Basics of Stock Options

Introduction

Options are very old instruments, going back, perhaps, to the time of Thales the Milesian (c. 624 BC to c. 547 BC).

Thales, according to Aristotle, purchased call options on the entire autumn olive harvest (or the use of the olive presses) and made a fortune.

Joseph de la Vega (in “Confusión de Confusiones,” 1688, 104 years before the NYSE was founded under the buttonwood tree) also wrote about how options were dominating trading on the Amsterdam stock exchange.

Dubofsky reports that options existed in ancient Greece and Rome, and that options were used during the tulipmania in Holland from 1624-1636.

In the U.S., options were traded as early as the 1800’s and were available only as customized OTC products until the CBOE opened on April 26, 1973.

What is an Option?

A call option is a financial instrument that gives the buyer the right, but not the obligation, to purchase the underlying asset at a pre-specified price on or before a specified date

A put option is a financial instrument that gives the buyer the right, but not the obligation, to sell the underlying asset at a pre-specified price on or before a specified date

A call option is like a rain check. Suppose you spot an ad in the newspaper for an item you really want. By the time you get to the store, the item is sold out. However, the manager offers you a rain check to buy the product at the sale price when it is back in stock. You now hold a call option on the product with the strike price equal to the sale price and an intrinsic value equal to the difference between the regular and sale prices. Note that you do not have to use the rain check. You do so only at your own option. In fact, if the price of the product is lowered further before you return, you would let the rain check expire and buy the item at the lower price.

Options are Contracts

The option contract specifies: The underlying instrument The quantity to be delivered The price at which delivery occurs The date that the contract expires

Three parties to each contract The Buyer The Writer (seller) The Clearinghouse

The Option Buyer

The purchaser of an option contract is buying the right to exercise the option against the seller. The timing of the exercise privilege depends on the type of option: American-style options can be exercised any time before

expiration European-style options may only be exercised during a short

window before expiration Purchasing this right conveys no obligations, the buyer

can let the option expire if they so desire. The price paid for this right is the option premium. Note

that the worst that can happen to an option buyer is that she loses 100% of the premium.

The Option Writer

The writer of an option contract is accepting the obligation to have the option exercised against her, and receiving the premium in return.

If the option is exercised, the writer must: If it is a call, sell the stock to the option buyer at the exercise price

(which will be lower than the market price of the stock). If it is a put, buy the stock from the put buyer at the exercise price

(which will be higher than the market price of the stock). Note that the option writer can potentially lose far more than the

option premium received. In some cases the potential loss is (theoretically) unlimited.

Writing and option contract is not the same thing as selling an option. Selling implies the liquidation of a long position, whereas the writer is a party to the contract.

The Role of the Clearinghouse

The clearinghouse (the Options Clearing Corporation) exists to minimize counter-party risk.

The clearinghouse is a buyer to each seller, and a seller to each buyer.

Because the clearinghouse is well diversified and capitalized, the other parties to the contract do not have to worry about default. Additionally, since it takes the opposite side of every transaction, it has no net risk (other than the small risk of default on a trade).

Also handles assignment of exercise notices.

Examples of Options

Direct options are traded on: Stocks, bonds, futures, currencies, etc.

There are options embedded in: Convertible bonds Mortgages Insurance contracts Most corporate capital budgeting projects etc.

Even stocks are options!

Option Terminology

Strike (Exercise) Price - this is the price at which the underlying security can be bought or sold.

Premium - the price which is paid for the option. For equity options this is the price per share. The total cost is the premium times the number of shares (usually 100).

Expiration Date – This is the date by which the option must be exercised. For stock options, this is usually the Saturday following the third Friday of the month. In practice, this means the third Friday.

Moneyness – This describes whether the option currently has an intrinsic value above 0 or not: In-the-Money –

for a call this is when the stock price exceeds the strike price, for a put this is when the stock price is below the strike price.

Out-of-the-Money – for a call this is when the stock price is below the strike price, for a put this is when the stock price exceeds the strike price.

American-style - options which can be exercised before expiration. European-style - options which cannot be exercised before expiration.

The Intrinsic Value of Options

The intrinsic value of an option is the profit (not net profit!) that would be received if the option were exercised immediately: For call options: IV = max(0, S - X) For put options: IV = max(0, X - S)

At expiration, the value of an option is its intrinsic value. Before expiration, the market value of an option is the sum of the

intrinsic value and the time value. Since options can always be sold (not necessarily exercised) before

expiration, it is almost never optimal to exercise them early. If you did so, you would lose the time value. You’d be better off to sell the option, collect the premium, and then take your position in the underlying security.

Profits from Buying a Call

-1000-500

0500

1000150020002500300035004000

0 20 40 60 80 100Stock Price at Expiration

Pro

fit

S = 50X = 50r = 5%t = 90 dayss = 30%Call Price = 3.27

Selling a Call

-4000-3500-3000-2500-2000-1500-1000-500

0500

1000


Pro

fit

S = 50X = 50r = 5%t = 90 dayss = 30%Call Price = 3.27

Profits from Buying a Put

-500

0

500

1000

1500

2000

2500

3000

3500

4000


Pro

fit

S = 50X = 50r = 5%t = 90 dayss = 30%Put Price = 2.65

Selling a Put

-4000

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500


Pro

fit

S = 50X = 50r = 5%t = 90 dayss = 30%Put Price = 2.65

Combination Strategies

We can construct strategies consisting of multiple options to achieve results that aren’t otherwise possible, and to create cash flows that mimic other securities

Some examples: Buy Write Straddle Synthetic Securities

The Buy-Write Strategy

This strategy is more conservative than simply owning the stock

It can be used to generate extra income from stock investments

In this strategy we buy the stock and write a call

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 20 40 60 80 100

Stock Price at Expiration

Pro

fit

Stock Profit Call Profit Strategy Profit

S = 50 X = 50r = 5% t = 90 dayss = 30%Call Price = 3.27

The Straddle

If we buy a straddle, we profit if the stock moves a lot in either direction

If we sell a straddle, we profit if the stock doesn’t move much in either direction

This straddle consists of buying (or selling) both a put and call at the money

-1000

-500

0

500

1000

1500

2000

2500

3000

3500

4000

0 20 40 60 80 100


Pro

fit

Put Profit Call Profit Strategy Profit

S = 50 X = 50r = 5% t = 90 dayss = 30%Call Price = 3.27Put Price = 2.65

Synthetic Securities

With appropriate combinations of the stock and options, we can create a set of cash flows that are identical to puts, calls, or the stock

We can create synthetic: Long Stock — Buy Call, Sell Put Long Call — Buy Put, Buy Stock Long Put — Buy Call, Sell Stock Short Stock — Sell Call, Buy Put Short Call — Sell Put, Sell Stock Short Put — Sell Call, Buy Stock

The reasons that this works requires knowledge of Put-Call Parity

Put-Call Parity

Put-Call parity defines the relationship between put prices and call prices that must exist to avoid possible arbitrage profits:

In other words, a put must sell for the same price as a long call, short stock and lending the present value of the strike price (why?).

By manipulating this equation, we can see how to create synthetic securities (in the above form it shows how to create a synthetic put option).

P C S Xe rt

Put-Call Parity Example

Assume that we find the following conditions: S = 100 X = 100 r = 10% t = 1 year C = 16.73 P = ?

Cash Flows At Expiration if Stock Price Is:

Action Cash Inflow 110 100 90Buy Call -16.73 10.00 0.00 0.00Sell Stock 100.00 -110.00 -100.00 -90.00Buy Bond -90.48 100.00 100.00 100.00Total -7.21 0.00 0.00 10.00

Put-Call Parity Example

Q: What is the value of the put? A: 7.21 To see that the put must be priced at 7.21, first note that the portfolio that we have

created results in exactly the same payoffs, under all conditions, as a put option with a strike price of 100. By the law of one price, all assets which provide the same cash flows must be priced the same or arbitrage will force them to be the same. The net cash outlay for our portfolio is 7.21, so the put price must be 7.21.

Suppose that the put was priced at 8.00. In this case we could purchase our portfolio (for 7.21) and sell the put (for 8). At expiration, if the stock is below 100 the stock would be put to us at 100 (a cash outflow), we would sell the bond for 100 (a cash inflow) and we would keep the premium of 8. Thus, our net profit is 0.79 on a strategy which should have earned 0.00!

If the stock was over 100 at expiration, we would exercise the call and purchase the stock for 100, sell the bond for 100 and keep the put premium of 8. This again results in a profit of 0.79 on a strategy which should have earned 0.00!

If the put was priced under 7.21, we could buy the put and sell our portfolio. This would also create arbitrage profits.

Synthetic Long Stock Position

We can create a synthetic long position in the stock by buying a call, selling a put, and lending the strike price at the risk-free rate until expiration -5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 20 40 60 80 100


Pro

fit

Put Profit Call Profit Lend at Risk-free Strategy Profit


S C P Xe rt

Synthetic Long Call Position

We can create a synthetic long position in a call by buying a put, buying the stock, and borrowing the strike price at the risk-free rate until expiration -5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 20 40 60 80 100


Pro

fit

Put Profit Stock Profit Borrow at Risk-free Strategy Profit


C P S Xe rt

Synthetic Long Put Position

We can create a synthetic long position in a put by buying a call, selling the stock, and lending the strike price at the risk-free rate until expiration -5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 20 40 60 80 100


Pro

fit

Stock Profit Call Profit Lend at Risk-free Strategy Profit


P C S Xe rt

Synthetic Short Stock Position

We can create a synthetic short position in the stock by selling a call, buying a put, and borrowing the strike price at the risk-free rate until expiration

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 20 40 60 80 100


Pro

fit

Put Profit Call Profit Borrow at Risk-free Strategy Profit


S P C Xe rt

Synthetic Short Call Position

We can create a synthetic short position in a call by selling a put, selling the stock, and lending the strike price at the risk-free rate until expiration -5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 20 40 60 80 100


Pro

fit

Put Profit Stock Profit Lend at Risk-free Strategy Profit


C P S Xe rt

Synthetic Short Put Position

We can create a synthetic short position in a put by selling a call, buying the stock, and borrowing the strike price at the risk-free rate until expiration

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 20 40 60 80 100


Pro

fit

Stock Profit Call Profit Borrow at Risk-free Strategy Profit


P S C Xe rt

Option Valuation

The value of an option is the present value of its intrinsic value at expiration. Unfortunately, there is no way to know this intrinsic value in advance.

The most famous (and first successful) option pricing model, the Black-Scholes OPM, was derived by eliminating all possibilities of arbitrage.

Note that the Black-Scholes models work only for European-style options.

Option Valuation Variables

There are five variables in the Black-Scholes OPM (in order of importance): Price of underlying security Strike price Annual volatility (standard deviation) Time to expiration Risk-free interest rate

Variables’ Affect on Option Prices

Call Options Direct Inverse Direct Direct Direct

Put Options Inverse Direct Direct Inverse Direct

Variable– Stock Price– Strike Price– Volatility– Interest Rate– Time

Option Valuation Variables: Underlying Price

The current price of the underlying security is the most important variable.

For a call option, the higher the price of the underlying security, the higher the value of the call.

For a put option, the lower the price of the underlying security, the higher the value of the put.

Option Valuation Variables: Strike Price

The strike (exercise) price is fixed for the life of the option, but every underlying security has several strikes for each expiration month

For a call, the higher the strike price, the lower the value of the call.

For a put, the higher the strike price, the higher the value of the put.

Option Valuation Variables: Volatility

Volatility is measured as the annualized standard deviation of the returns on the underlying security.

All options increase in value as volatility increases.

This is due to the fact that options with higher volatility have a greater chance of expiring in-the-money.

Option Valuation Variables: Time to Expiration

The time to expiration is measured as the fraction of a year.

As with volatility, longer times to expiration increase the value of all options.

This is because there is a greater chance that the option will expire in-the-money with a longer time to expiration.

Option Valuation Variables: Risk-free Rate

The risk-free rate of interest is the least important of the variables.

It is used to discount the strike price, but because the time to expiration is usually not more than 3 months, and interest rates are usually fairly low, the discount is small and has only a tiny effect on the value of the option.

The risk-free rate, when it increases, effectively decreases the strike price. Therefore, when interest rates rise, call options increase in value and put options decrease in value.

The Black-Scholes Call Valuation Model

At the top (right) is the Black-Scholes valuation model for calls. Below are the definitions of d1 and d2.

Note that S is the stock price, X is the strike price, s is the standard deviation, t is the time to expiration, and r is the risk-free rate.

C S Xe rt N d N d1 2

d

SX

rt t

t1

205

ln . s

s

d d t2 1 s


The Black-Scholes-Merton option pricing model says the value of a stock option is determined by six factors:

S, the current price of the underlying stocky, the dividend yield of the underlying stockK, the strike price specified in the option contractr, the risk-free interest rate over the life of the option contractT, the time remaining until the option contract expires, (sigma) which is the price volatility of the underlying stock


The price of a call option on a single share of common stock is: C = Se–yTN(d1) – Ke–rTN(d2)

The price of a put option on a single share of common stock is: P = Ke–rTN(–d2) – Se–yTN(–d1)

Tσdd

Tσ

T2σyrKSlnd

12

2

1


In the Black-Scholes-Merton formula, three common fuctions are used to price call and put option prices:

e-rt, or exp(-rt), is the natural exponent of the value of –rt (in common terms, it is a discount factor)

ln(S/K) is the natural log of the "moneyness" term, S/K.

N(d1) and N(d2) denotes the standard normal probability for the values of d1 and d2.

In addition, the formula makes use of the fact that:

N(-d1) = 1 - N(d1)

B-S Call Valuation Example

Assume a call with the following variables: S = 100 X = 100 r = 0.05 s = 0.10 t = 90 days = 0.25 years

C e 100 0 275 100 0 225 2 660 05 0 25* N . * N . .. * .

d1

100

1000 05 0 25 0 5 0 01 0 25

01 0 250 275

ln . * . . * . * .

. * ..

d2 0 275 01 0 25 0 225 . . * . .

The Black-Scholes Put Valuation Model

At right is the Black-Scholes put valuation model.

The variables are all the same as with the call valuation model.

Note: N(-d1) = 1 - N(d1)

P Xe Srt N d N d2 1

d

S

Xrt t

t1

205

ln . s

s

d d t2 1 s

B-S Put Valuation Example

Assume a put with the following variables: S = 100 X = 100 r = 0.05 s = 0.10 t = 90 days = 0.25 years

P e 100 0 225 100 0 275 1420 05 0 25* N . * N . .. * .

d1

100

1000 05 0 25 0 5 0 01 0 25

01 0 250 275

ln . * . . * . * .

. * ..

d2 0 275 01 0 25 0 225 . . * . .

Naked Options - Margin

Method Ia.Calculate the option premium for No. of sharesb.Compute 0.20 (MP per share) x (No. of shares)c.Compute the amount by which the contract is out of money

A+B -C

Method IINo.of shares x Option Premium per share + 0.10 (MP per share)x100

Basics of Stock Options

Documents

Transcript of Basics of Stock Options