Chapter 10: Options Markets
Tuesday March 22, 2011
By Josh Pickrell
Options: An Introduction
• Options limit downside risk, while retaining
the potential for upside gain.
• Options are used as a risk management tool.
• An option’s value is derived from some
underlying asset.
Options: An Introduction
• An option contract gives its owner the right,
but not the legal obligation, to conduct a
transaction involving an underlying asset at a
predetermined future date (the exercise date)
and at a predetermined price (the exercise or
strike price).
– Options give the option holder the right to decide
whether or not the trade will eventually take place.
– The seller of the option has the obligation to
perform if the buyer exercises the option.
Options: An Introduction
• American options may be exercised at any time
up to and including the contract’s expiration date.
• European options can be exercised only on the
contract’s expiration date.
• Before the expiration date, an American option
and a European option on the same asset with the
same strike price may have different values.
– Will an American option be worth less than an
European option? Explain
Options: An Introduction
• A call option gives its owner the right to buy
the underlying asset at a specific price.
• A put option gives its owner the right to sell
the underlying asset at a specific price.
• For every buyer of an option, there must be a
seller.
– The seller of the option is also called the option
writer.
Options: An Introduction
• To acquire these rights, owners of options must
buy them by paying a price called the option
premium to the seller of the option.
– Listed stock options contracts trade on exchanges
are normally for 100 shares of stock.
– After issuance, stock options are adjusted for stock
splits but not cash dividends.
Digital Options
• A digital option has a payoff at maturity which
is either a fixed amount or zero. As with any
other investment, only two things can be done
with an option: buy or sell.
Digital Options
• Examples:
– The payoff for a digital call option is $1 if the
stock price rises to $10, and $0 if the stock price
never rises to $10.
– The payoff for a digital put option is $1 if the stock
price falls below $10, and $0 if the stock price fails
to fall below $10.
– Suppose that we combine the put and call options.
What does the position look like? Draw the
diagram and write out the vector notation.
Digital Options: Vector Notation
• Write out the vector notation for the following
positions: long stock, short stock, buy call,
write call, buy put, write put.
• The advantage of the vector notation is that it
is a very simple tool that can be used to
understand how to solve problems in managing
risky positions.
Spot Market
• Cash / Spot markets  physical market where
assets transact
– Prices change frequently
– Two positions can be taken in the spot markets:
• A long position: purchase of an asset (e.g. stock) with
the anticipation that the price will rise.
• A short position: sell an asset (e.g. stock) with the
anticipation that the price will fall and you will be able
to purchase at a lower price in the future.
Derivatives Market
• The price of a derivative instrument is based
on the underlying spot price of some specified
asset
– Derivatives enable us to make the short bet with
ease, whereas the spot market may be difficult to
short.
Option Markets
• Exchange-trade or listed options are regulated,
standardized, liquid, and backed by the Option
Clearing Corporation for Chicago Board
Options Exchange transactions.
– Most options have expiration dates within two to
four months of the current date.
Option Markets
• Over the Counter (OTC) options on stocks for
the retail trade all but disappeared with the
growth of the organized exchanges in 1971.
– There is now, however, an active market in OTC
options on currencies, swaps, and equities.
– These options are largely unregulated, custom, and
involves counterparty risk.
Standard Options
• Although digital options do exist, it is more
relevant to discuss the standard option where
the payoff is equal to the difference between
the value of the stock and the strike price.
• Draw the profit diagrams (hockey sticks) for
the following six positions:
– (1) long stock, (2) short stock, (3) long call, (4)
write call, (5) long put, (6) write put.
Standard Options: The Notation
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•
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St = the price of the underlying stock at time t
X = the exercise price of the option
T = the time to expiration
Ct = the price of a call option at time t
Pt = the price of a put option at time t
Rfr = the risk free rate
Standard Options: Value
• An options value is the sum of the time value
and the intrinsic value.
• The intrinsic value is defined as the value that
you would receive if you immediately
exercised the option.
• The value of a standard option can never be
less than zero.
Standard Options - Moneyness
• Moneyness refers to whether an option is inthe-money or out-the-money.
– If immediate exercise of the option would generate
a positive payoff, it is in the money.
– If immediate exercise would result in a loss
(negative payoff), it is out of the money.
– When the current asset price equals the exercise
price, meaning exercise will generate neither a
gain nor loss, the option is at the money.
Standard Options- Moneyness
• The following describes the conditions for a call
option:
– In the money call option: S – X > 0
– Out of the money call option: S – X < 0
– At the money call option: S = X
• The following describes the conditions for a put
option:
– In the money put option: X – S > 0
– Out of the money put option: X – S < 0
– At the money call option: S = X
Option Basics
• An options intrinsic value is the amount by
which the option is in the money.
– An option has zero intrinsic value if it is at the
money or out of the money, regardless of whether
it is a call or a put option.
– The intrinsic value of a call option:
• C = max[0, S – X]
– The intrinsic value of a put option:
• P = max[0, X – S]
Option Basics
• The time value of an option is the amount by
which the option premium exceeds the
intrinsic value and is sometimes called the
speculative value of the option.
• Option value = intrinsic value + time value.
Option Basics
• At any point during the life of an option
contract, its value will typically be greater than
its intrinsic value.*
– Because there is some probability that the stock
price will change in an amount that gives the
option a positive payoff at expatriation.
– For American options the longer the time to
expiration, the greater the time value and, other
thing equal, the greater the option’s premium
(price).
Option Basics
• Lower bound: Theoretically, no option will sell
for less than its intrinsic value and no option can
take on a negative value.
– Call option: Ct = max [0, St – X]
– Put option: Pt = max[0, X – St]
• Upper bound:
– Call option: the maximum value is at any time t is the
time-t share price of the underlying stock. (St)
– Put option: the price for an American put cannot be
more than its strike price. (X)
Options: Profit Equations
• (long) Call Option:
Cprofit = max(0, S – X) – premium
• (long) Put Option:
Pprofit = max(0, X – S) – premium
• You only need to memorize these two
equations. The short (or written) positions are
simply the above equations multiplied by a
negative sign.
Put-Call Parity
• The put-call parity shows that the value of a
put option is related to the value of the call
option.
– The parity is based on the efficient market concept
that two investments with same cash flows and
same risks must have the same price. (Keep this in
mind as we work thorough this parity).
Put-Call Parity
• Strategy 1: buy a share of the underlying asset
(S), and buy a put (P) on this stock with an
exercise price of X that expires at time T.
• Strategy 2: buy a call on asset (S) with an
exercise price of X that expires at time T and
buy a discounted risk-free bond where the
maturity is equal to X expiring at time T.
Put-Call Parity
• The Put-Call Parity:
S + P = C + PV(X)
• At matuirty there are three possible outcomes:
(1) S > X, (2) S < X, and (3) S = X.
– What is the value of each portfolio under these
three outcomes?
Put-Call Parity
• Under every possible outcome, the two
strategies yield exactly the same outcome.
Remember that the efficient market hypothesis
says that if two assets have the same payoff
structure, then they should have the same
price.
• We can use the put-call parity to create a
synthetic put option and a synthetic call option.
Option Pricing: Binomial Tree
• A binomial tree is based on the assumption that
asset prices will move up or down at known
points in time.
• Each node in the tree represents the stock at a
given price and at a given time.
Option Pricing: Black-Scholes Model
• Black and Scholes (1973) developed a
mathematical formula to price options. This
formula can be used to price call and put
options.
– Note: the original formula was developed to price
European options with no dividends.
• See the formula on Page 268.
Option Pricing
• The price of call (or put) option depends on:
– stock price
– time to expiration
– volatility
– risk-free rate
– strike price
– dividends
Option Strategies
• Lets look at several examples of option
strategies:
– Long stock + write a call (covered call)
– Long stock + buy a put (protective put)
– Long stock + short call + long put (costless collar)
– Short call + long call (x2) (Call back spread)