Math 476 / 567
Actuarial Risk Theory
Fall 2015
University of Illinois at Urbana-Champaign
Professor Rick Gorvett
Options and Put-Call Parity
August 27, 2015
Why is this Option Stuff So Important?
Payoff
Of Call
Option
X
(Exercise price)
ST (Value of
Underlying Asset)
Insurance is an Option
Payment
Under
Insurance
Policy
X
(Deductible)
ST (Size of Loss)
Insurance is an Option
Payment
Under
Insurance
Policy
Ded.
ST (Size of Loss)
Policy Limit
A Sampling of
Options
and Other Derivatives
through History
Ancient Greece
“There is the anecdote of Thales the Milesian and his
financial device… He was reproached for his poverty,
which was supposed to show that philosophy was of no
use. According to the story, he knew by his skill in the
stars while it was yet winter that there would be a great
harvest of olives in the coming year; so, having a little
money, he gave deposits for the use of all the olivepresses in Chios and Miletus, which he hired at a low
price because no one bid against him. When the
harvest-time came, and many were wanted all at once
and of a sudden, he let them out at any rate which he
pleased, and made a quantity of money. Thus he
showed the world that philosophers can easily be rich if
they like, but that their ambition is of another sort…”
- Aristotle, Politics, Book One, Part XI
Phoenician Shipping
Merchants and ship-owners used options to
hedge their ships and cargoes
Mesopotamia
Mercantile forward contracts, written in
cuneiform on clay tablets, circa 1700 BC
China
Forward contracts on rice, entered into prior
to planting, circa 2000 BC
Belgium and The Netherlands
• Antwerp and Amsterdam
• Grain
• Herring
• Tulips
Tulip Bubble
• Mid-1630s
• Tulip demand exploded and prices skyrocketed
• Options and futures were used to ensure price
and supply
• Bubble burst in 1637
America
• 19th century
– “Privileges”
– Non-standardized / over-the-counter
• Synthetic loans
– Financier Russell Sage
– Put-call parity
– Get around usury laws
Chicago Board Options Exchange
• Began trading standardized options on April 26,
1973
• 911 contracts traded on first day (options on 16
different “underlying” companies)
CBOE and Options
“…any history of the excitement in finance in the
1960s and 1970s must mention the options pricing
work of Black and Scholes (1973) and Merton
(1973b). These are the most successful papers in
economics – ever – in terms of academic and
applied impact. Every Ph.D. student in economics is
exposed to this work, and the papers are the
foundation of a massive industry in financial
derivatives.”
- Eugene F. Fama, “My Life in Finance,” arXiv
Modeling Underlying Assets
“…distributions of stock returns are fat-tailed: there
are far more outliers than would be expected from
normal distributions – a fact reconfirmed in
subsequent market episodes, including the most
recent. Given the accusations of ignorance on this
score recently thrown our way in the popular media,
it is worth emphasizing that academics in finance
have been aware of the fat tails phenomenon in asset
returns for about 50 years.”
- Eugene F. Fama, “My Life in Finance,” arXiv
Options
and their
Characteristics
A Type of Derivative
• A forward is the obligation to buy or sell
something at a pre-specified time and at a
pre-specified price
• An option is the right to buy or sell
something at a pre-specified time (or during
a pre-specified time-period) and at a prespecified price
Types of Options
• Call option: the holder has a right to buy
the underlying asset
• Put option: the holder has a right to sell the
underlying asset
• Counterparties: parties to the option
agreement
• One can buy (long) or sell (short) an option
Question # 1
• Abby agrees to buy Ben’s car for $1,000 three
months from now
Abby’s position: ________ _________
(long or short)
(forward or option)
Ben’s position: ________ _________
(long or short)
(forward or option)
Question # 2
• Abby agrees to sell Ben her car for $500 three
months from now, if it is worth less than $500
Abby’s position: ________ _________
(long or short)
Ben’s position:
(forward or option)
________ _________
(long or short)
(forward or option)
“Parameters” of Options
• Exercise price = strike price = price at
which the holder of the option can exercise
the option (and thus buy or sell the
underlying asset)
• Premium = amount paid for the option
• Expiration date
• “Style”
– American option: can exercise any time up to and
including expiration date
– European option: can exercise only on expiration date
Examples of Options They’re Everywhere
• Traded options
– On stocks, indices, FX, interest rates, futures,
swaps, options,...
•
•
•
•
Warrants
Convertible bonds
Call provisions on bonds
On projects
– To expand, abandon, postpone
• Insurance
Value Of Options At
Expiration
C = Max [ST - X, 0]
C = Call option value (or payoff) at expiration
ST = Price of underlying asset at expiration
X = Exercise price
P = Max [X - ST, 0]
P = Put option value (or payoff) at expiration
Question # 3
• Abby sells Matthew a January European call option on
one share of ABC stock
• Suppose ABC stock is initially trading at 32.5
• Exercise price = 35
• Premium = 3
• In January, suppose:
ST=30
ST=40
(Total payoff [profit/loss])
Abby:
___ [ __ ]
___ [ __ ]
Matthew:
___ [ __ ]
___ [ __ ]
Option Values
• Prior to expiration:
– In-the-money
– At-the-money
– Out-of-the-money
Call
St > X
St = X
St < X
Put
St < X
St = X
St > X
• Intrinsic value: profit that could be made if the
option was immediately exercised
– Call: stock price - exercise price
– Put: exercise price - stock price
• Time value: the difference between the option price
and the intrinsic value
Option Values: Payoff Charts
• Call -- long position:
Payoff
ST
X
• Call -- short position:
X
• Put -- long position:
• Put -- short position:
X
X
ST
ST
ST
Payoff vs. Profit/Loss:
Long a Call Option
Payoff
Profit/Loss
ST
Call
Premium
X
Purposes of Derivatives
• Speculative
– Highly risky
– Highly leveraged
– Very volatile
• Hedging
– Combine with other securities
– Hedge (minimize) risk from other securities
Hedging
• “Hedge”: Take a position that offsets a
risk
• “Risk”:
“Uncertainty” regarding the value of
the underlying asset
• By hedging, one changes the risk inherent in
owning the underlying asset
• The return distribution of the underlying asset is
not changed
“Risk” vs “Uncertainty”
• “The term ‘risk,’ as loosely used in everyday speech and in
economic discussion, really covers two things which,
functionally at least, in their causal relations to the phenomena
of economic organization, are categorically different…. The
essential fact is that ‘risk’ means in some cases a quantity
susceptible of measurement, while at other times it is something
distinctly not of this character…. It will appear that a
measurable uncertainty, or ‘risk’ proper, as we shall use the
term, is so far different from an unmeasurable one that it is not
in effect an uncertainty at all. We shall accordingly restrict the
term “uncertainty” to cases of the non‐quantit(at)ive type.”
- Frank Knight, Risk, Uncertainty, and Profit, 1921
Using Options to Hedge
• Combine the underlying asset with an
option or options
• Can reduce or eliminate downside risk
while retaining upside potential
• Can protect against falls in held asset
values, or against increases in input prices
Option Strategies
• Protective put
– Own stock (long position)
– Own put (long position)
• Covered call
– Own stock (long position)
– Sell call (short position)
• Straddle
• Spread
Protective Put
• Investor owns asset
• Investor also buys (holds) a put on the asset
• Guarantees investment portfolio proceeds are
at least equal to the exercise price of the put
+
=
Question # 4
• Suppose you own a share of stock, and you
purchase a put option with an exercise price of
22.5 on that stock, for a premium of $ 0.75
ST :
Premium:
30
25
20
15
____
____
____
____
____
=====
____
____
=====
____
____
=====
____
Put Payoff: ____
=====
Overall:
____
Covered Call
• Investor purchases stock
• Investor also sells (writes) a call option on the
stock
• Option position is “covered” by owning the
underlying stock itself
• (vs. “naked option”)
• Provides additional (premium) income
+
=
Question # 5
• Suppose you own a share of stock, and you write
a call option with an exercise price of 35 on that
stock, for a premium of $ 2.00
ST :
Premium:
30
35
40
45
____
____
____
____
____
=====
____
____
=====
____
____
=====
____
Call Payoff: ____
=====
Overall:
____
Straddle
• (Long) Straddle: buy both a call and a put
on a stock
• Each option has the same exercise price and
expiration date
• Believe stock will be relatively volatile
• Worst-case: no movement in stock price
Spread
• Combination of options
– Two or more calls, or
– Two or more puts
• Horizontal spread: sale and purchase of options
with different expiration dates
• Vertical spread: simultaneous sale and purchase
of options with different exercise prices -- e.g.,
X2
X1
+
=
X1
X2
A “Spread” in the Context of
Insurance
Insurance
Policy Limit
Recovery
From Loss
Size of Loss
Ded.
Ded. +
Policy
Limit
“Exotic” Options
• Certain characteristics of “plain vanilla”
options are adjusted to produce “exotic
options”
• Some characteristics of plain vanilla
options:
– American or European
– Linear payoff
– Does not “disappear”
– Value of underlying at exercise
Put-Call Parity
General Concept
• Arbitrage implies a certain relationship between
put, call, and underlying asset prices
• Assuming same exercise prices and expiration
dates, and non-dividend-paying stock, two
portfolios have, at payoff, identical values:
– One European call option + cash of PV(X)
– One European put option + one share of stock
• C + PV(X) = P + S
Put-Call Parity
Specific Relationships
• McDonald text terminology:
Call – put = PV(forward price – strike price)
C( K , T )  P( K , T )  erT ( F0,T  K )
• For non-dividend-paying stock:
C ( K , T )  P( K , T )  S 0  e  rT K
Put-Call Parity
Specific Relationships (cont.)
• For dividend-paying stock:
C( K , T )  P( K , T )  S0  PV0,T ( Div )  e
• For index options:
C ( K , T )  P( K , T )  e T S 0  e  rT K
 rT
K
Put-Call Parity Example
• Find the value of the one-year (T = 1) put
with exercise price (X) of 110 when
C(110,1) = 10.16
S0 = 100
=0
r = 0.10
P(110,1) = 10.16 + 110 e -.10 × 1 - 100 = 9.69
Put-Call Parity Example (cont.)
• Find the value of the one-year (T = 1) put
with exercise price (X) of 110 when
C(110,1) = 10.16
S0 = 100
 = 0.02 (2% div yield)
r = 0.10
P(110,1) = 10.16 + 110 e -.10 × 1 - 100 e -.02 × 1
= 11.67