chapter 18 options the upside without the downside
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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
• 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
OPTION PAYOFFS
PAYOFF OF A CALL OPTION
PAYOFF OF ACALL OPTION
E (EXERCISE PRICE) STOCK PRICE
PAY OFF OF A PUT OPTION
PAYOFF OF APUT OPTION
E (EXERCISE PRICE) STOCK PRICE
PAYOFFS TO THE SELLER OF OPTIONSPAYOFF
E STOCK PRICE
(a) SELL A CALLPAYOFF
E STOCK PRICE
(b) SELL A PUT
OPTIONSBUYER/HOLDER SELLER/WRITER
RIGHTS/ 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
PUT CALL PARITY THEOREM - 1
Value of stock Buy stock Value of put Buy put position (S1) position (P1) E - Stock Stock E price (S1) E price (S1) Value of combination Buy a stock (S1) Value of borrow position (-E) E Buy a put (P1) E Combination (buy a call) C1= S1+ P1-E Stock price (S1) 0 E Stock price (S1) Borrow (-E) -E --------------------------------------- - E
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
OPTION STRATEGIES PROTECTIVE PUT
PROFITS STOCK PROTECTIVE PUT ST S0 = X
- P - S0
OPTION STRATEGIES COVERED CALL
A. STOCK
B. WRITTEN CALL
PAYOFF
C. COVERED CALLX
OPTION STRATEGIES STRADDLELONG STRADDLE : BUY A CALL AS WELL AS A PUT …SAME EXERCISE
PRICE
A : CALL B : PUT PAYOFF AND PROFIT PAYOFF AND PROFIT PAYOFF PROFIT PAYOFF ST ST PROFIT C : STRADDLE PAYOFF AND PROFIT PAYOFF PROFIT P+C X ST
OPTION STRATEGIES SPREADA 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 EXPIRATIONA : CALL HELD B : CALL WRITTEN PAYOFF PAYOFF ST ST PAYOFF AND PROFIT PAYOFF PROFIT X1 ST X2
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
OPTION VALUE : BOUNDS
UPPER AND LOWER BOUNDS FOR THE VALUE OF CALL OPTION
VALUE OF UPPER LOWER CALL OPTION BOUND (S0) BOUND ( S0 – E) STOCK PRICE
0 E
FACTORS DETERMINING THE OPTION VALUE
• EXERCISE PRICE
• EXPIRATION DATE
• STOCK PRICE
• STOCK PRICE VARIABILITY
• INTEREST RATE
C0 = f [S0 , E, 2, t , rf ] + - + + +
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 (u S - E, 0)
Cd = MAX (dS - E, 0)
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
ILLUSTRATION
S = 200, u = 1.4, d = 0.9E = 220, r = 0.10, R = 1.10
Cu = MAX (u S - 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
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
PIONEER STOCK
1. PROBABILITY OF RISE IN A RISK-NEUTRAL WORLD
RISE 40% TO 280FALL 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 OPTIONRS. 24
= RS. 21.82 1.10
BLACK - SCHOLES MODEL
E C0 = S0 N (d1) - N (d2)
ert
N (d) = VALUE OF THE CUMULATIVE NORMAL DENSITY FUNCTION
S0 1 ln 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
BLACK - SCHOLES MODEL ILLUSTRATION
S0 = RS.60 E = RS.56 = 0.30 t = 0.5 r = 0.14
STEP 1 : CALCULATE d1 AND d2
S0 2 ln 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) = 0.7768 N (d2) = N (0.5493) = 0.7086
STEP 3 : E 56 = = RS. 52.21 ert e0.14 x 0.5
STEP 4 : C0 = RS. 60 x 0.7768 - RS. 52.21 x 0.7086 = 46.61 - 37.00 = 9.61
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
ADJUSTMENT FOR DIVIDENDS
SHORT - TERM OPTIONS
Divt
ADJUSTED STOCK PRICE = S = (1 + r)t
E VALUE OF CALL = S N (d1) - N (d2)
ert
S 2 ln E + r + 2 t
d1 = t
ADJUSTMENT FOR DIVIDENDS - 2LONG - TERM OPTIONS
C = S e -yt N (d1) - E e -rt N (d2)
S 2 ln 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
PUT - CALL PARITY - REVISITED
JUST BEFORE EXPIRATION
C1 = S1 + P1 - E
IF THERE 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
INDEX OPTION ON S & P CNX NIFTY
CONTRACT SIZE 200 TIMES S & P CNX NIFTY
TYPE EUROPEAN
CYCLE ONE, TWO, AND THREE MONTHS
EXPIRY DAY LAST THURSDAY … EXPIRY MONTH
SETTLEMENT CASH - SETTLED
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
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
QUOTATIONS
CONTRACTS PREMIUM (QTY, VALUE, NO) EXPIRY (STRIKE PRICE) DATE
CALLRELIANCE (340) 5.50, 5.70 [26400, 9107, 44] 28.02.02
PUTRELIANCE (320) 14.15, 21.00 [5400, 18.25, 9] 28.02.02
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
• 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-call parity 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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