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1-1 Energy & Finance Track Futures and Options Prof. Christophe Pérignon (HEC) perignon@hec.fr www.hec.fr/perignon Spring 2011 - May 3, 2011

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1-2 Futures and Options Prof. Christophe Pérignon (HEC) perignon@hec.fr Part 1: Introduction and Background

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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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1-4 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)

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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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1-6 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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1-7 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% IntelMSFT LIBOR

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1-8 Microsoft Cash Flows ---------Millions of Dollars--------- LIBORFLOATINGFIXEDNet DateRateCash Flow Mar. 5, 20084.2% Sep. 5, 20084.8%+2.10–2.50–0.40 Mar. 5, 20095.3%+2.40–2.50–0.10 Sep. 5, 20095.5%+2.65–2.50+0.15 Mar. 5, 20105.6%+2.75–2.50+0.25 Sep. 5, 20105.9%+2.80–2.50+0.30 Mar. 5, 20116.4%+2.95–2.50+0.45

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1-9 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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1-10 2. Futures Contracts (II) The FUTURES PRICE (F 0 ) 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: S T - F 0 Terminal cash flow for SHORT position: F 0 - S T 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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1-11 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 S 0 = $1,250.4 F 0 (Nov 2010) = $1,251.3 Source: www.kitco.com Source: www.cmegroup.com

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1-12 Sources: www.onechicago.com and yahoo finance

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1-13 Quotes retrieved on September 7, 2010

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1-14 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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1-15 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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1-16 Example: Cisco Options (CBOE quotes) Option Cash Flows on the Expiration Date Cash flow at time T of a long call : Max(0, S T - K) Cash flow at time T of a long put : Max(0, K - S T ) 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 1-18

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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

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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 1-20

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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 1-21

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Losses Attributed to Derivatives: 1993-2008 CorporationDateInstrument Loss (US$ million) Société Générale Amaranth Hedge Fund Orange County, California Showa Shell Sekiyu, Japan Kashima Oil, Japan Metallgesellschaft, Germany Barrings, UK Allied Irish Bank, US Ashanti, Ghana China Aviation Oil, Singapore Yakult Honsha, Japan Calyon, France National Australia Bank, Aus. Codelco, Chile Procter & Gamble, US Natwest, UK Jan. 2008 Sep. 2006 Dec. 1994 Feb. 1993 Apr. 1994 Jan. 1994 Feb. 1995 Feb. 2002 Oct. 1999 Dec. 2004 Mar. 1998 Sept. 2007 Jan. 2004 Apr. 1994 Feb. 1997 Index Futures 7,100 Futures on Natural Gas 6,500 Reverse Repos 1,810 Currency Forwards 1,580 Currency Forwards 1,450 Oil Futures 1,340 Stock Index Futures 1,330 Currency Derivatives 691 Gold “Exotics” 570 Oil Derivatives 550 Stock Index Derivatives 523 Credit Derivatives 348 Currency Options 262 Copper Futures 200 Differential Swaps 157 Swaptions 127 1-22

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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 billion11. JPMorgan Chase $14.3 billion 2. Merrill Lynch $51.8 billion12. Deutsche Bank $10.8 billion 3. UBS $44.2 billion13. Credit Suisse $10.5 billion 4. HSBC $27.4 billion14. Wells Fargo $10 billion 5. Wachovia $22.5 billion15. Barclays $9.1 billion 6. Bank of America $21.2 billion16. Lehman Brothers $8.2 billion 7. IKB Deutsche $15.3 billion17. Credit Agricole $8 billion 8. Royal Bank of Scotland $14.9 billion18. Fortis $7.4 billion 9. Washington Mutual $14.8 billion19. HBOS $7.1 billion 10. Morgan Stanley $14.4 billion20. Societe Generale $6.8 billion 1-23

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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 1-24

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5. Credit Derivatives: (2) Collateralized Debt Obligations (CDO) Pool of Loans B 3% AA 8% AAA 15% 1-25

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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 1-26

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1-27 Delivery Most contracts are closed out before maturity : long 5 contracts at t 1 + short 5 contracts at t 2 > t 1 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

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1-28 Contract Specifications: Futures on CAC40 Index Contract CONTRAT À TERME FERME SUR L’INDICE CAC 40 (FCE) Underlying Asset CAC 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) NotionalValue of the index × 10 € Minimum Tick0,5 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 DateThird Friday of the month at 4PM Liquidation Settled in Cash. The terminal value of the index is the average value of the index between 3:40 and 4:00PM (41 observations). Margin Margin requirement is 225 points per contract Margin is reduced for trading on spread (long and short positions on contracts with different maturities) Transaction Cost Trading Fee (Euronext Paris) : 0,14 € Clearing Fee (LCH.Clearnet) : 0,13 €

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1-30 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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1-31 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

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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 1-32

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Losses Attributed to Derivatives: 1993-2008 CorporationDateInstrument Loss (US$ million) Société Générale Amaranth Hedge Fund Orange County, California Showa Shell Sekiyu, Japan Kashima Oil, Japan Metallgesellschaft, Germany Barrings, UK Allied Irish Bank, US Ashanti, Ghana China Aviation Oil, Singapore Yakult Honsha, Japan Calyon, France National Australia Bank, Aus. Codelco, Chile Procter & Gamble, US Natwest, UK Jan. 2008 Sep. 2006 Dec. 1994 Feb. 1993 Apr. 1994 Jan. 1994 Feb. 1995 Feb. 2002 Oct. 1999 Dec. 2004 Mar. 1998 Sept. 2007 Jan. 2004 Apr. 1994 Feb. 1997 Index Futures 7,100 Futures on Natural Gas 6,500 Reverse Repos 1,810 Currency Forwards 1,580 Currency Forwards 1,450 Oil Futures 1,340 Stock Index Futures 1,330 Currency Derivatives 691 Gold “Exotics” 570 Oil Derivatives 550 Stock Index Derivatives 523 Credit Derivatives 348 Currency Options 262 Copper Futures 200 Differential Swaps 157 Swaptions 127 1-33

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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 billion11. JPMorgan Chase $14.3 billion 2. Merrill Lynch $51.8 billion12. Deutsche Bank $10.8 billion 3. UBS $44.2 billion13. Credit Suisse $10.5 billion 4. HSBC $27.4 billion14. Wells Fargo $10 billion 5. Wachovia $22.5 billion15. Barclays $9.1 billion 6. Bank of America $21.2 billion16. Lehman Brothers $8.2 billion 7. IKB Deutsche $15.3 billion17. Credit Agricole $8 billion 8. Royal Bank of Scotland $14.9 billion18. Fortis $7.4 billion 9. Washington Mutual $14.8 billion19. HBOS $7.1 billion 10. Morgan Stanley $14.4 billion20. Societe Generale $6.8 billion 1-34

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1-36 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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1-37 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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1-38 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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1-39 Futures and Options Prof. Christophe Pérignon (HEC) perignon@hec.fr Part 2: Pricing

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1-40 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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1-41 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 S 0 exp(r T) < F 0

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1-42 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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1-43 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) = $410 ARBITRAGE PROFIT = $20 NOTE THAT ARBITRAGE PROFIT AS LONG AS S 0 exp(r T) > F 0 Therefore F 0 has to be equal to S 0 exp(r T) = $410

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1-44 Futures Price for an Investment Asset For any investment asset that provides no income and has no storage costs F 0 = S 0 e rT Immediate arbitrage opportunity if: F 0 > S 0 e rT short the Futures, long the asset F 0 < S 0 e rT long the Futures, short sell the asset

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1-45 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 F 0 = S 0 e cT For a consumption asset F 0 S 0 e cT The convenience yield, y, is the benefit provided when owning a physical commodity. It is defined as: F 0 = S 0 e (c – y )T

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1-46 Source: www.theoildrum.com Source: Quarterly Bulletin, Bank of England, 2006 Examples

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1-47 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( S T, K ) at the maturity of the options They must therefore be worth the same today: c + Ke -rT = p + S 0

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1-48 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 ƒ u S d ƒ d SƒSƒ 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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1-49 Consider the portfolio that is shares and short one option The portfolio is riskless when S u – ƒ u = S d – ƒ d or S u – ƒ u S d – ƒ d

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1-50 Value of the portfolio at time T is: S u – ƒ u or S d – ƒ d Value of the portfolio today is: (S u – ƒ u )e – rT Another expression for the portfolio value today is S – f Hence the option price today is: f = S – ( S u – ƒ u )e – rT Substituting for we obtain: f = [ p ƒ u + (1 – p )ƒ d ]e –rT where

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1-51 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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1-52 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 20 1.2823 22 18 24.2 3.2 19.8 0.0 16.2 0.0 2.0257 0.0 A B C D E F

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1-53 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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1-54 The Black-Scholes Formulas

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1-55 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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1-56 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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1-57 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, S 0 = €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 (c bs ) = €2.30 However, market price (c mkt ) = €2.89 Main result : Black-Scholes Price (c bs ) ≠ Market Price (c mkt )

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1-59 Implied Volatility: Definition Implied Volatility, or Implied Standard Deviation (ISD), is the volatility parameter ( ) 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{c bs ( ) – c mkt } { }

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1-60 Market-Level Implied Volatility (VIX)

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1 Introduction Chapter 1. 2 The Nature of Derivatives A derivative is an instrument whose value depends on the values of other more basic underlying variables.

1 Introduction Chapter 1. 2 The Nature of Derivatives A derivative is an instrument whose value depends on the values of other more basic underlying variables.

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