Options

1
Options

• Chapter 28

2
Background

• Put and call prices are affected by
• Price of underlying asset
• Options 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 assets price change does not
  impact the value of an option

3
Introduction 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 stocks 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

4
One-Period Binomial Call Pricing Formula

• Intrinsic ValueCall MAX0, 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 MAX0,50 - 40 10
• The stock price was 40.91 upon expiration the
  option would be valued at
• COPDown MAX0, 40.91 - 40 0.91

5
One-Period Binomial Call Pricing Formula

• 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)

6
One-Period Binomial Call Pricing Formula

• To find the options 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 (1risk-free rate
  amount borrowed)
• MAX 0, Price0Down exercise price COPDown
  ValueDown Down value of stock (1risk-free
  rate amount borrowed)
• Equations can be solved simultaneously to
  determine the hedge ratio

7
One-Period Binomial Call Pricing Formula

• 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 option

8
One-Period Binomial Call Pricing Formula

• The initial price (COP0) of the call option is
• Hedge ratio P0 B COP0
• Observations
• P0 is a major determinant of a call options
  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

9
Multi-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

10
Multi-Period Binomial Call Pricing Formula
11
Multi-Period Binomial Call Pricing Formula

• 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 Pascals triangle

12
Multi-Period Binomial Call Pricing Formula
Resembles a normal probability distribution as n
increases.
13
Multi-Period Binomial Call Pricing Formula

• Pascals triangle in tree form

14
Black and Scholes Call Option Pricing Model

• Black Scholes (BS) developed a formula to
  price call options
• Assume normally distributed rates of return

15
BS Call Valuation Formula

• Use a self-financing portfolio
• COP0 (P0h B)
• Assume a hedge ratio of N(x)
• Borrowings equal XPe(-RFR)dN(y)
• BS equation
• COP0 P0 N(x)- XPe(-RFR)dN(y)
• where

Fraction of year until call expires
Values of x and y have no intuitive meaning.
16
BS 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 BS model you need
• Table of natural logarithms (or a calculator)
• Table of cumulative normal distribution
  probabilities

17
Example

• 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

18
Example
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.

• 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

19
The Hedge Ratio

• Represents the fraction of a change in an
  options 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 BS 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

20
Risk Statistics and Option Values

• Investor normally estimates an assets standard
  deviation of returns and uses it as an input into
  the BS model
• However, can insert the calls current price into
  the model and compute the implied volatility of
  the underlying asset
• Risk statistics change over time

21
Put-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 Position Value When Options Expire
Portfolio of 3 positions in same stock If P lt XP If P gt 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
Portfolios two total values XP XP
Values are the same whether the stock is in or
out of the money when options expirethus
portfolio is perfectly hedged.
22
Put-Call Parity Formula

• This portfolio is worth the present value of the
  options exercise price or
• XP ? (1RFR)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 ? (1RFR)d

23
Pricing 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 ? (1RFR)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

24
Checking 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

25
Example

• Given information
• On 7/12/2000 KOs 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

26
The 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

27
The Effects of Cash Dividend Payments

• Impacts the value of an American call option

Price curve reflects the options price if it is
not exercised and not expired (alive).
If the options live value before ex-dividend gt
value ex dividend by more than dividend, call
should be exercised before it trades ex-dividend
to capture cash dividend (while embedded in
stocks price).
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 lt 1.
28
The Effects of Cash Dividend Payments

• The present value of the cash dividend payment
  should be considered in the BS option pricing
  model
• COPe P0 Div/(1RFR)N(x) XPe(-RFR)dN(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 1.8690.853 500.9770.796 10.69

The addition of the cash dividend has lowered
the call value by 1.60.
29
Options 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

30
Synthetic Positions Can Be Created 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

31
Example

• 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

32
Example

• 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 lt 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).
33
Synthetic Positions Can Be Created 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

34
Synthetic 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 gt 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

35
Writing Covered Calls

• Covered call
• Writing a call option against securities you
  already own
• Cover the writers 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

36
Writing Covered Calls

• 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

37
Writing Covered Calls

• 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

38
Straddles

• 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

39
Long 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

40
Short 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

41
Spreads

• 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

42
Spreads

• Diagonal spreads combine vertical and horizontal
  spreads
• Credit spreads
• Generate premium income exceeding related costs
• Debit spreads
• Generate an initial cash outflow

43
Strangles

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

Price of underlying asset Payoff from put Payoff from call Strangles total payoff
XPp ? P XPp P 0 XPp P
XPC gt P gt XPp 0 0 0
P gt XPC 0 P XPC P XPC
44
Strangles

• 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

45
Bull Spread

• Vertical spread involving two calls with same
  expiration date
• Debit transaction
• Used if believe price of underlying asset will
  rise, but not significantly

46
Bear 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

47
Butterfly Spreads

• Combination of a bull and bear spread on the same
  underlying security
• Long butterfly spread
• Will maximize profit if underlying assets price
  does not fluctuate from XPB
• Short butterfly spread
• Profitable if optioned asset experiences large up
  and/or down price fluctuations

48
The Bottom Line

• Binomial option pricing model
• Mathematically simple
• BS Option Pricing Model
• First closed-form option pricing model
• Binomial option pricing model is equivalent to
  BS if there are an infinite number of tiny time
  periods
• Put prices can be determined using put-call
  parity formula

49
The 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