Financial Options
In Depth
Zachary Emig
MBA Class of 2005
Ross School of Business Finance Club
1
Today’s Agenda
I. What are options?
II. Payoff diagrams and risk profiles
III. Option pricing: the intuition
IV. Binomial models and Black-Scholes
V. Implied volatility
VI. Other terminology
VII. Uses of options
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What Are Options?
Options are securities which give the holder
the right, but not the obligation, to buy or
sell another instrument at a fixed price
before or at a fixed expiration date.
Options that grant the holder the right to
buy are call options; those that grant the
holder the right to sell are put options.
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What Defines Options?
What would you need to know in order to identify an
option?
1. Whether it is a Call or a Put.
2. What the underlying security is. Options can be
written on almost anything: stock prices, stock
indexes, FX rates, interest rates, etc.
3. The expiration date (tomorrow? Next month? Next
year?)
4. The fixed price (called the strike price or
exercise price) you can buy or sell at.
4
What Else Defines Options?
There are many different styles of options, related to
when you can exercise.
European option: holder can only exercise on the
expiration date. Most common type.
American option: holder can exercise anytime up
through the expiration date.
Bermudan option: holder can exercise every n
months up until the expiration date.
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Today’s Agenda
I. What are options?
II. Payoff diagrams and risk profiles
III. Option pricing: the intuition
IV. Binomial models and Black-Scholes
V. Implied volatility
VI. Other terminology
VII. Uses of options
6
Payoff Diagrams
When speaking about options, using
“hockey stick” payoff diagrams greatly
simplifies things.
An option’s payoff takes into account only
what the holder gets at expiration (or
when exercised), not what they paid for
the option up front.
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An Example: Union Pacific (UNP)
http://finance.yahoo.com/q/bc?s=UNP
• Closed at $64.69 on Nov. 12
• Considerable volatility over past year
• All examples represent European options
8
Payoff Diagrams
Payoff diagrams plot option holder’s payoff versus
the price of the underlying security. K=Strike Price.
Call Option, K=$75
Put Option, K=$55
Option Payoff
Option Payoff
$0
$0
$75
Underlying
Price
Why Once
exercise
UNP
call
is and
above
pay$75,
$75, when
call holder
you buy
willUNP
exercise
on
theoption
markettofor
buy
less
UNP
than
at $75.
$75?
$K
Underlying
Price
With a put, on the other hand,
But say
UNPexercise
drops toright
$45,tothe
holder
won’t
will exercise
sell
sellput
at holder
$55 if they
can sell to
UNP
at $55, afordifference
of $10.
$65.
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Terminology
• An option is said to be in the money if, were it able to
•
•
be exercised immediately, the payoff would be positive.
An option is at the money if the underlying is at the
strike price.
An option is out of the money if the holder wouldn’t
execute immediately, were they able to.
Call Option, K=$75
In the
money
Option Payoff
Option Payoff
Out of the
money
Put Option, K=$55
In the
money
Out of the
money
$0
$0
$75
At the
money
Underlying
Price
$55
At the
money
Underlying
Price
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Too Good To Be True?!?
Payoff diagrams suggest that holding options has no
downside. Is this possibly true?
Call Option, K=$75
Option Profit
Payoff
$0
Option
Premium
$75
Underlying
Price
Of course not. To buy a
call or put option,
investors must pay a
premium, which must
equal the value of option.
This is reflected by
changing the payoff
diagram to a P&L
daigram.
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Risk Profiles
Think about the risk profile for the 4 basic option positions.
Long Call Position
Long Put Position
Option
P&L
Option
P&L
$0
$0
$75
Underlying
Price
$55
Short Call Position
Underlying
Price
Short Put Position
$0
$0
Underlying
Price
Option
P&L
Option
P&L
$75
$55
Underlying
Price
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Today’s Agenda
I. What are options?
II. Payoff diagrams and risk profiles
III. Option pricing: the intuition
IV. Binomial models and Black-Scholes
V. Implied volatility
VI. Other terminology
VII. Uses of options
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Example: Option Pricing
• Let’s say you bought UNP back in August at $55,
•
•
and have seen a 19% gain since
Now you want to lock in some profits, and protect
yourself in case it plummets
How much would you pay for a Feb ’05 $60 put?
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What Does the Put Do For You?
You are here!
If UNP is still above
K=$60 in February, the
put expires worthless, and
all you’ve lost is the
premium you paid for it.
If UNP is at the strike
on expiration, there’s
no point in exercising;
the put expires
worthless.
But if UNP has fallen to
$55 by the end of Feb.
’05, you’ll be able to
exercise the put and sell
at $60, recouping $5 of
that loss.
A long put position like “insurance” for a long stock position.
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Think About It: What Factors Go
Into Pricing An Option?
• What changes would make this put option
more valuable?
• What would make it less valuable?
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Option Pricing Factors
1. How does option value move in relation to
underlying price?
For a put option, it’s value goes up when the
underlying price goes down: the holder can sell at
price K something available at the market for < K
(for a call, the opposite is true).
Option Payoff
$0
$K
Underlying
Price
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Option Pricing Factors
2. How does option value move in relation to strike
price?
For a put option, the lower the strike price, the lower
it’s value (for a call, the opposite is true). It is less
likely the put will end up below the strike price.
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Option Pricing Factors
3. How does option value move in relation to time to
expiration?
Whether it’s a put or call, the longer the time to
expiration, the more valuable the option is. This is
intuitive from the graph below; the stock has less
chance of dropping into the money by the end of
next week than it does by the end of Feb. 05.
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Option Pricing Factors
4. How does option value move in relation to
underlying volatility?
For a both calls and puts, the greater the underlying
stock’s volatility, the more likely that the option will
move into the money.
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Option Pricing Factors
5. How does option value move in relation to interest
rates?
Options provide potential future payouts to holders;
the present value of those potential payouts is
discounted by the interest rate, so the higher the
rate, the less valuable those payouts (whether a call
or put).
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Summary of Pricing Factor
Relationships
Call Value
Put Value
Underlying Price Positive
Negative
Strike Price
Negative
Positive
Time to
Maturity
Underlying
Volatility
Interest Rates
Positive
Positive
Positive
Positive
Negative
Negative
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Today’s Agenda
I. What are options?
II. Payoff diagrams and risk profiles
III. Option pricing: the intuition
IV. Binomial models and Black-Scholes
V. Implied volatility
VI. Other terminology
VII. Uses of options
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Option Pricing: Binomial Models
Basic method for option pricing is through Binomial
Models: assume stock can either go up or down.
Period 2
Stock Price:
Option Value:
Method also
provides
hedging ratios
as a
convenient
side effect.
Period 1
Stock Price:
Option Value:
Hedge Ratio :
B:
Period 0
Stock Price:
Option Value:
Hedge Ratio :
B:
Hedge Ratio :
-
B:
-
Payoff
$73.33
$6.42
0.7122
-45.8013
Period 2
Stock Price:
Option Value:
$64.71
$3.15
0.3958
-22.4636
Period 1
Stock Price:
Option Value:
$83.09
$13.09
$64.71
$0.00
Hedge Ratio :
-
B:
-
Payoff
$57.11
$0.00
Hedge Ratio :
0.0000
B:
0.0000
Period 2
Stock Price:
Option Value:
Hedge Ratio :
B:
$50.40
$0.00
-
Payoff
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-
Option Pricing: Black-Scholes
Black-Scholes is a formula that incorporates statistical
methods to more accurately determine option value;
it, and customized variants, are widely used
throughout the street.
Do you have to memorize the
Black-Scholes formula for
interviews? No, but you should
know the 5 inputs (s=stock price,
r=riskfree rate, x=strike price,
t=time to maturity, =volatility),
how they affect value.
http://www.riskglossary.com/articles/black_scholes_1973.htm
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Today’s Agenda
I. What are options?
II. Payoff diagrams and risk profiles
III. Option pricing: the intuition
IV. Binomial models and Black-Scholes
V. Implied volatility
VI. Other terminology
VII. Uses of options
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How to Find Option Prices
Yahoo! Finance lists options prices for different
maturities and strike prices.
Here are the Nov. 15 UNP call prices for Dec. 04, Jan.
05, and Feb. 05 expiration.
maturity
= increasing
value.
As expected, increasing strike
= decreasing
value.
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http://finance.yahoo.com/q/op?s=UNP
Implied Volatility
Think about it; Black-Scholes is:
Poption=f(Sstockprice,Kstrike,Tmaturity,Vvolatility,Rriskfreerate)
Set by
the
market
Set by
the
market
Contract
specific
Contract
specific
?
Set by the
market
Knowing 5 of the 6 properties above, we can easily
solve for volatility. This is called the implied
volatility of the stock, because it is the volatility the
market is pricing in for the next T years.
In I-Banks, often times options traders are called
“volatility traders”, since going long an option implies
you are going long vol.
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Volatility Trading
CBOE (Chicago Board Option Exchange) created
Volatility Index (VIX) in 1993. VIX
“measures the market's expectation of 30-day volatility, but in a way
that conforms to the latest thinking and research among industry
practitioners. The New VIX is based on S&P 500 index option prices
and incorporates information from the volatility "skew" by using a
wider range of strike prices rather than just at-the-money series.”
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20
15
10
5
http://www.cboe.com/micro/vix/historical.aspx
http://www.cboe.com/micro/vix/faq.aspx
Nov 04
Oct 04
Oct 04
Sep 04
Sep 04
Aug 04
Aug 04
Jul 04
Jul 04
Jul 04
Jun 04
Jun 04
May 04
May 04
Apr 04
Apr 04
Mar 04
Mar 04
Feb 04
Feb 04
Jan 04
Jan 04
0
Jan 04
VIX futures trade
at a value of 10x
the VIX index
value. Also
known as the
“fear gauge”.
Daily VIX Closing Prices
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Volatility Surfaces
Most trading desks get daily reports with the charts
of the vol surface (implied volatility for different
maturities and strike prices). What should that look
like?
Unlike what might
be expected,
implied vol is not
flat; it shows both
smile and skew.
Read Hull for more
details.
Volatility Surface
70%
60%
50%
40%
60
30%
http://www.riskglossary.com/articles/volatility_skew.htm
1
2
3
Months
6 12 18
24
45
Strike
Prices
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Today’s Agenda
I. What are options?
II. Payoff diagrams and risk profiles
III. Option pricing: the intuition
IV. Binomial models and Black-Scholes
V. Implied volatility
VI. Other terminology
VII. Uses of options
31
The Greeks
Traders will often talk
about “the Greeks” in
relation to securities
with optionality; what
do they mean?
No, the Greeks are five
letters used to describe an
option’s behavior. For
interviews, it’s probably
enough just to know what
each represents:
•Delta – Sensitivity of option to underlying’s price changes
•Gamma – Sensitivity of Delta to underlying’s price changes
•Vega – Sensitivity of implied vol to underlying’s price changes
•Theta – Sensitivity of option to the passage of time
•Rho – Sensitivity of option to interest rate changes
http://www.riskglossary.com/articles/greeks.htm
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How are the Greeks used?
Trading desks with complex option positions will have
daily reports on their exposures to different risks.
Volatility/Variance swap
trader: sample Vega report
Book: S&P Volatility Swaps
Current S&P:
Maturity
Position
Vega
Z4
$811,223.00
1.23
F5
$1,941,578.00
1.27
G5
$1,756,614.00
1.34
H5
$423,014.00
1.36
J5
($148,914.00)
1.37
K5
$2,141,048.00
1.39
M5
($570,140.00)
1.45
N5
($104,184.00)
1.50
Q5
$98,014.00
1.51
U5
$234,189.00
1.59
V5
$174,911.00
1.63
X5
($80,141.00)
1.67
Z5
$560,002.00
1.69
F6
$284,014.00
1.77
1,181.94
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Other Concepts: Intrinsic Value
Option prices can be decomposed into the option’s
intrinsic value and time value.
Intrinsic value is the payoff value of the option
were it possible to exercise immediately; if out of the
money, zero, otherwise the difference between the
strike and current stock price.
Example: The Jan’05, $40
strike call is selling for $24.70.
The closing UNP price was
$64.47. So $64.47$40.00=$24.47 is the intrinsic
value of the option.
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Other Concepts: Time Value
Example: The Jan’05, $40 strike call is selling for $24.70. The
closing UNP price was $64.47. So $64.47-$40.00=$24.47 is
the intrinsic value of the option. The remaining $0.23 of the
option’s price is it’s time value, i.e. what you’ll pay for the
chance that over the next 2 months it’s payoff will increase.
For a deep in the money call option, this is small compared to
it’s [current] intrinsic value.
In comparison, the out of the
money $70 option has $0
intrinsic value. So its $0.35
price is all time value, i.e. the
chance that over 2 months the
stock price will rise to above
$70.
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Today’s Agenda
I. What are options?
II. Payoff diagrams and risk profiles
III. Option pricing: the intuition
IV. Binomial models and Black-Scholes
V. Implied volatility
VI. Other terminology
VII. Uses of options
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What’s the Big Deal?
So, why are options so important?
Because of the breakthrough of Black-Scholes in
1973, we now have a standard, accurate way of
valuing them. Almost every security either has
optionality in or can be represented with options.
• Mortgage securities are straight debt with the
homeowner holding a call option (repayment).
• Convertible bonds are straight corporate debt with
the owner holding a call option on the stock.
• Common stock can be thought of as a call option
on a firm’s assets with strike price of $0!
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Option Strategies - Straddle
Options also allow investors to take “quirky” views on
underlying securities.
Pure volatility play;
investor neutral on stock
appreciation/depreciation.
Long Put P&L Long Call P&L
$0
View: Believe that by
January UNP will either go
up or down by a lot. Go
long a call and a put.
P&L
Example: Straddles
$65
Underlying
Price
Net P&L
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Option Strategies – Covered Call
In a covered call, investor expects mild stock
appreciation but not a large increase.
http://biz.yahoo.com/opt/education.html
Net P&L
$0
See Yahoo! Finance for all
the basic options
strategies.
P&L
Investor goes long the
stock and then writes
(sells) an OTM (out of the
money) call option for its
premium.
$K
Long Stock
P&L
Underlying
Price
Short Call
P&L
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Conclusion
•Nearly every trading desk I sat on this summer used
options or option pricing in some way.
•What to know for interviews? The underlying
concepts: risk profiles, inputs to Black-Scholes, the
Greeks, basic option strategies.
•The jargon is tricky. Traders now a days talk about
going “long gamma” or “long vega”. Focus on what
that actually means (“long vega”=“long
volatility”=“holding an option”?).
•Remember: almost any security can be represented
as an option.
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The Next Steps
If you are set on going into sales and trading, you
will have to know this stuff. Courses to take:
•FIN 580 – Options & Futures – Very good
introduction to valuing options
•FIN 618 – Derivatives – Builds upon FIN 580
•FIN 615/645 – Valuations/Adv. Valuations – Touches
upon real options
•FIN 622 – Corporate Financial Engineering – Discuss
uses of options and derivatives for solving corporate
financing needs.
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