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Chapter 18 OPTIONS The Upside Without the Downside

Chapter 18 OPTIONS The Upside Without the Downside. OUTLINE Terminology Options and Their Payoffs Just Before Expiration Option Strategies Factors Determining Option Values Binomial Model for Option Valuation Black-Scholes Model Equity Options in India. TERMINOLOGY

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Chapter 18 OPTIONS The Upside Without the Downside

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  1. Chapter 18 OPTIONS The Upside Without the Downside

  2. OUTLINE • Terminology • Options and Their Payoffs Just Before Expiration • Option Strategies • Factors Determining Option Values • Binomial Model for Option Valuation • Black-Scholes Model • Equity Options in India

  3. TERMINOLOGY • CALL AND PUT OPTIONS • OPTION HOLDER AND OPTION WRITER • EXERCISE PRICE OR STRIKING PRICE • EXPIRATION DATE OR MATURITY DATE • EUROPEAN OPTION AND AMERICAN OPTION • EXCHANGE-TRADED OPTIONS AND OTC OPTIONS • AT THE MONEY, IN THE MONEY, AND OUT OF THE MONEY OPTIONS • INTRINSIC VALUE OF AN OPTION • TIME VALUE OF AN OPTION

  4. OPTION PAYOFFSPAYOFF OF A CALL OPTION PAYOFF OF A CALL OPTION E (EXERCISE PRICE) STOCK PRICEPAY OFF OF A PUT OPTION PAYOFF OF A PUT OPTION E (EXERCISE PRICE) STOCK PRICE

  5. PAYOFFS TO THE SELLER OF OPTIONS PAYOFF E STOCK PRICE (a) SELL A CALL PAYOFF E STOCK PRICE (b) SELL A PUT

  6. OPTIONSBUYER/HOLDER SELLER/WRITERRIGHTS/ BUYERS HAVE RIGHTS- SELLERS HAVE ONLYOBLIGATIONS NO OBLIGATIONS OBLIGATIONS-NO RIGHTS CALL RIGHT TO BUY/TO GO OBLIGATION TO SELL/GO LONG SHORT ON EXERCISE PUT RIGHT TO SELL/ TO OBLIGATION TO BUY/GO GO SHORT LONG ON EXERCISE PREMIUM PAID RECEIVED EXERCISE BUYER’S DECISION SELLER CANNOT INFLUENCE MAX. LOSS COST OF PREMUIM UNLIMITED LOSSESPOSSIBLE MAX. GAIN UNLIMITED PROFITS PRICE OF PREMIUMPOSSIBLE CLOSING • EXERCISE • ASSIGNMENT ON OPTIONPOSITION OF • OFFSET BY SELLING • OFFSET BY BUYING BACKEXCHANGE OPTION IN MARKET OPTION IN MARKETTRADED • LET OPTION LAPSE • OPTION EXPIRES AND KEEP WORTHLESS THE FULL PREMIUM

  7. PUT CALL PARITY THEOREM - 1

  8. PUT CALL PARITY THEOREM - 2 IF C1 IS THE TERMINAL VALUE OF THE CALL OPTION C1 = MAX [(S1 - E), 0] P1 = MAX [(E - S1 ), 0] S1 = TERMINAL VALUE E = AMOUNT BORROWED C1 = S1 + P1 - E

  9. OPTION STRATEGIES PROTECTIVE PUT

  10. OPTION STRATEGIES COVERED CALL A. STOCK B. WRITTEN CALL PAYOFFC. COVERED CALL X

  11. OPTION STRATEGIES STRADDLE LONG STRADDLE : BUY A CALL AS WELL AS A PUT …SAME EXERCISE PRICE

  12. OPTION STRATEGIES SPREAD A SPREAD INVOLVES COMBINING TWO OR MORE CALLS (OR PUTS) ON THE SAME STOCK WITH DIFFERING EXERCISE PRICES OR TIMES TO MATURITY PAYOFF AND PROFIT OF A VERTICAL SPREAD AT EXPIRATION

  13. COLLAR A collar is an options strategy that limits the value of a portfolio within two bounds An investor who holds an equity stock buys a put and sells a call on that stock. This strategy limits the value of his portfolio between two pre-determined bounds, irrespective of how the price of the underlying stock moves

  14. OPTION VALUE : BOUNDS UPPER AND LOWER BOUNDS FOR THE VALUE OF CALL OPTION

  15. FACTORS DETERMINING THE OPTION VALUE • EXERCISE PRICE • EXPIRATION DATE • STOCK PRICE • STOCK PRICE VARIABILITY • INTEREST RATE C0 = f [S0 , E, 2,t , rf ] + - + + +

  16. BINOMIAL MODELOPTION EQUIVALENT METHOD - 1 • A SINGLE PERIOD BINOMIAL (OR 2 - STATE) MODEL • S CAN TAKE TWO POSSIBLE VALUES NEXT YEAR, uS OR dS (uS > dS) • B CAN BE BORROWED .. OR LENT AT A RATE OF r, THE RISK-FREE RATE .. (1 + r) = R • d < R > u • E IS THE EXERCISE PRICE • Cu = MAX (uS - E, 0) • Cd = MAX (dS - E, 0)

  17. BINOMIAL MODEL : OPTION EQUIVALENTMETHOD - 2 PORTFOLIO SHARES OF THE STOCK AND B RUPEES OF BORROWING STOCK PRICE RISES :  uS - RB = Cu STOCK PRICE FALLS : dS - RB = Cd Cu - Cd SPREAD OF POSSIBLE OPTION PRICE = =S (u - d) SPREAD OF POSSIBLE SHARE PRICES dCu - uCd B = (u - d) R SINCE THE PORTFOLIO (CONSISTING OF  SHARES AND B DEBT) HAS THE SAME PAYOFF AS THAT OF A CALL OPTION, THE VALUE OF THE CALL OPTION IS C =  S - B

  18. ILLUSTRATION S = 200, u = 1.4, d = 0.9E = 220, r = 0.10, R = 1.10 Cu = MAX (uS - E, 0) = MAX (280 - 220, 0) = 60 Cd = MAX (dS - E, 0) = MAX (180 - 220, 0) = 0 Cu - Cd 60 = = = 0.6 (u - d) S 0.5 (200) dCu - uCd 0.9 (60) B = = = 98.18 (u - d) R 0.5 (1.10) 0.6 OF A SHARE + 98.18 BORROWING … 98.18 (1.10) = 108 REPAYT PORTFOLIO CALL OPTION WHEN u OCCURS 1.4 x 200 x 0.6 - 108 = 60 Cu = 60 WHEN d OCCURS 0.9 x 200 x 0.6 - 108 = 0 Cd = 0 C =  S - B = 0.6 x 200 - 98.18 = 21.82

  19. BINOMIAL MODEL RISK-NEUTRAL METHOD WE ESTABLISHED THE EQUILIBRUIM PRICE OF THE CALL OPTION WITHOUT KNOWING ANYTHING ABOUT THE ATTITUDE OF INVESTORS TOWARD RISK. THIS SUGGESTS … ALTERNATIVE METHOD … RISK-NEUTRAL VALUATION METHOD 1. CALCUL ATE THE PROBABILITY OF RISE IN A RISK NEUTRAL WORLD 2. CALCULATE THE EXPECTED FUTURE VALUE .. OPTION 3. CONVERT .. IT INTO ITS PRESENT VALUE USING THE RISK-FREE RATE

  20. PIONEER STOCK 1. PROBABILITY OF RISE IN A RISK-NEUTRAL WORLD RISE 40% TO 280 FALL 10% TO 180 EXPECTED RETURN = [PROB OF RISE x 40%] + [(1 - PROB OF RISE) x - 10%] = 10% p = 0.4 2. EXPECTED FUTURE VALUE OF THE OPTION STOCK PRICE Cu = RS. 60 STOCK PRICE Cd = RS. 0 0.4 x RS. 60 + 0.6 x RS. 0 = RS. 24 3. PRESENT VALUE OF THE OPTION RS. 24 = RS. 21.82 1.10

  21. BLACK - SCHOLES MODEL E C0 = S0 N (d1) - N (d2) ert N (d) = VALUE OF THE CUMULATIVE NORMAL DENSITY FUNCTION S0 1ln E+ r + 22 t d1 = t d2 = d1 -  t r = CONTINUOUSLY COMPOUNDED RISK - FREE ANNUAL INTEREST RATE  = STANDARD DEVIATION OF THE CONTINUOUSLY COMPOUNDED ANNUAL RATE OF RETURN ON THE STOCK

  22. BLACK - SCHOLES MODELILLUSTRATION S0 = RS.60 E = RS.56  = 0.30t = 0.5 r = 0.14 STEP 1 : CALCULATE d1 AND d2 S0 2ln E+ r + 2 t d1 = t .068 993 + 0.0925 = = 0.7614 0.2121 d2 = d1 -  t = 0.7614 - 0.2121 = 0.5493 STEP 2 : N (d1) = N (0.7614) = 0.7768N (d2) = N (0.5493) = 0.7086 STEP 3 : E 56 = = RS. 52.21erte0.14 x 0.5 STEP 4 : C0 = RS. 60 x 0.7768 - RS. 52.21 x 0.7086 = 46.61 - 37.00 = 9.61

  23. ASSUMPTIONS • THE CALL OPTION IS THE EUROPEAN OPTION • THE STOCK PRICE IS CONTINUOUS AND IS DISTRIBUTED LOGNORMALLY • THERE ARE NO TRANSACTION COSTS AND TAXES • THERE ARE NO RESTRICTIONS ON OR PENALTIES FOR SHORT SELLING • THE STOCK PAYS NO DIVIDEND • THE RISK-FREE INTEREST RATE IS KNOWN AND CONSTANT

  24. ADJUSTMENT FOR DIVIDENDS SHORT - TERM OPTIONS DivtADJUSTED STOCK PRICE = S =  (1 + r)t E VALUE OF CALL = SN (d1) - N (d2) ert S2ln E+ r + 2 t d1 = t

  25. ADJUSTMENT FOR DIVIDENDS - 2LONG - TERM OPTIONSC = S e -ytN (d1) - E e -rt N (d2) • S2ln E+ r - y + 2 t d1 = t • d2 = d1 -  t • THE ADJUSTMENT • DISCOUNTS THE VALUE OF THE STOCK TO THE PRESENT AT THE DIVIDEND YIELD TO REFLECT THE EXPECTED DROP IN VALUE ON ACCOUNT OF THE DIVIDEND PAYMENS • OFFSETS THE INTEREST RATE BY THE DIVIDEND YIELD TO REFLECT THE LOWER COST OF CARRYING THE STOCK

  26. PUT - CALL PARITY - REVISITED JUST BEFORE EXPIRATION C1 = S1 + P1 - E IFTHERE IS SOME TIME LEFT C0 = S0 + P0 - E e -rt THE ABOVE EQUATION CAN BE USED TO ESTABLISH THE PRICE OF A PUT OPTION & DETERMINE WHETHER THE PUT - CALL PARITY IS WORKING

  27. INDEX OPTION ON S & P CNX NIFTYCONTRACT SIZE 200 TIMES S & P CNX NIFTYTYPE EUROPEANCYCLE ONE, TWO, AND THREE MONTHSEXPIRY DAY LAST THURSDAY … EXPIRY MONTHSETTLEMENT CASH - SETTLED

  28. QUOTATION FEB. 12, 2002 CONTRACT (STRIKE PREMIUM [TRADED, OPEN EXPIRYPRICE) VALUE, NO, QTY, INT DATE RS. IN LAKH]NIFTY (1020) 114 [2000, 22.71, 10] 6400 28 - 02 - 02

  29. OPTIONS ON INDIVIDUAL SECURITIES CONTRACT SIZE … NOT LESS THAN RS.200,000 AT THE TIME OF INTRODUCTION TYPE AMERICAN TRADING CYCLE MAXIMUM THREE MONTHS EXPIRY LAST THURSDAY OF THE EXPIRY MONTH STRIKE PRICE THE EXCHANGE SHALL PROVIDE A MINIMUM OF FIVE STRIKE PRICES FOR EVERY OPTION TYPE (CALL & PUT) …2 (ITM), 2 (OTM), 1 (ATM) BASE PRICE BASE PRICE ON INTRODUCTION … THEORETICAL VALUE … AS PER B-S MODEL EXERCISE ALL ITM OPTIONS WOULD BE AUTOMATICALLY EXERCISED BY NSCCCL ON THE EXPIRATION DAY OF THE CONTRACT SETTLEMENT CASH-SETTLED

  30. QUOTATIONS CONTRACTS PREMIUM (QTY, VALUE, NO) EXPIRY (STRIKE PRICE) DATECALLRELIANCE (340) 5.50, 5.70 [26400, 9107, 44] 28.02.02PUTRELIANCE (320) 14.15, 21.00 [5400, 18.25, 9] 28.02.02

  31. SUMMING UP • An option gives its owner the right to buy or sell an asset on or before a • given date at a specified price. An option that gives the right to buy is called • a call option; an option that gives the right to sell is called a put option. • A European option can be exercised only on the expiration date whereas an • American option can be exercised on or before the expiration date. • The payoff of a call option on an equity stock just before expiration is equal • to: •   Stock Exercise • price - price, 0 • The payoff of a put option on an equity stock just before expiration is equal • to: • Exercise Stock • price - price, 0 Max Max

  32. Puts and calls represent basic options. They serve as building blocks for • developing more complex options. For example, if you buy a stock along • with a put option on it (exercisable at price E), your payoff will be E if the • price of the stock (S1) is less than E;otherwise your payoff will be S1. • A complex combination consisting of (i) buying a stock, (ii) buying a put • option on that stock, and (iii) borrowing an amount equal to the exercise • price, has a payoff that is identical to the payoff from buying a call option. • This equivalence is referred to as the put-callparity theorem. • The value of a call option is a function of five variables: (i) price of the • underlying asset, (ii) exercise price, (iii) variability of return, (iv) time left to • expiration, and (v) risk-free interest rate. • The value of a call option as per the binomial model is equal to the value of • the hedge portfolio (consisting of equity and borrowing) that has a payoff • identical to that of the call option. • The value of a call option as per the Black - Scholes model is: • E • C0 = S0 N (d1) - N (d2) • ert

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