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Mechanics of Options Markets Chapter 8

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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.

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

OPTIONS BASICS

A contingent claim:

The option’s 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 holderlong.

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 option’s expiration date.

PUT Buyer holderlong.

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 option’s expiration date.

Call Sellerwritershort.

In exchange for receiving the call’s premium, the

Call sellerhas 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 Sellerwritershort.

In exchange for receiving the put premium, the

Put sellerhas 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 asset’s 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 call’s 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 => 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 => expects the price of the underlying asset, St, to remain below or at the exercise price, K, during the option’s 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 => 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 => 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= $47.27share;K = $45/share;T-t = .5yrs)

Ct= $5.78/share

On date t, the BUYER of this call pays the call’s 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 => 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( S=$47.27share;K =$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 => hopes the price of the underlying asset, currently St = $47.27/share, remains below or at the exercise price, K = $45/share, during the option’s life of half a year and hence, keep the premium ct = $5.78/share

A numerical example: LONG PUT

p(S= $47.27share;K = $45/share;T-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 => 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 =$47.27share;K = $45/share;T-t = .5yrs)

pt= $2.25/share

On date t, the SELLER of this put receives the market premium pt=$2.25/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 => 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.

$

47.27K = 45

t = nowT = .5yrS

- More terminology
- Premium =The option Market Price
- Premium = [Intrinsic value + extrinsic value]
- Intrinsic value:
- CallsMax{0, St - K) ≥ 0
- PutsMax{0, K - St) ≥ 0

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

CallsPuts

St > K St < K

or:

St – K > 0 K – St > 0

The Intrinsic value of an option that is

in-the money is positive.

Out-of-the-money

CallsPuts

St < K St > K

or

St - K< 0 K – St < 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.

- 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

CLIENTES

BROKERS

The OCC’s

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 OCC’s 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 OCC’s guarantee, the market is anonymous and traders only have to offset their positionsin order to come out of the market.

The OCC has no control over the market prices. These are determined by trader’s supply and demand.

The OCC

The OCC’s 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.

- 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:
- [JANAPR JUL OCT]
- [FEBMAY AUG NOV]
- [MARJUN SEP DEC]

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 > 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.

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

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:

Xe

12

1002.70481382

10,0002.71814592

1,000,0002.71828046

In the limit 2.718281828…..

Recall that in our example:

n = 10 years.

R = 12%

P=$100.

So, P = $100 invested at a 12% annual rate,

continuously compounded for ten years will

grow to:

Continuous compounding yields the

highest return:

CompoundingmFactor

Simple13.105848208

Quarterly43.262037792

Monthly123.300386895

Daily3653.319462164

Continuously ∞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 bond’s 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 bill’s maturity. Hence, its current market price is the NPV of the bill’s Face Value:

Pt = NPV{the 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 thatinvestors also can borrow moneyat 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 = NPV{the T-bill Face-Value}.

Thus:

the current time, t, T-bill price, Pt , which pays FV upon its maturity on date T, is:

Pt = [FV]e-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 = [FV]e-r(T-t)

Pt = [$1,000]e-(.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 = [FV]e-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 stock’s 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 broker’s or a client’s account and sell it

in the investor’s name. The investor’s obligation is to hand over the shares some time in the future, or upon the broker’s 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 investor’s 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 sale’s 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 option’s 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 > 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 = Max{0, 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 < K

and the Cash flow/share = K - ST.

expires worthless if the PUT is

out-of-the money:ST> K

and the Cash flow/share = 0.

Algebraically:

Cash Flow/share = Max{0, ST – K}

- All parts of the strategy remain open till the option’s expiration.
4.All parts of the strategy are closed out at option’s 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+Max{0, ST -K}

Short call: ct - Max{0, ST -K}

Long put:-pt +Max{0, K- ST}

Short put: pt - Max{0, 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 ofmoney.

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+ Max{0, ST -K}

Short call: ct- Max{0, ST -K}

Long put:-pt+ Max{0, K- ST}

Short put: pt- Max{0, 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.