OPTIONS

THE TWO BASIC OPTIONS - PUT AND CALL

• Other options are just combinations of these.

• Options are “derivatives” and other derivatives may include options

• The price of an option is called a “premium” because options are equivalent to insurance and the price of insurance is called a premium.

CALL OPTION CONTRACT

Definition: The right to purchase 100 shares of a security at a specified exercise price (Strike) during a specific period.

EXAMPLE: A January 60 call on Microsoft (at 7 1/2)

This means the call is good until the third Friday of January and gives the holder the right to purchase the stock from the writer at $60 / share for 100 shares.

cost is $7.50 / share x 100 shares = $750 premium or

option contract price.

PUT OPTION CONTRACT

Definition: The right to sell 100 shares of a security at a specified exercise price during a specific

period.

EXAMPLE: A January 60 put on Microsoft (at 14 1/4)

This means the put is good until the third Friday of January and gives the holder the right to sell the stock to the writer for $60 / share for 100 shares.

cost $14.25 / share x 100 shares = $1425 premium.

INTRINSIC AND TIME VALUE

•An option's INTRINSIC value is its value if it were exercised immediately.

•An option's TIME value is its price minus its intrinsic value.

Microsoft Stock Price = 53 1/4 at the time - October 1987

QUESTION: Which Microsoft option has greater intrinsic value? - put

QUESTION: Which Microsoft option has greater time value? - call

BREAK EVEN

•CALL BREAKEVEN

The stock must rise $14.25 = (60 - 53.25) + 7.50 (premium) for the Call buyer to break even.

•PUT BREAKEVEN

The stock must fall $7.50 = (53.25 - 60) + 14.25 (premium) for the Put buyer to break even.

QUESTION: Which Microsoft option is a better deal? - depends your expectation

Look at Options Quotes: www.bigcharts.com

HOW TO CALCULATE PROFIT OR LOSS ON OPTIONS

TWO PARTS

•Cash flows from premiums•Cash flows when you close out positions

If you close out before expiration, then option value is its market value.

If you close out at expiration then:

Call value = Max[0, (Stock price - Exercise price)]Put value = Max[0, (Exercise price - Stock price)]

EXAMPLE: Both Options on the same stock

Sell a put for $4 - strike = $45Buy a call for $2 - strike = $55

Assume that the stock price ends at $58

premiums put 400 call -200

option values put 0at expiration call 300

Net 500

Assume that the stock price ends at $40

premiums put 400same as above call -200

option value put -500at expiration call 0

Net -300

Note: The value of the put is actually $500 but because you sold the put, it costs you $500 to buy it back.

WHAT ARE YOUR PERCENT RETURNS ON THE CALL?

Net investment = premium = 200

Stock goes to $58

Stock goes to $40

The holder of opposite positions earns opposite results - zero sum. This is because options are derivative securities created by one individual (the writer (seller)) and bought by another. Original issue of stocks does not have this.

Re .turn

300 200

2005 50%

Re turn

200

2001 100%

HOW TO CLOSE OUT A POSITION IN AN OPTION.

• sell (or buyback) option• exercise option• let option expire - worthless

OPTIONS ALLOW

• large leverage but limited downside• hedge a profit

CALL STRATEGIES

•Buy option alone

not sure when stock will move =>use long maturitymost leveraged - high strike price

•Hedge

if you own stock - sell stock - buy optionif you don't own stock - short stock - buy option with

high strike price

•Sell option

naked - bearish on stock - receive premium

PUT STRATEGIES

•Buy option

not sure when stock moves => use long maturitymost leverage - low strike price

•Hedge - own stock - buy put - protect profit

•Sell put naked - if bullish on stock

COMBINED STRATEGIES

• straddle - 1 call and 1 put

• strip - 1 call, 2 puts

• strap - 2 calls, 1 put

• money spread

• time spread

Illustrate complex options using

www.wolframalpha.com (click “Examples”, then Money and Finance” then under “Derivatives Valuation” click “Option” then click “more” and select a complex option)

Explain the payoff profiles.

USE OPTIONS TO CUT UP PRICE DISTRIBUTIONS - NOW CALLED "FINANCIAL ENGINEERING"

PresentStock Price

Stock Price Distribution

PresentStock Price

Put Exercise Price

CallExercise Price

If you believe the stock will soon rise (fall) sharply, buy the right (left) tail by purchasing a call (put). If you believe that the stock will eitherrise or fall sharply, but don’t know which, buy both tails by purchasinga call and a put. This combination is called a straddle. Selling bothtails is called a strangle.

PresentStock Price

Put Exercise Price

If you are longterm bullish on a stock but wish to avoid any near-termsharp decline, buy the stock and a put. This is called a protective put.Of course, the put costs money so your net gains will be smaller.

One way to pay the additional costs of the put in a protective put is toalso sell a call of similar value. This is called a collar. Of course, byselling the call you give up some gains if the stock price rises above the call’s exercise price.

Financial engineering involves combining purchases and sales ofstocks, options or futures to capture only the parts of a stock pricedistribution that you want. The possibilities are endless. If you buy only the stock you own the whole price distribution.

PresentStock Price

Put Exercise Price

CallExercise Price

BLACK - SCHOLES MODEL

CRUCIAL INSIGHT - it is possible to replicate the payoff to an option by some investment strategy involving the underlying asset and lending or borrowing.

We can do this because the option value and the stock value are perfectly correlated – we need to know the hedge ratio (how much the option price increase when the stock price increases).

Therefore, we should be able to derive the value of an option from the asset price and the interest rate.

THERE IS A HEDGE RATIO BETWEEN THE CALL AND STOCK THAT ALLOWS ONE TO EXACTLY REPLICATE AN OPTION

For a simplified approach to replication one can use the Binomial Model.

1. Assume that

S = Stock price today C = Call option price todayr = risk-free rate q = the probability the stock price will increase(1-q) = probability the stock price will decreaseu = the multiplicative stock price increase

(u > 1 + r > 1)d = multiplicative decrease (0 < d < 1 < 1 + r )Cu = call price if stock price increasesCd = call price if stock decreases

The hedge ratio specifies how one asset’s price moves for a given change in another’s.

Each period the stock can take on only two values; the stock can move up to uS or down to dS.

2. Construct a risk-free hedge portfolio composed of one share of stock and m call options written against the stock. This means the payoffs in the up or down moves will be the same so that

uS – mCu = dS – mCd

Solve for m, the hedge ratio of calls to be written on stock

m = S(u – d)/(Cu - Cd )

3. Because we constructed the portfolio to be risk-free, then

(1 + r)(S – mC) = uS – mCu

Or

4. Substituting for the hedge ratio m,

)1(

])1[(

rm

mCurSC u

r

duru

Cdudr

C

Cdu

1

)1()1(

Or to simplify let

and

So C = [pCu + (1 - p)Cd] / (1 + r)

Here, p is called the hedging probability, also called the risk-neutral probability. The potential option payoffs Cu and Cd are multiplied by the risk neutral probabilities and the sum is discounted at the risk-free rate. (Note: the risk-neutral probability for the up (down) move is less (more) than the objective probability that would be used if we discounted the payoffs with a risk-adjusted rate (say a CAPM rate based on option beta) because the value in the numerator must be smaller if we are discounting at the smaller risk-free rate r.

du

drp

)1(

du

rup

)1(

1

Example: Suppose that a stock’s price is S=100 and it can increase by 100% or decrease by 50%. If the risk-free rate is 8% and the exercise price for the call is $125, find the price of the call and the hedge ratio.

u = 2, d = 0.5, r = .08

Cu = Max [0, 200 – 125] = 75Cd = Max [0, 50 – 125] = 0

m = S(u – d)/(Cu - Cd ) = 100(2 – .5)/(75 - 0) = 2

85.2608.1

5.2)08.1(2

05.2

5.)08.1(75

C

Here, the option price moves half as much as the stock’s. Therefore, if you own one share of the stock in this example, you can hedge, that is, eliminate your risk, by selling two calls.

To see how hedging works, form a hedged portfolio by buying one share and selling 2 options and find its risk-free end-of-period value. (Why is this risk-free?)

Stock goes:Down Up

Own the stock 50 200Sold 2 options 0 -150

50 50

Find the present value of the portfolio’s end value by discounting at the risk-free rate.

In this case, 50/(1+.08)=46.30.

You borrow this amount of money and add (S – 46.30) = (100 – 46.30) = $53.70 of your own money to buy one share. This leveraged position in the stock should give the same return as owning two calls.

To see this note that in one year you pay off the loan and you will have

150 [= 200 - 46.30(1+.08)] if stock goes to 200 or 0 [= 50 - 46.30(1+.08)] if the stock goes to 50.

Set the present value of the hedged portfolio equal to its discounted risk-free value and solve for C.

Here, S - 2 C = 100 - 2C = 46.30 => C=26.85.

GET THE PUT VALUE - PUT/CALL PARITY FORMULA

Put Price = C - S + E/(1 + risk-free rate)t

For this case:Put = 26.85 - 100 + 125/(1 + .08)1 = 42.6.

This model shows that, to get an option value, ones needs to know the

• current stock price• option’s exercise price• risk-free rate• option maturity• stock price volatility.

BLACK-SCHOLES MODEL - A NEARLY EXACT OPTION PRICING MODEL

C0 = P0N(d1) - E e-rt N(d2)

where Price of Stock = P0 Exercise price = ERisk free rate = rTime until expiration in years = tNormal distribution function = N( )Exponential function (base of natural log) = e

Note: Here the hedge ratio is represented by N(d1) and N(d2) where:

where Standard deviation of stocks return = Natural log function = ln

dP E r t

t10

25

ln( / ) ( . )

dP E r t

td t2

02

1

5

ln( / ) ( . )

NOTE: the call price is a weighted average of the stock price and the present value of the exercise price (replicating strategy – buy N(d1) shares and sell N(d2) bonds).

The weights are cumulative probabilities of a normal distribution. These probabilities are sometimes described as “risk-adjusted” probabilities. For each (d), we have the term ln(P0/E) which is the percent by which the stock price exceeds the exercise price (i.e. is in the money). Clearly, if the stock price exceeds the exercise price by a large percentage, the more likely the option will be valuable (i.e. exercised) at expiration. But note that ln(P0/E) is divided by so that the probability is adjusted for the stock’s risk and the time to expiration. A call on a risky stock (relatively large ) in the money by a given percent has less probability of staying in the money.

TO GET THE VALUE OF THE CALL, C0

EXAMPLE: ASSUME

Price of Stock P0 = 36Exercise price E = 40Risk free rate r = .05time period 3 mo. t = .25Std Dev of stock return = .50

•Substitute into d1 and d2.

d1

236 40 05 5 50 25

50 2525

ln( / ) [. . (. ) ].

. ..

50.25.50.25.2

d

•Substitute d1, d2 and other variables in the main equation

C0 = 36N(-.25) - 40e-.05(.25)N(-.50)

•Look up in the normal table for d to get N(d).

here N(d1) = N(-.25) = .4013 and N(d2) = N(-.50) =.3085

•Substitute in the main equation

C e005 2536 4013 40 3085 2 26 (. ) (. ) .. (. )

USE PUT CALL PARITY FORMULA TO GET PUT PRICE

T0 = PUT PRICE

To see why this holds, look at the stock price distribution and how the put gives you the left tail of the distribution. Then see that shorting the stock and buying the call leaves you with the same left tail. Or see that payoff at time t=0 is equal on both sides no matter what price is.

EXAMPLE - use info above - you need the call price

= 2.26 - 36 + 39.5 = 5.76

T C P Ee rt0 0 0

T e005 252 26 36 40 . . (. )

•BUYING A CALL OPTION IS LIKE BUYING STOCK USING MARGIN MONEY

•BUYING A PUT IS LIKE SELLING A STOCK SHORT AND INVESTING (LENDING) PROCEEDS.

One difference is that with an option, the most you can lose is 100%.With margin you can lose more.

The attraction of an option is this limited lossfeature - this is why a premium is paid.

The more volatile the stock the more valuable thisfeature is. (see futuresource.com –S&P volatility term structure).

HOW MARGIN WORKS

EXAMPLES: USING MARGIN MONEY - ignore interest

Assume: Required margin = 60%Stock Price = 75Investment = 30,000

Buy without margin borrowingHow many shares can you buy? - 30,000/75 = 400

Suppose price goes to 100

% return = (400 x 100) - 30,000

30 000333 33 3%

,. .

Suppose price goes to 40

Buy with margin borrowing - M = investable funds = your funds/margin rate

How much do you have to invest counting margin borrowing?

M = 30,000/.60 = 50,000

=> you borrow 20,000 and buy 50,000/75 = 667 shares

% return = (400 x 40) - 30,000

30 000466 46 6%

,. .

Suppose price goes to 100

Note: Remember your investment at risk is still just 30,000 - must pay back 20,000 margin

Suppose price goes to 40

Selling short and having to put up 60% margin gives the same returns but opposite signs.

% return = (667 x 100) - 30,000 - 20,000

30 000556 556%

,. .

% return = (667 x 40) - 30,000 - 20,000

30 000777 77 7%

,. .

A stock is an option on the value of the firm if there is debt in the capital structure. the value of the debt is the strike price. shareholders exercise their option to own the firm if the firm's value exceeds to value of debt, otherwise, they default and give the firm to the debtholders.

Initial Values Firm Value FallsFirm value 10mm 3mmDebt value 5mm 3mmEQUITY value 5mm 0mm