Option Markets 232D. 1. Introduction. Daniel Andrei. Fall / 67

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1 Option Markets 232D 1. Introduction Daniel Andrei Fall / 67

2 My Background MScF 2006, PhD Lausanne, Switzerland Since July 2012: assistant professor of finance at UCLA Anderson I conduct research in the area of theoretical asset pricing, with a special focus on the role of information in financial markets More information about me at 2 / 67

3 Outline I Class Organization 4 II Introduction to Derivatives 10 Definitions 11 Buying and Short-Selling 17 Continuous Compounding 22 Forward Contracts 27 Call Options 33 Put Options 40 Moneyness, Put-Call Parity 46 Options Are Insurance 56 Various Strategies 58 3 / 67

4 Outline I Class Organization 4 II Introduction to Derivatives 10 Definitions 11 Buying and Short-Selling 17 Continuous Compounding 22 Forward Contracts 27 Call Options 33 Put Options 40 Moneyness, Put-Call Parity 46 Options Are Insurance 56 Various Strategies 58 4 / 67

5 Useful Information Class materials: Slides: Class website on myanderson Textbook: Robert McDonald, Derivatives Markets, Pearson Addison Wesley, second edition, My contact information: Office: C4.20 Office hours: Thursday (Sep 27 - Dec 6), 4-5 pm or by appointment Phone: (310) [email protected] TA: Nimesh Patel Office: C4.01 Office hours: Room C4.14, Tuesday (Oct 2 - Dec 11), 5-6 pm or by appointment [email protected] 5 / 67

6 Schedule: Thursdays, 8:30 11:20 AM, Cornell D-313 Week 1: Sep. 27 Week 2: Oct. 4 Week 3: Oct. 11 Week 4: Oct. 18 Week 5: Oct. 25 Week 6: Nov. 1 Introduction to derivatives Chapters 1, 2, 3, 9, Appendix B Binomial option pricing Chapters 10, 11 Black-Scholes, Greeks Chapters 12, 13 Week 7: Nov. 8 Week 8: Nov. 15 Forwards, futures, and swaps Chapters 5, 6, 7, 8 Thanksgiving: Thursday, Nov. 22 Week 9: Nov. 29 Volatility risk Chapters 11, 12, 13 Week 10: Dec. 6 Risk management, other topics Chapters 3, 4, 15,... Midterm: Week 7 (Thursday, Nov. 8), 1.5 hours, open book exam Final exam: Thursday, Dec. 13, D-313, 1.5 hours, open book exam 6 / 67

7 Evaluation There will be five quizzes, available via Equiz.me To register, please follow this link. Please use an address that explicitely shows your name (preferably your UCLA Anderson address). Once links are posted, they will be available online for two weeks. Once available, you can solve them as many times as you want. I strongly advise working in groups, but you will have to submit your answers individually. Grade formula: 40% final exam, 30% midterm, 20% quizzes, 10% class participation. 7 / 67

8 In addition to enrolling through the proper authorities, please send me an with the following information: name program and year in program major (if any) your background in finance and mathematics telephone number any other information you consider important 8 / 67

9 Ground Rules These rules help ensure that no one interferes with the learning of another: I will start promptly and I expect you to be on time If, because of some unusual circumstance, you come in late, please enter as quietly as possible If it is necessary for you to leave early, please sit next to a door You may leave the room briefly if it is an emergency As a general rule, you may not use notebook computer or other handheld computing or communication devices in class 9 / 67

10 Outline I Class Organization 4 II Introduction to Derivatives 10 Definitions 11 Buying and Short-Selling 17 Continuous Compounding 22 Forward Contracts 27 Call Options 33 Put Options 40 Moneyness, Put-Call Parity 46 Options Are Insurance 56 Various Strategies / 67

11 What is a Derivative? Definition An agreement between two parties which has a value determined by the price of something else. Types Options, futures, and swaps. Uses Risk management Speculation Reduce transaction costs Regulatory arbitrage 11 / 67

12 Three Different Perspectives End-users: Corporations Economic observers: Regulators Researchers Intermediaries: Market-makers Traders End-users: Investment managers End-users: Investors 12 / 67

13 Financial Engineering The construction of a financial product from other products New securities can be designed using existing securities Financial engineering principles Facilitate hedging of existing positions Enable understanding of complex positions Allow for creation of customized products Render regulation less effective Examples: In 2002, Universal Studios issued a $175 million catastrophe bond to cover its production studios against an earthquake in Southern California In 2003, Walt Disney issued a catastrophe bond to cover its theme park in Japan 13 / 67

14 Exchange Traded Derivatives Contracts outstanding (millions) ,985 1,990 1,995 2,000 2,005 2,010 source: Bank for International Settlements, Quarterly Review, June / 67

15 Exchange Traded Derivatives (cont d) Notional amount (billions) 80,000 60,000 40,000 20, ,985 1,990 1,995 2,000 2,005 2,010 source: Bank for International Settlements, Quarterly Review, June / 67

16 Over-the-Counter (OTC) Derivatives Notional amount (billions) 600, , ,000 1,998 2,000 2,002 2,004 2,006 2,008 2,010 2,012 source: Bank for International Settlements, Quarterly Review, June / 67

17 Buying and Selling a Financial Asset Brokers: commissions Market-makers: bid-ask spread (reflects the perspective of the market-maker) The price at which you can buy The price at which you can sell ask (offer) bid Example: Buy and sell 100 shares of XYZ XYZ: bid=$49.75, ask=$50, commission=$15 Buy: (100 $50) + $15 = $5,015 Sell: (100 $49.75) $15 = $4,960 Transaction cost: $5,015 $4960 = $55 What the marketmaker will sell for What the marketmaker pays 17 / 67

18 Problem 1.4: ABC stock has a bid price of $40.95 and an ask price of $ Assume that the brokerage fee is quoted as 0.3% of the bid or ask price. a. What amount will you pay to buy 100 shares? b. What amount will you receive for selling 100 shares? c. Suppose you buy 100 shares, then immediately sell 100 shares. What is your round-trip transaction cost? 18 / 67

19 Problem 1.4: ABC stock has a bid price of $40.95 and an ask price of $ Assume that the brokerage fee is quoted as 0.3% of the bid or ask price. a. What amount will you pay to buy 100 shares? ($ ) + ($ ) = $4, b. What amount will you receive for selling 100 shares? ($ ) ($ ) = $4, c. Suppose you buy 100 shares, then immediately sell 100 shares. What is your round-trip transaction cost? $4, $4, = $ / 67

20 Short-Selling When price of an asset is expected to fall First: borrow and sell and asset (get $$) Then: buy back and return the asset (pay $) If price fell in the mean time: Profit $ = $$ - $ The lender must be compensated for dividends received Example: Cash flows associated with short-selling a share of IBM for 90 days. Note that the short-seller must pay the dividend, D, to the share-lender. Day 0 Dividend Ex-Day Day 90 Borrow shares Return shares Action Sell shares Purchase shares Cash +S 0 D S / 67

21 VW s 348% Two-Day Gain Is Pain for Hedge Funds From the Wall Street Journal, 2008: In short squeezes, investors who borrowed and sold stock expecting its value to fall exit from the trades by buying those shares, or covering their positions. That can send a stock upward if shares are hard to come by. When shares are scarce, that can push a company-s market capitalization well beyond a reasonable valuation. [...] Indeed, the recent stock gains left Volkswagen s market value at about $346 billion, just below that of the world s largest publicly traded corporation, Exxon Mobil Corp. 20 / 67

22 Problem 1.6: Suppose you short-sell 300 shares of XYZ stock at $30.19 with a commission charge of 0.5%. Supposing you pay commission charges for purchasing the security to cover the short-sale, how much profit have you made if you close the short-sale at a price of $29.87? 21 / 67

23 Problem 1.6: Suppose you short-sell 300 shares of XYZ stock at $30.19 with a commission charge of 0.5%. Supposing you pay commission charges for purchasing the security to cover the short-sale, how much profit have you made if you close the short-sale at a price of $29.87? Initially, we will receive the proceeds form the sale of the asset, less the proportional commission charge: 300 ($30.19) 300 ($30.19) = $9, When we close out the position, we will again incur the commission charge, which is added to the purchasing cost: 300 ($29.87) ($29.87) = $9, Finally, we receive total profits of: $9, $9, = $ / 67

24 Continuous Compounding Terms often used to to refer to interest rates: Effective annual rate r: if you invest $1 today, T years later you will have (1 + r) T Annual rate r, compounded n times per year: if you invest $1 today, T years later you will have ( 1 + r ) nt n Annualized continuously compounded rate r: if you invest $1 today, T years later you will have ( e rt lim 1 + r ) nt n n 22 / 67

25 Continuous Compounding: Example Suppose you have a zero-coupon bond that matures in 5 years. The price today is $ for a bond that pays $100. The effective annual rate of return is ( ) 1/5 $100 1 = 0.10 $ The continuously compounded rate of return is ln($100/$62.092) 5 = = The continuously compounded rate of return of 9.53% corresponds to the effective annual rate of return of 10%. To verify this, observe that e = 1.10 or ( ln(1.10) = ln e ) = / 67

26 Continous Compounding When we multiply exponentials, exponents add. So we have e x e y = e x+y This makes calculations of average rate of return easier. When using continuous compounding, increases and decreases are symmetric. Moreover, continuously compounded returns can be less than -100% 24 / 67

27 Problem B.2: Suppose that over 1 year a stock price increases from $100 to $200. Over the subsequent year it falls back to $100. What is the effective return over the first year? What is the continuously compounded return? What is the effective return over the second year? The continuously compounded return? What do you notice when you compare the first- and second-year returns computed arithmetically and continuously? 25 / 67

28 Problem B.2: Suppose that over 1 year a stock price increases from $100 to $200. Over the subsequent year it falls back to $100. What is the effective return over the first year? What is the continuously compounded return? $200 $100 effective return = = 100% ( $100 ) $200 continuously compounded return = ln = 69.31% $100 What is the effective return over the second year? The continuously compounded return? $100 $200 effective return = = 50% ( $200 ) $100 continuously compounded return = ln = 69.31% $200 What do you notice when you compare the first- and second-year returns computed arithmetically and continuously? 25 / 67

29 A World Without Options Treasury Bills Stocks Payoff ($) Stock Price ($) Payoff ($) Stock Price ($) 26 / 67

30 Forward Contracts Definition: a binding agreement (obligation) to buy/sell an underlying asset in the future, at a price set today. A forward contract specifies: 1. The features and quantity of the asset to be delivered 2. The delivery logistics, such as time, date, and place 3. The price the buyer will pay at the time of delivery 27 / 67

31 Forward Contracts vs Futures Contracts Forward and futures contracts are essentially the same except for the daily resettlement feature of futures contracts, called marking-to-market. Because futures are exchange-traded, they are standardized and have specified delivery dates, locations, and procedures. Plenty of information is available from: 28 / 67

32 Reading Price Quotes Figure 1: Index futures price listings source: Wall Street Journal, September 11, / 67

33 Payoff (Value at Expiration) of a Forward Contract Every forward contract has both a buyer and a seller. The term long is used to describe the buyer and short is used to describe the seller. Payoff for Long forward = Spot price at expiration Forward price Short forward = Forward price Spot price at expiration Example: S&R index: Today: Spot price = $1, month forward price = $1,020 In 6 months at contract expiration: Spot price = $1,050 Long position payoff = $1,050 - $1,020 = $30 Short position payoff = $1,020 - $1,050 = -$30 30 / 67

34 Payoff Diagram for Forwards 200 Long forward 100 Payoff ($) $1, 020 Short forward ,000 1,100 1,200 S&R Index Price ($) 31 / 67

35 Problem 2.4.a: Suppose you enter in a long 6-month forward position at a forward price of $50. What is the payoff in 6 months for prices of $40, $45, $50, $55, and $60? 32 / 67

36 Problem 2.4.a: Suppose you enter in a long 6-month forward position at a forward price of $50. What is the payoff in 6 months for prices of $40, $45, $50, $55, and $60? The payoff to a long forward at expiration is equal to: Payoff to long forward = Spot price at expiration Forward price Therefore, we can construct the following table: Price of asset in 6 months Payoff ot the long forward / 67

37 Call Options A non-binding agreement (right but not an obligation) to buy an asset into the future, at a price set today Preserves the upside potential, while at the same time eliminating the downside The seller of a call option is obligated to deliver if asked 33 / 67

38 Definition and terminology A call option gives the owner the right but not the obligation to buy the underlying asset at a predetermined price during a predetermined time period Strike (or exercise) price: the amount paid by the option buyer for the asset if he/she decides to exercise Exercise: the act of paying the strike price to buy the asset Expiration: the date by which the option must be exercised or becomes worthless Exercise style: specifies when the option can be exercised European-style: can be exercised only at expiration date American-style: can be exercised at any time before expiration Bermudan-style: can be exercised during specified periods 34 / 67

39 Reading Price Quotes Figure 2: S&P 500 Index Call Options source: Chicago Board Options Exchange, September 11, / 67

40 Call Option Example Consider a call option on the S&R index with 6 months to expiration and strike price of $1,000. In six months at contract expiration: if spot price is $1,100 call buyer s payoff = $1,100 $1,000 = $100, call seller s payoff = -$100 $900 call buyer s payoff = $0, call seller s payoff = $0 The payoff of a call option is then C T = max [S T K,0] (1) where K is the strike price, and S T is the spot price at expiration. The option profit is computed as Call profit = max [S T K,0] future value of premium (2) 36 / 67

41 Diagrams for Purchased Call 200 Purchased call 200 Long forward Payoff ($) Profit ($) $95.68 Purchased call ,000 1,100 1,200 S&R Index Price ($) $1, ,000 1,100 1,200 S&R Index Price ($) 37 / 67

42 Diagrams for Written Call Payoff ($) Profit ($) $95.68 Written call Written call ,000 1,100 1,200 S&R Index Price ($) $1, 020 Short forward ,000 1,100 1,200 S&R Index Price ($) 38 / 67

43 Problem 2.10.a: Consider a call option on the S&R index with 6 months to expiration and strike price of $1,000. The future value of the option premium is $ For the figure below, which plots the profit on a purchased call, find the S&R index price at which the call option diagram intersects the x-axis Profit ($) $95.68 Purchased call ,000 1,100 1,200 S&R Index Price ($) 39 / 67

44 Problem 2.10.a: Consider a call option on the S&R index with 6 months to expiration and strike price of $1,000. The future value of the option premium is $ For the figure below, which plots the profit on a purchased call, find the S&R index price at which the call option diagram intersects the x-axis. 200 The profit of the long call option is: 100 max [0,S T $1,000] $95.68 Profit ($) $95.68 Purchased call To find the S&R index price at which the call option diagram intersects the x-axis, we have to set the above equation equal to zero. We get ,000 1,100 1,200 S&R Index Price ($) S T = $1, / 67

45 Put Options A put option gives the owner the right but not the obligation to sell the underlying asset at a predetermined price during a predetermined time period. The payoff of the put option is The option profit is computed as P T = max [K S T,0] (3) Put profit = max [K S T,0] future value of premium (4) 40 / 67

46 Reading Price Quotes Figure 3: S&P 500 Index Put Options source: Chicago Board Options Exchange, September 11, / 67

47 Diagrams for Purchased Put 200 Purchased put 200 Short forward Payoff ($) Profit ($) Purchased put $ ,000 1,100 1,200 S&R Index Price ($) $1, ,000 1,100 1,200 S&R Index Price ($) 42 / 67

48 Diagrams for Written Put Long forward Payoff ($) Profit ($) Written put $95.68 Written put ,000 1,100 1,200 S&R Index Price ($) $1, ,000 1,100 1,200 S&R Index Price ($) 43 / 67

49 Problem 2.14.a: Suppose the stock price is $40 and the effective annual interest rate is 8%. Draw payoff and profit diagrams for a 40-strike put with a premium of $3.26 and maturity of 1 year. 44 / 67

50 Problem 2.14.a: Suppose the stock price is $40 and the effective annual interest rate is 8%. Draw payoff and profit diagrams for a 40-strike put with a premium of $3.26 and maturity of 1 year. In order to be able to draw the profit diagram, we need to find the future value of the put premium: FV (premium) = $3.26 ( ) = $ We get the following payoff and profit diagram: Payoff (blue) or profit (red) ($) Payoff Profit $ Stock Price ($) 44 / 67

51 Potential Gain and Loss for Forward and Option Positions Position Maximum Loss Maximum Gain Long forward Forward price Unlimited Short forward Unlimited Forward Price Long call FV(premium) Unlimited Short call Unlimited FV(premium) Long put FV(premium) Strike price FV(premium) Short put FV(premium) Strike price FV(premium) 45 / 67

52 Moneyness In-the-money option: positive payoff if exercised immediately At-the-money option: zero payoff if exercised immediately Out-of-the-money option: negative payoff if exercised immediately 46 / 67

53 Put-Call Parity Suppose you are buying a call option and selling a put option on a non-dividend paying stock. Both options have maturity T and strike price K: Long call Combined position Payoff ($) Payoff ($) Short put K K Stock Price ($) Stock Price ($) 47 / 67

54 Put-Call Parity (cont d) Your payoff at maturity is C T P T = max [S T K,0] max [K S T,0] = max [S T K,0] + min[s T K,0] = S T K We have two strategies with the same payoff at maturity: Buy a call and sell a put, thus paying a premium of Ct P t today Buy a share of the stock and borrow PV (K), thus paying a premium of S t PV (K) today Positions that have the same payoff should have the same cost (Law of one price): C t P t = S t PV (K) (5) Equation (5) is known as put-call parity, and one of the most important relations in options. 48 / 67

55 Put-Call Parity (cont d) Parity provides a cookbook for the synthetic creation of options. It tells us that and that C t = P t + S t PV (K) (6) P t = C t S t + PV (K) (7) The first relation says that a call is equivalent to a leveraged position on the underlying asset, which is insured by the purchase of a put. The second relation says that a put is equivalent to a short position on the stock, insured by the purchase of a call Parity generally fails for American-style options, which may be exercised prior to maturity. 49 / 67

56 Why Does the Price of an At-the-Money call Exceed the Price of an At-the-Money put? Parity shows that the reason for the call being more expensive is the time value of money: C t P t = K PV (K) > 0 (8) A common erroneous explanation is that the profit on a call is unlimited, while the profit on a put can be no greater than the strike price, which seems to suggest that the call should be more expensive than the put. This argument also seems to suggest that every stock is worth more than its price! 50 / 67

57 Problem 3.8: The S&R index price is $1,000 and the effective 6-month interest rate is 2%. Suppose the premium on a 6-month S&R call is $ and the premium on a 6-month put with the same strike price is $ What is the strike price? 51 / 67

58 Problem 3.8: The S&R index price is $1,000 and the effective 6-month interest rate is 2%. Suppose the premium on a 6-month S&R call is $ and the premium on a 6-month put with the same strike price is $ What is the strike price? This question is a direct application of the Put-Call Parity: C t P t = S t PV (K) $ $60.18 = $1,000 K 1.02 K = $ / 67

59 Put-Call Parity for Dividend Paying Stocks If the stock is paying dividends over the lifetime of the option, the put-call parity becomes C t P t = [S t PV (Div)] PV (K) (9) where PV (Div) is the present value of the stream of dividends paid on the stock until maturity. Hence, we can write C t = P t + [S t PV (Div)] PV (K) (10) P t = C t [S t PV (Div)] + PV (K) (11) Equations (10) (11) help us to find maximum and minimum option prices. 52 / 67

60 Maximum and Minimum Option Prices: Call Price C t S t C t C t max [0,S t PV (Div) PV (K)] S t 53 / 67

63 Maximum and Minimum Option Prices: Call Price C t S t C t Call option price somewhere here C t max [0,S t PV (Div) PV (K)] S t 53 / 67

64 Maximum and Minimum Option Prices: Put Price P t K P t Put option price somewhere here P t max [0,PV (K) S t + PV (Div)] S t 54 / 67

65 Example: Equity-Linked CDs A 1,999 First Union National Bank CD promises to repay in 5.5 years initial invested amount and 70% of the gain in S&P 500 index (this is a principal protected equity-linked CD) Assume $10,000 invested when S&P 500 = 1,300 Final payoff is ( [ $10, max 0, S ]) final Payoff ($) 13,000 12,000 11,000 10,000 Payoff of Equity-Linked CD 9,000 $1,300 where S final = value of the 8,000 S&P 500 after 5.5 years ,000 1,200 1,400 1,600 1,800 S&P 500 Price ($) 55 / 67

66 Options are Insurance: Insuring a Long Position (Floors) A put option is combined with a position in the underlying asset Goal: to insure against a fall in the price of the underlying asset (when one has a long position in that asset) 2,000 Long S&R Index 2,000 Combined payoff 1,000 Long put 1,000 Payoff ($) 0 Payoff ($) 0 1,000 1,000 2, ,000 1,500 2,000 S&R Index Price ($) 2, ,000 1,500 2,000 S&R Index Price ($) 56 / 67

67 Options are Insurance: Insuring a Short Position (Caps) A call option is combined with a position in the underlying asset Goal: to insure against an increase in the price of the underlying asset (when one has a short position in that asset) 2,000 2,000 1,000 Long call 1,000 Payoff ($) 0 1,000 Short S&R Index Payoff ($) 0 1,000 Combined payoff 2, ,000 1,500 2,000 S&R Index Price ($) 2, ,000 1,500 2,000 S&R Index Price ($) 57 / 67

68 Various Strategies: Payoffs Bull Spread Collar Straddle Profit ($) Profit ($) Profit ($) Stock Price ($) Stock Price ($) Stock Price ($) Strangle Butterfly Spread Ratio Spread Profit ($) Profit ($) Profit ($) Stock Price ($) Stock Price ($) Stock Price ($) 58 / 67

69 Various Strategies: Positions Bull Spread Collar Straddle K low K ATM K high Call Buy Sell Put Call Put K low K ATM K high Sell Buy Call Put K low K ATM K high Buy Buy Strangle Butterfly Spread Ratio Spread Call Put K low K ATM K high Buy Buy K low K ATM K high Call Sell Buy Put Buy Sell K low K ATM K high Call Buy Sell (n) Put Note that you can achieve the same results with different combinations (but always at the same cost!) 59 / 67

70 Various Strategies: Rationales Bull Spread You believe a stock will appreciate buy a call option (forward position insured) You can lower the cost if you are willing to reduce your profit should the stock appreciate sell a call with higher strike Surprisingly, you can achieve the same result by buying a low-strike put and selling a high-strike put Opposite: bear spread Collar A collar is fundamentally a short position (resembling a short forward contract) Often used for insurance when we own a stock (collared stock) The collared stock looks like a bull spread; however, it arises from a different set of transactions Opposite: written collar Straddle A straddle can profit from stock price moves in both directions The disadvantage is that it has a high premium because it requires purchasing two options Opposite: written straddle Strangle To reduce the premium of a straddle, you can buy out-of-the-money options rather than at-the-money options. Opposite: written strangle Butterfly Spread A butterfly spread is a written straddle to which we add two options to safeguard the position: An out-of-the money put and an out-of-the money call. A butterfly spread can be thought of as a written straddle for the timid (or for the prudent!) Opposite: long iron butterfly Ratio Spread Ratio spreads involve buying one option and selling a greater quantity (n) of an option with a more out-of-the money strike The ratio (i.e., 1 by n ) is the number of short options divided by the number of long options The options are either both calls or both puts It is possible to construct ratio spreads with zero premium we can construct insurance that costs nothing if it is not needed! 60 / 67

71 A World Without Options Treasury Bills Stocks Payoff ($) Stock Price ($) Payoff ($) Stock Price ($) 61 / 67

72 Madoff Ran Vast Options Game From the Wall Street Journal, 2008: Mr. Madoff told clients he was using a fairly common options-trading strategy to generate modest but steady returns for more than two decades. The strategy involved buying stocks, while also trading options in a way designed to limit losses on the shares [...] People who analyzed client statements said Mr. Madoff s firm couldn t have bought and sold the options he claimed because those totals would have outstripped total trading volume those days. Reproduced from: Gregoriou and Lhabitant, Madoff: A riot of Red Flags, / 67

73 Collars in Acquisitions: WorldCom/MCI On October 1, 1997, WorldCom Inc. CEO (Bernard Ebbers) sent the following note to the CEO of MCI (Bert Roberts), and it was also released through the typical newswires: I am writing to inform you that this morning WorldCom is publicly announcing that it will be commencing an offer to acquire all the outstanding shares of MCI for $41.50 of WorldCom common stock per MCI share. The actual number of shares of WorldCom common stock to be exchanged for each MCI share in the exchange offer will be determined by dividing $41.50 by the 20-day average of the high and low sales prices for WorldCom common stock prior to the closing of the exchange offer, but will not be less than shares (if WorldCom s average stock price exceeds $40) or more than shares (if WorldCom s average stock price is less than $34). 63 / 67

74 Collars in Acquisitions: WorldCom/MCI (cont d) The payoff is contingent upon price of WorldCom s 20-day average stock price prior to the closing exchange offer: 50 Value of Offer Slope = Slope = $34 $ WCOM s Average Stock Price at Closing 64 / 67

75 Example: Another Equity-Linked Note In July 2004, Marshall & Ilsley Corp. (ticker symbol MI) raised $400 million by issuing mandatorily convertible bonds effectively maturing in August 2007 The bond pays an annual 6.5% coupon and at maturity makes payments in shares, with the number of shares dependent upon the firm s stock price. The specific terms of the maturity payment are in the table below Marshall & Ilsley Share Price S MI < S MI $25/S MI < S MI Number of Shares Paid to Bondholders 65 / 67

76 Example: Another Equity-Linked Note (cont d) The graph of the maturity payoff should remind us of a written collar: Bondholder payoff Payoff ($) Slope = Slope = $37.32 $ Marshall & Ilsley Stock Price ($) 66 / 67

77 Positions Consistent With Different Views on the Stock Price and Volatility Direction Volatility Will Increase No Volatility View Volatility Will Fall Price Will Fall Buy puts Sell underlying Sell calls No Price View Buy straddle Do nothing Sell straddle Price Will Increase Buy calls Buy underlying Sell puts 67 / 67

78 Option Markets 232D 2. Binomial Option Pricing Daniel Andrei Fall / 60

79 Outline I Discrete-Time Option Pricing: The Binomial Model 3 II Call Options 6 A One-Period Binomial Tree 7 The Binomial Solution 15 A Two-Period Binomial Tree 22 Many Binomial Periods 27 Self-Financing Strategy 28 III Put Options 30 IV Uncertainty in the Binomial Model 35 V Risk-Neutral Pricing 41 VI American Options 53 2 / 60

81 Discrete-Time Option Pricing: The Binomial Model Until now, we have looked only at some basic principles of option pricing We examined payoff and profit diagrams, and upper/lower bounds on option prices We saw that with put-call parity we could price a put or a call based on the prices of the combinations on instruments that make up the synthetic version of the put or call. What we need to be able to do is price a put or a call without the other instrument. In this section, we introduce a simple means of pricing an option. 4 / 60

82 Discrete-Time Option Pricing: The Binomial Model (cont d) The approach we take here is called the binomial tree. The word binomial refers to the fact that there are only two outcomes (we let the underlying price move to only one of two possible new prices). It may appear that this framework oversimplifies things, but the model can eventually be extended to encompass all possible prices. 5 / 60

84 The Binomial Option Pricing Model The binomial option pricing model assumes that, over a period of time, the price of the underlying asset can move only up or down by a specified amount that is, the asset price follows a binomial distribution: up u S S down d S 7 / 60

85 A Simple Example XYZ does not pay dividends and its current price is $41. In one year the price can be either $ or $32.903, i.e., u = and d = Consider a European call option on the stock of XYZ, with a $40 strike an 1 year to expiration. The continuously compounded risk-free interest rate is 8%. We wish to determine the option price: S = 41 C =? us = C u = max [0, ] = ds = C d = max [0, ] = 0 8 / 60

86 A Simple Example (cont d) Let us try to find a portfolio that mimics the option (replicating portfolio). We have two instruments: shares of stock and a position in bonds (i.e., borrowing or lending). To be specific, we wish to find a portfolio consisting of shares of stock and a dollar amount B in borrowing or lending, such that the portfolio imitates the option whether the stock rises or falls. The value of this replicating portfolio at maturity is: { e 0.08 B = e 0.08 B = 0 (1) 9 / 60

87 A Simple Example (cont d) The unique solution of this system of 2 equations with 2 unknowns is = 0.738, B = , (2) i.e., buy shares of XYZ and borrow $ at the risk-free rate. In computing the payoff for the replicating portfolio, we assume that we sell the shares at the market price and that we repay the borrowed amount, plus interest. Thus, we obtain that the option and the replicating portfolio have the same payoff: $ if the stock price goes up and $0 if the stock price goes down. 10 / 60

88 A Simple Example (cont d) By the law of one price, positions that have the same payoff should have the same cost. The price of the option must be C = $41 $ = $7.839 (3) 11 / 60

89 Arbitraging a Mispriced Option Suppose that the market price for the option is $8 instead of $7.839 (the option is overpriced). We can sell the option and buy a synthetic option at the same time (buy low and sell high). The initial cash flow is and there is no risk at expiration: $8.00 $7.839 = $0.161 (4) Stock Price in 1 Year $ $ Written call $0 -$ purchased shares $ $ Repay loan of $ $ $ Total payoff $0 $0 12 / 60

90 Arbitraging a Mispriced Option (cont d) Suppose that the market price for the option is $7.5 instead of $7.839 (the option is underpriced). We can buy the option and sell a synthetic option at the same time (buy low and sell high). The initial cash flow is and there is no risk at expiration: $7.839 $7.5 = $0.339 (5) Stock Price in 1 Year $ $ Purchased call $0 $ short-sold shares -$ $ Sell T-bill $ $ Total payoff $0 $0 13 / 60

91 A Remarkable Result So far we have not specified the probabilities of the stock going up and down. In fact, probabilities were not used anywhere in the option price calculations. This is a remarkable result: Since the strategy of holding shares and B bonds replicates the option whichever way the stock moves, the probability of an up or down movement in the stock is irrelevant for pricing the option. 14 / 60

92 The Binomial Solution Suppose that the stock has a continuous dividend yield of δ, which is reinvested in the stock. Thus, if you buy one share at time 0 and the length of a period is h, at time h you will have e δh shares. The up and down movements of the stock price reflect the ex-dividend price. We can write the stock price as us 0 when the stock goes up and ds 0 when the stock goes down. We can represent the tree for the stock and the option as follows: S 0 us 0 C u 1 C 0 ds 0 C d 1 15 / 60

93 The Binomial Solution (cont d) If the length of a period is h, the interest factor per period is e rh. A successful replicating portfolio will satisfy { S0 u e δh + B e rh = C u 1 S 0 d e δh + B e rh = C d 1 (6) This is a system of two equations in two unknowns and B. Solving for and B gives = e δh C 1 u C 1 d S 0 (u d) B = e rh C 1 d u C 1 ud u d ( = e rh C1 u S 0ue δh) (7) 16 / 60

94 The Binomial Solution (cont d) Given the expressions (7) for and B, we can derive a simple formula for the value of the option. The cost of creating the option is the net cash required to buy the shares and bonds. Thus, the cost of the option is C 0 = S 0 + B = e rh (C u 1 e (r δ)h d u d + C1 d u e (r δ)h ) u d (8) Note that if we are interested only in the option price, it is not necessary to solve for and B; that is just an intermediate step. If we want to know only the option price, we can use equation (8) directly. 17 / 60

95 The Binomial Solution (cont d) The assumed stock price movements, u and d, should not give rise to arbitrage opportunities. In particular, we require that d < e (r δ)h < u (9) Note that because is the number of shares in the replicating portfolio, it can also be interpreted as the sensitivity of the option to a change in the stock price. If the stock prices changes by $1, then the option price, S + B, changes by. 18 / 60

96 The Binomial Solution (cont d) In our example, δ = 0 (XYZ does not pay dividends) and h = 1 (the length of a period is 1 year). The solution for and B reduces to = C 1 u C 1 d B S 0 (u d) = e r C d 1 u C u 1 d u d = e r (C u 1 S 0u) (10) The option price further simplifies to C 0 = S 0 + B = e r (C u 1 e r d u d + C 1 d u e r ) u d (11) 19 / 60

97 Problem 10.1.a: Let S = $100, K = $105, r = 8% (continuously compounded), T = 0.5, and δ = 0. Let u = 1.3, d = 0.8, and the number of binomial periods n = 1. What are the premium,, and B for a European call? 20 / 60

98 Problem 10.1.a: Let S = $100, K = $105, r = 8% (continuously compounded), T = 0.5, and δ = 0. Let u = 1.3, d = 0.8, and the number of binomial periods n = 1. What are the premium,, and B for a European call? Using the formulas given in (7), we calculate the following values: = 0.5 B = Call price = / 60

99 Problem 10.21: Suppose that u < e (r δ)h. Show that there is an arbitrage opportunity. Now suppose that d > e (r δ)h. Show again that there is an arbitrage opportunity. 21 / 60

100 Problem 10.21: Suppose that u < e (r δ)h. Show that there is an arbitrage opportunity. Now suppose that d > e (r δ)h. Show again that there is an arbitrage opportunity. If u < e (r δ)h, we short a tailed position of the stock and invest the proceeds at the interest rate. There is an arbitrage opportunity: t = 0 state = d state = u Short stock +e δh S d S u S Lend money e δh S +e (r δ)h S +e (r δ)h S Total 0 >0 >0 If d > e (r δ)h, we buy a tailed position of the stock and borrow at the interest rate. There is an arbitrage opportunity: t = 0 state = d state = u Buy stock e δh S +d S +u S Borrow money e δh S e (r δ)h S e (r δ)h S Total 0 >0 >0 21 / 60

101 A Two-Period Binomial Tree We can extend the previous example to price a 2-year option, assuming all inputs are the same as before. S 0 = 41 C 0 =? S u 1 = C u 1 =? S d 1 = C d 1 =? S uu 2 = C uu 2 = S ud 2 = C ud 2 = S dd 2 = C dd 2 = 0 Note that an up move followed by a down move (S2 ud ) generates the same stock price as a down move followed by an up move (S2 du ). This is called a recombining tree. 22 / 60

102 A Two-Period Binomial Tree (cont d) To price the option when we have two binomial periods, we need to work backward through the tree. Suppose that in period 1 the stock price is S1 u = $ We can use equation (11) to derive the option price: C u 1 = e r ( C2 uu e r d u d + C 2 ud u e r ) = $ (12) u d Using equations (10), we can also solve for the composition of the replicating portfolio: = 1, B = (13) i.e., buy 1 share of XYZ and borrow $ at the risk-free rate, which costs 1 $ $ = $ / 60

103 A Two-Period Binomial Tree (cont d) Suppose that in period 1 the stock price is S1 d = $ We can use equation (11) to derive the option price: C d 1 = e r ( C2 ud e r d u d + C 2 dd u e r ) = $3.187 (14) u d Using equations (10), we can also solve for the composition of the replicating portfolio: = 0.374, B = (15) i.e., buy shares of XYZ and borrow $9.111 at the risk-free rate, which costs $ $9.111 = $ / 60

104 A Two-Period Binomial Tree (cont d) Move backward now at period 0. The stock price is S 0 = 41. We can use equation (11) to derive the option price: C 0 = e r (C u 1 e r d u d + C 1 d u e r ) = $ (16) u d Using equations (10), we can also solve for the composition of the replicating portfolio: = 0.734, B = (17) i.e., buy shares of XYZ and borrow $ at the risk-free rate, which costs $41 $ = $ / 60

105 A Two-Period Binomial Tree (cont d) The two-period binomial tree with the option price at each node as well as the details of the replicating portfolio is: S 0 = $41 C 0 = $ = B = $ S u 1 = $ C u 1 = $ = 1 B = $ S d 1 = $ C d 1 = $3.187 = B = $9.111 S uu 2 = $ C uu 2 = $ S ud 2 = $ C ud 2 = $8.114 S dd 2 = $ C dd 2 = $0 26 / 60

106 Many Binomial Periods: Three-Period Example Once we understand the two-period option it is straightforward to value an option using more than two binomial periods. The important principle is to work backward through the tree: S 0 = $41 C 0 = $ = B = $ S u 1 = $ C u 1 = $ = B = $ S d 1 = $ C d 1 = $4.685 = B = $ S uu 2 = $ C uu 2 = $ = 1 B = $ S ud 2 = $ C ud 2 = $ = B = $ S dd 2 = $ C dd 2 = $0 = 0 B = $0 S uuu 3 = $ C uuu 3 = $ S uud 3 = $ C uud 3 = $ S udd 3 = $ C udd 3 = $0 S ddd 3 = $ C ddd 3 = $0 27 / 60

107 Self-Financing Strategy In the two-period binomial tree example, suppose that the stock moves from S 0 = $41 to S u 1 = $ The replicating portfolio should be modified as follows: 1. Buy = shares of XYZ (increase the XYZ position from shares to 1 share), which costs $ = $ Increase borrowing from $ e 0.08 = $ to $36.925, which yields $ The amount necessary to buy shares is equal to the amount obtained from increased borrowing. Modifying the portfolio does not require additional cash. Thus, the replicating portfolio is self-financing. 28 / 60

108 Self-Financing Strategy (cont d) If the stock moves from S 0 = $41 to S1 d = $32.903, the replicating portfolio should be modified as follows: 1. Sell = shares of XYZ (decrease the XYZ position from shares to shares), which yields $ = $ Decrease borrowing from $ e 0.08 = $ to $9.111, which costs $ The amount obtained from selling shares is equal to the amount necessary to decrease borrowing. Modifying the portfolio does not require additional cash. Once again, the replicating portfolio is self-financing. 29 / 60

109 Outline I Discrete-Time Option Pricing: The Binomial Model 3 II Call Options 6 A One-Period Binomial Tree 7 The Binomial Solution 15 A Two-Period Binomial Tree 22 Many Binomial Periods 27 Self-Financing Strategy 28 III Put Options 30 IV Uncertainty in the Binomial Model 35 V Risk-Neutral Pricing 41 VI American Options / 60

110 Put Options We compute put option prices using the same stock price tree and in the same way as call option prices. The only difference with an European put option occurs at expiration: Instead of computing the price as max [0,S K], we use max [0,K S]. Here is a two-period binomial tree for an European put option with a $40 strike: 31 / 60

111 Put Options (cont d) S 0 = $41 P 0 = $3.823 = B = $ S1 u = $ P1 u = $0 = 0 B = $0 S1 d = $ P1 d = $7.209 = B = $ S uu 2 = $ P uu 2 = $0 S ud 2 = $ P ud 2 = $0 S dd 2 = $ P dd 2 = $ The proof that the replicating portfolio is self-financing is left as an exercise. 32 / 60

112 Put Options (Using the Parity Relationship) For non-dividend paying stocks, the basic parity relationship for European options with the same strike price and time to expiration is C t P t = S t PV (strike price) (18) We can use this relationship to find the put price at all nodes: S 0 = 41 C 0 = P 0 = S1 u = C1 u = P1 u = 0 S1 d = C1 d = P1 d = S uu 2 = C uu 2 = P uu 2 = 0 S ud 2 = C ud 2 = P ud 2 = 0 S dd 2 = C dd 2 = 0 P dd 2 = / 60

113 Problem 10.1.b: Let S = $100, K = $105, r = 8% (continuously compounded), T = 0.5, and δ = 0. Let u = 1.3, d = 0.8, and the number of binomial periods n = 1. What are the premium,, and B for a European put? 34 / 60

114 Problem 10.1.b: Let S = $100, K = $105, r = 8% (continuously compounded), T = 0.5, and δ = 0. Let u = 1.3, d = 0.8, and the number of binomial periods n = 1. What are the premium,, and B for a European put? Using the formulas given in (7), we calculate the following values: = 0.5 B = Put price = / 60

116 Uncertainty in the Binomial Model A natural measure of uncertainty about the stock return is the annualized standard deviation of the continuously compounded stock return, which we will denote by σ. If we split the year into n periods of length h (so that h = 1/n), the standard deviation over the period of length h, σ h, is (assuming returns are uncorrelated over time) σ h = σ h (19) In other words, the standard deviation of the stock return is proportional to the square root of time. 36 / 60

117 Uncertainty in the Binomial Model (cont d) We incorporate uncertainty into the binomial tree by modeling the up and down moves of the stock price. Without uncertainty, the stock price next period must equal: S t+h = S t e (r δ)h (20) To interpret this, without uncertainty, the rate of return on the stock must be the risk-free rate. Thus, the stock price must rise at the risk-free rate less the dividend yield, r δ. We now model the stock price evolution as us t = S t e (r δ)h e +σ h ds t = S t e (r δ)h e σ h (21) 37 / 60

118 Uncertainty in the Binomial Model (cont d) We can rewrite this as u = e (r δ)h+σ h d = e (r δ)h σ h (22) Return has two parts, one of which is certain [(r δ)h], and the other of which ( is uncertain and generates the up and down stock price moves σ ) h. Note that if we set volatility equal to zero, we are back to (20) and we have S t+h = us t = ds t = S t e (r δ)h. Zero volatility does not mean that prices are fixed; it means that prices are known in advance. 38 / 60

119 Uncertainty in the Binomial Model (cont d) In our example we assumed that u = and d = These correspond to an annual stock price volatility of 30%: u = e (0.08 0) = d = e (0.08 0) = (23) We will use equations (22) to construct binomial trees. This approach (called the forward tree approach) is very convenient because it never violates the no arbitrage restriction d < e (r δ)h < u 39 / 60

120 Problem 10.19: For a stock index, S = $100, σ = 30%, r = 5%, δ = 3%, and T = 3. Let n = 3. What is the price of a European call option with a strike of $95? What is the price of a European put option with a strike of $95? 40 / 60

121 Problem 10.19: For a stock index, S = $100, σ = 30%, r = 5%, δ = 3%, and T = 3. Let n = 3. What is the price of a European call option with a strike of $95? The price of a European call option with a strike of 95 is $ What is the price of a European put option with a strike of $95? The price of a European put option with a strike of 95 is $ / 60

123 Risk-Neutral Pricing There is a probabilistic interpretation of the binomial solution for the price of an option (equation 8, restated below): C 0 = e rh (C u 1 The terms e(r δ)h d u d follows from inequality 9). e (r δ)h d u d and u e(r δ)h u d + C1 d u e (r δ)h ) u d (24) sum to 1 and are both positive (this Thus, we can interpret these terms as probabilities. Equation (8) can be written as [ ] C 0 = e rh p C1 u + (1 p )C1 d (25) This expression has the appearance of an expected value discounted at the risk-free rate. Thus, we will call p e(r δ)h d u d the risk-neutral probability of an increase in the stock price. 42 / 60

124 Risk-Neutral Pricing (cont d) Pricing options using risk-neutral probabilities can be done in one step, no matter how large the number of periods. In our example, p = For the one-period European call option we have: The call price is Call Price in 1 Year Probability $ $ C 0 = e 0.08 ( $ $0) = $7.839 (26) 43 / 60

125 Risk-Neutral Pricing (cont d) For the two-period European call option we have: The call price is Call Price in 2 Years Probability $ p 2 = $ p (1 p ) = $0 (1 p ) 2 = C 0 = e ( $ $ $0) = $ (27) The probability of reaching any given node is the probability of one path reaching that node times the number of paths reaching that node. For example, the probability of reaching the node S ud 2 = $ is 2p (1 p ). It can be easily verified that the sum of probabilities in the table above is / 60

126 Risk-Neutral Pricing (cont d) For the three-period European call option we have: The call price is Call Price in 3 Years Probability $ p 3 = $ p 2 (1 p ) = $0 3p (1 p ) 2 = $0 (1 p ) 3 = C 0 = e = $ ! [ ] k!(3 k)! p k (1 p ) 3 k max S 0 u k d 3 k K,0 k=0 (28) It can be easily verified that the sum of probabilities in the table above is 1. It is left as an exercise to find the price of the two-period European put using risk-neutral probabilities. 45 / 60

127 Risk-Neutral Pricing (cont d) For an arbitrary number of periods n, the price of an European call option is given by C 0 = e rn n n! [ ] k!(n k)! p k (1 p ) n k max S 0 u k d n k K,0 k=0 (29) We will use this formula later on when talking about Black-Scholes: when the number of steps becomes great enough the price of the option appear to approach a limiting value. This value is given by the Black-Scholes formula. 46 / 60

128 Problem 11.12: Let S = $100, σ = 0.3, r = 0.08, T = 1, and δ = 0. Use equation (29) to compute the risk-neutral probability of reaching a terminal node and the price at that node for n = 3. Plot the risk-neutral distribution of year-1 stock prices. 47 / 60

129 Problem 11.12: Let S = $100, σ = 0.3, r = 0.08, T = 1, and δ = 0. Use equation (29) to compute the risk-neutral probability of reaching a terminal node and the price at that node for n = 3. Plot the risk-neutral distribution of year-1 stock prices. For n = 3, u and d are calculated as follows: u = e (0.08 0) 1/ /3 = d = e (0.08 0) 1/ /3 = It follows that p = , and n k Stock price Probability / 60

130 Understanding Risk-Neutral Pricing A risk-neutral investor is indifferent between a sure thing and a risky bet with an expected payoff equal to the value of the sure thing. A risk-averse investor prefers a sure thing to a risky bet with an expected payoff equal to the value of the sure thing. Formula (25) suggests that we are discounting at the risk-free rate, even though the risk of the option is at least as great as the risk of the stock. Thus, the option pricing formula, equation (25), can be said to price options as if investors are risk-neutral. Is this option pricing consistent with standard discounted cash flow calculations? 48 / 60

131 Understanding Risk-Neutral Pricing (cont d) Assume, as in our example, that a stock does not pay dividends and the length of a period is 1 year (δ = 0 and h = 1). Risk-Neutral Pricing The risk-free rate is the discount rate for any asset including the stock: S 0 = e r [p us 0 + (1 p )ds 0 ] Solving for p gives us p = er d u d Standard DCF calculation Suppose that the continuously compounded expected return on the stock is α. If p is the true probability of the stock going up, p must be consistent with u, d, and α. S 0 = e α [pus 0 + (1 p)ds 0 ] Solving for p gives us p = eα d u d 49 / 60

132 Understanding Risk-Neutral Pricing (cont d) Risk-Neutral Pricing We discount the option payoff at the risk-free rate, r. Standard DCF calculation At what rate do we discount the option payoff? Since an option is equivalent to holding a portfolio consisting of shares of stock and B bonds, the expected return on this portfolio is e γ = S 0 B S 0 + B eα + S 0 + B er where γ is the discount rate for the option. The option price is The option price is ] ] C 0 = e r [ p C u 1 + (1 p )C d 1 C 0 = e γ [ pc u 1 + (1 p)c d 1 50 / 60

133 Understanding Risk-Neutral Pricing (cont d) Are these two prices the same? Yes (proof left as an exercise). Note that it does not matter whether we have the correct value of α to start with. Any consistent pair of α and γ will give the same option price. Risk-neutral pricing is valuable because setting α = r results in the simplest pricing procedure. 51 / 60

134 Understanding Risk-Neutral Pricing (cont d) We need to emphasize that at no point are we assuming that investors are risk-neutral. Rather, risk-neutral pricing is an interpretation of the formulas above. 52 / 60

136 American Options An European option can only be exercised at the expiration date, whereas an American option can be exercised at any time. Because of this added flexibility, an American option must always be at least as valuable as an otherwise identical European option: C Amer (S,K,T ) C Eur (S,K,T ) P Amer (S,K,T ) P Eur (S,K,T ) Combining these statements, together with the maximum and minimum option prices (from the first Section), gives us S C Amer (S,K,T ) C Eur (S,K,T ) max [0,S PV (Div) PV (K)] K P Amer (S,K,T ) P Eur (S,K,T ) max [0,PV (K) S + PV (Div)] 54 / 60

137 American Call An American-style call option on a nondividend-paying stock should never be exercised prior to expiration (proof in class). For an American call on a dividend-paying stock it might be beneficial to exercise the option prior to expiration (by exercising the call, the owner will be entitled to dividend payments that she would not have otherwise received). Consider the previous example, with one exception: δ = 0.065, i.e., XYZ stock has a continuous dividend yield of 6.5% per year. 55 / 60

138 American Call (cont d) Because of dividends, early exercise is optimal at the node where the stock price is $ S 0 = $41 C 0 = $8.635 C 0,NO = $8.635 C 0,EX = $1 S u 1 = $ C u 1 = $ C u 1,NO = $ C u 1,EX = $ S d 1 = $ C d 1 = $2.761 C d 1,NO = $2.761 C d 1,EX = $0 C,NO = value of call if not exercised C,EX = value of call if exercised S uu 2 = $ C uu 2 = $ C uu 2,NO = $ C uu 2,EX = $ S ud 2 = $ C ud 2 = $7.029 C ud 2,NO = $7.029 C ud 2,EX = $2.249 S dd 2 = $ C dd 2 = $0 C dd 2,NO = $0 C dd 2,EX = $0 S uuu 3 = $ C uuu 3 = $ S uud 3 = $ C uud 3 = $ S udd 3 = $ C udd 3 = $0 S ddd 3 = $ C ddd 3 = $0 56 / 60

139 American Put When the underlying stock pays no dividend, a call will not be early-exercised, but a put might be. Suppose a company is bankrupt and the stock price falls to zero. Then a put that would not be exercised until expiration will be worth PV (K), which is smaller than K for a positive interest rate. Therefore, early exercise would be optimal in order to receive the strike price earlier. This can also be shown by using a parity argument. Consider the previous example, with δ = / 60

140 American Put (cont d) Early exercise is optimal at the node where the stock price is $ S 0 = $41 P 0 = $6.546 P 0,NO = $6.546 P 0,EX = $0 S u 1 = $ P u 1 = $2.314 P u 1,NO = $2.314 P u 1,EX = $0 S d 1 = $ P d 1 = $ P d 1,NO = $ P d 1,EX = $9.167 P,NO = value of put if not exercised PC,EX = value of put if exercised S uu 2 = $ P uu 2 = $0 P uu 2,NO = $0 P uu 2,EX = $0 S ud 2 = $ P ud 2 = $4.363 P ud 2,NO = $4.363 P ud 2,EX = $0 S dd 2 = $ P dd 2 = $ P dd 2,NO = $ P dd 2,EX = $ S uuu 3 = $ P uuu 3 = $0 S uud 3 = $ P uud 3 = $0 S udd 3 = $ P udd 3 = $8.228 S ddd 3 = $ P ddd 3 = $ / 60

141 Problem 10.20: For a stock index, S = $100, σ = 30%, r = 5%, δ = 3%, and T = 3. Let n = 3. What is the price of an American call option with a strike of $95? What is the price of an American put option with a strike of $95? 59 / 60

142 Problem 10.20: For a stock index, S = $100, σ = 30%, r = 5%, δ = 3%, and T = 3. Let n = 3. What is the price of an American call option with a strike of $95? The price of an American call option with a strike of 95 is $ What is the price of an American put option with a strike of $95? The price of an American put option with a strike of 95 is $ / 60

143 Understanding Early Exercise In deciding whether to early-exercise an option, the option holder compares the value of exercising immediately with the value of continuing to hold the option. Consider the cost and benefits of early exercise for a call option and a put option. By exercising, the option holder Call Option Receives the stock and therefore receives future dividends Bears the interest cost of paying the strike price prior to expiration Loses the insurance implicit in the call (the option holder is protected against the possibility that the stock price will be less than the strike price at expiration). Put Option Dividends are lost by giving up the stock Receives the strike price sooner rather than later Loses the insurance implicit in the put (the option holder is protected against the possibility that the stock price will be more than the strike price at expiration) 60 / 60

144 Option Markets 232D 3. Black-Scholes Formula and the Greeks Daniel Andrei Fall / 74

145 Outline I From Binomial Trees to the Black-Scholes Option Pricing Formula 3 Discrete Time vs Continuous Time 4 The Limiting Case of the Binomial Formula 5 Lognormality and the Binomial Model 11 Black-Scholes Assumptions 14 Inputs in the Binomial Model and in Black-Scholes 15 II Black-Scholes Formula 16 Black-Scholes Formula for a European Call Option 17 Black-Scholes Formula for a European Put Option 23 III Implied Volatility 27 IV WSJ reading 33 V Market-Maker Risk and Delta-Hedging 38 VI Option Greeks 46 VII Gamma-Neutrality 60 VIII Calendar Spreads 66 IX Appendix: Formulas for Option Greeks 70 2 / 74

146 Outline I From Binomial Trees to the Black-Scholes Option Pricing Formula 3 Discrete Time vs Continuous Time 4 The Limiting Case of the Binomial Formula 5 Lognormality and the Binomial Model 11 Black-Scholes Assumptions 14 Inputs in the Binomial Model and in Black-Scholes 15 II Black-Scholes Formula 16 Black-Scholes Formula for a European Call Option 17 Black-Scholes Formula for a European Put Option 23 III Implied Volatility 27 IV WSJ reading 33 V Market-Maker Risk and Delta-Hedging 38 VI Option Greeks 46 VII Gamma-Neutrality 60 VIII Calendar Spreads 66 IX Appendix: Formulas for Option Greeks 70 3 / 74

147 Discrete Time vs Continuous Time We refer to the structure of the binomial model as discrete time, which means that time moves in distinct increments This is much like looking at a calendar and observing only months, weeks, or days We know that time moves forward at a rate faster than one day at a time: hours, minutes, seconds, fractions of seconds, and fractions and fractions of seconds When we talk about time moving in the tiniest increments, we are talking about continuous time. Discrete time world Binomial model Continuous time world Black-Scholes model 4 / 74

148 The Limiting Case of the Binomial Formula An obvious objection to the binomial calculations thus far is that the stock can only have a few different values at expiration. It seems unlikely that the option price calculation will be accurate. The solution to this problem is to divide the time to expiration into more periods, generating a more realistic tree. To illustrate how to do this, we will re-examine the 1-year European call option, which has a $40 strike and initial stock price $41. Let there be 3 binomial periods. Since it is a 1-year call, this means that the length of a period is h = 1/3. We will assume that other inputs stay the same, so r = 0.08 and σ = / 74

149 The Limiting Case of the Binomial Formula (cont d) Since the length of the binomial period is shorter, u and d are closer to 1 than before ( and as opposed to and with h = 1). The risk-neutral probability of the stock price going up in a period is p = e(0.08 0) 1/ = The binomial model implicitly assigns probabilities to the various nodes. The risk-neutral probability for each final period node, together with the call value, is: Call Price in 1 Year (3 periods) Probability $ p 3 = $ p 2 (1 p ) = 0.34 $0 3p (1 p ) 2 = $0 (1 p ) 3 = / 74

150 The Limiting Case of the Binomial Formula (cont d) Thus, the price of the European call option is given by C 0 = e ( $ $12.814) = $ We can vary the number of binomial steps, holding fixed the time to expiration,t. The general formula is C 0 = e rt n n! [ ] k!(n k)! p k (1 p ) n k max S 0 u k d n k K,0 k=0 (1) (we also need to modify u, d, and p at each time). 7 / 74

151 The Limiting Case of the Binomial Formula (cont d) The following table computes binomial call option prices, using the same inputs as before. Number of steps (n) Binomial Call Price ($) 1 $ $ $ $ $ $ $ $6.960 $6.961 Changing the number of steps changes the option price, but once the number of steps becomes great enough we appear to approach a limiting value for the price, given by the Black-Scholes formula. 8 / 74

152 The Limiting Case of the Binomial Formula (cont d) This can be seen graphically below: Call Price Number of Steps (n) 9 / 74

153 Problem 12.2: Using the BinomCall function, compute the binomial approximations for the following call option: S 0 = $41, K = $40, σ = 0.3, r = 0.08, T = 0.25 (3 months), and δ = 0. Be sure to compute prices for n = 8,9,10,11, and 12. What do you observe about the behavior of the binomial approximation? 10 / 74

154 Problem 12.2: Using the BinomCall function, compute the binomial approximations for the following call option: S 0 = $41, K = $40, σ = 0.3, r = 0.08, T = 0.25 (3 months), and δ = 0. Be sure to compute prices for n = 8,9,10,11, and 12. What do you observe about the behavior of the binomial approximation? N Call Price The observed values are slowly converging towards the Black-Scholes value (3.399). Please note that the binomial solution oscillates as it approaches the Black-Scholes value. 10 / 74

155 Lognormality and the Binomial Model The lognormal distribution is the probability distribution that arises from the assumption that continuously compounded returns on the stock are normally distributed. In a binomial tree, as we increase the number of periods until expiration, continuously compounded returns approach a normal distribution. We can plot probabilities of outcomes (of stock returns) from the binomial tree for different values of n (2, 3, 6, and 10), as shown in the following figure. The 10-period binomial tree approaches fairly well the normal distribution. 11 / 74

156 Lognormality and the Binomial Model (cont d) / 74

157 Problem 11.13: Let S = $100, σ = 0.30, r = 0.08, t = 1, and δ = 0. Use equation (1) to compute the probability of reaching a terminal node and the price at that node and plot the risk-neutral distribution of year-1 stock prices for n = 50. What is the risk-neutral probability that S 1 < $80? S 1 > $120? 13 / 74

158 Problem 11.13: Let S = $100, σ = 0.30, r = 0.08, t = 1, and δ = 0. Use equation (1) to compute the probability of reaching a terminal node and the price at that node and plot the risk-neutral distribution of year-1 stock prices for n = 50. What is the risk-neutral probability that S 1 < $80? S 1 > $120? For n = 50, we have u = , d = , and p = We get the following diagram: To obtain the required probabilities we sum all probabilities for which the final stock price is below 80 or above 120 respectively. We obtain Pr(S < 80) = Pr(S > 120) = / 74

159 Black-Scholes Assumptions Assumptions about stock return distribution Continously compounded returns on the stocks are normally distributed and independent over time The volatility of continuously compounded returns is known and constant Future dividends are known, either as dollar amount or as a fixed dividend yield Assumptions about the economic environment The risk-free rate is known and constant There are no transaction costs or taxes It is possible to short-sell costlessly and to borrow at the risk-free rate 14 / 74

160 Inputs in the Binomial Model and in Black-Scholes There are seven inputs to the binomial model and six inputs to the Black-Scholes model: Inputs Binomial Model Current price of the stock, S 0 Strike price of the option, K Volatility of the stock, σ Continuously compounded risk-free interest rate, r Time to expiration, T Dividend yield on the stock, δ Number of binomial periods, n Black- Scholes 15 / 74

161 Outline I From Binomial Trees to the Black-Scholes Option Pricing Formula 3 Discrete Time vs Continuous Time 4 The Limiting Case of the Binomial Formula 5 Lognormality and the Binomial Model 11 Black-Scholes Assumptions 14 Inputs in the Binomial Model and in Black-Scholes 15 II Black-Scholes Formula 16 Black-Scholes Formula for a European Call Option 17 Black-Scholes Formula for a European Put Option 23 III Implied Volatility 27 IV WSJ reading 33 V Market-Maker Risk and Delta-Hedging 38 VI Option Greeks 46 VII Gamma-Neutrality 60 VIII Calendar Spreads 66 IX Appendix: Formulas for Option Greeks / 74

162 Black-Scholes Formula for a European Call Option The Black-Scholes formula for a European call option on a stock that pays dividends at the continuous rate δ is C 0 (S 0,K,σ,r,T,δ) = S 0 e δt N (d 1 ) Ke rt N (d 2 ) (2) where ( ) ( d 1 = ln S0 K + r δ σ2) T σ T (3) d 2 = d 1 σ T (4) N (x) is the cumulative normal distribution function, which is the probability that a number randomly drawn from a standard normal distribution will be less than x. 17 / 74

163 Black-Scholes Formula for a European Call Option (cont d) Let S 0 = $41, K = $40, σ = 0.3, r = 0.08, T = 1 year, and δ = 0. Computing the Black-Scholes call price, we obtain C 0 = $41 e 0 N ( ) $40 e 0.08 N ( ) = $41 e $40 e = $6.961 Note that this result corresponds to the limit obtained from the binomial model. 18 / 74

164 Black-Scholes Formula for a European Call Option (cont d) The following figure plots Black-Scholes call option prices (today and at expiration) for stock prices ranging from $20 to $ Call Price Stock Price 19 / 74

165 A Remarkable Result In the binomial model, we have not specified the probabilities of the stock going up and down (which would give us the expected return of the stock). The expected return on the stock does not appear in the Black-Scholes formula either. This raises a question: Consider a stock with a higher beta (and hence having a higher expected return). A call option on this stock would have a higher probability of settlement in-the-money, hence a higher option price. Why is this not the case? 20 / 74

166 A Remarkable Result (cont d) The Black-Scholes formula (2) provides the answer. This formula shows the composition of the replicating portfolio, which in the binomial case was C 0 = S 0 + B We can easily identify (the position in the risky asset) and B (the dollar amount of borrowing or lending) in the Black-Scholes formula: = e δt N (d 1 ) (5) B = Ke rt N (d 2 ) (6) 21 / 74

167 A Remarkable Result (cont d) If β S is the stock beta, then the option beta is β Option = S 0 + B β S (7) A high stock beta implies a high option beta, so the discount rate for the expected payoff of the option is correspondingly greater. The net result one of the key insights from the Black-Scholes analysis is that beta is irrelevant: The larger average payoff to options on high beta stocks is exactly offset by the larger discount rate. 22 / 74

168 Black-Scholes Formula for a European Put Option The Black-Scholes formula for a European put option is P 0 (S 0,K,σ,r,T,δ) = S 0 e δt N ( d 1 ) + Ke rt N ( d 2 ) (8) where d 1 and d 2 are given by equations (3) and (4). Put-call parity must hold: P 0 (S 0,K,σ,r,T,δ) = C 0 (S 0,K,σ,r,T,δ) + Ke rt Se δt (9) This follows from the formulas (2) and (8), together with the fact that for any x, N ( x) = 1 N (x). 23 / 74

169 Black-Scholes Formula for a European Put Option (cont d) Let S 0 = $41, K = $40, σ = 0.3, r = 0.08, T = 1 year, and δ = 0. Computing the Black-Scholes put price, we obtain P 0 = $41 e 0 N ( ) + $40 e 0.08 N ( ) = $41 e $40 e = $2.886 In the binomial model, if we fix the number of periods to n = 500, we obtain a price of $ Computing the price using put-call parity (equation 9) yields P 0 = $ $40e 0.08 $41 = $ / 74

170 Black-Scholes Formula for a European Put Option (cont d) The following figure plots Black-Scholes put option prices (today and at expiration) for stock prices ranging from $20 to $ Put Price Stock Price 25 / 74

171 Problem 12.20: Let S = $100, K = $90, σ = 30%, r = 8%, δ = 5%, and T = 1. a. What is the Black-Scholes call price? b. Now price a put where S = $90, K = $100, σ = 30%, r = 5%, δ = 8%, and T = / 74

172 Problem 12.20: Let S = $100, K = $90, σ = 30%, r = 8%, δ = 5%, and T = 1. a. What is the Black-Scholes call price? The Black-Scholes call price is $17.70 b. Now price a put where S = $90, K = $100, σ = 30%, r = 5%, δ = 8%, and T = 1. The Black-Scholes put price is $ / 74

174 Implied Volatility Volatility is unobservable Choosing a volatility to use in pricing an option is difficult but important Using history of returns is not the best approach, because history is not a reliable guide to the future. We can invert Black-Scholes formula to obtain implied volatility We cannot use implied volatility to assess whether an option price is correct, but implied volatility does tell us the market s assessment of volatility 28 / 74

175 Implied Volatility (cont d) Example: Let S = $100, K = $90, r = 8%, δ = 5%, and T = 1. The market option price for a call option is $ What is the volatility that gives this option price? We must invert the following formula $18.25 = BSCall(100,90,σ,0.08,1,0.05) We use the implied volatility function: We find that setting σ = 31.73% gives us a call price of $ / 74

176 Implied Volatility (cont d) The following figure plots implied volatilities for S&P 500 options, 10/28/2004. Option prices (ask) from CBOE. Assumes S = $1,127.44, δ = 1.85%, r = 2%, and expiration date 1/22/2005. Implied volatility Calls Puts 1,100 1,120 1,140 Strike 30 / 74

177 Implied Volatility (cont d) There is a systematic pattern of implied volatility across strike prices, called volatility skew The volatility skew is not related to whether an option is a put or a call, but rather to differences in the strike price and time to expiration Explaining these patterns is a challenge for option pricing theory 31 / 74

178 Using Implied Volatility Implied volatility is important for a number of reasons If you need to price an option for which you cannot observe a market price, you can use implied volatility to generate a price consistent with the price of traded options Implied volatility is often used as a quick way to describe the level of option prices on a given underlying asset. Option prices are quoted sometimes in terms of volatility, rather than as a dollar price Volatility skew provides a measure of how well option pricing models work Just as stock markets provide information about stock prices and permit trading stocks, option markets provide information about volatility, and, in effect, permit the trading of volatility. 32 / 74

180 Four Common Strategies in Unsettled Markets (Wall Street Journal, October 31, / 74

181 Four Common Strategies in Unsettled Markets (Wall Street Journal, October 31, 2012 (cont d) 35 / 74

185 Market-Maker Risk A market-maker stands ready to sell to buyers and to buy from sellers. Without hedging, an active market-maker will have an arbitrary position generated by fulfilling customer orders. This arbitrary portfolio has uncontrolled risk. Consequently, market-makers attempt to hedge the risk of their position. We will se here how they do so. 39 / 74

186 Market-Maker Risk (cont d) Suppose a customer wishes to buy 100 European call options with maturity of 91 days. The market-maker fills this order by selling 100 call options. To be specific, suppose that S = $41, K = $40, σ = 0.3, r = 0.08 (continuously compounded), and δ = 0. We will let T denote the expiration time of the option and t the present, so time to expiration is T t. Thus, T t = 91/365 = Suppose that the market-maker does not hedge the written option (naked position) and the stock has the following evolution over the next 5 days: Day Stock ($) / 74

187 Market-Maker Risk (cont d) We can measure the profit of the market-maker by marking-to-market the position: if we liquidated the position today, what would be the gain or loss? Day Stock ($) Call Position ($) Daily Profit ($) / 74

188 Market-Maker Risk (cont d) 200 Naked Position Daily Profit Day 42 / 74

189 Delta-Hedging Suppose the market-maker hedges the position with shares. At time 0, the delta of a call at a stock price of $41 is This suggests that a $1 increase in the stock price should increase the value of the option by approximatively $ The market-maker takes an offsetting position in shares, position that hedges the fluctuations in the option price. We say that such a position is delta-hedged. Then, the market-maker rebalances the portfolio each day, by computing the new delta of the call. The following table summarizes delta, the number of purchased shares, the net investment, and profit for each day for 5 days (interest expenses are ignored for simplicity). 43 / 74

190 Delta-Hedging (cont d) Day Stock ($) Option Delta Stock Position (# shares) Daily Profit (Call) Daily profit (Shares) Daily profit (Total) / 74

191 Delta-Hedging (cont d) Daily Profit Naked Position Delta Hedged Day Delta hedging prevents the position from reacting to small changes in the underlying stock. For large changes, we need to take into account the fluctuations in delta. 45 / 74

193 Option Greeks Option Greeks are formulas that express the change in the option price when an input to the formula changes, taking as fixed all other inputs. They are used to assess risk exposures. For example: A market-making bank with a portfolio of options would want to understand its exposure to stock price changes, interest rates, volatility, maturity, etc. A portfolio manager wants to know what happens to the value of a portfolio of stock index options if there is a change in the level of the stock index. An options investor would like to know how interest rate changes and volatility changes affect profit and loss. 47 / 74

194 Option Greeks (cont d) Before providing detailed definition of the Greeks, let s have some intuition on how changes in inputs affect option prices: Change in input Change in call price Change in put price S t C t P t σ C t P t T t (t ) C t generally P t ambiguous r C t P t δ C t P t Table 1 : Changes in Black-Scholes inputs and their effect on option prices. An increase in the stock price (S t ) raises the chance that the call will be exercised, thus raises the call option price. Conversely, it lowers the put option price. 48 / 74

195 Option Greeks (cont d) An increase in volatility raises the price of a call or put option, because it increases the expected value if the option is exercised. Options generally but not always become less valuable as time to expiration decreases, i.e., there is a time decay. There are exceptions, for example deep-in-the-money call options on an asset with high dividend yield and deep-in-the-money puts. A higher interest rate reduces the present value of the strike (to be paid by a call option holder), and thus increases the call price. The put option entitles the owner to receive the strike, whose present value is lower with a higher interest rate. Thus, a higher interest rate decreases the put price. A call entitle the holder to receive stock, but without dividends prior to expiration. Thus, the greater the dividend yield, the lower the call price. Conversely, a put option is more valuable when the dividend yield is greater. 49 / 74

196 Option Greeks (cont d) The Greeks are tools that let us to quantify these relationships: Input Greek Definition Mnemonic S t (Delta) Measures the option price change when the stock price increases by $1 S t Γ (Gamma) Measures the change in when the stock price increases by $1 S t Ω (Elasticity) Measures the percentage change in the option price when the stock price increases by 1% σ Vega Measures the option price change when there is an increase in volatility of 1% t θ (theta) Measures the option price change when there is a decrease in the time to maturity (increase in calendar time) of 1 day r ρ (rho) Measures the option price change when there is an increase in the interest rate of 1% (100 basis points) δ Ψ (Psi) Measures the option price change when there is an increase in the continuous dividend yield of 1% (100 basis points) vega volatility theta time rho r 50 / 74

197 Option Greeks (cont d) Let us come back to Table 1 and complete it with the proper signs of the Greeks: Input Call Option Put Option C t Call > 0 P t Put < 0 S t Call Γ Call > 0 Put Γ Put > 0 Ω Call 1 Ω Put 0 σ C t Vega Call > 0 P t Vega Put > 0 t C t gen. θ Call gen. < 0 P t ambig. θ Put any r C t ρ Call > 0 P t ρ Put < 0 δ C t Ψ Call < 0 P t Ψ Put > 0 51 / 74

198 Option Greeks: Example The Greeks are mathematical derivatives of the option price formula with respect to the inputs. Suppose that the stock price is S t = $41, the strike price is K = $40, volatility is σ = 0.3, the risk-free rate is r = 0.08, the time to expiration is T t = 1, and the dividend yield is δ = 0. The values for the Greeks are Input Call Option Put Option S t Γ Call = > 0 Γ Put = > 0 Call = > 0 Delta Put = < 0 Ω Call = Ω Put = σ Vega Call = > 0 Vega Put = > 0 t θ Call = gen. < 0 θ Put = any r ρ Call = > 0 ρ Put = < 0 δ Ψ Call = < 0 Ψ Put = > 0 52 / 74

199 Delta ( ) Measures the change in the option price for a $1 change in the stock price: 1 Call Put 0.5 Delta Stock Price 53 / 74

200 Gamma (Γ) Measures the change in delta when the stock price changes: 0.04 Call Put 0.03 Gamma Stock Price 54 / 74

201 Elasticity (Ω) Measures the percentage change in the option price relative to the percentage change in the stock price: 10 Call Put 5 Elasticity Stock Price 55 / 74

202 Vega Measures the change in the option price when volatility changes (divide by 100 for a change per percentage point): 0.15 Call Put 0.1 Vega Stock Price 56 / 74

203 Theta (θ) Measures the change in the option price with respect to calendar time, t, holding fixed the maturity date T. To obtain per-day theta, divide by Call Put Theta Stock Price 57 / 74

204 Rho (ρ) Measures the change in the option price when the interest rate changes (divide by 100 for a change per percentage point, or by 10,000 for a change per basis point): Call Put Rho Stock Price 58 / 74

205 Psi (Ψ) Measures the change in the option price when the continuous dividend yield changes (divide by 100 for a change per percentage point): 0.2 Call Put 0 Psi Stock Price 59 / 74

207 Gamma-Neutrality Gamma hedging is the construction of options positions that are hedged such that the total gamma of the position is zero. We cannot do this using just the stock, because the gamma of the stock is zero (the delta of a stock is constant and equal to 1). Hence, we must acquire another option in an amount that offsets the gamma of the written call. In addition to the 91-day call from the previous example, consider a 45-strike 120-day call. The ratio of the gamma of the two options is Γ K=40,T t=91 = = (10) Γ K=45,T t= Thus, we need to buy of the 45-strike options for every 40-strike option we have sold. 61 / 74

208 Gamma-Neutrality (cont d) The Greeks resulting form this position are in the last column of the following table: 40-Strike Call 45-Strike Call Total Position Price ($) Delta ( ) Gamma (Γ) Theta (θ) Since delta is , we need to buy 21.8 shares of stock to be both delta- and gamma-hedged. 62 / 74

209 Gamma-Neutrality (cont d) We can compare the delta-hedged position with the delta- and gamma-hedged position. The delta-hedged position has the problem that large moves always cause losses. The delta- and gamma-hedged position loses less if there is a large move down, and can make money if the stock price increases. This can be seen from the following figure. It compares the 1-day holding period profit for delta-hedged position described earlier and delta- and gamma hedged position. 63 / 74

210 Gamma-Neutrality (cont d) Delta-hedged Delta- and gamma-hedged Overnight Profit ($) Stock Price 64 / 74

211 Gamma-Neutrality (cont d) Delta-gamma hedging prevents the position from reacting to large changes in the underlying stock: Daily Profit Naked Position -hedged - and Γ-hedged Day 65 / 74

213 Calendar Spreads To protect against a stock price increase when selling a call, you can simultaneously buy a call option with the same strike and greater time to expiration. This purchased calendar spread exploits the fact that the written near-to-expiration option exhibits greater time decay than the purchased far-to-expiration option, and therefore is profitable if the stock price does not move. 67 / 74

214 Calendar Spreads (cont d) Suppose you sell a 40-strike call with 91 days to expiration and buy a 40-strike call with 1 year to expiration. Assume a stock price of $40, r = 8%, σ = 30%, and δ = 0. The premiums are $2.78 for the 91-day call and $6.28 for the 1-year call. Theta is more negative for the 91-day call ( ) than for the 1-year call ( ). Thus, if the stock price does not change over the course of 1 day, the position will make money since the written option loses more value than the purchased option. 68 / 74

215 Calendar Spreads (cont d) The profit diagram for this position for a holding period of 91 days is displayed below Profit in 91 days ($) 0 10 Short call Calendar spread Stock price ($) 69 / 74

217 Appendix: Formulas for Option Greeks Delta ( ) measures the change in the option price for a $1 change in the stock price: Call = C S = e δ(t t) N (d 1 ) (11) Put = P S = e δ(t t) N ( d 1 ) (12) Gamma (Γ) measures the change in delta when the stock price changes: Γ Call = 2 C S 2 = e δ(t t) N (d 1 ) Sσ T t (13) Γ Put = 2 P S 2 = Γ Call (14) 71 / 74

218 Appendix: Formulas for Option Greeks (cont d) Elasticity (Ω) measures the percentage change in the option price relative to the percentage change in the stock price: Ω Call = S t Call C t (15) Ω Put = S t Put P t (16) Vega measures the change in the option price when volatility changes (divide by 100 for a change per percentage point): Vega Call = C σ = Se δ(t t) N (d 1 ) T t (17) Vega Put = P σ = Vega Call (18) 72 / 74

219 Appendix: Formulas for Option Greeks (cont d) Theta (θ) measures the change in the option price with respect to calendar time, t, holding fixed the maturity date T. To obtain per-day theta, divide by 365. θ Call = C t = δse δ(t t) N (d 1 ) rke r(t t) N (d 2 ) Ke r(t t) N (d 2 )σ 2 T t (19) θ Put = P t = θ Call + rke r(t t) δse δ(t t) (20) Rho (ρ) measures the change in the option price when the interest rate changes (divide by 100 for a change per percentage point, or by 10,000 for a change per basis point): ρ Call = C r = (T t)ke r(t t) N (d 2 ) (21) ρ Put = P r = (T t)ke r(t t) N ( d 2 ) (22) 73 / 74

220 Appendix: Formulas for Option Greeks (cont d) Psi (Ψ) Measures the change in the option price when the continuous dividend yield changes (divide by 100 for a change per percentage point): Ψ Call = C δ = (T t)se δ(t t) N (d 1 ) (23) Ψ Put = P δ = (T t)se δ(t t) N ( d 1 ) (24) 74 / 74

221 Option Markets 232D 4. Forwards and Futures Daniel Andrei Fall / 62

222 Outline I Pricing Forwards 3 II Futures Contracts 11 III Financial Forwards and Futures 13 Index Futures 14 Currency Futures 24 IV Commodity Forwards and Futures 27 V Interest Rate Forwards and Futures 41 VI Options on Futures 55 2 / 62

223 Outline I Pricing Forwards 3 II Futures Contracts 11 III Financial Forwards and Futures 13 Index Futures 14 Currency Futures 24 IV Commodity Forwards and Futures 27 V Interest Rate Forwards and Futures 41 VI Options on Futures 55 3 / 62

224 Alternative ways to buy a stock 1. Outright Purchase: Pay S 0 Receive security Time 0 2. Forward: Pay F 0,T =? Receive security Time T A forward contract is an arrangement in which you both pay for the stock and receive it at time T, with the time T price specified at time 0. What should you pay for the stock in this case? Arbitrage ensures that there is a very close relationship between prices and forward prices 4 / 62

225 Pricing a Forward Contract Let S 0 be the spot price of an asset at time 0, and r the continuously compounded interest rate. Assume that dividends are continuous and paid at a rate δ. Then the forward price at a future time T must satisfy F 0,T = S 0 e (r δ)t (1) Suppose that F 0,T > S 0 e (r δ)t. Then an investor can execute the following trades at time 0 (buy low and sell high) and obtain an arbitrage profit: Cash Flows Transaction Time 0 Time T (expiration) Buy tailed position in stock (e δt units) S 0 e δt S T Borrow S 0 e δt +S 0 e δt S 0 e (r δ)t Short forward 0 F 0,T S T Total 0 F 0,T S 0 e (r δ)t > 0 5 / 62

226 Pricing a Forward Contract (cont d) Suppose that F 0,T < S 0 e (r δ)t. Then an investor can execute the following trades at time 0 (buy low and sell high) and obtain once again an arbitrage profit: Cash Flows Transaction Time 0 Time T (expiration) Short tailed position in stock (e δt units) S 0 e δt S T Lend S 0 e δt S 0 e δt S 0 e (r δ)t Long forward 0 S T F 0,T Total 0 S 0 e (r δ)t F 0,T > 0 Consequently, and assuming that the non-arbitrage condition holds, we have F 0,T = S 0 e (r δ)t 6 / 62

227 Payoff Diagram for a Forward Payoff ($) = Profit ($) Long forward S T F 0,T F 0,T Short forward F 0,T S T ,000 1,100 1,200 S T 7 / 62

228 Risk-Neutral Valuation The risk neutral technique also works for forward contracts. When the contract is agreed to initially, its value is 0. Its payoff at maturity is Therefore which yields F 0,T = S 0 e (r δ)t S T F 0,T 0 = S 0 e δt F 0,T e rt 8 / 62

229 Put-Call Parity and the Forward Price Consider a call and a put having both the same maturity and the same strike price. Let the strike price be equal to the forward price with the same maturity: K = F 0,T = S 0 e (r δ)t Write the put-call parity relationship: C 0 P 0 = S 0 e δt Ke rt = S 0 e δt S 0 e (r δ)t e rt = 0 The price of a call equals the price of a put with the same maturity when the strike price is equal to the forward price. 9 / 62

230 Example: Let S = $100, σ = 30%, r = 8%, δ = 5%, T = 1, and K = $100e ( ) 1 = $ What is the Black-Scholes call price? The Black-Scholes call price is $11.34 What is the Black-Scholes put price? The Black-Scholes put price is $ / 62

231 Outline I Pricing Forwards 3 II Futures Contracts 11 III Financial Forwards and Futures 13 Index Futures 14 Currency Futures 24 IV Commodity Forwards and Futures 27 V Interest Rate Forwards and Futures 41 VI Options on Futures / 62

232 Futures Contracts Exchange-traded forward contracts Typical features of futures contracts Matches buy and sell orders Keeps track of members obligations and payments After matching the trades, becomes counterparty Differences from forward contracts Settled daily through the mark-to-market process low credit risk Highly liquid easier to offset an existing position Highly standardized structure harder to customize 12 / 62

234 Index Futures Figure 1 : Listing of various index futures contracts from the Wall Street Journal, October 6-7, / 62

235 The S&P 500 Futures Contract Specifications for the S&P 500 index futures contract Underlying: S&P 500 index Where traded: Chicago Mercantile Exchange Size: $250 S&P 500 index Months: Mar, Jun, Sep, Dec Trading ends: Business day prior to determination of settlement price Settlement: Cash-settled, based upon opening price of S&P 500 on third Friday of expiration month Suppose the futures price is 1100 and you wish to enter into 8 long futures contracts. The notional value of 8 contracts is 8 $ = $2, = $2.2 million 15 / 62

236 The S&P 500 Futures Contract (cont d) Suppose that there is 10% margin and weekly settlement (in practice settlement is daily). The margin on futures contracts with a notional value of $2.2 million is $220,000. The margin balance today from long position in 8 S&P 500 futures contracts is Week Multiplier ($) Futures Price Price Change Margin Balance ($) , Over the first week, the futures price drops points to On a mark-to-market basis, we have lost $2,000 ( 72.01) = $144,020 Thus, if the continuously compounded interest rate is 6%, our margin balance after one week is $220,000 e /52 $144,020 = $76, / 62

237 The S&P 500 Futures Contract (cont d) Because we have a 10% margin, a 6.5% decline in the futures price results in a 65% decline in margin. The margin balance after the first week is Week Multiplier ($) Futures Price Price Change Margin Balance ($) , , The decline in margin balance means the broker has significantly less protection should we default. For this reason, participants are required to maintain the margin at a minimum level, called the maintenance margin. This is often set at 70% to 80% of the initial margin level. In this example, the broker would make a margin call, requesting additional margin. We can go on for a period of 10 weeks, assuming weekly marking-to-market and a continuously compounded risk-free rate of 6%. 17 / 62

238 The S&P 500 Futures Contract (cont d) The margin balance after a period of 10 weeks is Week Multiplier ($) Futures Price Price Change Margin Balance ($) , , , , , , , , , , , The 10-week profit on the position is obtained by subtracting from the final margin balance the future value of the original margin investment: $44, $220,000 e /52 = $177, / 62

239 The S&P 500 Futures Contract (cont d) What if the position had been forwarded rather than a futures position, but with prices the same? In that case, after 10 weeks our profit would have been ( ) $2,000 = $176,700 The futures and forward profits differ because of the interest earned on the mark-to-market proceeds (in the present cases, we have founded losses as they occurred and not at expiration, which explains the loss). 19 / 62

240 Quanto Index Contracts Specifications for the Nikkei 225 index futures contract Underlying: Nikkei 225 stock index Where traded: Chicago Mercantile Exchange Size: $5 Nikkei 225 index Months: Mar, Jun, Sep, Dec Trading ends: Business day prior to determination of settlement price Settlement: Cash-settled, based upon opening Osaka quotation of the Nikkei 225 index on the second Friday of expiration month There is one very important difference with the S&P 500 contract: Settlement of the contract is in a different currency (dollars) than the currency of denomination for the index (yen) Consequently, the contract insulates investors from currency risk, permitting them to speculate solely on whether the index rises or falls 20 / 62

241 Uses Of Index Futures Why buy an index futures contract instead of synthesizing it using the stocks in the index? Lower transaction costs Asset allocation: switching investments among asset classes Example: invested in the S&P 500 index and wish to temporarily invest in bonds instead of the index. What to do? Alternative #1: sell all 500 stocks and invest in bonds Alternative #2: take a short forward position in S&P 500 index 21 / 62

242 Uses Of Index Futures (cont d) Suppose that the current index price, S 0, is $100, and the effective 1-year risk-free rate is 10%. The forward price is therefore $110. The effect of owning the stock and selling forward is Cash Flows Transaction Today 1 year, S 1 = $80 1 year, S 1 = $130 Own $100 -$100 $80 $130 Short $110 0 $110 - $80 $110 - $130 Total -$100 $110 $ / 62

243 Uses Of Index Futures (cont d) General asset allocation: futures overlay, alpha-porting Cross-hedging: hedge portfolios that are not exactly the index Cross-hedging with perfect correlation Cross-hedging with imperfect correlation Risk management for stock-pickers 23 / 62

244 Currency Futures Figure 2 : Listing of various currency futures contracts from the Wall Street Journal, October 6-7, / 62

245 Currency Contracts Currency forwards and futures are widely used to manage foreign exchange risk Suppose that T years from today you want to buy 1 with dollars. Denote the yen-denominated interest rate by r y, the dollar-denominated interest rate by r, and the exchange rate today ($/ ) by x 0 The forward price for a yen is F 0,T = x 0 e (r ry )T (2) The forward currency rate will exceed the current exchange rate when the domestic risk-free rate is higher than the foreign risk-free rate Notice that equation (2) is just like equation (1), for stock index futures, with the foreign interest rate equal to the dividend yield. 25 / 62

246 Currency Contracts (cont d) We can prove equation (2) by absence of arbitrage and by building a synthetic currency forward: borrowing in one currency and lending in another, which creates the same cash-flow as the forward contract: Cash Flows Year 0 Year T Transaction $ $ Borrow x 0 e ry T dollar at r +x 0 e ry T x 0 e (r ry )T Convert to x 0 x 0 e ry T +e ry T Invest in yen-denominated bill at r y e ry T 1 Total 0 0 x 0 e (r ry )T 1 Example: Suppose that the yen-denominated interest rate is r y = 2% and the dollar-denominated rate is r = 6%. The current exchange rate rate is x 0 = dollars per yen. The 1-year forward rate is 0.009e = / 62

248 Commodity Forwards and Futures Figure 3 : Listing of various currency futures contracts from the Wall Street Journal, October 6-7, / 62

249 Introduction to Commodity Forwards and Futures Financial forward prices are described by the general formula F 0,T = S 0 e (r δ)t At a general level, commodity forward prices can be described by the same formula. There are, however, important differences: For financial assets δ is the dividend yield, whereas for commodities δ is the commodity lease rate While the dividend yield for a financial asset can typically be observed directly, the lease rate for a commodity can be estimated only by observing the forward price The formula for a commodity forward price is F 0,T = S 0 e (r δ l )T (3) 29 / 62

250 Forward Prices and The Lease Rate When we observe the forward price, we can infer the lease rate. Specifically, if the forward price is F 0,T, the annualized lease rate is δ l = r 1 T ln ( F0,T S 0 ) (4) If instead we use an effective annual interest rate, the effective annual lease rate is 1 + r δ l = 1 (5) 1/T (F 0,T /S 0 ) 30 / 62

251 The Forward Curve Commodities are complex because every commodity market differs in the details. For example: Storage is not possible for electricity Gold is durable and relatively inexpensive to store (compared to its value) Some commodities feature seasonality in production (for example, corn in the United States is harvested primarily in the fall) One way to observe and understand this heterogeneity is to build the forward curve (or the forward strip), i.e., the set of prices for different expiration dates for a given commodity If on a given date the forward curve is upward-sloping, then the market is in contango. If the forward curve is downward sloping, the market is in backwardation Forward curves can have portions in backwardation and portions in contango 31 / 62

252 Forward Curves for Various Commodities, May 5, 2004 Corn Futures Price (cents per bushel) Gasoline Futures Price (cents per gallon) Contango Backwardation Backwardation Months to Maturity Crude Oil Futures Price ($ per barrel) Backwardation Months to Maturity Months to Maturity Gold Futures Price ($ per ounce) Contango Months to Maturity 32 / 62

253 Futures Prices for Corn for the first Wednesday in June, Futures Prices (cents per bushel) Jul Sep Dec Mar May Jul Sep Dec Expiration Month 33 / 62

254 Oil Prices: Contango vs. Backwardation NPR news, Dec 17, 2008: Contango in Oil Markets Explained The supply glut in the oil market has led to a contango price structure, in which oil futures are priced higher than their spot price... Usually, the market goes into contango when there is a supply glut, when there s too much oil on the market. And this is what we ve seen happen just in the last few months as the recession fully set in and people realized the days of easy credit and high oil prices are gone... There s just a lot of oil floating around with no demand to take it up. Wall Street Journal, Oct 28, 2011: Oil Futures Return to Backwardation Crude oil traded on the New York Mercantile Exchange has been trading in contango, a commodity price structure in which futures contracts for near-term delivery are cheaper than further-out contracts. But that s changed in the last few days... Currently, we do have rising production from Libya, which does suggest more generous supply in the months ahead... Aside from that, we re also at a time of year when global demand is strong seasonally and low seasonal demand in the second quarter points to a potential surplus at that time a reason for prices to be softer then. 34 / 62

255 Nymex Crude Falls as Brent Divergence Widens From the Wall Street Journal, October 8, 2012: They re really in two different universes... In terms of supply and demand, they re just two different markets. Declining output in the North Sea, tension in the Middle East, and the Iranian oil embargo are having a disproportionate impact on the Brent market. In the U.S., the Energy Information Administration reported domestic crude-oil output rose to its highest level since 1996 during the prior week. The surge in production has been fueled by rising flows from U.S. shale-oil fields. 35 / 62

256 Gold Futures Specifications for the NYMEX gold futures contract Underlying: Refined gold bearing approved refiner stamp Where traded: New York Mercantile Exchange Size: 100 troy ounces Months: Feb, Apr, Aug, Oct, out two years. Jun, Dec, out 5 years Trading ends: Third-to-last business day of maturity month Delivery: Any business day of the delivery month 36 / 62

257 Suppose that on June 6, 2001, the gold spot price is $265.7 and the gold future price with maturity in December is $269. The June to December interest rate (annualized, effective) is %. What is the annualized 6-month gold lease rate? Using equation (5), the annualized 6-month lease rate is 6-month lease rate = = 1.456% 1/0.5 (269/265.7) 37 / 62

258 Gold Lease Rate Slides to Lowest on Record as European Banks Seek Dollars Bloomberg, December 8, 2011: The interest rate for lending gold in exchange for dollars plunged to the lowest on record this week as European banks sought ways to secure the U.S. currency amid the region s debt crisis European banks especially are having liquidity funding problems, which does see a lot of lending of gold and that s putting downward pressure on lease rates 38 / 62

259 Coffee Standoff Tests Growers Grit Source: Wall Street Journal, October 8, / 62

260 Gas Futures Tumble After Power Outages Sap Demand From the Wall Street Journal, October 31, 2012: Natural-gas futures tumbled 3% on Tuesday as traders bet on falling demand from gas-fired power plants after Storm Sandy cut power to millions of people along the East Coast. We re faced with the loss of demand in a well-supplied market, said Teri Viswanath, an energy analyst at BNP Paribas. Futures prices could see further declines if data later this week and next show falling demand and rising stockpiles. Natural gas for December delivery settled 11.2 cents lower at $3.691 per million British thermal units on New York Mercantile Exchange electronic trading. After tumbling to decade lows in April under $2/MMBtu, gas futures have made a steady climb amid rising demand from utilities. Many power producers switched to burning gas from coal to take advantage of low gas prices. 40 / 62

262 Bond Basics P(0,t 2 ) P(0,t 1 ) 1 [1+r(0,t 1 )] t 1 1 [1+r(0,t 2 )] t 2 $1 $1 0 t 1 years t 2 years P 0 (t 1,t 2 ) 1 [1+r 0 (t 1,t 2 )] t 2 t 1 $1 Notation: P(0,t 1 ) and P(0,t 2 ): zero coupon bond r(0,t1 ) and r(0,t 2 ): effective annual interest rate (yield to maturity) P0 (t 1,t 2 ): implied forward zero-coupon bond price (quoted at time 0) r0 (t 1,t 2 ): implied forward rate (prevailing at time 0) 42 / 62

263 Implied Forward Rates and Zero Coupon Bonds Suppose that today is 0 and we plan to borrow $1 at t 1 and pay back at t 2 We can synthetically create this by trading zero-coupon bonds Cash Flows Transaction Time 0 Time t 1 Time t 2 Sell P(0,t 1 )/P(0,t 2 ) z-c bonds t 2 +P(0,t 1 ) P(0,t1) P(0,t 2) Buy 1 z-c bond t 1 P(0,t 1 ) $1 Total 0 $1 P(0,t1) P(0,t 2) It follows that the implied forward zero-coupon bond price must be consistent with the implied forward interest rate and the zero-coupon bond prices with maturities t 1 and t 2 : P 0 (t 1,t 2 ) = 1 [1 + r 0 (t 1,t 2 )] t 2 t 1 = P(0,t 2) P(0,t 1 ) 43 / 62

264 Forward Rate Agreements Consider a firm expecting to borrow $100m for 91 days (3 months), beginning 120 days from today, in June (using our previous notation, t 1 = 120/360 is the borrowing date and t 2 = 211/360 is the loan repayment date). In June, the effective quarterly interest rate can be either 1.5% or 2%. The implied June 91-day forward rate (the rate from June to September) is 1.8%. Here is the risk faced by the borrower, assuming no hedging: September (t Transaction June (t 1 = 120/360) 2 = 211/360) r quarterly = 1.5% r quarterly = 2% Borrow $100m +$100m -$101.5m -$102.0m 44 / 62

265 Forward Rate Agreements (cont d) A forward rate agreement (FRA) is an over-the-counter contract that guarantees a borrowing or lending rate on a given notional principal amount. FRAs can be settled at maturity (in arrears) or at the initiation of the borrowing or lending transaction FRA settlement in arrears: Payment to the borrower = ( r quarterly r FRA ) notional principal FRA settlement at the borrowing date: Payment to the borrower = r quarterly r FRA 1 + r quarterly notional principal 45 / 62

266 Forward Rate Agreements (cont d) FRA settlement in arrears: Transaction June (t 1 = 120/360) September (t 2 = 211/360) r quarterly = 1.5% r quarterly = 2% Borrow $100m +$100m -$101.5m -$102.0m FRA payment if r quarterly = 1.5% -$0.3m FRA payment if r quarterly = 2% +$0.2m Total +$100m -$101.8m -$101.8m FRA settlement at the time of borrowing: Transaction June (t 1 = 120/360) September (t 2 = 211/360) r quarterly = 1.5% r quarterly = 2% Borrow $100m +$100m -$101.5m -$102.0m FRA payment if r quarterly = 1.5% $295, $0.3m FRA payment if r quarterly = 2% +$196, $0.2m Total +$100m -$101.8m -$101.8m 46 / 62

267 Eurodollar Futures Specifications for the Eurodollar futures contract Where traded: Chicago Mercantile Exchange Size: 3-month Eurodollar time deposit, $1 million principal Months: Mar, Jun, Sep, Dec, out 10 years, plus 2 serial months and spot month Trading ends: 5 A.M. (11 A.M. London) on the second London bank business day immediately preceding the third Wednesday of the current month Delivery: Cash settlement Settlement: British Banker s Association Futures Interest Settlement Rate for 3-Month Eurodollar Interbank Time Deposits. (This is a 3-month rate annualized by multiplying by 360/90.) 47 / 62

268 Eurodollar Futures (cont d) Figure 4 : Listing for the 3-month Eurodollar futures contract from the Wall Street Journal, October 17, / 62

269 Eurodollar Futures (cont d) Since the notional principal is $1 million, a change in annualized LIBOR of 1 basis point would change the borrowing cost by $1million = $25 The payoff at expiration of a long Eurodollar futures contract is [(100 r LIBOR ) Futures Price] 100 $25 Convert the change in the futures price to basis points Change in borrowing cost per basis point 49 / 62

270 Eurodollar Futures: Example Let s consider again the example in which we wish to guarantee a borrowing rate for a $100m loan from June to September Suppose the June Eurodollar futures price is Implied 3-month LIBOR is = 1.8% over 3 months As before, consider 2 possible 3-month borrowing rates in June: 1.5%, which correspond to a Eurodollar futures price in June of 94 2%, which correspond to a Eurodollar futures price in June of / 62

271 Eurodollar Futures: Example (cont d) We expect to borrow $100m for 91 days, beginning 120 days from today. Thus, we are short the interest rate. We wish to hedge this exposure we short 100 Eurodollar futures contracts The futures contract settles in June: [( ) 100 $25] 100 Transaction June (t 1 = 120/360) September (t 2 = 211/360) r quarterly = 1.5% r quarterly = 2% Borrow $100m +$100m -$101.5m -$102.0m Eurodollar payment if r quarterly = 1.5% $300,000 Eurodollar payment if r quarterly = 2% +$200,000 [( ) 100 $25] / 62

272 Eurodollar Futures: Example (cont d) This is like the payment on an FRA paid in arrears, except that the futures contract settles in June, but our interest expense is not paid until September. We need to tail the position in order to earn or pay interest on our Eurodollar gain or loss before we actually have to make the interest payment: Number of Eurodolar contracts = = Transaction June (t 1 = 120/360) September (t 2 = 211/360) r quarterly = 1.5% r quarterly = 2% Borrow $100m +$100m -$101.5m -$102.0m Eurodollar payment if r quarterly = 1.5% $294,695 -$299,116 Eurodollar payment if r quarterly = 2% +$196,464 +$200,393 Total +$100m -$ m -$ m 52 / 62

273 Convexity Bias The net borrowing cost appears to be a little less than 1.8% This is not due to rounding error. Actually, the settlement structure of the Eurodollar contract works systematically in favor of the borrower: When the interest rate turns out to be high (2%), the short Eurodollar contract has a positive payoff and the proceeds can be reinvested at the high realized rate When the interest rate turns out to be low (1.5%), the short Eurodollar contract has a negative payoff and we can found this loss until the loan payment date by borrowing at a low rate In order for the futures price to be fair to both the borrower and lender, the rate implicit in the Eurodollar futures price must be higher than a comparable FRA rate. This difference between the FRA rate and the Eurodollar rate is called convexity bias. 53 / 62

274 Suppose that on June 6, 2001, the gold spot price is $265.7 and the gold future price with maturity in December is $269. The June and September Eurodollar futures prices on this date are and What is the annualized 6-month gold lease rate? 54 / 62

275 Suppose that on June 6, 2001, the gold spot price is $265.7 and the gold future price with maturity in December is $269. The June and September Eurodollar futures prices on this date are and What is the annualized 6-month gold lease rate? The 3-month LIBOR from June to September is = 0.988%. The 3-month LIBOR from September to December is = 0.978% The June to December interest rate is therefore = %, or % annualized. Using equation (5), the annualized 6-month gold lease rate is 6-month lease rate = = 1.456% 1/0.5 (269/265.7) 54 / 62

277 Options on Futures The options we have considered so far, spot options, provide the holder with the right to buy or sell a certain asset by a certain date for a certain price For futures options, the exercise of the option gives the holder a position in the futures contract. Why trade option on futures rather than options on the underlying asset? Futures contracts are, in many circumstances, more liquid than the underlying asset A futures price is known immediately from trading on the futures exchange, whereas the spot price of the underlying may not be so readily available Futures on commodities are often easier to trade than the commodities themselves Futures options entail lower transactions costs than spot options in many situations 56 / 62

278 Options on Futures: Mechanics Call Futures Options (American Style) When a call futures option is exercised, the holder acquires 1. A long position in the futures 2. A cash amount equal to the excess of the most recent settlement futures price over the strike price Put Futures Options (American Style) When a put futures option is exercised, the holder acquires 1. A short position in the futures 2. A cash amount equal to the excess of the strike price over the most recent settlement futures price 57 / 62

279 Options on Futures: Mechanics (cont d) If we compare how the spot and futures price evolve on the tree, we observe a different solution for u and d in the case of future prices The solution is exactly what we would get for an option on a stock index if δ, the dividend yield, were equal to the risk-free rate: S t+h = S t e (r δ)h+σ h S t F t,t = S t e (r δ)nh F t+h,t = F t,t e σ h S t+h = S t e (r δ)h σ h F t+h,t = F t,t e σ h Thus, the nodes are constructed as u = e σ h d = e σ h 58 / 62

280 Options on Futures Contracts: Example An option has a gold futures contract as the underlying asset. Assume S = $300, K = $300, σ = 0.1, r = 0.05, T = 1 year, δ = 0.02, and h = 1/3. The following figure shows the tree for pricing an American call option on a gold futures contract. 59 / 62

281 Options on Futures Contracts: Example (cont d) S uuu 3 = F uuu 3 = $367.6 F u 1 = $327.5 F2 uu = $347.0 S2 uu = $343.5 F 0 = $309.1 S u 1 = $321.0 S uud 3 = F uud 3 = $327.5 S 0 = $300.0 F d 1 = $291.8 F2 ud = $309.1 S2 ud = $306.1 S d 1 = $286.0 S udd 3 = F udd 3 = $291.8 F2 dd = $272.7 S2 dd = $275.4 S ddd 3 = F ddd 3 = $ / 62

282 Options on Futures Contracts: Example (cont d) Binomial tree for pricing an American call option on a gold futures contract F 0 = $309.1 C 0 = $17.1 C 0,NO = $17.1 C 0,EX = $9.1 F u 1 = $327.5 C u 1 = $29.1 C u 1,NO = $29.1 C u 1,EX = $27.5 F d 1 = $291.8 C d 1 = $6.3 C d 1,NO = $6.3 C d 1,EX = $0 C,NO = value of call if not exercised C,EX = value of call if exercised F uu 2 = $347.0 C uu 2 = $47.0 C uu 2,NO = $46.2 C uu 2,EX = $47.0 F ud 2 = $309.1 C ud 2 = $13.1 C ud 2,NO = $13.1 C ud 2,EX = $9.1 F dd 2 = $275.4 C dd 2 = $0 C dd 2,NO = $0 C dd 2,EX = $0 F uuu 3 = $367.6 C uuu 3 = $67.6 F uud 3 = $327.5 C uud 3 = $27.5 F udd 3 = $291.8 C udd 3 = $0 F ddd 3 = $260.0 C ddd 3 = $0 61 / 62

283 Options on Futures Contracts: Example (cont d) Early exercise is optimal when the futures price is $347.0 (which corresponds to a $343.5 spot price for the gold). The intuition for early exercise is that when an option on a futures contract is exercised, the option holder pays nothing, is entered into a futures contract, and receives mark-to-market proceeds of the difference between the strike price and the futures price. The motive for exercise is the ability to earn interest on the mark-to-market proceeds. 62 / 62

284 Option Markets 232D 5. Other Topics Daniel Andrei Fall / 67

285 Outline I Swaps 3 Commodity Swaps 5 Interest Rate Swaps 11 Variance and Volatility Swaps 16 II Measurement and Behavior of Volatility 26 III Extending the Black-Scholes Model 35 Option Pricing When the Stock Price Can Jump 37 Stochastic Volatility 42 IV Revision 49 V About My Current Research 60 2 / 67

286 Outline I Swaps 3 Commodity Swaps 5 Interest Rate Swaps 11 Variance and Volatility Swaps 16 II Measurement and Behavior of Volatility 26 III Extending the Black-Scholes Model 35 Option Pricing When the Stock Price Can Jump 37 Stochastic Volatility 42 IV Revision 49 V About My Current Research 60 3 / 67

287 Introduction to Swaps A swap is a contract calling for an exchange of payments, on one or more dates, determined by the difference in two prices A swap provides a means to hedge a stream of risky payments A single-payment swap is the same thing as a cash-settled forward contract 4 / 67

288 An Example of a Commodity Swap An industrial producer, IP Inc., needs to buy 100,000 barrels of oil 1 year from today and 2 years from today The forward prices for delivery in 1 year and 2 years are $20 and $21/barrel The 1- and 2-year zero-coupon bond yields are 6% and 6.5% (annualized, effective) 5 / 67

289 An Example of a Commodity Swap (cont d) IP can guarantee the cost of buying oil for the next 2 years by entering into long forward contracts for 100,000 barrels in each of the next 2 years. The present value of this cost per barrel is $ $ = $ Thus, IP could pay an oil supplier any payment stream with a present value of $ Typically, a swap will call for equal payments in each year For example, the payment per year per barrel, x, will have to satisfy the following equation x x = $ We find x = $ and we say that the 2-year swap price is $ / 67

290 Computing the Commodity Swap Rate Suppose there are n swap settlements, occurring on dates t i, i = 1,...,n The price of a zero-coupon bond maturing on date t i is P(0,t i ) The fixed payment on a commodity swap is F = ni=1 P(0,t i )F 0,ti ni=1 P(0,t i ) (1) where F 0,ti is the forward price with maturity t i The commodity swap price is a weighted average of commodity forward prices, where zero-coupon bond prices are used to determine the weights 7 / 67

291 Computing the Commodity Swap Rate (cont d) A buyer with seasonally varying demand (e.g., someone buying gas for heating) might enter into a swap in which quantities vary over time The swap price with seasonally-varying quantities is F = ni=1 Q ti P(0,t i )F 0,ti ni=1 Q ti P(0,t i ) (2) where Q ti is the quantity of gas purchased at time t i when Q t = 1, the formula is the same as equation (1), when the quantity is not varying It is also possible for prices to be time-varying (e.g., a gas buyer who needs gas for heating can enter into a swap in which the summer price is fixed at a low value, and then winter price is then determined by the zero present value condition) 8 / 67

292 Physical Versus Financial Settlement $ Oil Buyer Oil Physical settlement Swap Counterparty Spot $ Swap Counterparty Oil Buyer Spot Oil Oil Seller Financial settlement 9 / 67

293 Swaps are nothing more than forward contracts coupled with borrowing and lending money Consider the swap price of $20.483/barrel. Relative to the forward curve price of $20 in 1 year and $21 in 2 years, we are overpaying by $0.483 in the first year, and we are underpaying by $0.517 in the second year Thus, by entering into the swap, we are lending the counterparty money for 1 year. The interest rate on this loan is = 7% Given 1- and 2- year zero-coupon bond yields of 6% and 6.5%, the 1-year implied forward yield from year 1 to year 2 is r 0 (1,2) = P(0,1) P(0,2) 1 = 7% 10 / 67

294 Interest Rate Swaps The notional principal of the swap is the amount on which the interest payments are based The life of the swap is the swap term or swap tenor If swap payments are made at the end of the period (when interest is due), the swap is said to be settled in arrears 11 / 67

295 An Example of an Interest Rate Swap XYZ Corp. has $200M of floating-rate debt at LIBOR, i.e., every year it pays that year s current LIBOR XYZ would prefer to have fixed-rate debt with 3 years to maturity XYZ could enter a swap, in which they receive a floating rate and pay the fixed rate, which is % 12 / 67

296 An Example of an Interest Rate Swap (cont d) Pay LIBOR Lender Borrower (XYZ) Pay % Receive LIBOR Swap Counterparty On net, XYZ pays % XYZ net payment = LIBOR + LIBOR % = % 13 / 67

297 Computing the Swap Rate Suppose there are n swap settlements, occurring on dates t i, i = 1,...,n The implied forward interest rate from date t i 1 to date t i, known at date 0, is r 0 (t i 1,t i ) The price of a zero-coupon bond maturing on date t i is P(0,t i ) The fixed swap rate, R, is R = ni=1 P(0,t i )r(t i 1,t i ) ni=1 P(0,t i ) (3) where n i=1 P(0,t i )r(t i 1,t i ) is the present value of interest payments implied by the strip of forward rates, and n i=1 P(0,t i ) is the present value of a $1 annuity when interest rates vary over time 14 / 67

298 Computing the Swap Rate (cont d) We can rewrite equation (3) to make it easier to interpret [ ] n P(0,t i ) R = nj=1 r(t i 1,t i ) (4) P(0,t i=1 j ) Thus, the fixed swap rate is a weighted average of the implied forward rates, where zero-coupon bond prices are used to determine the weights 15 / 67

299 Variance and Volatility Swaps: Why Trade Volatility? Just as stock investors think they know something about the direction of the stock market, or bond investors think they can foresee the probable direction of interest rates, so you may think you have insight into the level of future volatility What do you do if you simply want exposure to a stock s volatility? Stock options are impure: they provide exposure to both the direction of the stock price and its volatility The easy way to trade volatility is to use volatility swaps, sometimes called realized volatility forward contracts These products provide pure exposure to volatility (and only to volatility) 16 / 67

300 Volatility Swaps A stock volatility swap is a forward contract on annualized volatility. Its payoff at expiration is equal to where (σ R K vol ) N (5) σr is the realized stock volatility (quoted in annual terms) over the life of the contract Kvol is the annualized volatility delivery price, typically quoted as a volatility, for example 30% N is the notional amount of the swap in dollars per annualized volatility point, for example N = $250, 000/(volatility point) 17 / 67

301 Volatility Swaps (cont d) The holder of a volatility swap at expiration receives N dollars for every point by which the stock s realized volatility σ R has exceeded the volatility delivery price K vol He or she is swapping a fixed volatility K vol for the actual (floating) future volatility σ R The procedure for calculating the realized volatility should be clearly specified with respect to the following aspects: How frequently the return is measured Whether returns are continuously compounded or arithmetic Whether the variance is measured by subtracting the mean or by simply squaring the returns The period of time over which variance is measured How to handle days on which trading does not occur 18 / 67

302 Uses for Volatility Swaps Directional trading of volatility levels. Clients who want to speculate on the future levels of stock or index volatility can go long or short realized volatility with a swap Hedging implicit volatility exposure. Risk arbitrageurs, investors following active benchmarking stragegies, portfolio managers, equity funds are implicitly short volatility. They can hedge this exposure by going long realized volatility with a swap 19 / 67

303 Variance Swaps A variance swap is a forward contract on annualized variance, the square of the realized volatility. Its payoff at expiration is equal to (σ 2 R K var ) N (6) where σ 2 R is the realized stock variance (quoted in annual terms) over the life of the contract K var is the delivery price for variance, for example (30%) 2 N is the notional amount of the swap in dollars per annualized volatility point squared, for example N = $100,000/(volatility point) 2 20 / 67

304 Variance vs. Volatility Contracts 10 Variance Swap 5 Payoff 0 5 K vol 10 Volatility Swap σ R 21 / 67

305 Replicating Variance Swaps If you own a portfolio of options of all strikes, weighted in inverse proportion to the square of the strike level, you will obtain an exposure to variance that is independent of stock price, just what is needed to trade variance. There is no simple replication strategy for synthesizing a volatility swap. We can approximate a volatility swap by statically holding a suitably chosen variance contract. 22 / 67

306 The CBOE Volatility Index VIX In 1993, the Chicago Board Options Exchange (CBOE) introduced the CBOE Volatility Index, VIX, which was originally designed to measure the market s expectation of 30-day volatility implied by at-the-money S&P 100 Index option prices Ten years later in 2003, CBOE and Goldman Sachs updated the VIX to reflect a new way to measure expected volatility. The new VIX is based on the S&P 500 Index, and estimates expected volatility by averaging the weighted prices of puts and calls over a wide range of strike prices. 23 / 67

307 The VIX Calculation The CBOE utilizes a wide variety of strike prices for SPX puts and calls to calculate the VIX The generalized formula used in the VIX calculation is ˆσ 2 = 2 K i T K 2 e rt Put(K i ) + 2 K i K 0 i T K i K 2 K i >K 0 i e rt Call(K i ) 1 T [ ] 2 F0,T 1 K 0 where ˆσ is VIX/100 T is time to expiration (in order to arrive at a 30 day implied volatility value, the calculation blends options expiring on two different dates, with the result being an interpolated implied volatility number) F0,T is the forward index price Ki is the strike price of i th out-of-the-money option r is the risk-free interest rate to expiration 24 / 67

308 Why should we care about VIX? VIX provides important information about investor sentiment. Since volatility often signifies financial turmoil, the VIX is often referrred to as the investor fear gauge. Investors can use VIX options and VIX futures to hedge their portfolios. The VIX is a good hedging tool because it has a strong negative correlation to the S&P Jan 2004 Jan 2006 Jan 2008 Jan 2010 Jan / 67

309 Outline I Swaps 3 Commodity Swaps 5 Interest Rate Swaps 11 Variance and Volatility Swaps 16 II Measurement and Behavior of Volatility 26 III Extending the Black-Scholes Model 35 Option Pricing When the Stock Price Can Jump 37 Stochastic Volatility 42 IV Revision 49 V About My Current Research / 67

310 Volatility is Crucial in Finance for forecasting return for the pricing of derivatives for asset allocation (trade-off between return and risk) for risk management (evaluation of the risk of a portfolio) 27 / 67

311 The major problem with volatility is that it is not directly observable from returns Unconditional volatility is estimated as the sample standard deviation σ = 1 T (r t µ) 2 (7) T t=1 where r t is the log return on period t and µ is the sample mean over T periods However, volatility is actually not constant through time. Therefore, conditional volatility σ t is a more relevant measure of risk at time t. 28 / 67

312 Volatility is not constant through time (volatility clustering) 0.2 SP500 return / 67

313 Using squared daily returns to proxy volatility will produce a very noisy volatility estimator SP500 squared return / 67

314 An alternative measure of volatility is the absolute return 0.25 SP500 absolute return / 67

315 The GARCH Model The ARCH (AutoRegressive Conditional Heteroskedasticity) model has been introduced by Engle (1982) Basic idea: unexpected returns ε t are serially uncorrelated but dependant. The dependency in ε t is described by a quadratic function of its lagged values: r t = µ + ε t ε t = σ t z t σ 2 t = ω + α 1 ε 2 t α p ε 2 t p z t iid N(0,1) Due to the large persistence in volatility, the ARCH model often requires a large p to fit the data 32 / 67

316 The GARCH Model (cont d) In such cases, it is more parsimonious to use the GARCH (Generalized ARCH) model proposed by Bollerslev (1986). The GARCH(1,1) model is defined as r t = µ + ε t ε t = σ t z t σ 2 t = ω + αε 2 t 1 + βσ 2 t 1 z t iid N(0,1) 33 / 67

317 GARCH(1,1) volatility for the S&P 500 (weekly data) 0.8 Volatility estimated using a GARCH(1,1) model / 67

319 Extensions of the Black-Scholes Model Three extensions 1. The Merton jump diffusion model 2. The Constant Elasticity of Variance model (CEV) 3. The Heston model 36 / 67

320 Valuation Formula when the Underlying Can Jump Stock prices sometimes move more than would be expected from a lognormal distribution: If market volatility is 20% and the expected return is 15%, a one-day 5% drop in the market occurs about once every 2.5 million days A 20% drop (as in October 1987) is virtually impossible if prices are lognormally distributed Merton (1976) used the Poisson distribution to count the number of price jumps that occur over a period of time. The Poisson distribution is summarized by the parameter λ: λh is the probability that one event occurs over the short interval h. Thus, λ is like an annualized probability of the event occurring over a short interval. 37 / 67

321 Valuation Formula when the Underlying Can Jump (cont d) Let the jump magnitude be Y. That is, if S is the pre-jump price, Y S is the post-jump price Let us assume that Y is lognormally distributed (hence, this is called the Poisson-lognormal model): ( ) ln(y ) N α J,σJ 2 You can interpret ln(y ) as the continuously compounded stock return if there is a jump; ln(y ) can go from to + You can interpret Y 1 as the effective stock return if there is a jump; Y 1 can go from 1 to, or Y can go from 0 to Thus, the Merton formula for pricing with jumps requires 3 extra parameters: 1. The jump probability, λ 2. The expected size of a jump when one does occur, α J 3. The volatility of the jump magnitude, σ J 38 / 67

322 Valuation Formula when the Underlying Can Jump (cont d) The resulting process for the stock is { ds t 0, if there is no jump = (α λk)dt + σdz + S t Y 1, if there is a jump (8) Merton (1976) shows that with the stock following equation (8), and with jumps diversifiable, the price of an European call is e λ T (λ T ) i BSCall S,K, σ i! 2 iσ2 J T,r λk + iα J,T,δ (9) T i=0 39 / 67

323 Jump Risk and Implied Volatility Suppose that the stock can jump to zero (i.e., Y = 0) with λ = 0.5% probability per year In this case, the value of a European call becomes BSCall(S,K,σ,r + λ,t,δ) Suppose that other parameters are: S = $40, σ = 30%, r = 8%, T = 0.25, and δ = 0. The jump parameters are: λ = 0.005, α J = ln(0) =, and σ J = 0 We do the following experiment: generate correct option prices i.e., prices properly accounting for the jump for a variety of strikes We then ask what implied volatility we would compute for these options using the ordinary Black-Scholes formula 40 / 67

324 Jump Risk and Implied Volatility (cont d) 0.34 Call Jump = 10.67, Call Std = Implied Volatility Strike Price 41 / 67

325 Constant Elasticity of Variance (CEV) Model Cox (1975) proposed the constant elasticity of variance (CEV) model Volatility varies with the level of the stock price ds S = (α δ)dt + σs β 2 2 dz The instantaneous standard deviation of the stock return is σ(s) = σs β 2 2 If β < 2 volatility decreases with the stock price If β > 2 volatility increases with the stock price If β = 2 the CEV model reduces to the lognormal process 42 / 67

326 The CEV Call Price For the case β < 2, the CEV call price is ( Se [1 δt Q 2y,2 + 2 )] ( 2 β,2x Ke rt Q 2x, ) 2 2 β,2y (10) where k = 2(r δ) σ 2 (2 β)(e (r δ)(2 β)t 1) x = ks 2 β e (r δ)(2 β)t y = kk 2 β Q(a, b, c) dentoes the noncentral Chi-squared distribution function with b degrees of freedom and noncentrality parameter c, evaluated at a 43 / 67

327 The CEV Call Price (cont d) For the case β > 2, the CEV call price is ( Se [1 δt Q 2x, )] 2 β 2,2y ( Ke rt Q 2y,2 + 2 ) β 2,2x (11) When β < 2, the CEV model generates a Black-Scholes implied volatility skew Implied volatility decreases with the option strike price 44 / 67

328 Implied Volatility in the CEV Model Suppose the usual parameters: S = $40, σ = 30%, r = 8%, T = 0.25, and δ = 0. Additionally, β = 1 and σ = = Now we do the following experiment: generate correct option prices i.e., prices properly accounting for time-varying volatility for a variety of strikes We then ask what implied volatility we would compute for these options using the ordinary Black-Scholes formula 45 / 67

329 Implied Volatility in the CEV Model (cont d) 0.34 Call Jump = 10.67, Call Std = Implied Volatility Call CEV = 10.65, Call Std = Strike Price 46 / 67

330 The Heston Stochastic Volatility Model The model allows volatility to vary stochastically but still be correlated with the stock price ds S = (α δ)dt + v(t)dz 1 (12) dv(t) = k( v v(t))dt + σ v v(t)dz 2 (13) v(t) is the volatility (instantaneous standard deviation of the stock return) The brownians Z 1 and Z 2 can be correlated: E(dZ 1 dz 2 ) = ρdt Heston s stochastic volatility model has an integral solution that can be evaluated numerically 47 / 67

331 Implied Volatility in the Heston Model 48 / 67

333 Below is a profit diagram for a position. In the table below, record option quantities which construct the diagram Slope=1 Profit 0 50 Strike Shares Calls Slope= Stock Price in one year 50 / 67

334 Below is a profit diagram for a position. In the table below, record option quantities which construct the diagram Slope=1 Profit 0 50 Strike Shares Puts Slope= Stock Price in one year 51 / 67

335 Which of the following options will NOT be exercised early? (a) Put on a dividend paying stock (b) Call on a dividend paying stock (c) Put on a non-dividend paying stock (d) Call on a non-dividend paying stock 52 / 67

336 Why is an American call option rarely exercised early, and thus priced similar to European options? (a) Early exercise requires purchasing the stock (b) Most option sellers cannot deliver the stock (c) Option prices usually exceed intrinsic values (d) Short sellers disrupt delivery 53 / 67

337 Assume that an investor is currently holding a reverse straddle position (i.e. a short put and short call), which is currently a profitable investment. All else being equal, what would this investor like to happen to vega? (a) Decrease (b) Increase (c) Stay constant (d) Indifferent 54 / 67

338 Suppose the delta of an option is 5.15 (a) It is a call option (b) It is a put option (c) Could be either call or put 55 / 67

339 The price of an option is decreasing with the stock price (a) It is a call option (b) It is a put option (c) Could be either call or put 56 / 67

340 The price of an option is decreasing with the strike price (a) It is a call option (b) It is a put option (c) Could be either call or put 57 / 67

341 What actions are required to both delta-hedge and gamma-hedge a written option position? (a) No action is required (b) We must buy more bonds (c) We must buy more shares of the underlying (d) We must long a different option so as to offset the short call gamma. 58 / 67

342 During periods when measured volatility is high, the typical day tends to exhibit high volatility. This behavior is referred to as volatility (a) Clustering (b) Smile (c) Skew (d) Stochasticity 59 / 67

344 February 2010: Wall Street takes break for Tiger Woods apology Source: Bloomberg, By Michael Patterson and Eric Martin - February 19, :16 EST 61 / 67

345 We explain the strong positive relationship between investors attention and stock market volatility 0.4 S&P500 Volatility Focus on Economic News / 67

346 Bottom fishing: The quest for the bottoming-out point that suggests a return to more bullish days Source: Yu (2009) 63 / 67

347 Disagreement over the speed of the reversion of the economy drives volatility Stock s Diffusion σ ω σ F σ G σ δ σ Time 64 / 67

348 Volatility is clustered Returns Time / 67

349 May 1995: IBM secretary Lorraine Cassano was asked to photocopy some papers... May June / 67

350 May 1995: IBM secretary Lorraine Cassano was asked to photocopy some papers... May June / 67

351 May 1995: IBM secretary Lorraine Cassano was asked to photocopy some papers... May Takeover of Lotus by IBM (to be announced June 5) June / 67

352 May 1995: IBM secretary Lorraine Cassano was asked to photocopy some papers... Takeover of Lotus by IBM (to be announced June 5) May June The information percolated during 6 hours 66 / 67

353 May 1995: IBM secretary Lorraine Cassano was asked to photocopy some papers... Takeover of Lotus by IBM (to be announced June 5) May June The information percolated during 6 hours Investors: 25 Investment: $ Trading profits: $ / 67

354 May 1995: IBM secretary Lorraine Cassano was asked to photocopy some papers... Takeover of Lotus by IBM (to be announced June 5) May June The information percolated during 6 hours Investors: 25 Investment: $ Trading profits: $ Source: New York Daily News, p. 5, Thursday, May 27th U.S. SECURITIES AND EXCHANGE COMMISSION Litigation Release No , MAY 26, / 67

355 The data are never for the whole society given to a single mind (Hayek, 1945, p. 519) Prices aggregate knowledge Language aggregates knowledge 67 / 67

356 The data are never for the whole society given to a single mind (Hayek, 1945, p. 519) Prices aggregate knowledge Language aggregates knowledge Word-of-mouth communication generates momentum and reversal in asset returns Word-of-mouth communication generates persistent volatility 67 / 67