Options

Francis & Ibbotson Chapter 28: Options 11

Slides by:

Pamela L. Hall, Western Washington University

OptionsOptions

Chapter 28


BackgroundBackground

Put and call prices are affected by– Price of underlying asset

– Option’s exercise price

– Length of time until expiration of option

– Volatility of underlying asset

– Risk-free interest rate

– Cash flows such as dividends

Premiums can be derived from the above factors Investors’ expectations about the direction of the

underlying asset’s price change does not impact the value of an option


Introduction to Binomial Option PricingIntroduction to Binomial Option Pricing

A simple valuation model is used to determine the price for a call option– Assumes only two possible rates of return over

time period• The price could either rise or fall

– For instance, if a stock’s price is currently $45.45 and it can change by either ±10% over the next period, the possible prices are

» $45.45 x 1.10 = $50» $45.45 x 0.90 = $40.91

• Ignores taxes, commissions and margin requirements• Assumes investor can gain immediate use of short sale

funds• Assume no cash flows are paid


One-Period Binomial Call Pricing FormulaOne-Period Binomial Call Pricing Formula

Intrinsic ValueCall = MAX[0, {Stock Price – Exercise Price}]– If the an option has an exercise price of

$40 and• The stock price was $50 upon expiration the

option would be valued at– COPUp = MAX[0,$50 - $40] = $10

• The stock price was $40.91 upon expiration the option would be valued at

– COPDown = MAX[0, $40.91 - $40] = $0.91



If we borrowed the money needed to purchase the optioned security at the risk-free rate– We would not need to invest any money to get

started (AKA: self-financing portfolio)• If the stock price rose the ending value of the portfolio

would be– VUp = Value of stock – (1+ risk-free)(amount borrowed)

• If the stock price fell the ending value of the portfolio would be

– VDown = Value of stock – (1+ risk-free)(amount borrowed)



To find the option’s price you must find the values for the amount of stock and borrowed funds that will equate COPUp and COPDown to ValueUp and ValueDown or– MAX [0, Price0Up – exercise price] = COPUp =

ValueUp = Up value of stock – (1+risk-free rate × amount borrowed)

– MAX [0, Price0Down – exercise price] = COPDown = ValueDown = Down value of stock – (1+risk-free rate × amount borrowed)

• Equations can be solved simultaneously to determine the hedge ratio



The hedge ratio represents the number of shares of stock costing P0 financed by borrowing B* dollars– This will duplicate the expiration payoffs

from a call optionUp Down

Up Down

Hedge Ratio COP COP

P P

Up DownDown Up

Up Down

B* Risk free rate

COP COPPPP P



The initial price (COP0) of the call option is

– Hedge ratio × P0 – B* = COP0

Observations– P0 is a major determinant of a call option’s initial

price

– The probability of the price fluctuations do not impact COP0

– Model is risk-neutral• Call has the same value whether investor is risk-averse,

risk-neutral or risk-seeking


Multi-Period Binomial Call Pricing FormulaMulti-Period Binomial Call Pricing Formula

One-period model can be used to – Encompass multiple time periods

– Value common stock, bonds, mortgages

What if stock currently priced at $45.45 could either rise or fall in value by 10% over each of the next two time periods





This concept can be extended to any number of time periods

Can add cash flow payments to the branches When a large number of small time periods

are involved, we obtain Pascal’s triangle



Resembles a normal

probability distribution as n

increases.



Pascal’s triangle in tree form


Black and Scholes Call Option Pricing ModelBlack and Scholes Call Option Pricing Model

Black & Scholes (B&S) developed a formula to price call options– Assume normally distributed rates of

return


B&S Call Valuation FormulaB&S Call Valuation Formula

Use a self-financing portfolio– COP0 = (P0h – B)

Assume a hedge ratio of N(x) Borrowings equal XP[e(-RFR)d]N(y) B&S equation

– COP0 = P0 N(x)- XP[e(-RFR)d]N(y)• where

( ) [ ]dσ

d0.5VAR(r)RFRXPPlnx

0.50 ++

=

Fraction of year until call

expires

0.5y x σd

Values of xand y haveno intuitivemeaning.


B&S Call Valuation FormulaB&S Call Valuation Formula

N(x) is a cumulative normal-density function of x– Gives the probability that a value less than x will

occur in a normal probability distribution

To use the B&S model you need– Table of natural logarithms (or a calculator)

– Table of cumulative normal distribution probabilities


ExampleExample

Given the following information, calculate the value of the call– P0 = $60

– XP (strike or exercise price) = $50– d (time to expiration) = 4 months or 1/3 of

a year– Risk-free rate = 7%– Variance (returns) = 14.4%


ExampleExample

Substituting the values for N(x) and N(y) into the COP0 equation– COP0 = $60(0.853) - $50(0.977)(0.796) = $51.18 - $38.89 = $12.29

ln($60/$50) [0.07 0.5(0.144)](0.333) 0.2296 1.04825

(0.3794)(0.5773) 0.2190x

1.04825 (0.3794)(0.5773) 0.829y Looking this

value up in the table yields an N(x) of 0.853.Looking this

value up in the table yields an N(y) of 0.796.


The Hedge RatioThe Hedge Ratio

Represents the fraction of a change in an option’s premium caused by a $1 change in the price of the underlying asset– AKA delta, neutral hedge ratio, elasticity, equivalence ratio– Calls have a hedge ratio between 0 and 1

Hedgers would like a hedge ratio that will completely eliminate changes in their hedged portfolio– Is presented as N(x) in the B&S equation

• If x has a value of 1.65, N(x) has a value of 0.9505– Means that 95.05 shares of a stock should be sold short to

establish a perfect hedge against 100 shares in an offsetting position


Risk Statistics and Option ValuesRisk Statistics and Option Values

Investor normally estimates an asset’s standard deviation of returns and uses it as an input into the B&S model– However, can insert the call’s current

price into the model and compute the implied volatility of the underlying asset

• Risk statistics change over time


Put-Call Parity FormulaPut-Call Parity Formula

Formula represents an arbitrage-free relationship between put and call prices on the same underlying asset– If the two options have identical strike prices and times to

maturity

Consider the following:

Portfolio of 3 positions in same stock

Position Value When Options Expire

If P < XP If P > XP

1) Long position on underlying stock P P

2) Short position in call option (sell call) 0 XP – P

3) Long position in put option (buy put) XP – P 0

Portfolio’s two total values XP XP

Values are the same whether

the stock is in or out of the money

when options expire—thus portfolio is perfectly hedged.


Put-Call Parity FormulaPut-Call Parity Formula

This portfolio is worth the present value of the option’s exercise price or– XP (1+RFR)d under either outcome

The portfolio must also be worth– P + POP – COP

This leads to the Put-Call Parity equation– P + POP – COP = XP (1+RFR)d


Pricing Put OptionsPricing Put Options

We can use put-call parity to value a put after the value of a call on the same security has been determined– POP= COP + (XP (1+RFR)d) – P

• Example: Calculate the price of a put option on a stock with a current price of $60, a strike price of $50, 4 months remaining until expiration, a risk-free rate of 7% and a variance of 14.4% with a call valued at $12.29

– POP = $12.29 + (50 (1.07)0.333)-$60 = $1.18


Checking Alignment of Put and Call PricesChecking Alignment of Put and Call Prices

When prices for both puts and calls on the same underlying stock are available– Put-call parity can be used to determine if

the prices are properly aligned• If not, arbitrage profits can be earned


ExampleExample

Given information– On 7/12/2000 KO’s stock was selling for $57– Call options with a strike price of $60 and one month until

expiration were selling for $1.625– Puts were selling for $4.125– 3-month T-bills were yielding 6%

Plugging data into the put-call parity equation– 4.125 1.625 + 60/1.060.0833 – 57 – 4.125 4.3344

• Either puts were under priced by 21¢ or calls were over priced by 21¢

Ignores transaction costs


The Effects of Cash Dividend PaymentsThe Effects of Cash Dividend Payments

Ex-dividend date– First trading day after the cash dividend is

paid– Stock trades at a reduced price

• Reduced by the amount of the cash dividend– Stockholders are no longer entitled to the dividend,

therefore they should not pay for it

The ex-dividend stock price drop-off – Reduces value of call options– Increases value of put options



Impacts the value of an American call option

On the ex-dividend date the stock price drops from Pd to Pe.

Option prices usually do not drop by the same amount because the slope of

the price curve < +1.

Price curve reflects the option’s price if it is not exercised and not

expired (alive).

If the option’s live value before ex-

dividend > value ex dividend by more than dividend, call should be exercised before it trades ex-dividend to capture

cash dividend (while embedded in

stock’s price).



The present value of the cash dividend payment should be considered in the B&S option pricing model– COPe = [P0 – Div/(1+RFR)]N(x) – XP[e(-RFR)d]N(y)

– Example• P0 = $60

• XP (strike or exercise price) = $50

• d (time to expiration) = 4 months or 1/3 of a year

• Risk-free rate = 7%

• Variance (returns) = 14.4%

• Expected cash dividend of $2 in one year– Present value of dividend = $2/1.07 = $1.869

– COPe = [60 – 1.869]0.853 –50[0.977]0.796 = $10.69

The addition of the cash dividend

has lowered the call value by $1.60.


Options MarketsOptions Markets

Chicago Board Options Exchange (CBOE)– Founded in 1973 but is now the largest options exchange in

world

American Stock Exchange– Second largest options exchange

Many options transactions are cleared through– Options Clearing Corporation (OCC)

International Securities Exchange (ISE)– Opened in 2000

– Electronic exchange• Competes with CBOE, AMEX, PHIX, PSE


Synthetic Positions Can Be Synthetic Positions Can Be Created From OptionsCreated From Options

Buying a call and selling a put on the same security– Creates the same position as a buy-and-

hold position in the security• AKA synthetic long position


ExampleExample

Given information– Phelps’ stock is currently trading for $40 a

share• You buy a six-month call with a $40 exercise

price for a $5 cost

• You write an 8-month put with an exercise price of $40 for $5 in premium income


ExampleExample

Contrasting the actual and synthetic long positions Possible Price

of stock at option

expiration

Results from 6-month call

with $50 exercise price

Results from $5 put written with

$50 exercise price

Result from combined

option positions

Result from long position in stock (100

shares)

$30 -500 -500 -1,000 -1,000

$35 -500 0 -500 -500

$40 -500 +500 0 0

$45 0 +500 +500 +500

$50 +500 +500 +1,000 +1,000

$55 -1,000 +500 +1,500 +1,500

If the call and put prices , the

synthetic position the actual

position.

Put-call parity shows that the price of a put must be < the price of a similar call. Thus, to make the put price = call price, put had to have a longer time to expiration (8 months vs. 6 months).


Synthetic Positions Can Be Synthetic Positions Can Be Created From OptionsCreated From Options

Some investors prefer a synthetic long position to an actual long position– Requires smaller initial investment

• Creates more financial leverage

– Owner of a synthetic long position does not collect cash dividends or coupon interest from underlying securities as they do not actually own those securities

– Also, when options expire additional premiums must be paid to re-establish position


Synthetic Short PositionSynthetic Short Position

Can create a synthetic short position by– Selling (writing) a call and simultaneously buying a put with a

similar exercise price on the same underlying stock

Superior to an actual short position in the stock– The premium income from selling the call should be > premium

paid to buy the put

– Requires a smaller initial investment than an actual short sell

– Does not have to pay cash dividends on the optioned stock

Disadvantages of a synthetic short position– After expiration of option more money would have to be spent to

re-establish position

– Could accumulate unlimited losses if the stock price rose high enough


Writing Covered CallsWriting Covered Calls

Covered call– Writing a call option against securities you already

own• Cover the writer’s exposure to potential loss

If call owner exercises the option– Option-writer delivers the already owned securities

without having to buy them in the market

Not all covered call positions are profitable– If stock price falls

• Long position in underlying stock decreases• However, receive call premium income



Naked call writing– Occurs when call writer does not own the

underlying security• Risky if the price of the underlying security

increases

Initial margin of 15% or more required– Whereas a covered option writer does not

have to put up extra margin to write a covered call



Covered call writers– Gain the most when stock price remains at

exercise price and option expired unexercised

• Receive premium income and get to keep the stock

– If stock price increases significantly would have been better off not having written the option

• Will have to give security to exerciser


StraddlesStraddles

Straddle occurs when – Equal number of puts and calls are bought on the

same underlying asset• Must have same maturity and strike price

Long straddle position– Profit if optioned asset either

• Experiences a large increase in price• Experiences a large decrease in price• Experiences large increases and decreases in price

Useful for a stock experiencing great deal of volatility


Long Straddle PositionLong Straddle Position

Infinite number of break-even points for a long straddle position– Downside limit

• Sum of put and call prices

– Upside limit• Sum of put and call prices

Believe the underlying stock has potential for enough price movements to make the straddle profitable before expiration– Only a small probability of losing the aggregate

premium outlay


Short Straddle PositionShort Straddle Position

Symmetrically opposite to long straddle position

Believe stock price will not vary significantly before options expire– Probability that straddle will keep 100%

of premium income is small


SpreadsSpreads

The purchase of one option and sale of a similar but different option– Can be either puts or calls but not puts and

calls– Spread can occur based on

• Different strike prices (vertical spreads)

• Different expirations (horizontal spreads)

• Time spreads, calendar spreads


SpreadsSpreads

Diagonal spreads combine vertical and horizontal spreads

Credit spreads– Generate premium income exceeding

related costs

Debit spreads– Generate an initial cash outflow


StranglesStrangles

Involves a put and call with same expiration date but different strike prices– Involves smaller total outlay than a straddle

Price of underlying

assetPayoff

from putPayoff from

callStrangle’s

total payoff

XPp P XPp – P 0 XPp – P

XPC > P > XPp

0 0 0

P > XPC 0 P – XPC P – XPC


StranglesStrangles

Long strangle– Debit transaction

• No premiums from writing options are received

Short strangle– Credit transaction

• No outlays• Small premiums received but also small

chance options will be exercised against writer


Bull SpreadBull Spread

Vertical spread involving two calls with same expiration date– Debit transaction– Used if believe price of underlying asset

will rise, but not significantly


Bear SpreadBear Spread

Vertical spread involving two puts with same expiration date but different strike prices– Are profitable only if asset price declines

between the two exercise prices• Losses are limited if expectations are

incorrect


Butterfly SpreadsButterfly Spreads

Combination of a bull and bear spread on the same underlying security– Long butterfly spread

• Will maximize profit if underlying asset’s price does not fluctuate from XPB

– Short butterfly spread• Profitable if optioned asset experiences large

up and/or down price fluctuations


The Bottom LineThe Bottom Line

Binomial option pricing model– Mathematically simple

B&S Option Pricing Model– First closed-form option pricing model

• Binomial option pricing model is equivalent to B&S if there are an infinite number of tiny time periods

Put prices can be determined using put-call parity formula


The Bottom LineThe Bottom Line

Ex-dividend stock price drop-off decreases (increases) value of a call (put) option

Puts and calls can be assembled to build more complex investing positions– Can build a position that will allow investor to benefit if

price of underlying asset• Rises• Falls • Fluctuates up and down• Never changes

Options allow us to analyze securities in ways we might not have originally realized

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