the firm buys 100 futures contracts of oil Nov it can hedge its price risk totally…

• The current price $52.67• Assume the price of oil at maturity can

only be $50.67, $52.67 or $54.67 per barrel

Answer…no matter the price of oil at maturity the

proceeds are the same …

P=$50.67 P=$52.67 P=54.67

Costs from oil purchase

100,000*P=

$5,067,000

100,000*P=

$5,267,000

100,000*P=

$5,467,000

Profit on Futures contract

$2*100,000=

(Pt-Fo)*100,000

-$200,000

$ 0

(Pt-Fo)*100,000

(no difference with buy price)

-$2*100,000=

(Pt-Fo)*100,000

$200,000

Total costs $5,267,000 $5,267,000 $5,267,000

Cross Hedging

• Hedging a position using futures contracts on another asset(s)

• For instance: Hedging a position in an actively managed stock portfolio by using futures on the index (index futures)…

• What are the sources of risk for an investor using such cross hedging?

• (hint: what if the return on the market does not move perfectly with the return on the managed portfolio?)

Class assignment 5:Speculation with Futures

• You hold 100 ounces Gold• You also sell one futures gold contract

(also 100 ounces)• The spot price of gold today is $391• The future price gold (June) is $396• If tomorrow the spread between the spot

and futures price narrows to $ 4.50 (Future price $398.50 and spot $394) how can an investor benefit from this?

Answer…

• Gain on holding Gold 1 day $394-$391=$3

• Loss on gold future $398.50-$396=$2.50

• Net gain is the decrease in the spread ($5 - $4.50)

Speculation again…

• You hold Sept long contract and a June short contract ; the Sept future price increases with 5cts while the June futures price increases with 4cts

• Will you have a gain?

answer

• Your contract to buy increases in price with 5cts so you gain 5 cts

• Your contract to sell increases with 4 cts so you loose 4 cts

• The spread narrows 1ct this is your net gain…

Class Assignment 6:Future Pricing…

• We follow the same reasoning of replication as what we did when we were looking at the valuation of options…

• We create 2 strategies that have the same pay off…and thus their investments should be equal:

• Strategy A: buy gold (price= -So)» sell gold at T (price ST)

• Strategy B: enter long position (no investment today) » Sell at T: (ST-Fo)

• At same time invest : -Fo/(1+Rf)^T» Growing in T to: Fo

• Required: show that strategy A and B have the same pay off as a Futures contract and then derive from this Fo=Function(So)

Answer: Equivalence…

• Or in General:• Fo(1+Rf)^T=So so Fo=So(1+Rf)^T• The price of Future today= spot price today up-rated at the missed (opportunity cost) Risk free

rate that we could have made on a T Bill for T periods…so:

• If gold sells for $400 spot and Rf=0.5% per month then a 6 month maturity futures contract should have a future price Fo of:

• So(1+Rf)^T=$400(1+0.005)^6=$412.15

• A 12 month contract should be priced:• So(1+Rf)^T=$400(1+0.005)^12=$424.67

Cash Flow

Today

Future Pay Off

(assuming T=1)

Strategy A:

Buy Gold

- So +St

Strategy B:

Long (Buy) Future+

Lend Fo/(1+Rf)

0

-Fo/(1+Rf)

St-Fo

+Fo= +St

Implying that: -So=-Fo/(1+Rf) or

Fo= So/(1+Rf)

Like with options…

• Mispricing leads to arbitrage profits (risk-free)

• You sell the side that is overpriced

• You buy the side that is under priced

• So you make risk free profit…

Class assignment 7: Futures Arbitrage (over pricing)

• Assume that the Future contract 6 months (last slides) was mispriced at $413 instead of $412.15

• Recall that the spot of gold now is $400

• Recall that Rf=0.5% per month

• Show that with a zero investment you can make a risk free profit of $0.85 (just the amount of the mispricing)

Answer…action Initial cash flow Cash flow at T

Borrow $400 and repay with interest at T

+$400 -$400(1+0.5%)^6=$412,15

Buy gold for $400 (buy under priced side)

-$400 ST

Enter in short future position $413 (sell over priced side)

$ 0 (no cash out) $413-ST

TOTAL $ 0 (no investment) $ 0.85 riskless

Class Assignment 8: Futures Arbitrage (under pricing)

• If the 6 month future contract in last assignment were under priced at $411 instead $412.15 can you proof that you can make a risk free arbitrage profit of $1.15?

• Spot gold $400

• Rf=0.5% per month…

Answer…

action Initial cash flow Cash flow at T

Lend $400 and receive with interest at T

-$400 $400(1+0.5%)^6=$412,15

sell gold for $400 (sell over priced side)

$400 -ST

Enter in long future position $411 (buy under priced side)

$ 0 (no cash out) ST-$411

TOTAL $ 0 (no investment) $ 1.15 riskless

Future value/Price

• We have derived that the price of a future contract is: Fo=So(1+Rf)^T indicating the opportunity cost of investing in T Bills

• If the underlying asset generates an income (dividend on stocks) then the future price changes in: Fo=So(1+Rf-d)^T in which d is the dividend yield on the stocks…

To see this consider:

• Suppose the S&P500 index is at 1000 and an investor who holds $1000 in a mutual fund indexed to the S&P500 wishes to temporarily hedge her exposure to market risk. The indexed portfolio provides dividend of $20 over the year and all dividends are assumed to be paid end of year. Assume that the Futures price for year end delivery is $1010. The investor enters the short (selling) side of the contract.

• Show the pay off if at maturity the stock in the mutual fund reaches $970,$990……$1050,$1070….

Clarifying with example…

Fi nal Val ue Stock St $970 $990 $1, 010 $1, 030 $1, 050 $1, 070Payoff short Futures 40 20 0 -20 -40 -60Di vi dend 20 20 20 20 20 20Total $1, 030 $1, 030 $1, 030 $1, 030 $1, 030 $1, 030

Note that the pay off from the short Futures i s: Fo($1010)-St

Thi s posi ti on i s perfectl y hedged!

Rate of Return on thi s perfectl y hedged portfol i o= ((Fo+Di vi dend)-So)/ So

Si nce thi s i s a ri sk free hedge the return shoul d be the same as Rf thus:

((Fo+D)-So)/ So=Rf

So Fo= So(1+Rf-D/ So) where D/ So i s d (di vi dend yi el d)

Fo= So(1+Rf-d)

Class Assignment 9: Future Contract Pricing for Stock Index

• The Rf is 0.5% per month

• The dividend yield on the S&P500 stock index is 0.1%

• The spot price of the S&P500 is 1200

• What should be the 3 month future contract price? And the 6 month?

• If the S&P500 rises to 1210 what will be the future prices of these contracts?

Answer…

• Fo=So(1+Rf-d)^T so for 3 month contract: Fo=1200(1.004)^3=1214.46

• For 6 month contract: 1200(1.004)^6=1229.09

• For 3 month contract if S&P=1210: 1210(1.004)^3=1224.58

• For 6 month contract if S&P=1210 1210(1.004)^6=1239.33

Swaps: Plain Interest Vanilla Swaps

• You are manager of a large portfolio of $100 million par value of long term bonds with coupon rate 7%. However you believe looking forward that interest rates will rise and you want to benefit from this! You know it will be extremely expensive in terms of transaction costs to change the portfolio but you know you could SWAP your fixed interest income for a floating rate income if you can find a market party that have interest forecasts opposite to you.

• A SWAP-dealer (bank) may advertise that it is willing to swap a cash flow based on the 6-month LIBOR rate for one based on a fixed rate of 7%.

• The portfolio manager thus is expected to enter into a SWAP agreement with the dealer

• The future will learn if the portfolio manager made the right decision• He has exchanged 7% interest on $100 Million for LIBOR*$100M• If LIBOR moves to 6.5% the decision to SWAP was wrong since the

portfolio manager looses $0.5M interest income but if LIBOR moves up to 7.5% he will gain $0.5M

•

Swaps normally provide benefits for all…(win/win)

• Assume companies A (triple A rated) and B (triple B rated)

• Financing conditions:• Show that if companies A and B

enter into a Swap that both can benefit

• Company A currently is financed with fixed rate debt and company B with floating rate debt (LIBOR)

• Note: the Swap dealer want a 0.05% fee…

• Note: Company B is the weaker party and agreed to pay the fee of the dealer…

• However assume that both companies will absorb 50% of the benefit from the Swap

Company A Fixed Rate:

7%

Fixed Rate:

7.5%

Company B LIBOR LIBOR+0.25%

Realize from this case….

• Company A and B have a very different rating and thus a different cost of debt

• Company A and B MUST have opposite expectations of future interest rates otherwise they will not enter into the Swap

• Company A want to Swap to floating rate and thus assumes interest rates will drop

• Company B want to Swap to fixed rate debt and thus assumes interest rates to rise

SWAP DEAL…

Pay outside bank -7%

Receive from SWAP dealer

+7%

Pay SWAP dealer -(LIBOR-0.1%)

Total Pay LIBOR-0.1%

Receive from B 7.05%

Pay A -7%

Receive from A +(LIBOR-0.1%)

Pay B -(LIBOR-0.1%)

TOTAL FEE 0.05%

Company A Company BSWAP

Dealer

7% LIBOR+0.25%

7% 7%+0.05%

LIBOR-0.1% LIBOR-0.1%

Pay outside bank -(LIBOR+0.25%)

Receive from SWAP dealer

LIBOR-0.1%

Pay SWAP dealer -7.05%

Total Pay -(7.05%+0.35%)

Class Assignment: Plain Vanilla Interest Swap

• Assume Companies A and B have the following financing conditions:

• The dealer fee is 0.2%• A is initially fixed rate financed

and want to SWAP to floating rate

• B pays the SWAP dealer• A and B share the benefit of

the SWAP 70:30• Set up the SWAP and show

this SWAP has a clear benefit for both A and B….

Company A Company B

Fixed Rate 10% 14.6%

Floating Rate

LIBOR+0.3% LIBOR+0.7%

Answer: Before drawing…

• Note the difference in fixed rates between A and B: 4.6%

• The difference in floating rates: 0.4%• The difference of these differences is 4.2%• This is potentially the benefit that A and B can

share but the SWAP dealer want 0.2%• Thus A and B can share 4% net• A will claim 70% of the benefit (70%*4%=2.8%)• B claims the remaining 30% (30%*4%=1.2%)• Now you can set up the drawing of this SWAP

deal…

Answer: SWAP DEAL…

Pay outside bank -10%

Receive from SWAP dealer

+10%

Pay SWAP dealer -(LIBOR-2.5%)

Total Pay LIBOR-2.5%

Receive from B 7.05%

Pay A -7%

Receive from A +(LIBOR-0.1%)

Pay B -(LIBOR-0.1%)

TOTAL FEE 0.05%

Company A Company BSWAP

Dealer

10% LIBOR+0.7%

10% 10%+0.2%

Pay outside bank -(LIBOR+0.7%)

Receive from SWAP dealer

LIBOR-2.5%

Pay SWAP dealer -10.2%

Total Pay (10.2%+3.2%)

LIBOR-2.5% LIBOR-2.5%

Concluding

• So A now pays LIBOR-2.5% and this is 2.8% better than it’s own financing if they would change their contract with their outside bank (because then they pay LIBOR+0.3%) Thus A did indeed get 70% of the total benefit of 4%

• And B pays: 13.4% and this is indeed 1.2% better than the interest they would pay to their outside bank (because then they pay 14.6%)Thus B gets 30% of the total benefit of 4%....