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Options and Futures Finansiell ekonomi 723g28 Linköpings University 1

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What is a Derivative? A derivative is an instrument whose value depends on, or is derived from, the value of another asset. Examples: futures, forwards, swaps, options, exotics… Obs: you may jump to slide #21 to start direct with options. 2

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How Derivatives are Used To hedge risks To speculate (take a view on the future direction of the market) To lock in an arbitrage profit To change the nature of a liability To change the nature of an investment without incurring the costs of selling one portfolio and buying another 3

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Options vs. Futures/Forwards A futures/forward contract gives the holder the obligation to buy or sell at a certain price at a certain date in the future An option gives the holder the right, but not the obligation to buy or sell at a certain price at a certain date in the future 4

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Foreign Exchange Quotes for GBP, ( £) May 24, 2010 5 BidOffer Spot1.44071.4411 1-month forward1.44081.4413 3-month forward1.44101.4415 6-month forward1.44161.4422 The forward price may be different for contracts of different maturities (as shown by the table)

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Long position and short position The party that has agreed to buy has a long position The party that has agreed to sell has a short position 6

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Example On May 24, 2010 the treasurer of a corporation enters into a long forward contract to buy £1 million in six months at an exchange rate of 1.4422 This obligates the corporation to pay $1,442,200 for £1 million on November 24, 2010 What are the possible outcomes? 7

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Svenska termer Forwards och Terminer Spotkontrakt: en överenskommelse mellan två parter att utbyta något idag för ett specificerat pris, spotpriset. à vista marknad. Terminskontrakt: en överenskommelse (skyldighet) mellan två parter att utbyta något för ett specificerat pris, terminspriset, vid en specifik framtida tidpunkt, lösendagen. 8

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Profit from a Long Forward Position (K= delivery price=forward price at the time contract is entered into) 9 Profit Price of Underlying at Maturity, S T K Payoff diagram

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Profit from a Short Forward Position (K= delivery price=forward price at the time contract is entered into) 10 Profit Price of Underlying at Maturity, S T K

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Futures Contracts Agreement to buy or sell an asset for a certain price at a certain time Similar to forward contract a forward contract is traded over the counter (OTC) ( Skräddarsydd) a futures contract is standardized and traded on an exchange. CME Group NYSE Euronext, BM&F (Sao Paulo, Brazil) TIFFE (Tokyo) 11

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Key Points About Futures They are settled daily Closing out a futures position involves entering into an offsetting trade Most contracts are closed out before maturity 12

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Margins A margin is cash or marketable securities deposited by an investor with his or her broker The balance in the margin account is adjusted to reflect daily settlement Margins minimize the possibility of a loss through a default on a contract 13

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Pricing of forward Guld (commodities): F = (1 + r f + s) · S 0 Finansiella tillgångar: F = (1 + r f ) · S 0 S 0 is the spot price. S is the storage cost r f is risk free interest rate F is the forward price 14

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Examples of Futures Contracts Agreement to: –Buy 100 oz. of gold @ US$1400/oz. in December –Sell £62,500 @ 1.4500 US$/£ in March –Sell 1,000 bbl. of oil @ US$90/bbl. in April 15 Oz: ounce Bbl: barrel

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Example : An Arbitrage Opportunity? Suppose that: The spot price of gold is US$1,400 The 1-year forward price of gold is US$1,500 The 1-year US$ interest rate is 5% per annum Q: What should be the 1-year forward price? Is there an arbitrage opportunity? 16

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The Forward Price of Gold If the spot price of gold is S and the forward price for a contract deliverable in T years is F, then F = S (1+r ) T where r is the 1-year (domestic currency) risk-free rate of interest. In our examples, S = 1400, T = 1, and r =0.05 so that F = 1400(1+0.05) = 1470 17

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Hedging Examples 1.An investor owns 1,000 Microsoft shares currently worth $28 per share. A two-month put with a strike price of $27.50 costs $1. The investor decides to hedge by buying 10 contracts 18

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Value of Microsoft Shares with and without Hedging 19

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Some Terminology Open interest: the total number of contracts outstanding –equal to number of long positions or number of short positions Settlement price: the price just before the final bell each day –used for the daily settlement process Trading Volume : the number of trades in one day 20

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Forward Contracts vs Futures Contracts 21 Contract usually closed out Private contract between 2 partiesExchange traded Non-standard contractStandard contract Usually 1 specified delivery dateRange of delivery dates Settled at end of contractSettled daily Delivery or final cash settlement usually occursprior to maturity FORWARDSFUTURES Some credit risk Virtually no credit risk

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Options The right but not the obligation…

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Options A call option is an option to buy a certain asset by a certain date for a certain price (the strike price) A put option is an option to sell a certain asset by a certain date for a certain price (the strike price) 23

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Option Obligations: the writer of the option 24

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American vs. European Options An American option can be exercised at any time during its life A European option can be exercised only at maturity The time value will be lost when you exercise prematurely. 25

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Option Value: Example Option values given an exercise price of $720 What are the payoff limits for call option buyers? Sellers? What are the payoff limits for put option buyers? Sellers?

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Call Option Value Call option value (buyer) given a $720 exercise price. Share Price Call option value 720 840 $120

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23-28 Call Option Profit $20 call option (buyer) given a $720 exercise price Share Price Call option value 720 840 $100

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23-29 Call Option Value Call option payoff (seller) given a $720 exercise price. Share Price Call option $ payoff 720 840 $-120

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Call Option Profit $20 call option (seller) given a $720 exercise price: Share Price Call option $ payoff 720 840 $-120 $-100

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Call Option: Example How much must the stock be worth at expiration in order for a call holder to break even if the exercise price is $50 and the call premium was $4?

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Put Option Value Put option value (buyer) given a $720 exercise price: Share Price Put option value 600 720 $120

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Put Option Profit $30 put option (buyer) given a $720 exercise price: Share Price Put option value 600 720 $90

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23-34 Put Option Value Put option payoff (seller) given a $720 exercise price. Share Price Put option $ payoff 600 720 -$120

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Put Option Profit $30 put option (seller) given a $720 exercise price. Share Price Put option $ payoff 600 720 -$90

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Put Options: Example What is your return on exercising a put option which was purchased for $10 with an exercise price of $85? The stock price at expiration is $81.

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Options Value Stock Price Upper Limit Lower Limit (Stock price - exercise price) or 0 which ever is higher 37

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Option Value 38

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Option Value Point A -When the stock is worthless, the option is worthless. Point B -When the stock price becomes very high, the option price approaches the stock price less the present value of the exercise price. Point C -The option price always exceeds its minimum value (except at maturity or when stock price is zero). The value of an option increases with both the variability of the share price and the time to expiration. 39

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Option Value Components of the Option Price 1 - Underlying stock price 2 - Strike or Exercise price 3 - Volatility of the stock returns (standard deviation of annual returns) 4 - Time to option expiration 5 - Time value of money (discount rate) 40

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Call Option Value

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Put-Call Parity: No Dividends Consider the following 2 portfolios: – Portfolio A: call option on a stock + zero-coupon bond (or a deposit) that pays K at time T – Portfolio B: Put option on the stock + the stock 42

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Values of Portfolios are the same at expiration (förfalldag) 43 S T > KS T < K Portfolio ACall optionS T − K0 Zero-coupon bondKK TotalSTST K Portfolio BPut Option0K− S T ShareSTST STST TotalSTST K

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The Put-Call Parity Result Both are worth max( S T, K ) at the maturity of the options They must therefore be worth the same today. This means that c + Ke -rT = p + S 0 44

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Suppose that What are the put option price? c + Ke -rT = p + S 0 p = c- S 0 +Ke -rT =3-31+30*EXP(-0,1*0,25) = 1,259 45 Ex: put-call parity c = 3 S 0 = 31 T = 0.25 r = 10% K =30

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Bounds for European and American Put Options (No Dividends) 46

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Synthetic options 47 Two or more options combines together creates exotic options

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Option Value: profit diagram for a straddle Straddle - Long call and long put - Strategy for profiting from high volatility Share Price Position Value Straddle Long put Long call 48

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Option Value Straddle - Long call and long put - Strategy for profiting from high volatility Share Price Position Value Straddle 49 An investor may take a long straddle position if he thinks the market is highly volatile, but does not know in which direction it is going to move.

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Exotic options: a butterfly option A butterfly 50 A long butterfly position will make profit if the future volatility is lower than the implied volatility. The spread is created by buying a call with a relatively low strike (x 1 ), buying a call with a relatively high strike (x 3 ), and shorting two calls with a strike in between (x 2 ). x2x2 x3x3 x1x1

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Long Call Profit from buying one European call option: option price = $5, strike price = $100, option life = 2 months 51 30 20 10 0 -5 708090100 110120130 Profit ($) Terminal stock price ($)

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Short Call Profit from writing one European call option: option price = $5, strike price = $100 52 -30 -20 -10 0 5 708090100 110120130 Profit ($) Terminal stock price ($)

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Long Put Profit from buying a European put option: option price = $7, strike price = $70 53 30 20 10 0 -7 706050408090100 Profit ($) Terminal stock price ($)

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Short Put Profit from writing a European put option: option price = $7, strike price = $70 54 -30 -20 -10 7 0 70 605040 8090100 Profit ($) Terminal stock price ($)

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Payoffs from Options What is the Option Position in Each Case? K = Strike price, S T = Price of asset at maturity 55 Payoff STST STST K K STST STST K K

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The Black-Scholes-Merton Formulas 56

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Real options With the limited liability of the modern corporations, the shareholders´ equity can be regarded as a real option on the assets of the firm. The shareholder value of equity value is max(V T −D, 0) where V T is the value of the firm and D is the debt repayment required. Thus the company can be considered as a call option on the firm value V at the strike price of D. 57

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Options on Real Assets Real Options - Options embedded in real assets Option to Expand Option to Abandon

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Options on Financial Assets Executive Stock Options Warrants Convertible Bonds Callable Bonds

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