Chapter 17 Options

Chapter 17

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

Index Option On S & P CNX Nifty

Contract size 200 times s & p cnx nifty

Type European

Cycle upto five years

Expiry day last Thursday … expiry month

Settlement cash - settled

Contracts Premium (Rs.) Open Int No. of

(Type. Stk. Price – Expiry) Open High Low Close (‘000) Cont

CE-5250.00 – Oct 130.00 130.00 60.00 65.90 313 3419

CE-5300.00 – Oct 110.00 110.00 45.00 53.55 907 15357

CE-5350.00 – Oct 85.00 90.00 32.05 42.65 15 214

CE-5400.00 – Oct 70.00 74.00 25.25 29.55 425 7989

CE-5450.00 – Oct 40.00 50.00 18.00 22.80 174 2897

CE-5150.00 – Nov 189.00 189.00 189.00 189.00 1 1

CE-5200.00 – Nov 218.20 218.20 169.00 187.60 208 306

CE-5300.00 – Nov 176.20 176.20 132.05 142.95 194 584

CE-5350.00 – Nov 121.00 139.95 121.00 139.95 0 3

CE-5400.00 – Nov 130.00 130.00 89.00 89.65 13 93

CE-5000.00 – Dec 400.00 400.00 300.00 310.15 0 3

CE-5200.00 – Dec 201.00 296.00 201.00 296.00 1 5

PE-4700.00 – Oct 33.00 54.00 33.00 46.85 1772 9551

PE-4750.00 – Oct 55.50 63.00 47.00 51.05 57 135

PE-4800.00 – Oct 40.05 74.95 40.05 62.00 1902 12647

PE-4850.00 – Oct 74.00 90.00 66.00 77.25 158 160

PE-4900.00 – Oct 65.00 109.90 63.00 92.40 1872 19661

PE-4300.00 – Nov 40.80 40.80 38.00 38.15 43 100

PE-4400.00 – Nov 59.95 72.00 59.95 69.00 17 46

PE-4500.00 – Nov 73.95 93.00 73.95 81.20 404 606

PE-4600.00 – Nov 85.00 117.90 85.00 100.00 121 241

PE-4700.00 – Nov 119.90 128.95 115.00 121.60 108 61

PE-4800.00 – Nov 98.00 162.00 98.00 150.00 106 309

PE-4900.00 – Dec 224.00 225.00 221.00 222.25 1 9

PE-5000.00 – Dec 289.00 299.00 245.00 245.00 0 4

Quotations of Nifty Options

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 NSCCL on the expiration day of the contract

Settlement Cash-settled

Quotations of Nifty Options(Bharti Airtel)

Source: The Economic Times February 4,2012

Contracts Premium (Rs.) Open Int No. of

(type, Stk Price-expiry)

Open High Low Close (‘000) Contr

CE-360-Feb 42 45.5 35 35.95 140 42

PE-360-Feb 5.30 5.80 4.00 5.45 412 521

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 long Obligation to sell/go short on exercise

Put Right to sell/ to go short Obligation to buy/go long on

exercise

Premium Paid Received

Exercise Buyer’s decision Seller cannot influence

Max. Loss Cost of premium Unlimited lossespossible

Max. Gain Unlimited profits Price of premiumpossible

Closing • Exercise • Assignment of 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

Exotic Options

Asian Options: Asian options are options whose payoffs depend on the average price of the underlying asset during some portion of the life of the option.

Barrier Options: The payoff of a barrier option depends not only on what the price of the underlying asset is at the time of option expiration but also on whether the price of the underlying asset has crossed some barrier.

Binary Options: A binary option provides a fixed payoff, depending on the fulfillment of some condition.

Lookback Options: The payoff of a lookback option depends on the maximum or minimum price of the underlying asset during the life of the option.

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 expiration

A : 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

Strategies with Stock Index OptionsYou can use stock index options the way you use individual stock options:

• If you expect the market to rise, buy calls on the stock index. If the market does rise, the gain from your calls can be substantial, far greater than the premium paid for the calls.

• If you expect the market to fall, buy puts on the stock index. If the market does fall, the gain from your puts can be considerable, far greater than the premium paid for the puts.

• If you own a diversified portfolio of stocks, you can hedge your position by buying puts on the stock index. If the market falls, the erosion in your portfolio value will be offset by the gains on the stock index puts. In effect, you are buying a form of market insurance. Of course, if your portfolio is not perfectly correlated with the market index, you may not achieve complete protection against market decline. As long as your portfolio is adequately diversified with the market index, you will achieve substantial protection against market decline.

Option Value: BoundsUpper 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 ]

+ - + + +

Variability and Call Option Value

So fundamental is this point that it calls for another illustration. Consider the probability distribution of the price of two stocks, P and Q, just before the call option (with an exercise price of 80) on them expires.

P QPrice Probability Price Probability60 0.5 50 0.580 0.5 90 0.5 While the expected price of stock Q is same as that of stock P, the variance of Q is higher than that of P. The call option (exercise price: 80) on stock P is worthless as there is no likelihood that the price of stock P will exceed 80. However, the call option on stock Q is valuable because there is a distinct possibility that the stock price will exceed the exercise price.

Variability and Call Option Value

Remember that there is a basic difference between holding a stock

and holding a call option on the stock. If you are a risk-averse

investor you try to avoid buying a high variance stock, as it exposes

you to the possibility of negative returns. However, you will like to

buy a call option on that stock because you receive the profit from

the right tail of the probability distribution, while avoiding the loss

on the left tail. Thus, regardless of your risk disposition, you will

find a high variance in the underlying stock desirable.

Basic Idea The standard DCF (discounted cash flow) procedure involves two steps, viz. estimation of expected future cash flows and discounting of these cash flows using an appropriate cost of capital. There are problems in applying this procedure to option valuation. While it is difficult (though feasible) to estimate expected cash flows, it is impossible to determine the opportunity cost of capital because the risk of an option is virtually indeterminate as it changes every time the stock price varies.

Since options cannot be valued by the standard DCF method, financial economists struggled to develop a rigorous method for valuing options for many years. Finally, a real breakthrough occurred when Fisher Black and Myron Scholes published their famous model in 1973. The basic idea underlying their model is to set up a portfolio which imitates the call option in its payoff. The cost of such a portfolio, which is readily observed, must represent the value of the call option.

The key insight underlying the Black and Scholes model may be illustrated through a single-period binomial (or two-state) model.

Binomial Model

Option 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 ModelOption Equivalent Method - 1

PortfolioPortfolio 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

IllustrationS = 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 equilibrium price of the call option without

knowing anything about the attitude of investors toward risk. This

suggests … alternative method … risk-neutral valuation method

1. Calculate 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 Stock1. 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 ModelIllustration

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

Step 2: Finding N(d1) and N (d2)The simplest way to find N(d1) and N(d2) is to use the Excel function NORMSDIST.

N(d1) = N (0.7614) = 0.7768N(d2) = N (0.5493) = 0.7086

If you don’t have easy access to the excel function NORMSDIST, you can get a very close approximation by using the Normal Distribution given in Table A.5 in Appendix A at the end of the book. The procedure for doing that may be illustrated with respect to N (0.7614) as follows

1. 0.7614 lies between 0.75 and 0.80.2. According to the table N (0.75) = 1 – 0.2264

= 0.7736 and N (0.80) = 1 – 0.2119 = 0.78813. For a difference of 0.05 (0.80 – 0.75) the cumulative probability increases by 0.0145 (0.7881 – 0.7736)4. The difference between 0.7614 and 0.75 is 0.01145 So, N (0.7614) = N (0.75) + 0.0114 x 0.0145

0.05 = 0.7736 + 0.0033 = 0.7769

 This value is indeed a close approximation for the true value 0.7768.

Sources of Discrepancy Suppose the observed call price in the above example were Rs. 12 rather than Rs. 9.61 does it mean that the market has mispriced the option? Before jumping to such a conclusion, let us look at two possible sources of discrepancy.

First, the Black-Scholes model, like all models, is based on certain simplifying assumptions which may not be satisfied in the real world. That makes the formula only approximately valid.

Second, the parameters used in the formula may not be accurately measured. Recall that there are five parameters in the model: SO, E, r, σ, and t. Out of these, SO, E, r, and t – the stock price, the exercise price, the risk-free interest rate, and the time to expiration – are given accurately as they are directly observable. σ (the standard deviation), however, is not directly observable and must be estimated by relying on historical data, or scenario analysis, or some other method. An error in estimating the standard deviation can result in a discrepancy between the price of an option and its Black Scholes value.

Assumptions

• The call option is the European option

• The stock price is continuous and is distributed log

normally

• 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

Implied Volatility

Indeed, market participants often look at an option valuation from a different angle. Instead of calculating a Black-Scholes option value using a given standard deviation, they ask: “What standard deviation justifies the observed option price, according to the Black-Scholes formula? This is referred to as the implied volatility of the option as this is the volatility level implied by the prevailing option price. If investors believe that the actual standard deviation is more (less) than the implied volatility, than the option’s fair price is deemed to be greater (lesser) than its observed price.

Another variant of this theme is to compare two options on the same stock with the same expiration date but different exercise prices. It makes sense to buy the option with the lower implied volatility and write the option with the higher implied volatility.

Volatility Index

The recently announced volatility index or VIX on the NSE – an

indication of investor perception of future volatility – was inspired

by CBOE’s VIX. It is a measure of the amount by which an

underlying index is expected to fluctuate in the near future

(calculated as annualised volatility, denoted in percentage for

instance 20%) based on the order book of the underlying index

options. India VIX is based on option prices of components of the

Nifty 50 Index.

  Replicating Portfolio for Calls and Puts

Replicating Portfolio

Option Position Binomial Black –Scholes Model Model

Buy Call Option Borrow B Borrow Ee–rt N (d2)

Buy shares of stock Buy N (d1) shares of stockSell Call Option Lend B Lend Ee–rt N (d2)

Sell short shares Sell short N (d1) sharesBuy Put Option Lend B’ Lend Ee–rt (1– N (d2))

Sell short shares Sell short (1– N (d1)) sharesSell Put Option Borrow B’ Borrow Ee–rt(1– N (d2))

Buy shares Buy (1– N (d1)) shares 

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 – 2 Long-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

payments

• 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

d1 = ln

60

+0.08

+

0.16

0.2550 2

0.40 0.25

=0.1823 + 0.04

= 1.11150.20

d2 = d1 - σ t

= 1.1115 – 0.20 = 0.9115

E=

Rs 50=

Rs 50= Rs 49.01

ert e 0.08 x 0.25 1.0202

TYPE EXERCISE STYLE GREEKS

Call Put American EuropeanTYPE: CALL

STYLE: AMERICAN

DELTA :

STRIKE PRICE SPOT PRICE GAMMA :

INTEREST RATE DIVIDEND YIELD THETA :

NO OF DAYS VEGA :

• VOLATILITY Premium

RHO :

NOTE : All calculations for European style are done using

BLACK-SCHOLES FORMULA

All calculations for American style are done using

Binomial method (255 level)

ResetCalculate

Option Calculators

Option Greeks

Option Greeks (represented as they are by Greek alphabets) reflect the sensitivity of option price to changes in the value of the underlying factors. The following are the commonly used option Greeks:

Delta: Delta represents the change in the value of an option for a given change in the value of the underlying share.

Gamma: Gamma represents the change in delta for a given movement in the share price.

Vega: Vega stands for a change in option value with respect to the change in the volatility of the underlying price.

Theta: Theta stands for the change in option value with respect to time to expiration.

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 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 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

ert

N (d2) C0 = S0 N (d1) -