OPTIONS

1
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.

2
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.
3
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.
4
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

5
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

6
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 Max0, (Stock price - Exercise
  price)
• Put value Max0, (Exercise price - Stock
  price)

7
EXAMPLE Both Options on the same stock Sell a
put for 4 - strike 45 Buy a call for 2 -
strike 55 Assume that the stock price ends
at 58 premiums put 400
call -200 option values put
0 at expiration call 300 Net 500
8
Assume that the stock price ends at
40 premiums put 400 same as above
call -200 option value put -500 at
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.
9
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.
10

• 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

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CALL STRATEGIES

• Buy option alone
• not sure when stock will move gtuse long
  maturity
• most leveraged - high strike price
• Hedge
• if you own stock - sell stock - buy option
• if you don't own stock - short stock - buy
  option with high strike price
• Sell option
• naked - bearish on stock - receive premium

12
PUT STRATEGIES

• Buy option
• not sure when stock moves gt use long maturity
• most leverage - low strike price
• Hedge - own stock - buy put - protect profit
• Sell put naked - if bullish on stock

13
COMBINED STRATEGIES

• straddle - 1 call and 1 put
• strip - 1 call, 2 puts
• strap - 2 calls, 1 put
• money spread
• time spread

14
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.
15
USE OPTIONS TO CUT UP PRICE DISTRIBUTIONS - NOW
CALLED "FINANCIAL ENGINEERING"
16
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.
17
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
today r risk-free rate q the probability
the stock price will increase (1-q)
probability the stock price will decrease u
the multiplicative stock price increase (u gt 1
r gt 1) d multiplicative decrease (0 lt d lt 1
lt 1 r ) Cu call price if stock price
increases Cd call price if stock decreases
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The hedge ratio specifies how one assets price
moves for a given change in anothers.
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 )
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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,
20
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.
21
Example Suppose that a stocks price is S100
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 75 Cd Max 0, 50
125 0 m S(u d)/(Cu - Cd ) 100(2
.5)/(75 - 0) 2
22
Here, the option price moves half as much as the
stocks. 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 50 200 Sold 2 options 0
-150 50 50
23
Find the present value of the portfolios 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.
24
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 gt
C26.85.
25
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
• options exercise price
• risk-free rate
• option maturity
• stock price volatility.

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BLACK-SCHOLES MODEL - A NEARLY EXACT OPTION
PRICING MODEL
C0 P0N(d1) - E e-rt N(d2) where Price of
Stock P0 Exercise price E Risk free rate
r Time until expiration in years t Normal
distribution function N( ) Exponential
function (base of natural log) e
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Note Here the hedge ratio is represented by
N(d1) and N(d2) where where
Standard deviation of stocks return
s Natural log function ln
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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 stocks 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.
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TO GET THE VALUE OF THE CALL, C0

• EXAMPLE ASSUME
• Price of Stock P0 36
• Exercise price E 40
• Risk free rate r .05
• time period 3 mo. t .25
• Std Dev of stock return s .50
• Substitute into d1 and d2.

30

• 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

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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 t0 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
32

• 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 loss
• feature - this is why a premium is paid.
• The more volatile the stock the more valuable
  this
• feature is. (see futuresource.com SP
  volatility term structure).

33
HOW MARGIN WORKS
EXAMPLES USING MARGIN MONEY - ignore interest
Assume Required margin 60 Stock Price
75 Investment 30,000 Buy without margin
borrowing How many shares can you buy? -
30,000/75 400 Suppose price goes to 100
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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 gt you borrow 20,000 and buy
50,000/75 667 shares
35
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.
36
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
Falls Firm value 10mm 3mm Debt value
5mm 3mm EQUITY value 5mm 0mm