FIN3600_15- Option Valuation

8/11/2019 FIN3600_15- Option Valuation

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Basics of Stock Options

Timothy R. Mayes, Ph.D.FIN 3600: Chapter 15


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Introduction

Options are very old instruments, going back, perhaps, to the time ofThales the Milesian (c. 624 BC to c. 547 BC).Thales, according to Aristotle, purchased call options on the entireautumn olive harvest (or the use of the olive presses) and made a

fortune.Joseph de la Vega (in Confusin de Confusiones, 1688, 104 years

before the NYSE was founded under the buttonwood tree) alsowrote about how options were dominating trading on the Amsterdamstock exchange.Dubofsky reports that options existed in ancient Greece and Rome,

and that options were used during the tulipmania in Holland from1624-1636.In the U.S., options were traded as early as the 1800s and wereavailable only as customized OTC products until the CBOE openedon April 26, 1973.


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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 dateA 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 dateA call option is like a rain check. Suppose you spot an ad in thenewspaper for an item you really want. By the time you get to thestore, the item is sold out. However, the manager offers you a raincheck 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 equalto the sale price and an intrinsic value equal to the difference between the regular and sale prices. Note that you do not have touse the rain check. You do so only at your own option. In fact, ifthe price of the product is lowered further before you return, youwould let the rain check expire and buy the item at the lower price.


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Options are Contracts

The option contract specifies:The underlying instrument

The quantity to be deliveredThe price at which delivery occursThe date that the contract expires

Three parties to each contractThe BuyerThe Writer (seller)The Clearinghouse


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The Option Buyer

The purchaser of an option contract is buying the right toexercise the option against the seller. The timing of theexercise privilege depends on the type of option:

American-style options can be exercised any time beforeexpiration

European-style options may only be exercised during a shortwindow 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. Notethat the worst that can happen to an option buyer is thatshe loses 100% of the premium.


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The Option Writer

The writer of an option contract is accepting the obligation to havethe option exercised against her, and receiving the premium inreturn.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 theoption premium received. In some cases the potential loss is

(theoretically) unlimited.Writing and option contract is not the same thing as selling anoption. Selling implies the liquidation of a long position, whereasthe writer is a party to the contract.


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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 sellerto each buyer.Because the clearinghouse is well diversified andcapitalized, the other parties to the contract do not haveto worry about default. Additionally, since it takes theopposite 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.


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Examples of Options

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

There are options embedded in:Convertible bondsMortgagesInsurance contracts

Most corporate capital budgeting projectsetc.

Even stocks are options!


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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.


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Profits from Buying a Call

-1000-500

0500

1000150020002500

300035004000

0 20 40 60 80 100Stock Price at Expiration

P r o

f i t

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


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Selling a Call

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

0500

1000


P r o

f i t

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


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Profits from Buying a Put

-5000

500

1000150020002500

300035004000


P r o

f i t

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


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Selling a Put

-4000

-3500-3000

-2500-2000

-1500-1000

-5000

500


P r o

f i t

S = 50X = 50r = 5%t = 90 days

= 30%Put Price = 2.65


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Combination Strategies

We can construct strategies consisting ofmultiple options to achieve results that arent

otherwise possible, and to create cash flows thatmimic other securitiesSome examples:

Buy WriteStraddleSynthetic Securities


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The Straddle

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

If we sell a straddle, we profit if the stockdoesnt move much ineither directionThis straddle consists of

buying (or selling) botha put and call at themoney

-1000

-500

0

500

1000

1500

20002500

3000

3500

4000

0 20 40 60 80 100

Stock Price at Expiration

P r o f i t

Put Profit Call Profit Strategy Profit

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


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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 PutLong Call Buy Put, Buy StockLong Put Buy Call, Sell StockShort Stock Sell Call, Buy PutShort Call Sell Put, Sell StockShort Put Sell Call, Buy Stock

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


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Put-Call Parity

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

In other words, a put must sell for the same price as along call, short stock and lending the present value of thestrike price (why?).By manipulating this equation, we can see how to createsynthetic securities (in the above form it shows how tocreate a synthetic put option).

P C S Xe rt


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Put-Call Parity Example

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

Cash Flows At Expiration ifStock Price Is:

Action Cash Inflow 110 100 90

Buy 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


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Synthetic Long Stock Position

We can create asynthetic long

position in thestock by buying acall, selling a put,and lending thestrike price at the

risk-free rate untilexpiration -5000

-4000

-3000

-2000

-1000

0

10002000

3000

4000

5000

0 20 40 60 80 100


P r o f i t

Put Profit Call Profit Lend at Risk-free Strategy Profit


S C P Xe rt


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Synthetic Long Call Position

We can create asynthetic long

position in a call by buying a put, buying thestock, and

borrowing thestrike price atthe risk-free rateuntil expiration -5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 20 40 60 80 100


P r o f i t

Put Profit Stock Profit Borrow at Risk-free Strategy Profit

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

Put Price = 2.65

C P S Xe rt


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Synthetic Short Stock Position

We can createa syntheticshort position

in the stock by selling acall, buying a

put, and borrowing the

strike price atthe risk-freerate untilexpiration

-5000-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 20 40 60 80 100


P r o f i t

Put Profit Call Profit Borrow at Risk-free Strategy Profit


S P C Xe rt


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Synthetic Short Call Position

We can create asynthetic short

position in a call by selling a put,selling the stock,and lending thestrike price at the

risk-free rateuntil expiration -5000-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 20 40 60 80 100


P r o

f i t

Put Profit Stock Profit Lend at Risk-free Strategy Profit


C P S Xe rt


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Option Valuation

The value of an option is the present value of itsintrinsic value at expiration. Unfortunately,there is no way to know this intrinsic value inadvance.The most famous (and first successful) option

pricing model, the Black-Scholes OPM, wasderived by eliminating all possibilities ofarbitrage.

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


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Option Valuation Variables

There are five variables in the Black-ScholesOPM (in order of importance):

Price of underlying securityStrike priceAnnual volatility (standard deviation)Time to expiration

Risk-free interest rate


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Option Valuation Variables: Underlying Price

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

For a call option, the higher the price of theunderlying security, the higher the value of thecall.For a put option, the lower the price of theunderlying security, the higher the value of the

put.


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Option Valuation Variables: Risk-free Rate

The risk-free rate of interest is the least important of thevariables.It is used to discount the strike price, but because the

time to expiration is usually less than 9 months (with theexception of LEAPs), and interest rates are usually fairlylow, the discount is small and has only a tiny effect onthe value of the option.The risk-free rate, when it increases, effectivelydecreases the strike price. Therefore, when interest ratesrise, call options increase in value and put optionsdecrease in value.


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Note

The following few slides on the Black-Scholesmodel will not be tested. I consider the use of

these models to be beyond the scope of thiscourse.I am including this information only for thoseinterested.


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B-S Call Valuation Example

Assume a call with the following variables:S = 100 X = 100r = 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 . .. * .

d 1

100100 0 05 0 25 0 5 0 01 0 25

01 0 250275

ln . * . . * . * .

. * ..

d 2 0 275 01 0 25 0 225 . . * . .


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The Black-Scholes Put Valuation Model

At right is the Black-Scholes put valuationmodel.The variables are all thesame as with the callvaluation model.

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

P Xe Srt N d N d2 1

dS

Xrt t

t1

20 5

ln . s

s

d d t2 1 s


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FIN3600_15- Option Valuation

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