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