OPTIONS
 An option is a contract giving the option purchaser (the buyer) the right, but not the
obligation, to buy or sell an asset for a specified price for a specified time period (until the
expiration date).
 There are two basic types of options:
 A call option grants the holder the right to purchase an asset at a specified price
(called the exercise or strike price) on or before some specified expiration date.
 A put option grants the holder the right to sell an asset at a specified price (the
exercise or strike price) on or before some specified expiration date.
 A buyer of an option is termed the holder; the seller of an option is referred to as the
writer or grantor.
American and European Options:
 An American option gives the buyer the right to exercise the option at any time between the
date of writing and the expiration or maturity date.
 A European option can be exercised only on the expiration date, not before.
 Exchange-traded options expire on the third Friday of the expiration month.
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PRICE ELEMENTS OF AN OPTION
Every option has three different price elements:
1. The exercise or strike price (X), which is the price at which the asset can be purchased
(call) or sold (put).
2. The underlying or actual price (S) in the market.
3. The premium, cost, price, or value of the option itself (usually paid in advance by the buyer
to the seller).
More Terminology
 An option whose exercise price is the same as the market price of the underlying
asset is said to be at the money (ATM).
 An option that would be profitable if exercised immediately is said to be in the
money (ITM).
 An option that would not be profitable if exercised immediately is referred to as
out of the money (OTM).
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CALL OPTIONS
 A call option is
 in the money if market price > exercise price (i.e. S>X)
o Call option premiums will be higher when (market price – exercise price) is
larger
 out of the money if market price < exercise price (i.e. S<X)
 Speculators who expect an asset's price to rise can purchase call options on that asset.
 Profit = Market price – (Exercise price + option premium)
 They may also sell (write) call options on an asset that they expect to depreciate.
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PUT OPTIONS
 A put option is
 in the money if market price < exercise price (i.e. S<X),
o Put option premiums will be higher when (exercise price – market price) is larger
 out of the money if market price > exercise price (i.e. S>X).
 Speculators who expect an asset's price to drop can purchase put options on that asset.
 Profit = Exercise price – (Market price + option premium)
 They may also sell (write) put options on an asset that they expect to increase in price.
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SUMMARIZING CALLS AND PUTS
Call holder
Call writer
Put holder
Put writer
Now
Pays the call premium and gets
the right to exercise
At Expiration
If S > X holder exercises
Net gain = S - X – C
If S < X holder does not exercise
Holder's loss = call premium
Receives the call premium and
agrees to deliver the asset at the
strike price if the holder
demands it any time before
expiration
If S < X holder does not exercise
Writer's gain = call
premium C
If S > X holder exercises
Writer’s net gain = X + C - S
Now
Pays the put premium and gets
the right to exercise
At Expiration
If S < X holder exercises
Net gain = X - S – P
If S > X holder does not exercise
Holder's loss = put premium
If S > X holder does not exercise
Writer's gain = put premium
If S < X holder exercises
Writer’s net gain = S + P - X
Receives the put premium and
agrees to buy the asset at the
strike price if the holder
demands it any time before
expiration
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ATTRACTIONS OF OPTIONS
 Options allow owners to change the payoff patterns of the underlying assets.
 Option owners can sell or exercise their options. They can also choose to let their options
expire. At most, they will lose the premiums they paid for their options.
 Example 1:
o Buying a call option allows the owner to realize the profit associated with a stock
without the high leverage associated with purchasing the stock
o Assume stock XYZ initially sells for S=$100. At the end of 6 months the stock is
worth $150. Purchasing the stock requires a $100 investment in order to gain the
$50 profit.
o If the owner had bought a call option that was at the money (i.e. X=$100 when
S=$100), the required investment would have been just the price of the call
option. This option could have been exercised at the end of the 6 months, giving
the holder $50 profit less the call premium.
o If the stock had decreased in value from the initial $100, the loss would have been
just the call premium.
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 Example 2:
o Buying a put option along with a share of stock guarantees minimum returns
from the stock investment
o Assume stock XYZ initially sells for S=$100. While the expected return is
positive, there is a risk that the stock’s price could decline over the next 6 months
when you plan to sell it. To insure against incurring an unacceptable loss, you
can purchase a put on the stock with X=$100. Should the stock price go down
then you would exercise the option and the maximum loss you would incur
when you purchase an option at the money would be P, the price of the put. If
the stock price goes up, then you would not exercise the option and the profit
would be the gain in the stock less P.
 Writers use options as a financial instrument where they hope that the option will
expire worthless and they will receive the premium as their profit
o Writing covered calls is one way to minimize the risk of writing a call:
 The seller writes a call on a stock that was previously purchased by the writer
at price Z<=S0. If the seller writes a call that is currently out of the money (so
X> S0>Z), there is a reasonable chance that the option will not be exercised by
the expiration date and the profit will be the call premium.
 If the stock price increases to a level that the call becomes in the money before
or at expiration (i.e. X< St ), then the holder will exercise the option and the
seller will have to sell the stock at X. The seller’s profit in this case is C + X – Z,
which is more profitable than the case where the seller did not already own the
stock and the profit would have been C + X – St.
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WHAT IS THE VALUE OF AN OPTION?
Intrinsic value: represents the payoff that could be obtained by immediate exercise of the option.
a.
Call: max (S - X, 0)
b.
Put: max (X - S, 0)
Determinants of Option Value given that there is still time to exercise the option:
1.
Variables relating to the underlying asset
a.
Current value (S)
b. Variance (2)
2.
Variables relating to the option characteristics
a.
Exercise price (X)
b. Time to expiration (t)
3.
Variables relating to financial markets
a.
Risk-free interest rate corresponding to life of option (r)
S
X
r
t

Factor
Higher
Higher
Higher
Longer
Higher
C
Higher
Lower
Higher
Higher
Higher
P
Lower
Higher
Lower
Higher
Higher
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BLACK-SCHOLES OPTION PRICING MODEL
• Assumes that stock prices are lognormally distributed.
distributed.
Then stock returns are normally
 The price of a call option (C) or put option (P) can be determined by the following formulae:
C  SNd1   Xe rt Nd 2 
P  Xe rt N d 2   SN d1 
where
and
ln S / X    r   2 / 2 t


d1 
 t
d 2  d1   t
 Xe  rt represents the present value of the exercise price (with continuous compounding).
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 You can view the N(d) terms as the risk-adjusted probabilities that the option will expire in the
money.
 Put-call parity says that a put and a call with the same expiration date t and exercise price X
written on the same asset will have the following relationships:
C  P  S  Xe rt
P  C  S  Xe rt
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INPUTS FOR BLACK-SCHOLES OPTION PRICING MODEL
 How to calculate time to expiration (t)?
1.
Count number of calendar days from current date to expiration date. Divide by
365.
2.
Count number of working days from current date to expiration date. Divide by #
working days per year.
 How to estimate risk-free rate?
Calculate the rate of return on U.S. Treasury bills
 How do we estimate volatility?
1.
Historical volatility – the volatility that is determined from a past series of prices
2.
Implied volatility – the volatility that makes the model price for an option equal to its
market price
3.
Expected volatility – the volatility that is anticipated by option traders
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