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Energy & Finance Track
Futures and OptionsProf. Christophe Pérignon (HEC)[email protected]/perignon
Spring 2011 - May 3, 2011
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Futures and OptionsProf. Christophe Pérignon (HEC)[email protected]
Part 1:Introduction and Background
1-3
The Nature of Derivatives
• A derivative is a financial asset whose value depends on the value of another asset, called underlying asset
• Examples of derivatives include Futures, Forwards, Options, Swaps, Credit Derivatives (CDS)
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Historical Facts
• Derivatives, while seemingly new, have been used for thousands years
* Aristotle, 350 BC (Olive)
* Netherlands, 1600s (Tulips)
* USA, 1800s (Grains, Cotton)
* Spectacular growth since 1970’s
• Increase in volatility (Liberalization, International trade, End of Bretton Woods, Oil price shocks)
• Black-Scholes model
• Derivatives Exchanges + Over The Counter (OTC)
1-5
Examples of Underlying Assets
• Stocks • Bonds• Exchange rates• Interest rates• Commodities/metals• Energy• Number of bankruptcies
among a group of companies
• Pool of mortgages
• Temperature, quantity of rain/snow
• Real-estate price index• Loss caused by an
earthquake/hurricane• Dividends• Volatility • Derivatives• etc
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Ways Derivatives are Used
• Hedge risks: reducing the risk
• Speculate: betting on the future direction of the market
• Lock in an arbitrage profit: taking advantage of a mispricing
Net effect for society?
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1. Interest Rate Swap
• Consider a 3-year swap initiated on 5 March 2008 between Microsoft and Intel.
• Microsoft agrees to pay to Intel an interest rate of 5% per annum on a notional principal of $100 million.
• In return, Intel agrees to pay Microsoft the 6-month LIBOR on the same notional principal.
• Payments are to be exchanged every 6 months, and the 5% interest rate is quoted with semi-annual compounding.
5%
Intel MSFT
LIBOR
1-8
Microsoft Cash Flows
---------Millions of Dollars---------
LIBOR FLOATING FIXED Net
Date Rate Cash Flow Cash Flow Cash Flow
Mar. 5, 2008 4.2%
Sep. 5, 2008 4.8% +2.10 –2.50 –0.40
Mar. 5, 2009 5.3% +2.40 –2.50 –0.10
Sep. 5, 2009 5.5% +2.65 –2.50 +0.15
Mar. 5, 2010 5.6% +2.75 –2.50 +0.25
Sep. 5, 2010 5.9% +2.80 –2.50 +0.30
Mar. 5, 2011 6.4% +2.95 –2.50 +0.45
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2. Futures Contracts
• A FUTURES contract is an agreement to buy or sell an asset at a certain time in the future for a certain price
• By contrast in a SPOT contract there is an agreement to buy or sell an asset immediately
• The party that has agreed to buy has a LONG position (initial cash-flow = 0)
• The party that has agreed to sell has a SHORT position (initial cash-flow = 0)
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2. Futures Contracts (II)
• The FUTURES PRICE (F0) for a particular contract is the price at which you agree to buy or sell
• It is determined by supply and demand in the same way as a spot price
• Terminal cash flow for LONG position: ST - F0
• Terminal cash flow for SHORT position: F0 - ST
Futures are traded on organized exchanges:• Chicago Board of Trade, Chicago Mercantile Exch. (USA)• Montreal Exchange (Canada)• EURONEXT.LIFFE (Europe)• Eurex (Europe)• TIFFE (Japan)
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Example: Gold
Sept 06, 2010(09.23 NY Time)
Oct 2010 $1,250.8
Nov 2010 $1,251.3
Dec 2010 $1,252.6
Dec 2011 $1,260.6
S0 = $1,250.4 F0 (Nov 2010) = $1,251.3 Source: www.kitco.com Source: www.cmegroup.com
1-12 Sources: www.onechicago.com and yahoo finance
1-13Quotes retrieved on September 7, 2010
Oct-1
0
Mar
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Aug-1
1
Jan-
12
Jun-
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Sep-1
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Feb-1
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May
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Oct-1
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Mar
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Jan-
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Nov-1
7
Apr-1
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Sep-1
83.5
4
4.5
5
5.5
6
6.5
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CME Natural Gas Futures Prices
Sep
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Oct
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Nov
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Dec
-10
Jan-
11F
eb-1
1M
ar-1
1A
pr-1
1M
ay-1
1Ju
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Jul-1
1A
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1S
ep-1
1O
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1N
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1D
ec-1
1Ja
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Feb
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Mar
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Apr
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May
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Jun-
12Ju
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Aug
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3.45
3.46
3.47
3.48
3.49
3.5
3.51
3.52
CME Copper Futures Prices
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3. Forward Contracts
• Forward contracts are similar to futures except that they trade on the over-the-counter market (not on exchanges)
• Forward contracts are popular on currencies and interest rates
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4. Options
• A call option is an option to buy a certain asset by a certain date for a certain price (the strike price K)
• A put option is an option to sell a certain asset by a certain date for a certain price (the strike price K)
American vs. European Options
• An American option can be exercised at any time during its life. Early exercise is possible.
• A European option can be exercised only at maturity
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Example: Cisco Options (CBOE quotes)
Option Cash Flows on the Expiration Date
• Cash flow at time T of a long call : Max(0, ST - K)• Cash flow at time T of a long put : Max(0, K - ST)
From NASDAQ :
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Total outstanding notional amount : $688 trillion(OTC = $615 ; Exchanges = $73, BIS, December 2009)
Annual U.S. Growth National Product : $14 trillion (US Department of Commerce, Year 2010)
Total Value of global stocks: $48 trillion (World Federation of Exchange Members, December 2009)
Total Value of global bonds : $26 trillion (BIS, June 2010)
Size of the Global Derivative Market
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Trading Activity for Derivatives
Contracts outstanding, Table 23B, BIS June 2010:
Futures: Interest Rates 68%, Currency 7%, Equity 25%
Options: Interest Rates 40%, Currency 2%, Equity 58% 1-19
International Evidence on Financial Derivatives Usage” by Bartram, Brown and Fehle (2008)
7,319 non-financial firms from 50 countries, 2000-2001
60% of the firms use derivatives in general 45% use currency derivatives 33% use interest rate derivatives 10% use commodity price derivatives
Factors Determining Derivatives Usage:
Size of the local derivatives market
Level of risk and financial sophistication
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In today’s derivatives markets, any type of financial payoff one can think of can be obtained at a price
For instance, if a corporation wants to receive a payment that is a function of the square of the yen/dollar exchange rate if the volatility of the S&P 500 index exceeds 35% during a month, it can do so
When anything is possible, but one does not have the required knowledge or experience, it is easy to make mistakes
Derivatives and Risk
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Losses Attributed to Derivatives: 1993-2008
Corporation Date Instrument Loss (US$ million)
Société GénéraleAmaranth Hedge FundOrange County, CaliforniaShowa Shell Sekiyu, JapanKashima Oil, JapanMetallgesellschaft, GermanyBarrings, UKAllied Irish Bank, USAshanti, GhanaChina Aviation Oil, SingaporeYakult Honsha, JapanCalyon, FranceNational Australia Bank, Aus.Codelco, ChileProcter & Gamble, USNatwest, UK
Jan. 2008Sep. 2006Dec. 1994Feb. 1993Apr. 1994Jan. 1994Feb. 1995Feb. 2002Oct. 1999Dec. 2004Mar. 1998Sept. 2007Jan. 2004Jan. 2004Apr. 1994Feb. 1997
Index Futures 7,100Futures on Natural Gas 6,500Reverse Repos 1,810Currency Forwards 1,580Currency Forwards 1,450Oil Futures 1,340Stock Index Futures 1,330Currency Derivatives 691Gold “Exotics” 570Oil Derivatives 550Stock Index Derivatives 523Credit Derivatives 348Currency Options 262Copper Futures 200 Differential Swaps 157Swaptions 127
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Banks' Subprime Writedowns & Losses Top 20
Source: Bloomberg, http://www.bloomberg.com/apps/news?pid=20601087&sid=aSKLfqh2qd9o&refer=worldwide
1. Citigroup $55.1 billion
11. JPMorgan Chase $14.3 billion
2. Merrill Lynch $51.8 billion
12. Deutsche Bank $10.8 billion
3. UBS $44.2 billion
13. Credit Suisse $10.5 billion
4. HSBC $27.4 billion
14. Wells Fargo $10 billion
5. Wachovia $22.5 billion 15. Barclays $9.1 billion
6. Bank of America $21.2 billion
16. Lehman Brothers $8.2 billion
7. IKB Deutsche $15.3 billion 17. Credit Agricole $8 billion
8. Royal Bank of Scotland $14.9 billion
18. Fortis $7.4 billion
9. Washington Mutual $14.8 billion 19. HBOS $7.1 billion
10. Morgan Stanley $14.4 billion
20. Societe Generale $6.8 billion
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5. Credit Derivatives: (1) Credit Default Swap
Default protection buyer
Default protection seller
CDS spread
Payment if default by reference entity
• Provides insurance against the risk of default by a particular company
• The buyer has the right to sell bonds issued by the company for their face value when a credit event occurs.
• The buyer of the CDS makes periodic payments to the seller until the end of the life of the CDS or a credit event occurs
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5. Credit Derivatives: (2) Collateralized Debt Obligations (CDO)
Pool of Loans
B 3%
AA 8%
AAA15%
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It's a joke that we are in markets like this. We are playing the dollar against the Swiss franc until 2042.”Cedric Grail, City of Saint Etienne CEO, quoted by Business Week (2010)
Loan features:• Notional: EUR20m• Maturity: 15 years• coupon rate:
Y1-2: 3.80%
Y3-15: 3.80% + Max(1.9700 – GBPCHF)
Capped at 24%
Market evolution:• GBPCHF at time of trade inception: 2.0700
=> expected coupon of 3.80% per year• GBPCHF today: 1.5215
=> current coupon level of 24% per year (it would be 45% without the cap…)
6. Toxic Loans of Local Authorities
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Delivery
• Most contracts are closed out before maturity : long 5 contracts at t1 + short 5 contracts at t2 > t1
• If a contract is not closed out before maturity, it usually settled by delivering the assets underlying the contract. When there are alternatives about what is delivered, where it is delivered, and when it is delivered, the party with the short position chooses.
• A few contracts (for example, those on stock indices) are settled in cash
1-281-28
Contract Specifications: Futures on CAC40 Index
ContractCONTRAT À TERME FERME SUR L’INDICE CAC 40
(FCE)
Underlying AssetCAC 40 stock index, made of 40 French blue chip
companies, computed by Euronext Paris SA, released every 30 seconds (value of 1000 on Dec. 31, 1987)
Notional Value of the index × 10 €
Minimum Tick 0,5 index point (5 €)
Maximum Price Fluctuation
+/- 200 points with respect to last closing price.As soon as the futures price exceed this limit, trading is
suspended
Maturity Date Third Friday of the month at 4PM
LiquidationSettled in Cash. The terminal value of the index is the
average value of the index between 3:40 and 4:00PM (41 observations).
MarginMargin requirement is 225 points per contract
Margin is reduced for trading on spread (long and short positions on contracts with different maturities)
Transaction CostTrading Fee (Euronext Paris) : 0,14 €Clearing Fee (LCH.Clearnet) : 0,13 €
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Default Risk with Futures
• Two investors agree to trade an asset in the future• One investor may:
– regret and leave– not have the financial resources
• Margins and Daily Settlement
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Margins
• A margin is cash (or liquid securities) deposited by an investor with his broker
• The balance in the margin account is adjusted to reflect daily gains or losses: “Daily Settlement” or “Marking to Market”
• If the balance on the margin account falls below a pre-specified level called maintenance margin, the investor receives a margin call
• If the investor is unable to meet a margin call, the position is closed
• Margins minimize the possibility of a loss through a default on a contract
In today’s derivatives markets, any type of financial payoff one can think of can be obtained at a price
For instance, if a corporation wants to receive a payment that is a function of the square of the yen/dollar exchange rate if the volatility of the S&P 500 index exceeds 35% during a month, it can do so
When anything is possible, but one does not have the required knowledge or experience, it is easy to make mistakes
Derivatives and Risk
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Losses Attributed to Derivatives: 1993-2008
Corporation Date Instrument Loss (US$ million)
Société GénéraleAmaranth Hedge FundOrange County, CaliforniaShowa Shell Sekiyu, JapanKashima Oil, JapanMetallgesellschaft, GermanyBarrings, UKAllied Irish Bank, USAshanti, GhanaChina Aviation Oil, SingaporeYakult Honsha, JapanCalyon, FranceNational Australia Bank, Aus.Codelco, ChileProcter & Gamble, USNatwest, UK
Jan. 2008Sep. 2006Dec. 1994Feb. 1993Apr. 1994Jan. 1994Feb. 1995Feb. 2002Oct. 1999Dec. 2004Mar. 1998Sept. 2007Jan. 2004Jan. 2004Apr. 1994Feb. 1997
Index Futures 7,100Futures on Natural Gas 6,500Reverse Repos 1,810Currency Forwards 1,580Currency Forwards 1,450Oil Futures 1,340Stock Index Futures 1,330Currency Derivatives 691Gold “Exotics” 570Oil Derivatives 550Stock Index Derivatives 523Credit Derivatives 348Currency Options 262Copper Futures 200 Differential Swaps 157Swaptions 127
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Banks' Subprime Writedowns & Losses Top 20 (Aug 2008)
Source: Bloomberg, http://www.bloomberg.com/apps/news?pid=20601087&sid=aSKLfqh2qd9o&refer=worldwide
1. Citigroup $55.1 billion
11. JPMorgan Chase $14.3 billion
2. Merrill Lynch $51.8 billion
12. Deutsche Bank $10.8 billion
3. UBS $44.2 billion
13. Credit Suisse $10.5 billion
4. HSBC $27.4 billion
14. Wells Fargo $10 billion
5. Wachovia $22.5 billion 15. Barclays $9.1 billion
6. Bank of America $21.2 billion
16. Lehman Brothers $8.2 billion
7. IKB Deutsche $15.3 billion 17. Credit Agricole $8 billion
8. Royal Bank of Scotland $14.9 billion
18. Fortis $7.4 billion
9. Washington Mutual $14.8 billion 19. HBOS $7.1 billion
10. Morgan Stanley $14.4 billion
20. Societe Generale $6.8 billion
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1-351-35
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Are Derivatives “Financial Weapons of Mass Destruction” ?
• “Derivatives are financial weapons of mass destruction, carrying dangers that, while now latent, are potentially lethal.” Warren Buffet
• Numerous losses caused by (mis)using derivatives
• Credit derivative losses
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Should We Fear Derivatives?
• “The answer is no. We should have a healthy respect for them. We do not fear planes because they may crash and do not refuse to board them because of that risk. Instead, we make sure that planes are as safe as it makes economic sense for them to be. The same applies to derivatives. Typically, the losses from derivatives are localized, but the whole economy gains from the existence of derivatives markets.”Rene Stulz (Ohio State University)
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Regulation of Derivatives Markets
• Exchange-based trades are transparent and cleared• OTC trades are less transparent and less frequently
cleared• Most OTC derivatives are arranged with a dealer (below)• Systemic risk concerns• Current derivatives reform proposals:
– Migration of OTC trading to exchanges– Centralized clearing for OTC products– Improved price/position transparency– Speculation position limits– Improved corporate governance in
financial risk management
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1. Corn: An Arbitrage Opportunity?
• Suppose that:– The spot price of corn is US$390 (for 1,000
bushels)– The quoted 1-year futures price of corn is US$425– The 1-year US$ interest rate is 5% per annum– No income or storage costs for corn
• Is there an arbitrage opportunity?
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• NOW– Borrow $390 from the bank– Buy corn at $390– Short position in a futures contract
• IN ONE YEAR– Sell corn at $425 (the futures price)– reimburse 390 exp(0.05) = $410
ARBITRAGE PROFIT = $15
NOTE THAT ARBITRAGE PROFIT AS LONG AS
S0 exp(r T) < F0
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2. Corn: Another Arbitrage Opportunity?
• Suppose that:– The spot price of corn is US$390– The quoted 1-year futures price of corn is
US$390– The 1-year US$ interest rate is 5% per
annum– No income or storage costs for corn
• Is there an arbitrage opportunity?
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• NOW– Short sell corn and receive $390– Make a $390 deposit at the bank– Long position in a futures contract
• IN ONE YEAR– Buy corn at $390 (the futures price)– Terminal value on the bank account 390 exp(0.05) =
$410ARBITRAGE PROFIT = $20
NOTE THAT ARBITRAGE PROFIT AS LONG AS
S0 exp(r T) > F0
Therefore F0 has to be equal to S0 exp(r T) = $410
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Futures Price for an Investment Asset
For any investment asset that provides no
income and has no storage costs
F0 = S0erT
Immediate arbitrage opportunity if:
F0 > S0erT short the Futures, long the asset
F0 < S0erT long the Futures, short sell the asset
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The Cost of Carry
• The cost of carry, c, is the storage cost plus the interest costs less the income earned
• For an investment asset F0 = S0ecT
• For a consumption asset F0 S0ecT
• The convenience yield, y, is the benefit provided when owning a physical commodity.
• It is defined as:
F0 = S0 e(c–y )T
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Source: www.theoildrum.com Source: Quarterly Bulletin, Bank of England, 2006
Examples
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Relation Between European Call and Put Prices (c and p)
• Consider the following portfolios:
Portfolio A : European call on a stock +
present value of the strike price in cash (Ke -rT )
Portfolio B : European put on the stock + the stock
• Both are worth Max(ST , K ) at the maturity of the options
• They must therefore be worth the same today:
c + Ke -rT = p + S0
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The Binomial Model of Cox, Ross and Rubinstein
• An option maturing in T years written on a stock that is currently worth S
S u ƒu
S d ƒd
Sƒ
where u is a constant > 1
: option price in the upper state
where d is a constant < 1
: option price in the lower state
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• Consider the portfolio that is D shares and short one option
• The portfolio is riskless when S u D – ƒu = S d D – ƒd or
dSuS
fƒ du
S u D – ƒu
S d D – ƒd
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• Value of the portfolio at time T is:
S u D – ƒu or S d D – ƒd
• Value of the portfolio today is:
(S u D – ƒu )e–rT
• Another expression for the portfolio value today is S D – f• Hence the option price today is:
f = S D – (S u D – ƒu )e–rT
• Substituting for D we obtain:
f = [ p ƒu + (1 – p )ƒd ]e–rT
wherepe d
u d
rT
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A Two-Step Example
• Each time step is 3 months• The tree is recombining (u = 1.1 and d = 0.9 are
constant)
20
22
18
24.2
19.8
16.2
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Valuing a Call Option (K=21, T=0.5):
• Value at node B= e–0.12×0.25(0.6523×3.2 + 0.3477×0) = 2.0257
• Value at node A = e–0.12×0.25(0.6523×2.0257 + 0.3477×0)= 1.2823
201.2823
22
18
24.23.2
19.80.0
16.20.0
2.0257
0.0
A
B
C
D
E
F
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Application: BIN Pricing
• Pricing an 18-month European call option using a 3 time-step binomial model
• Do the same for an 18-month European put option
• Check your results using the put-call parity
• Assume now that the put option is American. Would the price be any different?
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The Black-Scholes Formulas
TdT
TrKSd
T
TrKSd
dNSdNeKp
dNeKdNScrT
rT
10
2
01
102
210
)2/2()/ln(
)2/2()/ln( where
)( )(
)( )(
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The N(x) Function
• N(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x
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Application: BS Pricing
• Using the Black-Scholes model to compute the value of a 3-month European call option (K = 54 Euros) written on one share of TOTAL
• Assume the firm is not going to pay any dividend
over the next three months
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Implied Volatility: An Example
• Price an American Call option written on TOTAL• Date: June 30, 2008• Next dividend is in more than 3 months
• T = 0.25, K = €54, S0 = €53.86, r = 4%
• Option Pricing Model : Black-Scholes • Data: Past 62 end-of-the-day prices (Apr 1 - Jun 27, 2008)• Annualized volatility = 19.63%
• Black-Scholes Price (cbs) = €2.30
• However, market price (cmkt) = €2.89
Main result : Black-Scholes Price (cbs) ≠ Market Price (cmkt)
1-58
So 53,86 53,86 53,86 53,86 53,86 53,86 53,86 53,86 53,86 53,86 53,86 53,86 53,86K 54 54 54 54 54 54 54 54 54 54 54 54 54r 0,04 0,04 0,04 0,04 0,04 0,04 0,04 0,04 0,04 0,04 0,04 0,04 0,04sigma 0,18 0,19 0,20 0,21 0,22 0,23 0,24 0,25 0,26 0,27 0,28 0,29 0,30T 0,250 0,250 0,250 0,250 0,250 0,250 0,250 0,250 0,250 0,250 0,250 0,250 0,250d1 0,127 0,125 0,124 0,123 0,122 0,122 0,122 0,122 0,122 0,122 0,123 0,124 0,124d2 0,037 0,030 0,024 0,018 0,012 0,007 0,002 -0,003 -0,008 -0,013 -0,017 -0,021 -0,026N(d1) 0,551 0,550 0,549 0,549 0,549 0,549 0,548 0,548 0,549 0,549 0,549 0,549 0,549N(d2) 0,515 0,512 0,510 0,507 0,505 0,503 0,501 0,499 0,497 0,495 0,493 0,491 0,490c_bs 2,13 2,24 2,34 2,45 2,56 2,66 2,77 2,88 2,98 3,09 3,20 3,30 3,41c_market 2,89 2,89 2,89 2,89 2,89 2,89 2,89 2,89 2,89 2,89 2,89 2,89 2,89
0,00
0,50
1,00
1,50
2,00
2,50
3,00
3,50
4,00
0,18 0,19 0,20 0,21 0,22 0,23 0,24 0,25 0,26 0,27 0,28 0,29 0,30
Volatility
Bla
ck-S
cho
les
Pri
ce -
Cal
l T
OT
AL
Sep
08 K
=54
Black-Scholes Price
Market Price
1-59
Implied Volatility: Definition
• Implied Volatility, or Implied Standard Deviation (ISD), is the volatility parameter (s) for which the Black-Scholes price of the option is equal to the market price of the option
• Unlike for the option price, there is no closed-form solution for the implied volatility
• ISD needs to be estimated numerically:
Min{cbs(s) – cmkt}
{s}
1-60
Market-Level Implied Volatility (VIX)
