Options : A Primer

By A.V. Vedpuriswar

An option contract gives its owner the right, but not the legal obligation, to conduct a transaction involving an underlying asset at a predetermined future date (the exercise date) and at a predetermined price (the exercise or strike price).

An option gives the option buyer the right to decide whether or not the trade will eventually take place.

The seller of the option has the obligation to perform if the buyer exercises the option.

To acquire these rights, owner of the option must pay a price called the option premium to the seller of the option.

Introduction

Types of options

American options may be exercised at any time up to an including the contract's expiration date.

European options can be exercised only on the contract’s expiration date.

If two options are identical (maturity, underlying stock, strike price, etc.), the value of the American option will equal or exceed the value of the European option.

The owner of a call option has the right to purchase the underlying asset at a specific price for a specified time period.

The owner of a put option has the right to sell the underlying asset at a specific price for a specified time period.

If immediate exercise of the option would generate a positive payoff, it is in the money.

If immediate exercise would result in a loss (negative payoff), it is out of the money.

When the current asset price equals the exercise price, exercise will generate neither a gain nor loss, and the option is at the money.

In the money, Out of the money

In the money call options

If S – X > 0, a call option is in the money. S – X is the amount of the payoff a call holder would

receive from immediate exercise, buying a share for X and selling it in the market for a great price S.

If S – X < 0, a call option is out of the money. If S = X, a call option is said to be at the money.

If X – S > 0, a put option is in the money.

X – S is the amount of the payoff from immediate exercise, buying a share for S and exercising the put to receive X for the share.

If X – S < 0, a put option is out of the money.

If S = X, a put option is said to be at the money.

In the money put options

An option’s intrinsic value is the amount by which the option is in-the-money.

It is the amount that the option owner would receive if the option were exercised.

An option has zero intrinsic value if it is at the money or out of the money, regardless of whether it is a call or a put option.

The intrinsic value of a call option is the greater of (S-X) or 0. That is: C = Max[0, S – X]

Similarly, the intrinsic value of a put is (X – S) or 0, whichever is greater. That is

P = Max[0, X – S]

Intrinsic value

Problem

A call option has an exercise price of 40 and the underlying stock is trading at 37. What is the intrinsic value?

Solution

If we exercise the option, loss = 3

The stock is 3 out of the money. So it does not have any intrinsic value.

Problem

A put option has an exercise price of 40 and the underlying stock is trading at 37. What is the intrinsic value?

Solution

I can buy from the market at 37 and sell to the option writer for 40.

So the intrinsic value is 3.

Problem

I own a call option on the S&P 500 with an exercise price of 900. During expiration, the index was trading at 912. If the multiplier is 250, what is the profit I make?Solution

Notionally I can buy at 900 and sell at 912.

Profit = (912 – 900) (250) = $ 3000

Problem

Calculate the lowest possible price for an American put option with a strike price of 65, if the stock is trading at 63 and the risk free rate is 5%. The expiration of the option is after 4 months.

Solution

The minimum price = 65 – 63 = 2.

Otherwise risk free profits can be made by arbitraging.

Problem

Repeat the earlier problem if it is a European Put.

Solution

Present value of strike price = 65/(1+.05)0.33 .

= 63.96

So pay off = 63.96 – 63 = .96

Problem

A $35 call on a stock trading at $38 is priced at $5. What is the time value?

Solution

Intrinsic value = 38 – 35 = 3

Total value = 5

Time value = 5 – 3 = 2

Problem

A call option with exercise price 40, has a premium of 3. What is the pay off if the stock price = 38, 40, 42, 44?Solution

Stock Price Pay off

38 -3

40 -3

42 -3 + (42 – 40) = - 1

44 -3 + (44 – 40) = 1

Problem

A put option with exercise price 40 has a premium of 3. What is the pay off if the stock price = 38, 40, 42, 44?

Solution

Stock Price Pay off

38 -3 + (40 – 38) = - 1

40 -3

42 -3

44 -3

A put option with exercise price 40 has a premium of 3. What is the pay off if the stock price = 38, 40, 42, 44?

38

40

42

44-1

-3 -3 -3

35

36

37

38

39

40

41

42

43

44

45

1 2 3 4

-5.5

-3.5

-1.5

0.5

2.5

4.5

Stock price Pay Off

-3+ (40-38) = -1

Solution:

Problem

Suppose you have bought a $40 call and a $40 put each with premium of 3. What is the pay off is the stock price = 36, 38, 40, 42, 44?Solution

Stock Price Pay off

36 -3 + (40 – 36) - 3 = - 2

38 -3 + (40 – 38) – 3 = - 4

40 -3 – 3 = - 6

42 -3 – 3 + (42 - 40) = - 4

44 -3 – 3 + (44 – 40) = -2

Suppose you have bought a $40 call and a $40 put each with premium of 3. What is the pay off if stock price = 36, 38, 40, 42, 44?

Solution:

36 38 40 42 44-2

-4

-6

-4

-2

0

5

10

15

20

25

30

35

40

45

50

1 2 3 4 5

-7.5

-5.5

-3.5

-1.5

0.5

2.5

4.5

Stock price Pay Off

-3+ (40-38)-3 = - 2

-3+ (40-38)-3 = - 4

- 3 - 3 = - 6

-3-3+ (33-40) = -2

-3-3+ (42-40) = - 4

Problem

A trader adopts a combination of the following strategies:

a) Purchase of call option

Strike price = $1.40/Euro

Premium = $0.32

b) Sale of call option

Strike price = $1.60/Euro

Premium = $0.28

Determine the pay off.

Solution

a) Spot price < 1.40; Options will not be exercised.

Pay off = - .32 + .28 = - .04

b) 1.40 < spot price < 1.60; $1.40 call option will be exercised

Pay off = - .04 + S – 1.40 = S – 1.44

C) Spot price > 1.60; Both options will be exercised

Pay off = - .04 + S – 1.40 – (S-1.60)

= - .04 + S – 1.40 – S + 1.60

= .16

Problem

A trader buys the following options simultaneously construct the pay off table.

Put option: Strike price = 1.71 premium = 0.10

Call option: Strike price = 1.75 premium = 0.05

Solution

Spot price ≤ 1.71, only put option is exercised

Pay off = - 0.10 – 0.05 + 1.71 – S

= 1.56 – S

1.71 ≤ spot price ≤ 1.75 no option is exercised pay off

= - 0 .15

Spot price > 1.75 , only call option is exercised pay off

= - 0.15 + S – 1.75

= S – 1.90

Problem

A stock trades at 108 and there are two European options currently available.

Strike Price Premium

Put A 113 4

Put B 118 10

Explain how arbitraging can take place.

Solution

Buy Put A and Sell Put B

Certain cash flows = 10 – 4 = 6

S < 113 , Both options are exercised.

Pay off = (113 – S) – (118 – S) + 6

= 1

113 < S < 118 , only Put B is exercised

Pay off = 6 – (118 – S) = S – 112

S > 118, neither option is exercised

Pay off = 6

Problem

The following call options are trading

Option Strike Price Premium

Put A 113 4

Put B 118 10

Explain how arbitraging can take place. Solution

Sell B, Buy A

S < 30 No option is exercised , profit = 10 - 4 = 6

30 ≤ S 35 only A is exercised , profit = 6 + (S-30) = S - 24

S > 35 both options are exercised , profit

= 6+(S-30)- (S-35) = 11

Problem

Suppose you bought a put on a stock selling for $60 with a strike price of $55, for a $5 premium. What is the maximum gain possible?

Solution

Maximum gain = - 5 + (55-0)

Problem

I write a covered call on a $40 stock with an exercise price of $50 for a premium of $2. what will be my maximum gain?

Solution

Covered call means writing a call and buying the stock.

Premium received = 2; Cash paid for buying stock = 40

Maximum gain will be when the option is not exercised and the stock price reaches 50.

Then stock can be sold for 50 – 40 = 10

So Maximum gain = 10 + 2 = 12

Problem

What will be the maximum loss in the previous problem?Solution

If stock price falls to zero, pay off

= 2 + 0 = 2

Cash paid for buying stock = 40

Maximum loss = 2 – 40

= - 38

Bond options are most often based on Treasury bonds because of their active trading.

Index options settle in cash, nothing is delivered, and the payoff is made directly to the option holder’s account.

Options on futures sometimes called futures options, give the holder the right to buy or sell a specified futures contract on or before a given date at a given futures rice, the strike price.

Call options on futures contracts give the holder the right to enter into the long side of a future contract at a given futures price.

Put options on futures contracts give the holder the option to take on a short futures position at a future price equal to the strike price.

Specialised options

Interest rate options are similar to stock options except that the exercise price is an interest rate and the underlying asset is a reference a rate such as LIBOR.

Interest rate options are also similar to FRAs .

They are settled in cash, in an amount that is based on a notional amount and the spread between the strike a rate and the reference rate.

Most interest options are European options.

Interest rate options

Consider a long position in a LIBOR-based interest rate call option with a notional amount of $1,000,000 and a strike rate of 5%.

If at expiration, LIBOR is greater than 5%, the option can be exercised and the owner will receive $1,000,000 x (LIBOR – 5%).

If LIBOR is less than %, the option expires worthless and the owner receives nothing.

Let’s consider a LIBOR-based interest rate put option with the same features as the call that we just discussed.

Assume the option has a strike rate of 5% and notional amount of $1,000,000.

If at expiration, LIBOR falls below 5% the option writer (short) must pay the put holder an amount equal to $1,000,000 x (5% - LIBOR).

If at expiration, LIBOR is greater than 5%, the option expires worthless and the put writer makes no payments.

Problem

I have bought a call option on 90 day LIBOR with a notional principal of $2 million and a strike rate of 4%. At the expiration of the option, if LIBOR is 5%, what is the compensation I will receive?

Solution

(2,000,000) (.05 - .04) (90/360) = $5000

This compensation will be received 90 days after expiration.

Caps

An interest rate cap is a series of interest rate call options, having expiration dates that correspond to the reset dates on a floating-rate loan.

Caps are often used to protect a floating-rate borrower from an increase in interest rates.

Caps place a maximum (upper limit) on the interest payments on a floating-rate loan.

A cap may be structured to cover a certain number of periods or for the entire life of a loan.

The cap will make a payment at any future interest payment due date whenever the reference rate exceeds the cap rate.

An interest rate floor is a series of interest rate put options, having expiration dates that correspond to the reset dates on a floating-rate loan.

Floors are often used to protect a floating-rate lender from a decline in interest rates.

Floors place a minimum (lower limit) on the interest payments that are received from a floating-rate loan.

Floors

Collars

An interest rate collar combines a cap and a floor. A borrower with a floating-rate loan may buy a cap for

protection against rates above the cap and sell a floor in order to defray some of the cost of the cap.

Lower bound. Theoretically, no option will sell for less than its intrinsic value and no option can take on a negative value.

This means that the lower bound for any option is zero for both American and European options.

Upper bound. The maximum value of either an American or a European call option at any time t is the time-t share price of the underlying stock.

This makes sense because no one would pay a price for the right to buy an asset that exceeded the asset’s value. It would be cheaper to simply buy the underlying asset.

Call Option value

Upper bound for put options. The price for an American put option cannot be more than its strike price.

This is the exercise value in the event the underlying stock price goes to zero.

However, since European puts cannot be exercised prior to expiration, the maximum value is the present value of the exercise price discounted at the risk-free rate.

Even if the stock price goes to zero, and is expected to stay at zero, the intrinsic value, X, will not be received until the expiration date.

Put Option value bounds

For a European call option, construct the following portfolio:

A long at-the money European call option with exercise price X, expiring at time t = T

A long discount bond priced to yield the risk-free rate that pays X at option expiration.

A short position in one share of the underlying stock priced at S0 = X

The current value of this portfolio is c0 – S0 + X/(1+RFR)T

Valuing call options

At expiration time, t = T, this portfolio will pay cT – ST + X.

That is, we will collect cT = Max[0, ST – X) on the call option, pay ST to cover our short stock position, and collect X from the maturing bond.

If ST ≥ X, the call is in-the-money, and the portfolio will have a zero payoff because the call pays ST – X, the bond pays +X, and we pay – ST to cover our short position.

That is, the time t = T payoff is: ST – X + X – ST = 0.

If X > ST the call is out-of-the-money, and the portfolio has a positive payoff equal to X – ST because the call value, cT is zero, we collect X on the bond, a pay - ST to cover the short position.

So, the time t = T payoff is: 0 + X – ST = X - ST

Note that no matter whether the option expires in-the-money, at-the-money, or out-of-the-money, the portfolio value will be equal to or greater than zero. We will never have to make a payment.

To prevent arbitrage, any portfolio that has no possibility of a negative payoff cannot have a negative value. Thus, we can state the value of the portfolio at time t = 0 as:

c0 – S0 + X / (1+RFR)T ≥ 0

Which allows us to conduct that: c0 ≥ S0 – X/(1+RFR)T

Given two puts that are identical in all respects except exercise price, the one with the higher exercise price will have at least as much value as the one with the lower exercise price.

This is because the underlying stock can be sold at a higher price.

Similarly, given two calls that are identical in every respect except exercise price, the one with the lower exercise price will have at least as much value as the one with the higher exercise price.

This is because be underlying stock can be purchased at a lower price.

For American options and in most cases for European options, the longer the time to expiration, the greater the time value and, other things equal, the greater the option’s premium (price).

For far out-of-the-money options, the extra time may have no effect, but we can say the longer-term option will be no less valuable that the shorter-term option.

Option value and time to expiration

The case that doesn’t fit this pattern is the European put.

The minimum value of an in-the-money European put at any time t prior to expiration is X/(1+RFR)T-t – St.

While longer time to expiration increases option value through increased volatility, it decreases the present value of any option payoff at expiration.

For this reason, we cannot state positively that the value of a longer European put will greater than the value of a shorter-term put.

If volatility is high and the discount rate low, the extra time value will be the dominant factor and the longer-term put will be more valuable.

Low volatility and high interest rates have the opposite effect and the value of a longer-term in-the-money put option can be less than the value of a shorter-term put option.

Put Call Parity

Our derivation of put-call parity is based on the payoffs of two portfolio combinations, a fiduciary call and a protective put.

Fiduciary call

A fiduciary call is a combination of a pure-discount, riskless bond that pays X at maturity and a call with exercise price X.

The payoff for a fiduciary call at expiration is X when the call is out of the money, and X + (S – X) = S when the call is in the money.

A protective put is a share of stock together with a put option on the stock.

The expiration date payoff for a protective put is (X-S) + S = X when the put is in the money, and S when the put is out of the money.

When the put is the money, the call is out of the money, both portfolios pay X at expiration.

Similarly, when the put is out of the money and the call is in the money, both portfolios pay S at expiration.

Protective put

Problem

A stock is selling at $40, 3 month $50 put is selling for $11, a 3 month $50 is selling $1. The risk free rate is 6%. How much can be made on arbitrage.

Solution

Portfolio 1 : Fiduciary call

Buy Call, Invest in Bond

Investment = 1 + 50/(1+.06).25 = 50.28

Portfolio 2 : Protective put

Buy stock, Buy put

Investment = 40 + 11 = 51

So profit from arbitrage = 51 – 50.28 = 0.72

Problem

The current stock price is $52 and the risk free rate is i5%. A 3month $50 put is quoting at $1.50. Estimate the price for a 3 month $50 call.

Solution

Fiduciary call : C + 50 / (1+.05).25

Protective put : 52 + 1.5

To prevent arbitrage, we write:

C + 50/(1+.05).25 = 52 + 1.5

Or C = 53.5 – 40.39

= 4.11

Problem

The current stock price is $53 and the risk free rate is 5%. A 3 month European $50 call is quoting $3. What is the price of a 3 month $50 put?

Solution

To prevent arbitrage, we write:

C + 50/(1+.05).25 = 53 + P

Or P = 53 – 3 - 49.39

= 0.61

Options trading in India

NSE introduced trading in index options on June 4, 2001.

The options contracts are European style and cash settled and are based on the popular market benchmark S&P CNX Nifty index.

S&P CNX Nifty options contracts have 3 consecutive monthly contracts, additionally 3 quarterly months of the cycle March / June / September / December and 5 following semi-annual months of the cycle June / December would be available, so that at any point in time there would be options contracts with at least 3 year tenure available.

On expiry of the near month contract, new contracts (monthly/quarterly/ half yearly contracts as applicable) are introduced at new strike prices for both call and put options, on the trading day following the expiry of

the near month contract. S&P CNX Nifty options contracts expire on the last

Thursday of the expiry month. If the last Thursday is a trading holiday, the contracts

expire on the previous trading day.

New contracts with new strike prices for existing expiration date are introduced for trading on the next working day based on the previous day's index close values, as and when required.

In order to decide upon the at-the-money strike price, the index closing value is rounded off to the nearest applicable strike interval.

The in-the-money strike price and the out-of-the-money strike price are based on the at-the-money strike price.

The value of the option contracts on Nifty may not be less than Rs. 2 lakhs at the time of introduction.

The permitted lot size for futures contracts & options contracts shall be the same for a given underlying or such lot size as may be stipulated by the Exchange from time to time.

The price step in respect of S&P CNX Nifty options contracts is Re.0.05.

Base price of the options contracts, on introduction of new contracts, would be the theoretical value of the options contract arrived at based on Black-Scholes model of calculation of options premiums.

The base price of the contracts on subsequent trading days, will be the daily close price of the options contracts. The closing price shall be calculated as follows:

If the contract is traded in the last half an hour, the closing price shall be the last half an hour weighted average price.

If the contract is not traded in the last half an hour, but traded during any time of the day, then the closing price will be the last traded price (LTP) of the contract.

If the contract is not traded for the day, the base price of the contract for the next trading day is arrived at based on Black-Scholes model of calculation of options premiums.