Mechanics of Options MarketsChapter 8

OPTIONS ARE CONTRACTS Two parties Seller

and buyer A contract Specifying the rights and

obligations of the two parties. An underlying

asset a financial asset, a commodity or a

security, that is the basis of the contract.

Assets UnderlyingExchange-Traded Options(p. 190)

- Stocks
- Foreign Currency
- Stock Indices
- Futures
- Options
- Bonds

OPTIONS BASICS A contingent claim The options

value is contingent upon the value of the

underlying asset Two Types of Options A

Call THE RIGHT TO BUY THE UNDERLYING ASSET A

Put THE RIGHT TO SELL THE UNDERLYING ASSET

CALL Buyer ? holder ? long. In exchange for

making a payment of money, the call premium, the

call buyer has the right to BUY a specified

quantity of the underlying asset for the exercise

(strike) price before the options expiration

date.

PUT Buyer ? holder ? long. In exchange for

making a payment of money, the put premium, the

put buyer has the right to SELL a specified

quantity of the underlying asset for the exercise

(strike) price before the options expiration

date.

Call Seller ? writer ? short. In exchange for

receiving the calls premium, the Call seller

has the obligation to SELL the underlying asset

for the predetermined exercise (strike) price

upon being served with an exercise notice during

the life of the option, I.e., before the option

expires.

Put Seller ? writer ? short. In exchange for

receiving the put premium, the Put seller has

the obligation to BUY the underlying asset for

the predetermined exercise (strike) price upon

being served with an exercise notice during the

life of the option, I.e., before the option

expires.

- The two main types of Options (PUTS and CALLS)
- American Options
- exercisable any time before expiration
- European Options
- exercisable only on expiration date

OPTIONS NOTATIONS S The underlying assets

market price K - The exercise (strike)

price t The current date T The expiration

date T -t The time till expiration c, p

European call, put premiums C, P

American call, put premiums

Options definitions using the above

notation LONG CALL On date t, the BUYER of a

call option pays the calls market price, ct, Ct,

and holds the right to buy the underlying asset

at the strike price, K, before the call expires

on date T. (or on T, if the call is European).

Thus gt the call holder expects the price of

the underlying asset, St, to increase during the

life of the option contract.

SHORT CALL On date t, the SELLER of a call

option receives ct, Ct, and must sell the

underlying asset for K, if the option is

exercised by its holder before the option expires

on date T. Thus gt expects the price of the

underlying asset, St, to remain below or at the

exercise price, K, during the options life. This

way the writer keeps the premium.

LONG PUT On date t, the BUYER of a put option

pays pt, Pt, and holds the right to sell the

underlying asset for K before the put expires on

date T. Thus gt expects the market price of the

underlying asset, St, to decrease during the life

of the put.

SHORT PUT On date t, the SELLER of a put

receives pt, Pt, and must buy the underlying

asset for K if the put is exercised by its holder

before the put expires on date T. Thus gt

expects the market price of the underlying asset,

St, to remain at or above K during the life of

the put. This way the put writer keeps the

premium.

A numerical example LONG CALL C(S

47.27shareK 45/shareT-t .5yrs) Ct

5.78/share On date t, the BUYER of this call

pays the calls market price, 5.78/share, and

holds the right to buy the underlying asset at

the strike price, K 45/share, before the call

expires at T, half a year from now (at T, if the

call is European). Thus gt the call holder

expects the price of the underlying asset, St

47.27/share, to increase during the life of the

option contract.

A numerical example SHORT CALL C(

S47.27shareK 45/share T-t .5yrs) Ct

5.78/share On date t, the SELLER of this call

receives 5.78/share and must sell the underlying

asset for K 45/share, if the option is

exercised by its holder before the option expires

at T, half a year from now. Thus gt hopes the

price of the underlying asset, currently St

47.27/share, remains below or at the exercise

price, K 45/share, during the options life of

half a year and hence, keep the premium ct

5.78/share

A numerical example LONG PUT p(S 47.27shareK

45/shareT-t .5yrs) pt 2.25/share On

date t, the BUYER of this put pays the market

price of pt 2.25/share and holds the right to

sell the underlying asset for K 45/share

before the put expires half a year from now, at

T. Thus gt expects the market price of the

underlying asset, St 47.27/share, to decrease

during the half a year life span of the put.

A numerical example SHROT PUT p(S 47.27shareK

45/shareT-t .5yrs) pt 2.25/share On date

t, the SELLER of this put receives the market

premium pt2.25/share and must buy the

underlying asset for K 45 if the put is

exercised by its holder before the put expires

half a year from now at T. Thus gt expects the

market price of the underlying asset, St

47.27/share to remain at or above K 45 during

the life span of the put and to keep the premium,

pt 2.25/share.

47.27 K 45 t now

T .5yr S

More terminology Premium The option Market

Price Premium Intrinsic value extrinsic

value Intrinsic value Calls Max0, St - K)

0 Puts Max0, K - St) 0 Extrinsic value

(time value) Premium Intrinsic value

At-the-money St K In this case the intrinsic

value for both calls and puts is zero St - K

K - St 0 and the premium consists of the

Extrinsic (time) value only. PREMIUM 0

extrinsic value

In-the-money Calls Puts St gt K

St lt K or St K gt 0 K St

gt 0 The Intrinsic value of an option that

is in-the money is positive.

Out-of-the-money Calls Puts St

lt K St gt K or St - Klt 0

K St lt 0 In this case the intrinsic value is

zero and the premium consists of the extrinsic

(time) value only. PREMIUM 0 extrinsic value

The next table shows the market prices (premiums)

of calls and puts on IBM On Friday NOV 30 2007

t When IBM was trading at St 105/share. Notice

that there where options traded for several

expiration dates and for a wide range of strike

prices. Blanks mean that the option did not trade

on NOV 30 2007 OR did not exist.

S105 CALLS CALLS CALLS CALLS CALLS CALLS PUTS PUTS PUTS PUTS PUTS PUTS

K DEC07 JAN08 APR08 JUL08 JAN09 JAN10 DEC07 JAN08 APR08 JUL08 JAN09 JAN10

55 48.90

60 43.40 48.30 .05 1.05

65 39.00 40.10 52.60 .10 .30 .50 1.75

70 33.70 32.20 37.60 48.20 .15 .55 .70 1.90

75 33.28 30.80 32.30 33.50 .20 .60 1.10 2.40

80 23.30 25.60 28.60 28.80 32.38 .25 .75 1.85 3.97

85 18.90 20.20 24.40 24.90 25.70 .05 .30 1.40 2.45 4.38

90 17.36 18.00 18.40 20.40 22.90 27.00 .10 .60 1.95 3.40 5.70 8.70

95 11.40 11.60 14.70 18.10 18.25 .30 1.30 3.20 5.90 6.80

100 6.30 8.20 11.00 13.60 18.00 21.50 .90 2.30 4.80 6.10 8.90 11.83

105 2.90 4.80 8.20 10.55 13.60 2.35 4.10 6.90 8.00 11.00

110 .95 2.70 5.90 7.70 11.80 16.80 5.50 6.90 9.10 9.80 12.80 18.50

115 .20 1.31 3.95 6.75 10.30 9.80 10.60 10.80 15.00 15.60

120 .05 .60 2.60 5.20 7.10 12.50 16.30 15.00 14.50 18.00 18.50 23.50

125 .25 1.65 3.70 10.30 16.90 18.80 24.80

130 .15 1.15 1.80 4.30 9.60 22.60 22.20 27.50 31.20

135 .10 .80 1.75 33.10

140 .37 .95 3.10 7.20 32.70 38.70

145 .30 .80

150 .15 1.65 4.70

160 1.20 4.00

170 .50

- Options Markets
- OTC options
- Over the counter (OTC)
- Meaning
- Not on an organized exchange.
- 2. Exchange traded options
- An organized exchange
- Options clearing corporation (OCC)

WHEN OPTIONS ARE TRADED ON THE OTC TRADERS

BEAR Credit risk Operational risk Liquidity

risk

Credit Risk Does the other party have the means

to pay? Operational Risk Will the other party

deliver the commodity? Will the other party pay?

Liquidity Risk. Liquidity the speed (ease) with

which investors can buy or sell securities

(commodities) in the market. In case either party

wishes to get out of its side of the contract,

what are the obstacles? How to find another

counterparty? It may not be easy to do that. Even

if you find someone who is willing to take your

side of the contract, the other party may not

agree.

THE Option Clearing Corporation (OCC)(p. 198) The

exchanges understood that there will exist no

efficient options markets without contracts

standardization and an absolute guarantee to

the options holders that the market is

default-free, so they have created the

OPTIONS CLEARING CORPORATION (OCC) The OCC is

a nonprofit corporation

THE OPTION CLEARING CORPORATION PLACE IN THE

MARKET

EXCHANGE CORPORATION

OPTIONS CLEARING CORPORATION

CLEARING MEMBERS

NONCLEARING MEMEBRS

OCC MEMBER

BROKERS

CLIENTES

The OCCs absolute guarantee The holders of

calls and puts will always be able to exercise

their options if they so wish to do!!!

The absolute guarantee The OCCs absolute

guarantee provides traders with a default-free

market. Thus, any investor who wishes to engage

in options buying knows that there will be no

operational default.

The OCC Also, clears all options trading.

Maintains the list of all long and short

positions. Matches all long positions with short

positions. Hence, the total sum of all options

traders positions must be ZERO at all times.

The OCC Maintains the accounting books of all

trades. Charges fees to cover costs Assigns

Exercise notices Given the OCCs guarantee, the

market is anonymous and traders only have to

offset their positions in order to come out of

the market. The OCC has no control over the

market prices. These are determined by traders

supply and demand.

The OCC The OCCs absolute guarantee together

with matching all short and long trading makes

the market very liquid. 1 traders are not

afraid to enter the market 2 traders can quit

the market at any point in time by OFFSETTING

their original position.

CLIENT A BUY ORDER INFO TRADERS LONG SHORT OPTIONS PRICE OPEN INTEREST VOLUME

INFO TRADERS LONG SHORT OPTIONS PRICE

BROKER INFO TRADERS LONG SHORT OPTIONS PRICE

INFO TRADERS LONG SHORT OPTIONS PRICE

OCC MEMBER INFO TRADERS LONG SHORT OPTIONS PRICE

PRICE INFO TRADERS LONG SHORT OPTIONS PRICE

THE TRADING FLOOR TRADE INFO TRADERS LONG SHORT OPTIONS PRICE THE OCC

PRICE INFO TRADERS LONG SHORT OPTIONS PRICE

OCC MEMBER INFO TRADERS LONG SHORT OPTIONS PRICE MARGINS

INFO TRADERS LONG SHORT OPTIONS PRICE

BROKER INFO TRADERS LONG SHORT OPTIONS PRICE MARGINS

INFO TRADERS LONG SHORT OPTIONS PRICE

SELL ORDER CLIENT B INFO TRADERS LONG SHORT OPTIONS PRICE MARGINS

- OFFSETTING POSITIONS
- A trader with a LONG position who wishes to get

out of the market MAY - Exercise, or
- open a SHORT position with equal number of the

same options. - Example Suppose
- LONG 5, SEP, 85, IBM puts p0 4/share
- This position must be offset by
- SHORT 5, SEP, 85, IBM puts p1 3/share
- Cash flows -2,000 1,500 -500.

OFFSETTING POSITIONS A trader with a SHORT

position who wishes to get out of the market MUST

open a LONG position with equal number of the

same options. Example Suppose SHORT 25,

JAN, 75, BA calls c 7/share This position

must be offset by LONG 25, JAN, 75, BA

calls c 5/share Cash flows 17,500 - 12,500

5,000.

THE OCC Standardization Contract size the

number of units of the underlying asset

covered in one option. Exersice

prices Mostly, increments of 2.5, 5.00

and 10.00. Exercise notice and assignment

procedures Delivery sequence.

- THE OCC Standardization
- Expiration dates Saturday, immediately

following the third Friday of the

expiration month. - The basic expiration cycles
- JAN ? APR ? JUL ? OCT
- FEB ? MAY ? AUG ? NOV
- MAR ? JUN ? SEP ? DEC

A Review of Some Financial Economics Principles

- Arbitrage A market situation whereby an investor

can make a profit with - no equity and no risk.
- Efficiency A market is said to be efficient if

prices are such that there exist no arbitrage

opportunities. - Alternatively, a market is said to be inefficient

if prices present arbitrage opportunities for

investors in this market.

- Valuation The current market value (price) of

any project or investment is the net present

value of all the future expected cash flows from

the project. - One-Price Law Any two projects whose cash flows

are equal in every possible state of the world

have the same market value. - Domination Let two projects have equal cash

flows in all possible states of the world but

one. The project with the higher cash flow in

that particular state of the world has a higher

current market value and thus, is said to

dominate the other project.

- The Holding Period Rate of Return (HPRR)
- Buy shares of a stock on date t and sell
- them later on date T. While holding the
- shares, the stock has paid a cash dividend in
- the amount of D/share.
- The Holding Period Rate of Return HPRR is

Example St 50/share ST 51.5/share DT-t

1/share T t 73days.

- Risk-Free Asset is a security of investment

whose return carries no risk. Thus, the return

on this security is known and guaranteed in

advance. - Risk-Free Borrowing And Landing By purchasing

the risk-free asset, investors lend their capital

and by selling the risk-free asset, investors

borrow capita at the risk-free rate.

- The One-Price Law
- There exists only one risk-free rate in an

efficient economy. - Proof By contradiction. Suppose two risk-free
- rates exist in a market and R gt r. Since both are
- free of risk, ALL investors will try to borrow at

r - and invest the money borrowed in R, thus assuring
- themselves the difference. BUT, the excess demand
- For borrowing at r and excess supply of lending
- (investing) at R will change them. Supply

demand - only when R r.

Compounded Interest (p. 76)

- Any principal amount, P, invested at an
- annual interest rate, R, compounded
- annually, for n years would grow to
- An P(1 R)n.
- If compounded Quarterly
- An P(1 R/4)4n.

- In general
- Invest P dollars in an account which pays an
- annual interest rate R with m compounding
- periods every year.
- The rate in every period is R/m.
- The number of compounding periods is nm.
- Thus, P grows to
- An P(1 R/m)mn.

- An P(1 R/m)mn.
- Monthly compounding becomes
- An P(1 R/12)12n
- and daily compounding yields
- An P(1 R/365)365n.

- EXAMPLES
- n 10 years R 12 P 100
- 1.Simple compounding, m 1, yields
- A10 100(1 .12)10 310.5848
- 2.Monthly compounding, m 12, yields
- A10 100(1 .12/12)120 330.0387
- 3.Daily compounding, m 365, yields
- A10 100(1 .12/365)3,650 331.9462.

- Notations
- The annual rate R will be stated as Rm in order
- to make clear how many times a year it is
- compounded.
- For the annual rate is 10 with quarterly
- compounding, the corresponding formula is
- An P(1 .10/4)4n
- For the same annual rate with monthly
- compounding the corresponding formula is
- An P(1 .10/12)12n

- DISCOUNTING
- The Present Value today, date t, of a future cash

flow, FVT, on a future date T, - is given by DISCOUNTING

- DISCOUNTING the general case
- Let cji, j 1,2,3,m, be a sequence of m cash
- flows paid in year i, i 1,2,3,,n.
- Let Rm be the annual rate during these years.
- DISCOUNTING these cash flows yields the
- Present Value

- CONTINUOUS COMPOUNDING
- In the early 1970s, banks came up with the
- following economic reasoning Since the
- bank has depositors money all the time, this
- money should be working for the depositor
- all the time! This idea, of course, leads to the
- concept of continuous compounding.
- We want to apply this idea to the formula

CONTINUOUS COMPOUNDING As m increases the time span of every compounding period diminishes CONTINUOUS COMPOUNDING As m increases the time span of every compounding period diminishes CONTINUOUS COMPOUNDING As m increases the time span of every compounding period diminishes

Compounding m Time span

Yearly 1 1 year

Daily 365 1 day

Hourly 8760 1 hour

Every second 3,153,600 One second

Continuously 8 Infinitesimally small

- CONTINUOUS COMPOUNDING
- This reasoning implies that in order to impose

the concept of continuous time on the above

compounding expression, we need to solve

This expression may be rewritten as

- Recall that the number e is

X e 1 2 100 2.70481382 10,000 2.7181459

2 1,000,000 2.71828046 In the limit

2.718281828..

- Recall that in our example
- n 10 years.
- R 12
- P100.
- So, P 100 invested at a 12 annual rate,
- continuously compounded for ten years will
- grow to

Continuous compounding yields the highest

return Compounding m Factor Simple 1 3.105

848208 Quarterly 4 3.262037792 Monthly 12 3.

300386895 Daily 365 3.319462164 Continuously

8 3.320116923

Continuous Discounting (p. 77)

This expression may be rewritten as

Continuous Discounting

This expression may be rewritten as

Recall that in our example P 100 n 10

years and R 12 Thus, 100 invested at an

annual rate of 12 , continuously compounded for

ten years will grow to 332.0117.

Therefore, we can write the continuously

discounted value of 320.0117

- Equivalent Interest Rates (p.77)
- Rm The annual rate with m compounding

periods - every year.

- Equivalent Interest Rates (p.77)
- rc The annual rate with continuous

compounding

- Equivalent Interest Rates (p.77)
- Rm The annual rate with m
- compounding periods every year.
- rc The annual rate with continuous

compounding. - Definition Rm and rc are said to be

equivalent - if

- Equivalent Interest Rates (p.77)

- Equivalent Interest Rates (p.77)
- The same method applies to any two rates with

different periods of compounding. Thus, if we

have Rm1 and another Rm2 then the relationship

between the two rates is

- Risk-free lending and borrowing
- Treasury bills are zero-coupon bonds, or pure

discount bonds, issued by the Treasury. - A T-bill is a promissory paper which promises its

holder the payment of the bonds Face Value (Par-

Value) on a specific future maturity date. - The purchase of a T-bill is, therefore, an

investment that pays no cash flow between the

purchase date and the bills maturity. Hence, its

current market price is the NPV of the bills

Face Value - Pt NPVthe T-bill Face-Value
- We will only use continuous compounding

- Risk-free lending and borrowing
- Risk-Free Asset is a security whose return is a

known constant and it carries no risk. - T-bills are risk-free LENDING assets. Investors

lend money to the Government by purchasing

T-bills (and other Treasury notes and bonds) - We will assume that investors also can borrow

money at the risk-free rate. I.e., investors may

write IOU notes, promising the risk-free rate to

their buyers, thereby, raising capital at the

risk-free rate.

- Risk-free lending and borrowing
- LENDING
- By purchasing the risk-free asset,
- investors lend capital.
- BORROWING
- By selling the risk-free asset, investors borrow

capital. - Both activities are at the
- risk-free rate.

- We are now ready to calculate the current value

of a T-Bill. - Pt NPVthe T-bill Face-Value.
- Thus
- the current time, t, T-bill price, Pt , which

pays FV upon its maturity on date T, is - Pt FVe-r(T-t)
- r is the risk-free rate in the economy.

- EXAMPLE Consider a T-bill that promises its

holder FV 1,000 when it matures in 276 days,

with a risk-free yield of 5 Inputs for the

formula - FV 1,000 r .05 T-t 276/365yrs
- Pt FVe-r(T-t)
- Pt 1,000e-(.05)276/365
- Pt 962.90.

- EXAMPLE Calculate the yield-to -maturity of a

bond which sells for 965 and matures in 100

days, with FV 1,000. - Pt 965 FV 1,000 T-t 100/365yrs.
- Solving for r
- Pt FVe-r(T-t)

- SHORT SELLING STOCKS (p. 97)
- An Investor may call a broker and ask to sell a

particular stock short. - This means that the investor does not own shares

of the stock, but wishes to sell it anyway. The

investor speculates that the stocks share price

will fall and money will be made upon buying the

shares back at a - lower price. Alas, the investor does not own

shares of the stock. The broker will lend the

investor shares from the brokers or a clients

account and sell it - in the investors name. The investors obligation

is to hand over the shares some time in the

future, or upon the brokers request.

- SHORT SELLING STOCKS
- Other conditions
- The proceeds from the short sale cannot be used

by the short seller. Instead, they are deposited

in an escrow account in the investors name until

the investor makes - good on the promise to bring the shares back.

Moreover, the investor must deposit an additional

amount of at least 50 of the short sales

proceeds in the escrow account. This additional

amount guarantees that there - is enough capital to buy back the borrowed shares

and hand them over back to the broker, in case

the shares price increases.

- SHORT SELLING STOCKS
- There are more details associated with short

selling stocks. For example, if the stock pays

dividend, the short seller must pay the dividend

to the broker. Moreover, the short seller does

not gain interest on the amount deposited in the

escrow account, etc. - We will use stock short sales in many of

strategies associated with options trading. In

all of these strategies, we will assume that no

cash flow occurs from the time the strategy is

opened with the stock short sale until the time

the strategy terminates and the stock is

repurchased.

- SHORT SELLING STOCKS
- In terms of cash flows per share
- St is the cash flow/share from selling

the stock short thereby, opening a SHORT

POSITION on date t. - -ST is the cash flow from purchasing the

stock back on date T (and delivering it to

the lender thereby, closing the SHORT

POSITION.)

- Options Risk-Return Tradeoffs at expiration
- PROFIT PROFILE OF A STRATEGY
- A graph of the profit/loss as a function of all

possible market prices of the underlying asset - We will begin with profit profiles at the

options expiration I.e., an instant before the

option expires.

- Options Risk-Return Tradeoffs at Expiration
- 1. Only at expiration ? T-t 0
- No time value! Only intrinsic value!
- The CALL at Expiration
- is exercised if the CALL is
- in-the money ST gt K
- and the Cash flow/share ST K.
- expires worthless if the CALL is
- out-of-the money ST ? K
- and the Cash flow/share 0.
- Algebraically
- Cash Flow/share Max0, ST K

- Options Risk-Return Tradeoffs at Expiration
- 1. Only at expiration ? T-t 0
- No time value! Only intrinsic value!
- The PUT at Expiration
- is exercised if the PUT is
- in-the money ST lt K
- and the Cash flow/share K - ST.
- expires worthless if the PUT is
- out-of-the money ST gt K
- and the Cash flow/share 0.
- Algebraically
- Cash Flow/share Max0, ST K

- All parts of the strategy remain open till the

options expiration. - 4. All parts of the strategy are closed out at

options expiration. - 5. A Table Format
- The analysis of every strategy is done with a

table of cash flows. - Every row is one part (leg) of the strategy.
- Every row is analyzed separately.
- The cash flow of the entire strategy is the

vertical sum of the rows.

- The algebraic expressions of cash flows per

share ICF(t) CF at Expiration(T) - Long stock -St ST
- Short stock St - ST
- Long call -ct Max0, ST -K
- Short call ct - Max0, ST -K
- Long put -pt Max0, K- ST
- Short put pt - Max0, K - ST
- The profit/loss per share is the cash flow at

expiration plus the initial cash flow of the

strategy, disregarding the time value of money.

- The algebraic expressions of P/L per share at
- expiration
- P/L per share at Expiration
- Long stock -St ST
- Short stock St - ST
- Long call -ct Max0, ST -K
- Short call ct - Max0, ST -K
- Long put -pt Max0, K- ST
- Short put pt - Max0, K - ST

- 6. A Graph of the profit/loss profile at

expiration - The P/L per share from the strategy as a

function of all possible prices of the underlying

asset at expiration.