Option Markets 232D
1. Introduction
Daniel Andrei
Fall 2012
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My Background
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MScF 2006, PhD 2012. Lausanne, Switzerland
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Since July 2012: assistant professor of finance at UCLA Anderson
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I conduct research in the area of theoretical asset pricing, with a
special focus on the role of information in financial markets
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More information about me at www.danielandrei.net
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Outline
I Class Organization
II Introduction to Derivatives
Definitions
Buying and Short-Selling
Continuous Compounding
Forward Contracts
Call Options
Put Options
Moneyness, Put-Call Parity
Options Are Insurance
Various Strategies
4
10
11
17
22
27
33
40
46
56
58
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Outline
I Class Organization
II Introduction to Derivatives
Definitions
Buying and Short-Selling
Continuous Compounding
Forward Contracts
Call Options
Put Options
Moneyness, Put-Call Parity
Options Are Insurance
Various Strategies
4
10
11
17
22
27
33
40
46
56
58
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Useful Information
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Class materials:
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My contact information:
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Slides: Class website on myAnderson
Textbook: Robert McDonald, Derivatives Markets, Pearson Addison
Wesley, second edition, 2006.
Office: C4.20
Office hours: Thursday (Sep 27 - Dec 6), 4-5 pm or by appointment
Phone: (310) 825-3544
Email: daniel.andrei@anderson.ucla.edu
TA: Nimesh Patel
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Office: C4.01
Office hours: Room C4.14, Tuesday (Oct 2 - Dec 11), 5-6 pm or by
appointment
Email: nimesh.patel.2015@anderson.ucla.edu
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Schedule: Thursdays, 8:30–11:20 AM, Cornell D-313
Week
Week
Week
Week
Week
Week
Week
Week
1:
2:
3:
4:
5:
6:
7:
8:
Sep. 27
Oct. 4
Oct. 11
Oct. 18
Oct. 25
Nov. 1
Nov. 8
Nov. 15
Week 9: Nov. 29
Week 10: Dec. 6
Introduction to derivatives
Chapters 1, 2, 3, 9,
Appendix B
Binomial option pricing
Chapters 10, 11
Black-Scholes, Greeks
Chapters 12, 13
Forwards, futures, and swaps
Chapters 5, 6, 7, 8
Thanksgiving: Thursday, Nov. 22
Volatility risk
Chapters 11, 12, 13
Risk management, other topics Chapters 3, 4, 15, ...
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Midterm: Week 7 (Thursday, Nov. 8), 1.5 hours, open book exam
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Final exam: Thursday, Dec. 13, D-313, 1.5 hours, open book exam
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Evaluation
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There will be five quizzes, available via Equiz.me
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To register, please follow this link. Please use an email address that
explicitely shows your name (preferably your UCLA Anderson email
address).
Once links are posted, they will be available online for two weeks.
Once available, you can solve them as many times as you want. I
strongly advise working in groups, but you will have to submit your
answers individually.
Grade formula: 40% final exam, 30% midterm, 20% quizzes, 10%
class participation.
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In addition to enrolling through the proper authorities,
please send me an email with the following information:
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name
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program and year in program
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major (if any)
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your background in finance and mathematics
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telephone number
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any other information you consider important
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Ground Rules
These rules help ensure that no one interferes with the learning of another:
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I will start promptly and I expect you to be on time
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If, because of some unusual circumstance, you come in late, please
enter as quietly as possible
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If it is necessary for you to leave early, please sit next to a door
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You may leave the room briefly if it is an emergency
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As a general rule, you may not use notebook computer or other
handheld computing or communication devices in class
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Outline
I Class Organization
II Introduction to Derivatives
Definitions
Buying and Short-Selling
Continuous Compounding
Forward Contracts
Call Options
Put Options
Moneyness, Put-Call Parity
Options Are Insurance
Various Strategies
4
10
11
17
22
27
33
40
46
56
58
10 / 67
What is a Derivative?
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Definition
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Types
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An agreement between two parties which has a value determined by
the price of something else.
Options, futures, and swaps.
Uses
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Risk management
Speculation
Reduce transaction costs
Regulatory arbitrage
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Three Different Perspectives
Economic
observers:
I Regulators
End-users:
I Researchers
I Corporations
Intermediaries:
I Market-makers
I Traders
End-users:
I Investment managers
End-users:
I Investors
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Financial Engineering
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The construction of a financial product from other products
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New securities can be designed using existing securities
Financial engineering principles
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Facilitate hedging of existing positions
Enable understanding of complex positions
Allow for creation of customized products
Render regulation less effective
Examples:
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In 2002, Universal Studios issued a $175 million catastrophe bond to
cover its production studios against an earthquake in Southern
California
In 2003, Walt Disney issued a catastrophe bond to cover its theme
park in Japan
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Contracts outstanding (millions)
Exchange Traded Derivatives
200
100
0
85
1,9
1,9
90
95
1,9
2,0
00
05 ,010
2
2,0
source: Bank for International Settlements, Quarterly Review, June 2012.
http://www.bis.org/statistics/extderiv.htm
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Notional amount (billions)
Exchange Traded Derivatives (cont’d)
80,000
60,000
40,000
20,000
0
85
1,9
1,9
90
95
1,9
2,0
00
05
2,0
10
2,0
source: Bank for International Settlements, Quarterly Review, June 2012.
http://www.bis.org/statistics/extderiv.htm
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Notional amount (billions)
Over-the-Counter (OTC) Derivatives
600,000
400,000
200,000
98 00 02 04 06 08 10 12
1,9 2,0 2,0 2,0 2,0 2,0 2,0 2,0
source: Bank for International Settlements, Quarterly Review, June 2012.
http://www.bis.org/statistics/derstats.htm
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Buying and Selling a Financial Asset
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Brokers: commissions
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Market-makers: bid-ask spread (reflects the perspective of the
market-maker)
The price at which
you can buy
The price at which
you can sell
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ask (offer)
bid
What the marketmaker will sell for
What the marketmaker pays
Example: Buy and sell 100 shares of XYZ
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XYZ: bid=$49.75, ask=$50, commission=$15
Buy: (100 × $50) + $15 = $5, 015
Sell: (100 × $49.75) − $15 = $4, 960
Transaction cost: $5, 015 − $4960 = $55
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Problem 1.4: ABC stock has a bid price of $40.95 and an
ask price of $41.05. Assume that the brokerage fee is
quoted as 0.3% of the bid or ask price.
a. What amount will you pay to buy 100 shares?
b. What amount will you receive for selling 100 shares?
c. Suppose you buy 100 shares, then immediately sell 100 shares. What is
your round-trip transaction cost?
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Problem 1.4: ABC stock has a bid price of $40.95 and an
ask price of $41.05. Assume that the brokerage fee is
quoted as 0.3% of the bid or ask price.
a. What amount will you pay to buy 100 shares?
($41.05 × 100) + ($41.05 × 100) × 0.003 = $4, 117.32
b. What amount will you receive for selling 100 shares?
($40.95 × 100) − ($40.95 × 100) × 0.003 = $4, 082.72
c. Suppose you buy 100 shares, then immediately sell 100 shares. What is
your round-trip transaction cost?
$4, 117.32 − $4, 082.72 = $34.6
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Short-Selling
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When price of an asset is expected to fall
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First: borrow and sell and asset (get $$)
Then: buy back and return the asset (pay $)
If price fell in the mean time: Profit $ = $$ - $
The lender must be compensated for dividends received
Example: Cash flows associated with short-selling a share of IBM for
90 days. Note that the short-seller must pay the dividend, D, to the
share-lender.
Action
Cash
Day 0
Borrow shares
Sell shares
+S0
Dividend Ex-Day
—
—
−D
Day 90
Return shares
Purchase shares
−S90
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VW’s 348% Two-Day Gain Is Pain for Hedge Funds
From the Wall Street Journal, 2008:
In short squeezes, investors who borrowed
and sold stock expecting its value to fall
exit from the trades by buying those
shares, or covering their positions. That
can send a stock upward if shares are hard
to come by. When shares are scarce, that
can push a company-s market
capitalization well beyond a reasonable
valuation. [...] Indeed, the recent stock
gains left Volkswagen’s market value at
about $346 billion, just below that of the
world’s largest publicly traded corporation,
Exxon Mobil Corp.
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Problem 1.6: Suppose you short-sell 300 shares of XYZ
stock at $30.19 with a commission charge of 0.5%.
Supposing you pay commission charges for purchasing the
security to cover the short-sale, how much profit have you
made if you close the short-sale at a price of $29.87?
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Problem 1.6: Suppose you short-sell 300 shares of XYZ
stock at $30.19 with a commission charge of 0.5%.
Supposing you pay commission charges for purchasing the
security to cover the short-sale, how much profit have you
made if you close the short-sale at a price of $29.87?
Initially, we will receive the proceeds form the sale of the asset, less the
proportional commission charge:
300 × ($30.19) − 300 × ($30.19) × 0.005 = $9, 011.72
When we close out the position, we will again incur the commission
charge, which is added to the purchasing cost:
300 × ($29.87) + 300 × ($29.87) × 0.005 = $9, 005.81
Finally, we receive total profits of: $9, 011.72 − $9, 005.81 = $5.91.
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Continuous Compounding
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Terms often used to to refer to interest rates:
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Effective annual rate r : if you invest $1 today, T years later you will
have
(1 + r )T
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Annual rate r , compounded n times per year: if you invest $1
today, T years later you will have
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1+
r nT
n
Annualized continuously compounded rate r : if you invest $1
today, T years later you will have
r nT
1+
n→∞
n
e rT ≡ lim
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Continuous Compounding: Example
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Suppose you have a zero-coupon bond that matures in 5 years. The
price today is $62.092 for a bond that pays $100.
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The effective annual rate of return is
1/5
$100
− 1 = 0.10
$62.092
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The continuously compounded rate of return is
ln ($100/$62.092) 0.47655
=
= 0.09531
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5
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The continuously compounded rate of return of 9.53% corresponds to
the effective annual rate of return of 10%. To verify this, observe that
e 0.09531 = 1.10
or
ln (1.10) = ln e 0.09531 = 0.09531
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Continous Compounding
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When we multiply exponentials, exponents add. So we have
e x e y = e x +y
This makes calculations of average rate of return easier.
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When using continuous compounding, increases and decreases are
symmetric.
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Moreover, continuously compounded returns can be less than -100%
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Problem B.2: Suppose that over 1 year a stock price
increases from $100 to $200. Over the subsequent year it
falls back to $100.
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What is the effective return over the first year? What is the continuously
compounded return?
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What is the effective return over the second year? The continuously
compounded return?
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What do you notice when you compare the first- and second-year returns
computed arithmetically and continuously?
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Problem B.2: Suppose that over 1 year a stock price
increases from $100 to $200. Over the subsequent year it
falls back to $100.
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What is the effective return over the first year? What is the continuously
compounded return?
$200 − $100
= 100%
$100 $200
= 69.31%
continuously compounded return = ln
$100
effective return =
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What is the effective return over the second year? The continuously
compounded return?
$100 − $200
= −50%
$200 $100
= −69.31%
continuously compounded return = ln
$200
effective return =
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What do you notice when you compare the first- and second-year returns
computed arithmetically and continuously?
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A World Without Options
Stock Price ($)
Stocks
Payoff ($)
Payoff ($)
Treasury Bills
Stock Price ($)
26 / 67
Forward Contracts
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Definition: a binding agreement (obligation) to buy/sell an underlying
asset in the future, at a price set today.
A forward contract specifies:
1. The features and quantity of the asset to be delivered
2. The delivery logistics, such as time, date, and place
3. The price the buyer will pay at the time of delivery
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Forward Contracts vs Futures Contracts
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Forward and futures contracts are essentially the same except for the
daily resettlement feature of futures contracts, called
marking-to-market.
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Because futures are exchange-traded, they are standardized and have
specified delivery dates, locations, and procedures.
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Plenty of information is available from: www.cmegroup.com
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Reading Price Quotes
Figure 1: Index futures price listings
source: Wall Street Journal, September 11, 2012.
29 / 67
Payoff (Value at Expiration) of a Forward Contract
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Every forward contract has both a buyer and a seller.
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The term long is used to describe the buyer and short is used to
describe the seller.
Payoff for
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Long forward = Spot price at expiration – Forward price
Short forward = Forward price – Spot price at expiration
Example: S&R index:
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Today: Spot price = $1,000. 6-month forward price = $1,020
In 6 months at contract expiration: Spot price = $1,050
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Long position payoff = $1,050 - $1,020 = $30
Short position payoff = $1,020 - $1,050 = -$30
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Payoff Diagram for Forwards
200
Long forward
Payoff ($)
100
0
−100
$1, 020
−200
800
900
1,000
Short forward
1,100
1,200
S&R Index Price ($)
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Problem 2.4.a: Suppose you enter in a long 6-month
forward position at a forward price of $50. What is the
payoff in 6 months for prices of $40, $45, $50, $55, and
$60?
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Problem 2.4.a: Suppose you enter in a long 6-month
forward position at a forward price of $50. What is the
payoff in 6 months for prices of $40, $45, $50, $55, and
$60?
The payoff to a long forward at expiration is equal to:
Payoff to long forward = Spot price at expiration − Forward price
Therefore, we can construct the following table:
Price of asset in 6 months
40
45
50
55
60
Payoff ot the long forward
-10
-5
0
5
10
32 / 67
Call Options
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A non-binding agreement (right but not an obligation) to buy an
asset into the future, at a price set today
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Preserves the upside potential, while at the same time eliminating the
downside
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The seller of a call option is obligated to deliver if asked
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Definition and terminology
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A call option gives the owner the right but not the obligation to buy
the underlying asset at a predetermined price during a predetermined
time period
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Strike (or exercise) price: the amount paid by the option buyer for the
asset if he/she decides to exercise
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Exercise: the act of paying the strike price to buy the asset
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Expiration: the date by which the option must be exercised or
becomes worthless
Exercise style: specifies when the option can be exercised
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European-style: can be exercised only at expiration date
American-style: can be exercised at any time before expiration
Bermudan-style: can be exercised during specified periods
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Reading Price Quotes
Figure 2: S&P 500 Index Call Options
source: Chicago Board Options Exchange, September 11, 2012.
35 / 67
Call Option Example
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Consider a call option on the S&R index with 6 months to expiration
and strike price of $1,000.
In six months at contract expiration: if spot price is
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$1,100 ⇒ call buyer’s payoff = $1,100 – $1,000 = $100, call seller’s
payoff = -$100
$900 ⇒ call buyer’s payoff = $0, call seller’s payoff = $0
The payoff of a call option is then
CT = max [ST − K , 0]
(1)
where K is the strike price, and ST is the spot price at expiration.
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The option profit is computed as
Call profit = max [ST − K , 0] − future value of premium
(2)
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Diagrams for Purchased Call
200
200
Long forward
Purchased call
100
Profit ($)
Payoff ($)
100
0
0
Purchased
call
−$95.68
−100
−100
−200
−200
$1, 020
800
900
1,000
1,100
S&R Index Price ($)
1,200
800
900
1,000
1,100
1,200
S&R Index Price ($)
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Diagrams for Written Call
200
200
100
100
Profit ($)
Payoff ($)
$95.68
0
−100
Written
call
0
−100
Written call
−200
800
900
1,000
1,100
S&R Index Price ($)
$1, 020
−200
1,200
800
900
1,000
Short forward
1,100
1,200
S&R Index Price ($)
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Problem 2.10.a: Consider a call option on the S&R index
with 6 months to expiration and strike price of $1,000.
The future value of the option premium is $95.68. For the
figure below, which plots the profit on a purchased call,
find the S&R index price at which the call option diagram
intersects the x -axis.
200
Profit ($)
100
0
Purchased
call
−$95.68
−100
−200
800
900
1,000
1,100
1,200
S&R Index Price ($)
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Problem 2.10.a: Consider a call option on the S&R index
with 6 months to expiration and strike price of $1,000.
The future value of the option premium is $95.68. For the
figure below, which plots the profit on a purchased call,
find the S&R index price at which the call option diagram
intersects the x -axis.
200
The profit of the long call option is:
max [0, ST − $1, 000] − $95.68
Profit ($)
100
0
Purchased
call
−$95.68
−100
−200
800
900
1,000
1,100
1,200
To find the S&R index price at which
the call option diagram intersects the
x -axis, we have to set the above
equation equal to zero. We get
ST = $1, 095.68
S&R Index Price ($)
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Put Options
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A put option gives the owner the right but not the obligation to sell
the underlying asset at a predetermined price during a predetermined
time period.
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The payoff of the put option is
PT = max [K − ST , 0]
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(3)
The option profit is computed as
Put profit = max [K − ST , 0] − future value of premium
(4)
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Reading Price Quotes
Figure 3: S&P 500 Index Put Options
source: Chicago Board Options Exchange, September 11, 2012.
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Diagrams for Purchased Put
200
200
Purchased put
Short forward
100
Profit ($)
Payoff ($)
100
0
0
−100
−100
−200
−200
Purchased
put
−$95.68
$1, 020
800
900
1,000
1,100
S&R Index Price ($)
1,200
800
900
1,000
1,100
1,200
S&R Index Price ($)
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200
200
100
100
Profit ($)
Payoff ($)
Diagrams for Written Put
0
−100
Long forward
$95.68
0
Written
put
−100
Written put
−200
800
900
$1, 020
−200
1,000
1,100
S&R Index Price ($)
1,200
800
900
1,000
1,100
1,200
S&R Index Price ($)
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Problem 2.14.a: Suppose the stock price is $40 and the
effective annual interest rate is 8%. Draw payoff and profit
diagrams for a 40-strike put with a premium of $3.26 and
maturity of 1 year.
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Problem 2.14.a: Suppose the stock price is $40 and the
effective annual interest rate is 8%. Draw payoff and profit
diagrams for a 40-strike put with a premium of $3.26 and
maturity of 1 year.
FV (premium) = $3.26 × (1 + 0.08)
= $3.5208
We get the following payoff and profit
diagram:
40
Payoff (blue) or profit (red) ($)
In order to be able to draw the profit
diagram, we need to find the future
value of the put premium:
30
Payoff
20
Profit
10
0
−10
−$3.5208
0
20
40
60
Stock Price ($)
80
44 / 67
Potential Gain and Loss for Forward and Option Positions
Position
Long forward
Short forward
Long call
Short call
Long put
Short put
Maximum Loss
– Forward price
Unlimited
– FV(premium)
Unlimited
– FV(premium)
FV(premium) – Strike price
Maximum Gain
Unlimited
Forward Price
Unlimited
FV(premium)
Strike price – FV(premium)
FV(premium)
45 / 67
Moneyness
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In-the-money option: positive payoff if exercised immediately
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At-the-money option: zero payoff if exercised immediately
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Out-of-the-money option: negative payoff if exercised immediately
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Put-Call Parity
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Suppose you are buying a call option and selling a put option on a
non-dividend paying stock. Both options have maturity T and strike
price K :
Combined position
Payoff ($)
Payoff ($)
Long call
Short put
K
Stock Price ($)
K
Stock Price ($)
47 / 67
Put-Call Parity (cont’d)
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Your payoff at maturity is
CT − PT = max [ST − K , 0] − max [K − ST , 0]
= max [ST − K , 0] + min [ST − K , 0]
= ST − K
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We have two strategies with the same payoff at maturity:
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Buy a call and sell a put, thus paying a premium of Ct − Pt today
Buy a share of the stock and borrow PV (K ), thus paying a premium
of St − PV (K ) today
Positions that have the same payoff should have the same cost (Law
of one price):
Ct − Pt = St − PV (K )
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(5)
Equation (5) is known as put-call parity, and one of the most
important relations in options.
48 / 67
Put-Call Parity (cont’d)
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Parity provides a cookbook for the synthetic creation of options. It
tells us that
Ct = Pt + St − PV (K )
(6)
Pt = Ct − St + PV (K )
(7)
and that
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The first relation says that a call is equivalent to a leveraged position
on the underlying asset, which is insured by the purchase of a put.
The second relation says that a put is equivalent to a short position
on the stock, insured by the purchase of a call
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Parity generally fails for American-style options, which may be
exercised prior to maturity.
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Why Does the Price of an At-the-Money call Exceed the
Price of an At-the-Money put?
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Parity shows that the reason for the call being more expensive is the
time value of money:
Ct − Pt = K − PV (K ) > 0
(8)
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A common erroneous explanation is that the profit on a call is
unlimited, while the profit on a put can be no greater than the strike
price, which seems to suggest that the call should be more expensive
than the put.
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This argument also seems to suggest that every stock is worth more
than its price!
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Problem 3.8: The S&R index price is $1,000 and the
effective 6-month interest rate is 2%. Suppose the
premium on a 6-month S&R call is $109.20 and the
premium on a 6-month put with the same strike price is
$60.18. What is the strike price?
51 / 67
Problem 3.8: The S&R index price is $1,000 and the
effective 6-month interest rate is 2%. Suppose the
premium on a 6-month S&R call is $109.20 and the
premium on a 6-month put with the same strike price is
$60.18. What is the strike price?
This question is a direct application of the Put-Call Parity:
Ct − Pt = St − PV (K )
K
$109.20 − $60.18 = $1, 000 −
1.02
K = $970.00
51 / 67
Put-Call Parity for Dividend Paying Stocks
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If the stock is paying dividends over the lifetime of the option, the
put-call parity becomes
Ct − Pt = [St − PV (Div)] − PV (K )
(9)
where PV (Div) is the present value of the stream of dividends paid
on the stock until maturity.
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Hence, we can write
Ct = Pt + [St − PV (Div)] − PV (K )
(10)
Pt = Ct − [St − PV (Div)] + PV (K )
(11)
Equations (10)–(11) help us to find maximum and minimum option
prices.
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Maximum and Minimum Option Prices: Call Price
Ct
St ≥ Ct
Ct ≥ max [0, St − PV (Div) − PV (K )]
St
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Maximum and Minimum Option Prices: Call Price
Ct
St ≥ Ct
Ct ≥ max [0, St − PV (Div) − PV (K )]
St
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Maximum and Minimum Option Prices: Call Price
Ct
St ≥ Ct
Ct ≥ max [0, St − PV (Div) − PV (K )]
St
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Maximum and Minimum Option Prices: Call Price
Ct
St ≥ Ct
Call option
price somewhere here
Ct ≥ max [0, St − PV (Div) − PV (K )]
St
53 / 67
Maximum and Minimum Option Prices: Put Price
Pt
K ≥ Pt
Put option
price somewhere here
Pt ≥ max [0, PV (K ) − St + PV (Div)]
St
54 / 67
Example: Equity-Linked CDs
I
A 1,999 First Union National Bank CD promises to repay in 5.5 years
initial invested amount and 70% of the gain in S&P 500 index (this is
a principal protected equity-linked CD)
I
Assume $10,000 invested when
S&P 500 = 1,300
Payoff ($)
13,000
I
Final payoff is
12,000
Payoff of Equity-Linked CD
11,000
Sfinal
−1
$10, 000 × 1 + 0.7 × max 0,
1300
h
i
10,000
9,000
where Sfinal = value of the
S&P 500 after 5.5 years.
$1,300
8,000
600
800
1,000 1,200 1,400 1,600 1,800
S&P 500 Price ($)
55 / 67
Options are Insurance: Insuring a Long Position (Floors)
I
A put option is combined with a position in the underlying asset
I
Goal: to insure against a fall in the price of the underlying asset
(when one has a long position in that asset)
2,000
2,000
Long S&R Index
Long put
1,000
Payoff ($)
1,000
Payoff ($)
Combined payoff
0
−1,000
0
−1,000
−2,000
−2,000
0
500
1,000
1,500
S&R Index Price ($)
2,000
0
500
1,000
1,500
2,000
S&R Index Price ($)
56 / 67
Options are Insurance: Insuring a Short Position (Caps)
I
A call option is combined with a position in the underlying asset
I
Goal: to insure against an increase in the price of the underlying asset
(when one has a short position in that asset)
2,000
2,000
Long call
1,000
Payoff ($)
Payoff ($)
1,000
0
−1,000
0
Combined payoff
−1,000
Short S&R Index
−2,000
−2,000
0
500
1,000
1,500
S&R Index Price ($)
2,000
0
500
1,000
1,500
2,000
S&R Index Price ($)
57 / 67
Various Strategies: Payoffs
Bull Spread
Straddle
Profit ($)
Profit ($)
Profit ($)
Collar
Stock Price ($)
Strangle
Butterfly Spread
Ratio Spread
Stock Price ($)
Profit ($)
Profit ($)
Stock Price ($)
Profit ($)
Stock Price ($)
Stock Price ($)
Stock Price ($)
58 / 67
Various Strategies: Positions
Collar
Bull Spread
Klow
Call
Put
KATM
Khigh
Buy
Sell
Strangle
Klow
Call
Put
KATM
Call
Put
KATM
Straddle
Khigh
Sell
Buy
Klow
Call
Put
Butterfly Spread
Khigh
Buy
Buy
Klow
Call
Put
Klow
KATM
Khigh
Buy
Buy
Sell
Sell
KATM
Khigh
Buy
Buy
Ratio Spread
Klow
Call
Put
KATM
Buy
Khigh
Sell (n)
Note that you can achieve the same results with different combinations
(but always at the same cost!)
59 / 67
Various Strategies: Rationales
Bull Spread
I
I
I
I
You believe a stock will
appreciate ⇒ buy a call option
(forward position insured)
You can lower the cost if you
are willing to reduce your profit
should the stock appreciate ⇒
sell a call with higher strike
Surprisingly, you can achieve
the same result by buying a
low-strike put and selling a
high-strike put
Opposite: bear spread
Strangle
Collar
I
A collar is fundamentally a
short position (resembling a
short forward contract)
I
Often used for insurance when
we own a stock (collared stock)
I
The collared stock looks like a
bull spread; however, it arises
from a different set of
transactions
I
Opposite: written collar
Butterfly Spread
I
I
To reduce the premium of a
straddle, you can buy
out-of-the-money options rather
than at-the-money options.
I
I
Opposite: written strangle
I
A butterfly spread is a written
straddle to which we add two
options to safeguard the
position: An out-of-the money
put and an out-of-the money
call.
A butterfly spread can be
thought of as a written straddle
for the timid (or for the
prudent!)
Opposite: long iron butterfly
Straddle
I
A straddle can profit from stock
price moves in both directions
I
The disadvantage is that it has
a high premium because it
requires purchasing two options
I
Opposite: written straddle
Ratio Spread
I
Ratio spreads involve buying
one option and selling a greater
quantity (n) of an option with a
more out-of-the money strike
I
The ratio (i.e., “1 by n”) is the
number of short options divided
by the number of long options
I
The options are either both
calls or both puts
I
It is possible to construct ratio
spreads with zero premium ⇒
we can construct insurance that
costs nothing if it is not needed!
60 / 67
A World Without Options
Stock Price ($)
Stocks
Payoff ($)
Payoff ($)
Treasury Bills
Stock Price ($)
61 / 67
Madoff Ran Vast Options Game
From the Wall Street Journal, 2008:
“Mr. Madoff told clients he was using a fairly common options-trading strategy to
generate modest but steady returns for more than two decades. The strategy involved
buying stocks, while also trading options in a way designed to limit losses on the shares
[...] People who analyzed client statements said Mr. Madoff’s firm couldn’t have bought
and sold the options he claimed because those totals would have outstripped total
trading volume those days”.
Reproduced from: Gregoriou and Lhabitant, Madoff: A riot of Red Flags, 2009
62 / 67
Collars in Acquisitions: WorldCom/MCI
I
On October 1, 1997, WorldCom Inc. CEO (Bernard Ebbers) sent the
following note to the CEO of MCI (Bert Roberts), and it was also
released through the typical newswires:
I
“I am writing to inform you that this morning WorldCom is publicly
announcing that it will be commencing an offer to acquire all the
outstanding shares of MCI for $41.50 of WorldCom common stock
per MCI share. The actual number of shares of WorldCom common
stock to be exchanged for each MCI share in the exchange offer will
be determined by dividing $41.50 by the 20-day average of the high
and low sales prices for WorldCom common stock prior to the closing
of the exchange offer, but will not be less than 1.0375 shares (if
WorldCom’s average stock price exceeds $40) or more than 1.2206
shares (if WorldCom’s average stock price is less than $34).”
63 / 67
Collars in Acquisitions: WorldCom/MCI (cont’d)
The payoff is contingent upon price of WorldCom’s 20-day average
stock price prior to the closing exchange offer:
50
Value of Offer
I
Slope = 1.0375
40
Slope = 1.2206
30
$34
20
$40
20
25
30
35
40
45
50
55
WCOM’s Average Stock Price at Closing
64 / 67
Example: Another Equity-Linked Note
I
In July 2004, Marshall & Ilsley Corp. (ticker symbol MI) raised $400
million by issuing mandatorily convertible bonds effectively
maturing in August 2007
I
The bond pays an annual 6.5% coupon and at maturity makes
payments in shares, with the number of shares dependent upon the
firm’s stock price. The specific terms of the maturity payment are in
the table below
Marshall & Ilsley Share
Price
SMI < 37.32
37.32 ≤ SMI ≤ 46.28
46.28 < SMI
Number of Shares Paid
to Bondholders
0.6699
$25/SMI
0.5402
65 / 67
Example: Another Equity-Linked Note (cont’d)
The graph of the maturity payoff should remind us of a written collar:
50
40
Bondholder payoff
Payoff ($)
I
30
Slope = 0.5402
20
Slope = 0.6699
10
$37.32
0
0
$46.28
20
40
60
80
Marshall & Ilsley Stock Price ($)
66 / 67
Positions Consistent With Different Views on the Stock
Price and Volatility Direction
Price Will Fall
No Price View
Price Will Increase
Volatility Will Increase
Buy puts
Buy straddle
Buy calls
No Volatility View
Sell underlying
Do nothing
Buy underlying
Volatility Will Fall
Sell calls
Sell straddle
Sell puts
67 / 67
Option Markets 232D
2. Binomial Option Pricing
Daniel Andrei
Fall 2012
1 / 60
Outline
I Discrete-Time Option Pricing: The Binomial Model
3
II Call Options
A One-Period Binomial Tree
The Binomial Solution
A Two-Period Binomial Tree
Many Binomial Periods
Self-Financing Strategy
6
7
15
22
27
28
III Put Options
30
IV Uncertainty in the Binomial Model
35
V Risk-Neutral Pricing
41
VI American Options
53
2 / 60
Outline
I Discrete-Time Option Pricing: The Binomial Model
3
II Call Options
A One-Period Binomial Tree
The Binomial Solution
A Two-Period Binomial Tree
Many Binomial Periods
Self-Financing Strategy
6
7
15
22
27
28
III Put Options
30
IV Uncertainty in the Binomial Model
35
V Risk-Neutral Pricing
41
VI American Options
53
3 / 60
Discrete-Time Option Pricing: The Binomial Model
I
Until now, we have looked only at some basic principles of option
pricing
I
I
We examined payoff and profit diagrams, and upper/lower bounds on
option prices
We saw that with put-call parity we could price a put or a call based on
the prices of the combinations on instruments that make up the
synthetic version of the put or call.
I
What we need to be able to do is price a put or a call without the
other instrument.
I
In this section, we introduce a simple means of pricing an option.
4 / 60
Discrete-Time Option Pricing: The Binomial Model
(cont’d)
I
The approach we take here is called the binomial tree.
I
The word “binomial” refers to the fact that there are only two
outcomes (we let the underlying price move to only one of two
possible new prices).
I
It may appear that this framework oversimplifies things, but the
model can eventually be extended to encompass all possible prices.
5 / 60
Outline
I Discrete-Time Option Pricing: The Binomial Model
3
II Call Options
A One-Period Binomial Tree
The Binomial Solution
A Two-Period Binomial Tree
Many Binomial Periods
Self-Financing Strategy
6
7
15
22
27
28
III Put Options
30
IV Uncertainty in the Binomial Model
35
V Risk-Neutral Pricing
41
VI American Options
53
6 / 60
The Binomial Option Pricing Model
I
The binomial option pricing model assumes that, over a period of
time, the price of the underlying asset can move only up or down by a
specified amount—that is, the asset price follows a binomial
distribution:
up
u×S
S
dow
n
d ×S
7 / 60
A Simple Example
I
XYZ does not pay dividends and its current price is $41. In one year
the price can be either $59.954 or $32.903, i.e., u = 1.4623 and
d = 0.8025.
I
Consider a European call option on the stock of XYZ, with a $40
strike an 1 year to expiration. The continuously compounded risk-free
interest rate is 8%.
I
We wish to determine the option price:
uS = 59.954
S = 41
Cu = max [0, 59.954 − 40] = 19.954
C =?
dS = 32.903
Cd = max [0, 32.903 − 40] = 0
8 / 60
A Simple Example (cont’d)
I
Let us try to find a portfolio that mimics the option (replicating
portfolio).
I
We have two instruments: shares of stock and a position in bonds
(i.e., borrowing or lending).
I
To be specific, we wish to find a portfolio consisting of ∆ shares of
stock and a dollar amount B in borrowing or lending, such that the
portfolio imitates the option whether the stock rises or falls.
I
The value of this replicating portfolio at maturity is:
(
59.954∆ + e 0.08 B
32.903∆ + e 0.08 B
= 19.954
=0
(1)
9 / 60
A Simple Example (cont’d)
I
The unique solution of this system of 2 equations with 2 unknowns is
∆ = 0.738, B = −22.405,
(2)
i.e., buy 0.738 shares of XYZ and borrow $22.405 at the risk-free rate.
I
In computing the payoff for the replicating portfolio, we assume that
we sell the shares at the market price and that we repay the borrowed
amount, plus interest.
I
Thus, we obtain that the option and the replicating portfolio have the
same payoff: $19.954 if the stock price goes up and $0 if the stock
price goes down.
10 / 60
A Simple Example (cont’d)
I
By the law of one price, positions that have the same payoff should
have the same cost.
I
The price of the option must be
C = 0.738 × $41 − $22.405 = $7.839
(3)
11 / 60
Arbitraging a Mispriced Option
I
Suppose that the market price for the option is $8 instead of $7.839
(the option is overpriced).
I
We can sell the option and buy a synthetic option at the same time
(buy low and sell high). The initial cash flow is
$8.00 − $7.839 = $0.161
(4)
and there is no risk at expiration:
Written call
0.738 purchased shares
Repay loan of $22.405
Total payoff
Stock Price in 1 Year
$32.903
$59.954
$0
-$19.954
$24.271
$44.225
-$24.271
-$24.271
$0
$0
12 / 60
Arbitraging a Mispriced Option (cont’d)
I
Suppose that the market price for the option is $7.5 instead of $7.839
(the option is underpriced).
I
We can buy the option and sell a synthetic option at the same time
(buy low and sell high). The initial cash flow is
$7.839 − $7.5 = $0.339
(5)
and there is no risk at expiration:
Purchased call
0.738 short-sold shares
Sell T-bill
Total payoff
Stock Price in 1 Year
$32.903
$59.954
$0
$19.954
-$24.271
-$44.225
$24.271
$24.271
$0
$0
13 / 60
A Remarkable Result
I
So far we have not specified the probabilities of the stock going up
and down.
I
In fact, probabilities were not used anywhere in the option price
calculations.
I
This is a remarkable result: Since the strategy of holding ∆ shares
and B bonds replicates the option whichever way the stock
moves, the probability of an up or down movement in the stock
is irrelevant for pricing the option.
14 / 60
The Binomial Solution
I
Suppose that the stock has a continuous dividend yield of δ, which is
reinvested in the stock. Thus, if you buy one share at time 0 and the
length of a period is h, at time h you will have e δh shares.
I
The up and down movements of the stock price reflect the
ex-dividend price.
I
We can write the stock price as uS0 when the stock goes up and dS0
when the stock goes down. We can represent the tree for the stock
and the option as follows:
uS0
S0
C1u
C0
dS0
C1d
15 / 60
The Binomial Solution (cont’d)
I
If the length of a period is h, the interest factor per period is e rh .
I
A successful replicating portfolio will satisfy
(
∆ × S0 × u × e δh + B × e rh
∆ × S0 × d × e δh + B × e rh
I
= C1u
= C1d
(6)
This is a system of two equations in two unknowns ∆ and B. Solving
for ∆ and B gives
C u −C d

∆
1
= e −δh S01(u−d)
B
= e −rh
C1d u−C1u d
u−d
= e −rh C1u − ∆S0 ue δh
(7)
16 / 60
The Binomial Solution (cont’d)
I
Given the expressions (7) for ∆ and B, we can derive a simple
formula for the value of the option. The cost of creating the option is
the net cash required to buy the shares and bonds. Thus, the cost of
the option is
C0 = ∆S0 + B
=e
I
−rh
e
C1u
(r −δ)h − d
u−d
+ C1d
u − e (r −δ)h
u−d
!
(8)
Note that if we are interested only in the option price, it is not
necessary to solve for ∆ and B; that is just an intermediate step. If
we want to know only the option price, we can use equation (8)
directly.
17 / 60
The Binomial Solution (cont’d)
I
The assumed stock price movements, u and d, should not give rise to
arbitrage opportunities. In particular, we require that
d < e (r −δ)h < u
I
(9)
Note that because ∆ is the number of shares in the replicating
portfolio, it can also be interpreted as the sensitivity of the option to
a change in the stock price. If the stock prices changes by $1, then
the option price, ∆S + B, changes by ∆.
18 / 60
The Binomial Solution (cont’d)
I
In our example, δ = 0 (XYZ does not pay dividends) and h = 1 (the
length of a period is 1 year).
I
The solution for ∆ and B reduces to
I

∆
=
B
=
C1u −C1d
S0 (u−d)
C d u−C u d
e −r 1 u−d 1
(10)
= e −r (C1u − ∆S0 u)
The option price further simplifies to
C0 = ∆S0 + B = e
−r
u − er
−d
+ C1d
u−d
u−d
e
C1u
r
(11)
19 / 60
Problem 10.1.a: Let S = $100, K = $105, r = 8%
(continuously compounded), T = 0.5, and δ = 0. Let
u = 1.3, d = 0.8, and the number of binomial periods
n = 1. What are the premium, ∆, and B for a European
call?
20 / 60
Problem 10.1.a: Let S = $100, K = $105, r = 8%
(continuously compounded), T = 0.5, and δ = 0. Let
u = 1.3, d = 0.8, and the number of binomial periods
n = 1. What are the premium, ∆, and B for a European
call?
Using the formulas given in (7), we calculate the following values:
∆ = 0.5
B = −38.4316
Call price = 11.5684
20 / 60
Problem 10.21: Suppose that u < e (r −δ)h . Show that
there is an arbitrage opportunity. Now suppose that
d > e (r −δ)h . Show again that there is an arbitrage
opportunity.
21 / 60
Problem 10.21: Suppose that u < e (r −δ)h . Show that
there is an arbitrage opportunity. Now suppose that
d > e (r −δ)h . Show again that there is an arbitrage
opportunity.
I
If u < e (r −δ)h , we short a tailed position of the stock and invest the
proceeds at the interest rate. There is an arbitrage opportunity:
t = 0 state = d
state = u
−δh
Short stock +e S
−d × S
−u × S
Lend money −e −δh S +e (r −δ)h S +e (r −δ)h S
Total
0
>0
>0
I
If d > e (r −δ)h , we buy a tailed position of the stock and borrow at
the interest rate. There is an arbitrage opportunity:
t = 0 state = d
state = u
Buy stock
−e −δh S
+d × S
+u × S
−δh
(r
−δ)h
Borrow money
e S −e
S −e (r −δ)h S
Total
0
>0
>0
21 / 60
A Two-Period Binomial Tree
I
We can extend the previous example to price a 2-year option,
assuming all inputs are the same as before.
S2uu = 87.669
S1u = 59.954
S0 = 41
C0 =?
C1u =?
C2uu = 47.669
S2ud = 48.114
S1d = 32.903
C2ud = 8.114
C1d =?
S2dd = 26.405
C2dd = 0
I
Note that an up move followed by a down move (S2ud ) generates the
same stock price as a down move followed by an up move (S2du ). This
is called a recombining tree.
22 / 60
A Two-Period Binomial Tree (cont’d)
I
To price the option when we have two binomial periods, we need to
work backward through the tree.
I
Suppose that in period 1 the stock price is S1u = $59.954. We can use
equation (11) to derive the option price:
C1u
I
=e
−r
u − er
−d
+ C2ud
u−d
u−d
e
C2uu
r
= $23.029
(12)
Using equations (10), we can also solve for the composition of the
replicating portfolio:
∆ = 1, B = −36.925
(13)
i.e., buy 1 share of XYZ and borrow $36.925 at the risk-free rate,
which costs 1 × $59.954 − $36.925 = $23.029.
23 / 60
A Two-Period Binomial Tree (cont’d)
I
Suppose that in period 1 the stock price is S1d = $32.903. We can use
equation (11) to derive the option price:
C1d = e −r C2ud
I
u − er
er − d
+ C2dd
u−d
u−d
= $3.187
(14)
Using equations (10), we can also solve for the composition of the
replicating portfolio:
∆ = 0.374, B = −9.111
(15)
i.e., buy 0.374 shares of XYZ and borrow $9.111 at the risk-free rate,
which costs 0.374 × $32.903 − $9.111 = $3.187.
24 / 60
A Two-Period Binomial Tree (cont’d)
I
Move backward now at period 0. The stock price is S0 = 41. We can
use equation (11) to derive the option price:
C0 = e
I
−r
u − er
−d
+ C1d
u−d
u−d
e
C1u
r
= $10.737
(16)
Using equations (10), we can also solve for the composition of the
replicating portfolio:
∆ = 0.734, B = −19.337
(17)
i.e., buy 0.734 shares of XYZ and borrow $19.337 at the risk-free
rate, which costs 0.734 × $41 − $19.337 = $10.737.
25 / 60
A Two-Period Binomial Tree (cont’d)
I
The two-period binomial tree with the option price at each node as
well as the details of the replicating portfolio is:
S2uu = $87.669
S1u = $59.954
C2uu = $47.669
C1u = $23.029
S0 = $41
C0 = $10.737
∆ = 0.734
B = −$19.337
∆=1
B = −$36.925
S1d = $32.903
S2ud = $48.114
C2ud = $8.114
C1d = $3.187
∆ = 0.374
B = −$9.111
S2dd = $26.405
C2dd = $0
26 / 60
Many Binomial Periods: Three-Period Example
I
I
Once we understand the two-period option it is straightforward to
value an option using more than two binomial periods.
The important principle is to work backward through the tree:
uuu
= $128.198
uuu
= $88.198
S3
uu
S2
uu
C2
∆=1
u
B = −$36.925
C1 = $26.258
S0 = $41
∆ = 0.798
B = −$19.899
= $50.745
u
S1 = $59.954
C0 = $12.799
= $87.669
∆ = 0.981
B = −$32.580
d
S1 = $32.903
d
C1 = $4.685
C3
ud
S2
= $48.114
ud
C2
= $11.925
uud
= $70.356
uud
= $30.356
S3
C3
∆ = 0.956
B = −$34.086
∆ = 0.549
dd
S2
= $26.405
B = −$13.390
dd
C2
= $0
udd
= $38.612
udd
= $0
ddd
= $21.191
ddd
= $0
S3
C3
∆=0
B = $0
S3
C3
27 / 60
Self-Financing Strategy
I
I
In the two-period binomial tree example, suppose that the stock
moves from S0 = $41 to S1u = $59.954.
The replicating portfolio should be modified as follows:
1. Buy 1 − 0.734 = 0.266 shares of XYZ (increase the XYZ position from
0.734 shares to 1 share), which costs 0.266 × $59.954 = $15.977.
2. Increase borrowing from $19.337 × e 0.08 = $20.947 to $36.925, which
yields $15.977.
I
The amount necessary to buy shares is equal to the amount obtained
from increased borrowing.
I
Modifying the portfolio does not require additional cash. Thus, the
replicating portfolio is self-financing.
28 / 60
Self-Financing Strategy (cont’d)
I
If the stock moves from S0 = $41 to S1d = $32.903, the replicating
portfolio should be modified as follows:
1. Sell 0.734 − 0.374 = 0.360 shares of XYZ (decrease the XYZ position
from 0.734 shares to 0.374 shares), which yields
0.360 × $32.903 = $11.836.
2. Decrease borrowing from $19.337 × e 0.08 = $20.947 to $9.111, which
costs $11.836.
I
The amount obtained from selling shares is equal to the amount
necessary to decrease borrowing.
I
Modifying the portfolio does not require additional cash. Once again,
the replicating portfolio is self-financing.
29 / 60
Outline
I Discrete-Time Option Pricing: The Binomial Model
3
II Call Options
A One-Period Binomial Tree
The Binomial Solution
A Two-Period Binomial Tree
Many Binomial Periods
Self-Financing Strategy
6
7
15
22
27
28
III Put Options
30
IV Uncertainty in the Binomial Model
35
V Risk-Neutral Pricing
41
VI American Options
53
30 / 60
Put Options
I
We compute put option prices using the same stock price tree and in
the same way as call option prices.
I
The only difference with an European put option occurs at expiration:
Instead of computing the price as max [0, S − K ], we use
max [0, K − S].
I
Here is a two-period binomial tree for an European put option with a
$40 strike:
31 / 60
Put Options (cont’d)
S2uu = $87.669
S1u = $59.954
P2uu = $0
P1u = $0
S0 = $41
∆=0
P0 = $3.823
B = $0
∆ = −0.266
S1d = $32.903
B = $14.749
P1d = $7.209
S2ud = $48.114
P2ud = $0
∆ = −0.626
B = $27.814
S2dd = $26.405
P2dd = $13.595
I
The proof that the replicating portfolio is self-financing is left as an
exercise.
32 / 60
Put Options (Using the Parity Relationship)
I
For non-dividend paying stocks, the basic parity relationship for
European options with the same strike price and time to expiration is
Ct − Pt = St − PV (strike price)
I
(18)
We can use this relationship to find the put price at all nodes:
S2uu = 87.669
C2uu = 47.669
S1u = 59.954
S0 = 41
C1u
= 23.029
P1u
=0
S1d
C1d
= 32.903
S2ud = 48.114
C2ud = 8.114
C0 = 10.737
P0 = 3.823
P2uu = 0
= 3.187
P1d = 7.209
P2ud = 0
S2dd = 26.405
C2dd = 0
P2dd = 13.595
33 / 60
Problem 10.1.b: Let S = $100, K = $105, r = 8%
(continuously compounded), T = 0.5, and δ = 0. Let
u = 1.3, d = 0.8, and the number of binomial periods
n = 1. What are the premium, ∆, and B for a European
put?
34 / 60
Problem 10.1.b: Let S = $100, K = $105, r = 8%
(continuously compounded), T = 0.5, and δ = 0. Let
u = 1.3, d = 0.8, and the number of binomial periods
n = 1. What are the premium, ∆, and B for a European
put?
Using the formulas given in (7), we calculate the following values:
∆ = −0.5
B = 62.4513
Put price = 12.4513
34 / 60
Outline
I Discrete-Time Option Pricing: The Binomial Model
3
II Call Options
A One-Period Binomial Tree
The Binomial Solution
A Two-Period Binomial Tree
Many Binomial Periods
Self-Financing Strategy
6
7
15
22
27
28
III Put Options
30
IV Uncertainty in the Binomial Model
35
V Risk-Neutral Pricing
41
VI American Options
53
35 / 60
Uncertainty in the Binomial Model
I
A natural measure of uncertainty about the stock return is the
annualized standard deviation of the continuously compounded
stock return, which we will denote by σ.
I
If we split the year into n periods of length h (so that h = 1/n), the
standard deviation over the period of length h, σh , is (assuming
returns are uncorrelated over time)
√
(19)
σh = σ h
I
In other words, the standard deviation of the stock return is
proportional to the square root of time.
36 / 60
Uncertainty in the Binomial Model (cont’d)
I
We incorporate uncertainty into the binomial tree by modeling the
up and down moves of the stock price. Without uncertainty, the
stock price next period must equal:
St+h = St e (r −δ)h
(20)
I
To interpret this, without uncertainty, the rate of return on the stock
must be the risk-free rate. Thus, the stock price must rise at the
risk-free rate less the dividend yield, r − δ.
I
We now model the stock price evolution as
uSt = St e (r −δ)h e +σ
dSt = St e (r −δ)h e −σ
√
√
h
h
(21)
37 / 60
Uncertainty in the Binomial Model (cont’d)
I
We can rewrite this as
u = e (r −δ)h+σ
d = e (r −δ)h−σ
√
√
h
h
(22)
I
Return has two parts, one of which is certain [(r − δ) h], and the other
of which
and generates the up and down stock price
is√uncertain
moves σ h .
I
Note that if we set volatility equal to zero, we are back to (20) and we
have St+h = uSt = dSt = St e (r −δ)h . Zero volatility does not mean
that prices are fixed; it means that prices are known in advance.
38 / 60
Uncertainty in the Binomial Model (cont’d)
I
In our example we assumed that u = 1.4623 and d = 0.8025. These
correspond to an annual stock price volatility of 30%:
u = e (0.08−0)×1+0.3×
d = e (0.08−0)×1−0.3×
I
√
√
1
= 1.4623
1
= 0.8025
(23)
We will use equations (22) to construct binomial trees. This approach
(called the forward tree approach) is very convenient because it
never violates the no arbitrage restriction
d < e (r −δ)h < u
39 / 60
Problem 10.19: For a stock index, S = $100, σ = 30%,
r = 5%, δ = 3%, and T = 3. Let n = 3.
I
What is the price of a European call option with a strike of $95?
I
What is the price of a European put option with a strike of $95?
40 / 60
Problem 10.19: For a stock index, S = $100, σ = 30%,
r = 5%, δ = 3%, and T = 3. Let n = 3.
I
What is the price of a European call option with a strike of $95?
The price of a European call
option with a strike of 95 is
$24.0058
I
What is the price of a European put option with a strike of $95?
The price of a European put
option with a strike of 95 is
$14.3799
40 / 60
Outline
I Discrete-Time Option Pricing: The Binomial Model
3
II Call Options
A One-Period Binomial Tree
The Binomial Solution
A Two-Period Binomial Tree
Many Binomial Periods
Self-Financing Strategy
6
7
15
22
27
28
III Put Options
30
IV Uncertainty in the Binomial Model
35
V Risk-Neutral Pricing
41
VI American Options
53
41 / 60
Risk-Neutral Pricing
I
There is a probabilistic interpretation of the binomial solution for the
price of an option (equation 8, restated below):
C0 = e −rh
(r −δ)h
u − e (r −δ)h
e (r −δ)h − d
+ C1d
C1u
u−d
u−d
(r −δ)h
!
(24)
I
The terms e u−d−d and u−eu−d
follows from inequality 9).
I
Thus, we can interpret these terms as probabilities. Equation (8) can
be written as
h
sum to 1 and are both positive (this
C0 = e −rh p ∗ C1u + (1 − p ∗ ) C1d
I
i
(25)
This expression has the appearance of an expected value discounted
(r −δ)h
at the risk-free rate. Thus, we will call p ∗ ≡ e u−d−d the risk-neutral
probability of an increase in the stock price.
42 / 60
Risk-Neutral Pricing (cont’d)
I
Pricing options using risk-neutral probabilities can be done in one
step, no matter how large the number of periods. In our example,
p ∗ = 0.4256. For the one-period European call option we have:
Call Price in 1 Year
$19.954
$0
I
Probability
0.4256
0.5744
The call price is
C0 = e −0.08 (0.4256 × $19.954 + 0.5744 × $0) = $7.839
(26)
43 / 60
Risk-Neutral Pricing (cont’d)
I
I
For the two-period European call option we have:
Call Price in 2 Years
Probability
$47.669
p ∗2 = 0.1811
$8.114
2p ∗ (1 − p ∗ ) = 0.4889
$0
(1 − p ∗ )2 = 0.3300
The call price is
C0 = e −0.08×2 (0.1811 × $47.669 + 0.4889 × $8.114 + 0.3300 × $0)
= $10.737
(27)
I
The probability of reaching any given node is the probability of one
path reaching that node times the number of paths reaching that
node. For example, the probability of reaching the node
S2ud = $48.114 is 2p ∗ (1 − p ∗ ).
I
It can be easily verified that the sum of probabilities in the table
above is 1.
44 / 60
Risk-Neutral Pricing (cont’d)
I
I
For the three-period European call option we have:
Call Price in 3 Years
Probability
$88.198
p ∗3 = 0.0771
∗2
$30.356
3p (1 − p ∗ ) = 0.3121
$0
3p ∗ (1 − p ∗ )2 = 0.4213
$0
(1 − p ∗ )3 = 0.1896
The call price is
C0 = e −0.08×3
3
X
h
i
3!
p ∗k (1 − p ∗ )3−k max S0 u k d 3−k − K , 0
k! (3 − k)!
(28)
k =0
= $12.799
I
It can be easily verified that the sum of probabilities in the table
above is 1.
I
It is left as an exercise to find the price of the two-period European
put using risk-neutral probabilities.
45 / 60
Risk-Neutral Pricing (cont’d)
I
For an arbitrary number of periods n, the price of an European call
option is given by
C0 = e −rn
I
n
X
h
i
n!
p ∗k (1 − p ∗ )n−k max S0 u k d n−k − K , 0 (29)
k! (n − k)!
k=0
We will use this formula later on when talking about Black-Scholes:
when the number of steps becomes great enough the price of the
option appear to approach a limiting value. This value is given by the
Black-Scholes formula.
46 / 60
Problem 11.12: Let S = $100, σ = 0.3, r = 0.08, T = 1,
and δ = 0. Use equation (29) to compute the risk-neutral
probability of reaching a terminal node and the price at
that node for n = 3. Plot the risk-neutral distribution of
year-1 stock prices.
47 / 60
Problem 11.12: Let S = $100, σ = 0.3, r = 0.08, T = 1,
and δ = 0. Use equation (29) to compute the risk-neutral
probability of reaching a terminal node and the price at
that node for n = 3. Plot the risk-neutral distribution of
year-1 stock prices.
For n = 3, u and d are calculated as
follows:
√
u = e (0.08−0)×1/3+0.3× 1/3 = 1.2212
√
d = e (0.08−0)×1/3−0.3× 1/3 = 0.8637
It follows that p ∗ = 0.4568, and
n−k
0
1
2
3
Stock price
182.14
128.81
91.10
64.43
Probability
0.0953
0.3400
0.4044
0.1603
0.4
0.3
0.2
0.1
64.43
91.10 128.81 182.14
47 / 60
Understanding Risk-Neutral Pricing
I
A risk-neutral investor is indifferent between a sure thing and a risky
bet with an expected payoff equal to the value of the sure thing.
I
A risk-averse investor prefers a sure thing to a risky bet with an
expected payoff equal to the value of the sure thing.
I
Formula (25) suggests that we are discounting at the risk-free rate,
even though the risk of the option is at least as great as the risk of
the stock.
I
Thus, the option pricing formula, equation (25), can be said to price
options as if investors are risk-neutral.
I
Is this option pricing consistent with standard discounted cash
flow calculations?
48 / 60
Understanding Risk-Neutral Pricing (cont’d)
I
Assume, as in our example, that a stock does not pay dividends and
the length of a period is 1 year (δ = 0 and h = 1).
Risk-Neutral Pricing
I
The risk-free rate is the discount
rate for any asset including the
stock:
S0 = e −r [p ∗ uS0 + (1 − p ∗ ) dS0 ]
Solving for p ∗ gives us
er − d
p =
u−d
Standard DCF calculation
I
Suppose that the continuously
compounded expected return on
the stock is α. If p is the true
probability of the stock going up, p
must be consistent with u, d, and
α.
S0 = e −α [puS0 + (1 − p) dS0 ]
∗
Solving for p gives us
p=
eα − d
u−d
49 / 60
Understanding Risk-Neutral Pricing (cont’d)
Risk-Neutral Pricing
I
We discount the option payoff at
the risk-free rate, r .
Standard DCF calculation
I
At what rate do we discount the
option payoff? Since an option is
equivalent to holding a portfolio
consisting of ∆ shares of stock and
B bonds, the expected return on
this portfolio is
eγ =
I
The option price is
h
i
C0 = e −r p ∗ C1u + (1 − p ∗ ) C1d
I
B
S0 ∆
eα +
er
S0 ∆ + B
S0 ∆ + B
where γ is the discount rate for the
option.
The option price is
h
i
C0 = e −γ pC1u + (1 − p) C1d
50 / 60
Understanding Risk-Neutral Pricing (cont’d)
I
Are these two prices the same? Yes (proof left as an exercise).
I
Note that it does not matter whether we have the “correct” value of
α to start with.
I
Any consistent pair of α and γ will give the same option price.
I
Risk-neutral pricing is valuable because setting α = r results in the
simplest pricing procedure.
51 / 60
Understanding Risk-Neutral Pricing (cont’d)
I
We need to emphasize that at no point are we assuming that
investors are risk-neutral.
I
Rather, risk-neutral pricing is an interpretation of the formulas
above.
52 / 60
Outline
I Discrete-Time Option Pricing: The Binomial Model
3
II Call Options
A One-Period Binomial Tree
The Binomial Solution
A Two-Period Binomial Tree
Many Binomial Periods
Self-Financing Strategy
6
7
15
22
27
28
III Put Options
30
IV Uncertainty in the Binomial Model
35
V Risk-Neutral Pricing
41
VI American Options
53
53 / 60
American Options
I
An European option can only be exercised at the expiration date,
whereas an American option can be exercised at any time.
I
Because of this added flexibility, an American option must always be
at least as valuable as an otherwise identical European option:
CAmer (S, K , T ) ≥ CEur (S, K , T )
PAmer (S, K , T ) ≥ PEur (S, K , T )
I
Combining these statements, together with the maximum and
minimum option prices (from the first Section), gives us
S ≥ CAmer (S, K , T ) ≥ CEur (S, K , T ) ≥ max [0, S − PV (Div ) − PV (K )]
K ≥ PAmer (S, K , T ) ≥ PEur (S, K , T ) ≥ max [0, PV (K ) − S + PV (Div )]
54 / 60
American Call
I
An American-style call option on a nondividend-paying stock should
never be exercised prior to expiration (proof in class).
I
For an American call on a dividend-paying stock it might be beneficial
to exercise the option prior to expiration (by exercising the call, the
owner will be entitled to dividend payments that she would not have
otherwise received).
I
Consider the previous example, with one exception: δ = 0.065, i.e.,
XYZ stock has a continuous dividend yield of 6.5% per year.
55 / 60
American Call (cont’d)
I
Because of dividends, early exercise
is optimal at the node where the
stock price is $76.982.
S1u = $56.181
C1u = $18.255
u
C1,NO
u
C1,EX
S3uuu = $105.485
S2uu = $76.982
C2uu = $36.982
uu
C2,NO
= $35.213
uu
C2,EX
= $36.982
= $18.255
S2ud = $42.249
= $16.181
C2ud = $7.029
C0,NO = $8.635
S1d = $30.833
ud
C2,NO
= $7.029
C0,EX = $1
C1d = $2.761
ud
C2,EX
= $2.249
S0 = $41
C0 = $8.635
d
C1,NO
= $2.761
d
C1,EX
= $0
C·,NO = value of call if not exercised
C·,EX = value of call if exercised
C3uuu = $65.485
S2dd = $23.187
C2dd
dd
C2,NO
= $0
dd
C2,EX
= $0
S3uud = $57.892
C3uud = $17.892
S3udd = $31.772
C3udd = $0
= $0
S3ddd = $17.437
C3ddd = $0
56 / 60
American Put
I
When the underlying stock pays no dividend, a call will not be
early-exercised, but a put might be.
I
Suppose a company is bankrupt and the stock price falls to zero.
Then a put that would not be exercised until expiration will be worth
PV (K ), which is smaller than K for a positive interest rate.
I
Therefore, early exercise would be optimal in order to receive the
strike price earlier.
I
This can also be shown by using a parity argument.
I
Consider the previous example, with δ = 0.065.
57 / 60
American Put (cont’d)
I
Early exercise is optimal at the
node where the stock price is
$23.187.
S1u = $56.181
P1u = $2.314
S0 = $41
P0 = $6.546
u
P1,NO
u
P1,EX
S3uuu = $105.485
S2uu = $76.982
P2uu = $0
uu
P2,NO
= $0
uu
P2,EX
= $0
= $2.314
S2ud = $42.249
= $0
P2ud = $4.363
P0,NO = $6.546
S1d = $30.833
ud
P2,NO
= $4.363
P0,EX = $0
P1d = $10.630
ud
P2,EX
= $0
d
P1,NO
= $10.630
d
P1,EX
= $9.167
P·,NO = value of put if not exercised
PC·,EX = value of put if exercised
P3uuu = $0
S2dd = $23.187
P2dd
dd
P2,NO
= $16.813
dd
P2,EX
= $16.813
S3uud = $57.892
P3uud = $0
S3udd = $31.772
P3udd = $8.228
= $15.197
S3ddd = $17.437
P3ddd = $22.563
58 / 60
Problem 10.20: For a stock index, S = $100, σ = 30%,
r = 5%, δ = 3%, and T = 3. Let n = 3.
I
What is the price of an American call option with a strike of $95?
I
What is the price of an American put option with a strike of $95?
59 / 60
Problem 10.20: For a stock index, S = $100, σ = 30%,
r = 5%, δ = 3%, and T = 3. Let n = 3.
I
What is the price of an American call option with a strike of $95?
The price of an American call
option with a strike of 95 is
$24.1650
I
What is the price of an American put option with a strike of $95?
The price of an American put
option with a strike of 95 is
$15.2593
59 / 60
Understanding Early Exercise
I
In deciding whether to early-exercise an option, the option holder
compares the value of exercising immediately with the value of
continuing to hold the option.
I
Consider the cost and benefits of early exercise for a call option and a
put option. By exercising, the option holder
Call Option
Put Option
I
Receives the stock and therefore
receives future dividends
I
Dividends are lost by giving up the
stock
I
Bears the interest cost of paying
the strike price prior to expiration
I
Receives the strike price sooner
rather than later
I
Loses the insurance implicit in the
call (the option holder is protected
against the possibility that the
stock price will be less than the
strike price at expiration).
I
Loses the insurance implicit in the
put (the option holder is protected
against the possibility that the
stock price will be more than the
strike price at expiration)
60 / 60
Option Markets 232D
3. Black-Scholes Formula and the Greeks
Daniel Andrei
Fall 2012
1 / 74
Outline
I From Binomial Trees to the Black-Scholes Option Pricing Formula
Discrete Time vs Continuous Time
The Limiting Case of the Binomial Formula
Lognormality and the Binomial Model
Black-Scholes Assumptions
Inputs in the Binomial Model and in Black-Scholes
II Black-Scholes Formula
Black-Scholes Formula for a European Call Option
Black-Scholes Formula for a European Put Option
III Implied Volatility
IV WSJ reading
V Market-Maker Risk and Delta-Hedging
VI Option Greeks
VII Gamma-Neutrality
VIII Calendar Spreads
IX Appendix: Formulas for Option Greeks
3
4
5
11
14
15
16
17
23
27
33
38
46
60
66
70
2 / 74
Outline
I From Binomial Trees to the Black-Scholes Option Pricing Formula
Discrete Time vs Continuous Time
The Limiting Case of the Binomial Formula
Lognormality and the Binomial Model
Black-Scholes Assumptions
Inputs in the Binomial Model and in Black-Scholes
II Black-Scholes Formula
Black-Scholes Formula for a European Call Option
Black-Scholes Formula for a European Put Option
III Implied Volatility
IV WSJ reading
V Market-Maker Risk and Delta-Hedging
VI Option Greeks
VII Gamma-Neutrality
VIII Calendar Spreads
IX Appendix: Formulas for Option Greeks
3
4
5
11
14
15
16
17
23
27
33
38
46
60
66
70
3 / 74
Discrete Time vs Continuous Time
I
We refer to the structure of the binomial model as discrete time,
which means that time moves in distinct increments
I
This is much like looking at a calendar and observing only months,
weeks, or days
I
We know that time moves forward at a rate faster than one day at a
time: hours, minutes, seconds, fractions of seconds, and fractions and
fractions of seconds
I
When we talk about time moving in the tiniest increments, we are
talking about continuous time.
Discrete time world
Binomial model
Continuous time world
Black-Scholes model
4 / 74
The Limiting Case of the Binomial Formula
I
An obvious objection to the binomial calculations thus far is that the
stock can only have a few different values at expiration. It seems
unlikely that the option price calculation will be accurate.
I
The solution to this problem is to divide the time to expiration into
more periods, generating a more realistic tree.
I
To illustrate how to do this, we will re-examine the 1-year European
call option, which has a $40 strike and initial stock price $41.
I
Let there be 3 binomial periods. Since it is a 1-year call, this means
that the length of a period is h = 1/3. We will assume that other
inputs stay the same, so r = 0.08 and σ = 0.3.
5 / 74
The Limiting Case of the Binomial Formula (cont’d)
I
Since the length of the binomial period is shorter, u and d are closer
to 1 than before (1.2212 and 0.8637 as opposed to 1.4623 and 0.8025
with h = 1).
I
The risk-neutral probability of the stock price going up in a period is
p∗ =
I
e (0.08−0)×1/3 − 0.8637
= 0.4568
1.2212 − 0.8637
The binomial model implicitly assigns probabilities to the various
nodes. The risk-neutral probability for each final period node,
together with the call value, is:
Call Price in 1 Year (3 periods)
$34.678
$12.814
$0
$0
Probability
p ∗3 = 0.0953
3p ∗2 (1 − p ∗ ) = 0.34
3p ∗ (1 − p ∗ )2 = 0.4044
(1 − p ∗ )3 = 0.1603
6 / 74
The Limiting Case of the Binomial Formula (cont’d)
I
Thus, the price of the European call option is given by
C0 = e 0.08×1 (0.0953 × $34.678 + 0.34 × $12.814)
= $7.0739
I
We can vary the number of binomial steps, holding fixed the time to
expiration,T . The general formula is
C0 = e
−rT
n
X
h
i
n!
p ∗k (1 − p ∗ )n−k max S0 u k d n−k − K , 0 (1)
k! (n − k)!
k=0
(we also need to modify u, d, and p ∗ at each time).
7 / 74
The Limiting Case of the Binomial Formula (cont’d)
I
The following table computes binomial call option prices, using the
same inputs as before.
Number of steps (n)
1
2
3
4
10
50
100
500
∞
I
Binomial Call Price ($)
$7.839
$7.162
$7.074
$7.160
$7.065
$6.969
$6.966
$6.960
$6.961
Changing the number of steps changes the option price, but once the
number of steps becomes great enough we appear to approach a
limiting value for the price, given by the Black-Scholes formula.
8 / 74
The Limiting Case of the Binomial Formula (cont’d)
This can be seen graphically below:
Call Price
I
7.5
7
0
2
4
6
8 10 12 14
Number of Steps (n)
16
18
20
9 / 74
Problem 12.2: Using the BinomCall function, compute
the binomial approximations for the following call option:
S0 = $41, K = $40, σ = 0.3, r = 0.08, T = 0.25 (3
months), and δ = 0. Be sure to compute prices for
n = 8, 9, 10, 11, and 12. What do you observe about the
behavior of the binomial approximation?
10 / 74
Problem 12.2: Using the BinomCall function, compute
the binomial approximations for the following call option:
S0 = $41, K = $40, σ = 0.3, r = 0.08, T = 0.25 (3
months), and δ = 0. Be sure to compute prices for
n = 8, 9, 10, 11, and 12. What do you observe about the
behavior of the binomial approximation?
N
8
9
10
11
12
Call Price
3.464
3.361
3.454
3.348
3.446
The observed values are slowly converging towards the Black-Scholes value
(3.399). Please note that the binomial solution oscillates as it approaches
the Black-Scholes value.
10 / 74
Lognormality and the Binomial Model
I
The lognormal distribution is the probability distribution that arises
from the assumption that continuously compounded returns on the
stock are normally distributed.
I
In a binomial tree, as we increase the number of periods until
expiration, continuously compounded returns approach a normal
distribution.
I
We can plot probabilities of outcomes (of stock returns) from the
binomial tree for different values of n (2, 3, 6, and 10), as shown in
the following figure.
I
The 10-period binomial tree approaches fairly well the normal
distribution.
11 / 74
Lognormality and the Binomial Model (cont’d)
0.4
0.6
0.3
0.5
0.2
0.4
0.1
-0.22
0.38
0.3
-0.44
-0.09
0.25
0.60
0.2
0.2
0.1
0.1
0
-0
.8
-0 7
.6
-0 8
.4
-0 9
.3
-0 0
.1
0. 1
0
0. 8
2
0. 7
4
0. 6
65
0.
8
1. 4
03
.1
6
0.
08
0.
32
0.
57
0.
81
1
-0
.4
-0
-0
.6
5
0
12 / 74
Problem 11.13: Let S = $100, σ = 0.30, r = 0.08, t = 1,
and δ = 0. Use equation (1) to compute the probability of
reaching a terminal node and the price at that node and
plot the risk-neutral distribution of year-1 stock prices for
n = 50. What is the risk-neutral probability that S1 < $80?
S1 > $120?
13 / 74
Problem 11.13: Let S = $100, σ = 0.30, r = 0.08, t = 1,
and δ = 0. Use equation (1) to compute the probability of
reaching a terminal node and the price at that node and
plot the risk-neutral distribution of year-1 stock prices for
n = 50. What is the risk-neutral probability that S1 < $80?
S1 > $120?
For n = 50, we have u = 1.0450,
d = 0.9600, and p ∗ = 0.4894. We get
the following diagram:
To obtain the required probabilities we
sum all probabilities for which the final
stock price is below 80 or above 120
respectively. We obtain
0.1
0.05
Pr (S < 80) = 0.2006
Pr (S > 120) = 0.2829
0
0
200 400 600 800
13 / 74
Black-Scholes Assumptions
I
Assumptions about stock return distribution
I
I
I
I
Continously compounded returns on the stocks are normally distributed
and independent over time
The volatility of continuously compounded returns is known and
constant
Future dividends are known, either as dollar amount or as a fixed
dividend yield
Assumptions about the economic environment
I
I
I
The risk-free rate is known and constant
There are no transaction costs or taxes
It is possible to short-sell costlessly and to borrow at the risk-free rate
14 / 74
Inputs in the Binomial Model and in Black-Scholes
I
There are seven inputs to the binomial model and six inputs to the
Black-Scholes model:
Inputs
Current price of the stock, S0
Strike price of the option, K
Volatility of the stock, σ
Continuously compounded risk-free
interest rate, r
Time to expiration, T
Dividend yield on the stock, δ
Number of binomial periods, n
Binomial
Model
X
X
X
X
BlackScholes
X
X
X
X
X
X
X
X
X
15 / 74
Outline
I From Binomial Trees to the Black-Scholes Option Pricing Formula
Discrete Time vs Continuous Time
The Limiting Case of the Binomial Formula
Lognormality and the Binomial Model
Black-Scholes Assumptions
Inputs in the Binomial Model and in Black-Scholes
II Black-Scholes Formula
Black-Scholes Formula for a European Call Option
Black-Scholes Formula for a European Put Option
III Implied Volatility
IV WSJ reading
V Market-Maker Risk and Delta-Hedging
VI Option Greeks
VII Gamma-Neutrality
VIII Calendar Spreads
IX Appendix: Formulas for Option Greeks
3
4
5
11
14
15
16
17
23
27
33
38
46
60
66
70
16 / 74
Black-Scholes Formula for a European Call Option
I
The Black-Scholes formula for a European call option on a stock that
pays dividends at the continuous rate δ is
C0 (S0 , K , σ, r , T , δ) = S0 e −δT N (d1 ) − Ke −rT N (d2 )
(2)
where
S0
K
+ r − δ + 21 σ 2 T
√
d1 =
σ T
√
d2 = d1 − σ T
ln
I
(3)
(4)
N (x ) is the cumulative normal distribution function, which is the
probability that a number randomly drawn from a standard normal
distribution will be less than x .
17 / 74
Black-Scholes Formula for a European Call Option (cont’d)
I
Let S0 = $41, K = $40, σ = 0.3, r = 0.08, T = 1 year, and δ = 0.
Computing the Black-Scholes call price, we obtain
C0 = $41 × e −0 × N (0.49898) − $40 × e −0.08 × N (0.19898)
= $41 × e −0 × 0.6911 − $40 × e −0.08 × 0.5789
= $6.961
I
Note that this result corresponds to the limit obtained from the
binomial model.
18 / 74
Black-Scholes Formula for a European Call Option (cont’d)
The following figure plots Black-Scholes call option prices (today and
at expiration) for stock prices ranging from $20 to $60.
25
20
Call Price
I
15
10
5
0
20
30
40
50
Stock Price
60
19 / 74
A Remarkable Result
I
In the binomial model, we have not specified the probabilities of the
stock going up and down (which would give us the expected return of
the stock).
I
The expected return on the stock does not appear in the
Black-Scholes formula either.
I
This raises a question: Consider a stock with a higher beta (and
hence having a higher expected return). A call option on this stock
would have a higher probability of settlement in-the-money, hence a
higher option price. Why is this not the case?
20 / 74
A Remarkable Result (cont’d)
I
The Black-Scholes formula (2) provides the answer. This formula
shows the composition of the replicating portfolio, which in the
binomial case was
C0 = ∆S0 + B
I
We can easily identify ∆ (the position in the risky asset) and B (the
dollar amount of borrowing or lending) in the Black-Scholes formula:
∆ = e −δT N (d1 )
(5)
B = −Ke −rT N (d2 )
(6)
21 / 74
A Remarkable Result (cont’d)
I
If βS is the stock beta, then the option beta is
βOption =
∆
βS
∆S0 + B
(7)
I
A high stock beta implies a high option beta, so the discount rate for
the expected payoff of the option is correspondingly greater.
I
The net result—one of the key insights from the Black-Scholes
analysis—is that beta is irrelevant: The larger average payoff to
options on high beta stocks is exactly offset by the larger
discount rate.
22 / 74
Black-Scholes Formula for a European Put Option
I
The Black-Scholes formula for a European put option is
P0 (S0 , K , σ, r , T , δ) = −S0 e −δT N (−d1 ) + Ke −rT N (−d2 )
(8)
where d1 and d2 are given by equations (3) and (4).
I
Put-call parity must hold:
P0 (S0 , K , σ, r , T , δ) = C0 (S0 , K , σ, r , T , δ) + Ke −rT − Se −δT
I
(9)
This follows from the formulas (2) and (8), together with the fact
that for any x , N (−x ) = 1 − N (x ).
23 / 74
Black-Scholes Formula for a European Put Option (cont’d)
I
Let S0 = $41, K = $40, σ = 0.3, r = 0.08, T = 1 year, and δ = 0.
Computing the Black-Scholes put price, we obtain
P0 = −$41 × e −0 × N (−0.49898) + $40 × e −0.08 × N (−0.19898)
= −$41 × e −0 × 0.3089 + $40 × e −0.08 × 0.4211
= $2.886
I
In the binomial model, if we fix the number of periods to n = 500, we
obtain a price of $2.885.
I
Computing the price using put-call parity (equation 9) yields
P0 = $6.961 + $40e −0.08 − $41
= $2.886
24 / 74
Black-Scholes Formula for a European Put Option (cont’d)
The following figure plots Black-Scholes put option prices (today and
at expiration) for stock prices ranging from $20 to $60.
20
15
Put Price
I
10
5
0
20
30
40
50
Stock Price
60
25 / 74
Problem 12.20: Let S = $100, K = $90, σ = 30%,
r = 8%, δ = 5%, and T = 1.
a. What is the Black-Scholes call price?
b. Now price a put where S = $90, K = $100, σ = 30%, r = 5%, δ = 8%,
and T = 1.
26 / 74
Problem 12.20: Let S = $100, K = $90, σ = 30%,
r = 8%, δ = 5%, and T = 1.
a. What is the Black-Scholes call price?
The Black-Scholes call price is
$17.70
b. Now price a put where S = $90, K = $100, σ = 30%, r = 5%, δ = 8%,
and T = 1.
The Black-Scholes put price is
$17.70
26 / 74
Outline
I From Binomial Trees to the Black-Scholes Option Pricing Formula
Discrete Time vs Continuous Time
The Limiting Case of the Binomial Formula
Lognormality and the Binomial Model
Black-Scholes Assumptions
Inputs in the Binomial Model and in Black-Scholes
II Black-Scholes Formula
Black-Scholes Formula for a European Call Option
Black-Scholes Formula for a European Put Option
III Implied Volatility
IV WSJ reading
V Market-Maker Risk and Delta-Hedging
VI Option Greeks
VII Gamma-Neutrality
VIII Calendar Spreads
IX Appendix: Formulas for Option Greeks
3
4
5
11
14
15
16
17
23
27
33
38
46
60
66
70
27 / 74
Implied Volatility
I
Volatility is unobservable
I
Choosing a volatility to use in pricing an option is difficult but
important
I
Using history of returns is not the best approach, because history is
not a reliable guide to the future.
I
We can invert Black-Scholes formula to obtain implied volatility
I
We cannot use implied volatility to assess whether an option price is
correct, but implied volatility does tell us the market’s assessment of
volatility
28 / 74
Implied Volatility (cont’d)
I
Example: Let S = $100, K = $90, r = 8%, δ = 5%, and T = 1. The
market option price for a call option is $18.25. What is the volatility
that gives this option price?
I
We must invert the following formula
$18.25 = BSCall (100, 90, σ, 0.08, 1, 0.05)
I
We use the implied volatility function:
We find that setting σ = 31.73%
gives us a call price of $18.25
29 / 74
Implied Volatility (cont’d)
The following figure plots implied volatilities for S&P 500 options,
10/28/2004. Option prices (ask) from CBOE. Assumes
S = $1, 127.44, δ = 1.85%, r = 2%, and expiration date 1/22/2005.
Calls
Puts
Implied volatility
I
0.15
0.14
1,100
1,120
1,140
Strike
30 / 74
Implied Volatility (cont’d)
I
There is a systematic pattern of implied volatility across strike prices,
called volatility skew
I
The volatility skew is not related to whether an option is a put or a
call, but rather to differences in the strike price and time to expiration
I
Explaining these patterns is a challenge for option pricing theory
31 / 74
Using Implied Volatility
I
Implied volatility is important for a number of reasons
I
I
I
I
If you need to price an option for which you cannot observe a market
price, you can use implied volatility to generate a price consistent with
the price of traded options
Implied volatility is often used as a quick way to describe the level of
option prices on a given underlying asset. Option prices are quoted
sometimes in terms of volatility, rather than as a dollar price
Volatility skew provides a measure of how well option pricing models
work
Just as stock markets provide information about stock prices and
permit trading stocks, option markets provide information about
volatility, and, in effect, permit the trading of volatility.
32 / 74
Outline
I From Binomial Trees to the Black-Scholes Option Pricing Formula
Discrete Time vs Continuous Time
The Limiting Case of the Binomial Formula
Lognormality and the Binomial Model
Black-Scholes Assumptions
Inputs in the Binomial Model and in Black-Scholes
II Black-Scholes Formula
Black-Scholes Formula for a European Call Option
Black-Scholes Formula for a European Put Option
III Implied Volatility
IV WSJ reading
V Market-Maker Risk and Delta-Hedging
VI Option Greeks
VII Gamma-Neutrality
VIII Calendar Spreads
IX Appendix: Formulas for Option Greeks
3
4
5
11
14
15
16
17
23
27
33
38
46
60
66
70
33 / 74
Four Common Strategies in Unsettled Markets (Wall
Street Journal, October 31, 2012
34 / 74
Four Common Strategies in Unsettled Markets (Wall
Street Journal, October 31, 2012 (cont’d)
35 / 74
Four Common Strategies in Unsettled Markets (Wall
Street Journal, October 31, 2012 (cont’d)
36 / 74
Four Common Strategies in Unsettled Markets (Wall
Street Journal, October 31, 2012 (cont’d)
37 / 74
Outline
I From Binomial Trees to the Black-Scholes Option Pricing Formula
Discrete Time vs Continuous Time
The Limiting Case of the Binomial Formula
Lognormality and the Binomial Model
Black-Scholes Assumptions
Inputs in the Binomial Model and in Black-Scholes
II Black-Scholes Formula
Black-Scholes Formula for a European Call Option
Black-Scholes Formula for a European Put Option
III Implied Volatility
IV WSJ reading
V Market-Maker Risk and Delta-Hedging
VI Option Greeks
VII Gamma-Neutrality
VIII Calendar Spreads
IX Appendix: Formulas for Option Greeks
3
4
5
11
14
15
16
17
23
27
33
38
46
60
66
70
38 / 74
Market-Maker Risk
I
A market-maker stands ready to sell to buyers and to buy from
sellers.
I
Without hedging, an active market-maker will have an arbitrary
position generated by fulfilling customer orders. This arbitrary
portfolio has uncontrolled risk.
I
Consequently, market-makers attempt to hedge the risk of their
position.
I
We will se here how they do so.
39 / 74
Market-Maker Risk (cont’d)
I
Suppose a customer wishes to buy 100 European call options with
maturity of 91 days. The market-maker fills this order by selling 100
call options. To be specific, suppose that S = $41, K = $40, σ = 0.3,
r = 0.08 (continuously compounded), and δ = 0. We will let T
denote the expiration time of the option and t the present, so time to
expiration is T − t. Thus, T − t = 91/365 = 0.249.
I
Suppose that the market-maker does not hedge the written option
(naked position) and the stock has the following evolution over the
next 5 days:
Day
0
1
2
3
4
5
Stock ($)
41
42.5
39.5
37
40
40
40 / 74
Market-Maker Risk (cont’d)
I
We can measure the profit of the market-maker by
marking-to-market the position: if we liquidated the position
today, what would be the gain or loss?
Day
0
1
2
3
4
5
Stock ($)
41
42.5
39.5
37
40
40
Call
Position ($)
-339.47
-441.04
-246.31
-129.49
-271.04
-269.27
Daily
Profit ($)
-101.57
194.73
116.82
-141.55
1.77
41 / 74
Market-Maker Risk (cont’d)
Naked Position
Daily Profit
200
100
0
−100
1
2
3
Day
4
5
42 / 74
Delta-Hedging
I
Suppose the market-maker hedges the position with shares. At time
0, the delta of a call at a stock price of $41 is 0.645.
I
This suggests that a $1 increase in the stock price should increase the
value of the option by approximatively $0.645.
I
The market-maker takes an offsetting position in shares, position that
hedges the fluctuations in the option price. We say that such a
position is delta-hedged.
I
Then, the market-maker rebalances the portfolio each day, by
computing the new delta of the call.
I
The following table summarizes delta, the number of purchased
shares, the net investment, and profit for each day for 5 days (interest
expenses are ignored for simplicity).
43 / 74
Delta-Hedging (cont’d)
Day
Stock
($)
Option
Delta
0
1
2
3
4
5
41
42.5
39.5
37
40
40
0.645
0.730
0.548
0.373
0.581
0.580
Stock
Position
(# shares)
64.54
73.03
54.81
37.27
58.06
58.01
Daily
Profit
(Call)
Daily
profit
(Shares)
Daily
profit
(Total)
-101.57
194.73
116.82
-141.55
1.77
96.81
-219.10
-137.02
111.82
0.00
-4.77
-24.38
-20.20
-29.73
1.77
44 / 74
Delta-Hedging (cont’d)
Naked Position
Delta Hedged
Daily Profit
200
100
0
−100
1
I
2
3
Day
4
5
Delta hedging prevents the position from reacting to small changes in
the underlying stock. For large changes, we need to take into
account the fluctuations in delta.
45 / 74
Outline
I From Binomial Trees to the Black-Scholes Option Pricing Formula
Discrete Time vs Continuous Time
The Limiting Case of the Binomial Formula
Lognormality and the Binomial Model
Black-Scholes Assumptions
Inputs in the Binomial Model and in Black-Scholes
II Black-Scholes Formula
Black-Scholes Formula for a European Call Option
Black-Scholes Formula for a European Put Option
III Implied Volatility
IV WSJ reading
V Market-Maker Risk and Delta-Hedging
VI Option Greeks
VII Gamma-Neutrality
VIII Calendar Spreads
IX Appendix: Formulas for Option Greeks
3
4
5
11
14
15
16
17
23
27
33
38
46
60
66
70
46 / 74
Option Greeks
I
I
Option Greeks are formulas that express the change in the option
price when an input to the formula changes, taking as fixed all
other inputs.
They are used to assess risk exposures. For example:
I
I
I
A market-making bank with a portfolio of options would want to
understand its exposure to stock price changes, interest rates, volatility,
maturity, etc.
A portfolio manager wants to know what happens to the value of a
portfolio of stock index options if there is a change in the level of the
stock index.
An options investor would like to know how interest rate changes and
volatility changes affect profit and loss.
47 / 74
Option Greeks (cont’d)
I
Before providing detailed definition of the Greeks, let’s have some
intuition on how changes in inputs affect option prices:
Change in input
St ↑
σ↑
T − t ↓ (t ↑)
r ↑
δ↑
Change in call price
Ct ↑
Ct ↑
Ct generally ↓
Ct ↑
Ct ↓
Change in put price
Pt ↓
Pt ↑
Pt ambiguous
Pt ↓
Pt ↑
Table 1 : Changes in Black-Scholes inputs and their effect on option prices.
I
An increase in the stock price (St ) raises the chance that the call will
be exercised, thus raises the call option price. Conversely, it lowers
the put option price.
48 / 74
Option Greeks (cont’d)
I
An increase in volatility raises the price of a call or put option,
because it increases the expected value if the option is exercised.
I
Options generally—but not always—become less valuable as time to
expiration decreases, i.e., there is a time decay. There are
exceptions, for example deep-in-the-money call options on an asset
with high dividend yield and deep-in-the-money puts.
I
A higher interest rate reduces the present value of the strike (to be
paid by a call option holder), and thus increases the call price. The
put option entitles the owner to receive the strike, whose present
value is lower with a higher interest rate. Thus, a higher interest rate
decreases the put price.
I
A call entitle the holder to receive stock, but without dividends prior
to expiration. Thus, the greater the dividend yield, the lower the call
price. Conversely, a put option is more valuable when the dividend
yield is greater.
49 / 74
Option Greeks (cont’d)
I
The Greeks are tools that let us to quantify these relationships:
Input
Greek
Definition
Mnemonic
St
∆ (Delta)
St
Γ (Gamma)
St
Ω (Elasticity)
σ
Vega
t
θ (theta)
vega ↔
volatility
theta ↔
time
r
ρ (rho)
δ
Ψ (Psi)
Measures the option price change when
the stock price increases by $1
Measures the change in ∆ when the
stock price increases by $1
Measures the percentage change in the
option price when the stock price increases by 1%
Measures the option price change when
there is an increase in volatility of 1%
Measures the option price change when
there is a decrease in the time to maturity (increase in calendar time) of 1 day
Measures the option price change when
there is an increase in the interest rate
of 1% (100 basis points)
Measures the option price change when
there is an increase in the continuous
dividend yield of 1% (100 basis points)
rho ↔ r
50 / 74
Option Greeks (cont’d)
I
Let us come back to Table 1 and complete it with the proper signs of
the Greeks:
Input
St ↑
σ↑
t ↑
r ↑
δ↑
Call Option
Ct ↑
∆Call > 0
∆Call ↑
ΓCall > 0
ΩCall ≥ 1
Ct ↑
VegaCall > 0
Ct gen. ↓ θCall gen. < 0
Ct ↑
ρCall > 0
Ct ↓
ΨCall < 0
Put Option
Pt ↓
∆Put < 0
∆Put ↑
ΓPut > 0
ΩPut ≤ 0
Pt ↑
VegaPut > 0
Pt ambig.
θPut any
Pt ↓
ρPut < 0
Pt ↑
ΨPut > 0
51 / 74
Option Greeks: Example
I
The Greeks are mathematical derivatives of the option price formula
with respect to the inputs.
I
Suppose that the stock price is St = $41, the strike price is K = $40,
volatility is σ = 0.3, the risk-free rate is r = 0.08, the time to
expiration is T − t = 1, and the dividend yield is δ = 0. The values for
the Greeks are
Input
St
σ
t
r
δ↑
Call Option
∆Call = 0.691
>0
ΓCall = 0.029
>0
ΩCall = 4.071
≥1
VegaCall = 0.144
>0
θCall = −0.011
gen. < 0
ρCall = 0.214
>0
ΨCall = −0.283
<0
Put Option
DeltaPut = −0.309
ΓPut = 0.029
ΩPut = −4.389
VegaPut = 0.144
θPut = −0.003
ρPut = −0.156
ΨPut = 0.127
<0
>0
≤0
>0
any
<0
>0
52 / 74
Delta (∆)
Measures the change in the option price for a $1 change in the stock price:
Call
Put
1
Delta
0.5
0
−0.5
−1
20
30
40
Stock Price
50
60
53 / 74
Gamma (Γ)
Measures the change in delta when the stock price changes:
0.04
Call
Put
Gamma
0.03
0.02
0.01
20
30
40
Stock Price
50
60
54 / 74
Elasticity (Ω)
Measures the percentage change in the option price relative to the
percentage change in the stock price:
Call
Put
10
Elasticity
5
0
−5
20
30
40
Stock Price
50
60
55 / 74
Vega
Measures the change in the option price when volatility changes (divide by
100 for a change per percentage point):
Call
Put
0.15
Vega
0.1
0.05
0
20
30
40
Stock Price
50
60
56 / 74
Theta (θ)
Measures the change in the option price with respect to calendar time, t,
holding fixed the maturity date T . To obtain per-day theta, divide by 365.
Call
Put
Theta
0.01
0
−0.01
−0.01
20
30
40
Stock Price
50
60
57 / 74
Rho (ρ)
Measures the change in the option price when the interest rate changes
(divide by 100 for a change per percentage point, or by 10,000 for a
change per basis point):
0.4
Call
Put
Rho
0.2
0
−0.2
−0.4
20
30
40
Stock Price
50
60
58 / 74
Psi (Ψ)
Measures the change in the option price when the continuous dividend
yield changes (divide by 100 for a change per percentage point):
Call
Put
0.2
Psi
0
−0.2
−0.4
−0.6
20
30
40
Stock Price
50
60
59 / 74
Outline
I From Binomial Trees to the Black-Scholes Option Pricing Formula
Discrete Time vs Continuous Time
The Limiting Case of the Binomial Formula
Lognormality and the Binomial Model
Black-Scholes Assumptions
Inputs in the Binomial Model and in Black-Scholes
II Black-Scholes Formula
Black-Scholes Formula for a European Call Option
Black-Scholes Formula for a European Put Option
III Implied Volatility
IV WSJ reading
V Market-Maker Risk and Delta-Hedging
VI Option Greeks
VII Gamma-Neutrality
VIII Calendar Spreads
IX Appendix: Formulas for Option Greeks
3
4
5
11
14
15
16
17
23
27
33
38
46
60
66
70
60 / 74
Gamma-Neutrality
I
Gamma hedging is the construction of options positions that are
hedged such that the total gamma of the position is zero.
I
We cannot do this using just the stock, because the gamma of the
stock is zero (the delta of a stock is constant and equal to 1).
I
Hence, we must acquire another option in an amount that offsets the
gamma of the written call.
I
In addition to the 91-day call from the previous example, consider a
45-strike 120-day call.
I
The ratio of the gamma of the two options is
ΓK =40,T −t=91
0.0606
=
= 1.1213
ΓK =45,T −t=120 0.0540
I
(10)
Thus, we need to buy 1.1213 of the 45-strike options for every
40-strike option we have sold.
61 / 74
Gamma-Neutrality (cont’d)
I
The Greeks resulting form this position are in the last column of the
following table:
Price ($)
Delta (∆)
Gamma (Γ)
Theta (θ)
I
40-Strike Call
3.395
0.645
0.061
-0.018
45-Strike Call
1.707
0.381
0.054
-0.014
Total Position
-1.481
-0.218
0
0.002
Since delta is -0.218, we need to buy 21.8 shares of stock to be both
delta- and gamma-hedged.
62 / 74
Gamma-Neutrality (cont’d)
I
We can compare the delta-hedged position with the delta- and
gamma-hedged position. The delta-hedged position has the problem
that large moves always cause losses.
I
The delta- and gamma-hedged position loses less if there is a large
move down, and can make money if the stock price increases.
I
This can be seen from the following figure. It compares the 1-day
holding period profit for delta-hedged position described earlier and
delta- and gamma hedged position.
63 / 74
Gamma-Neutrality (cont’d)
Delta-hedged
Delta- and gamma-hedged
Overnight Profit ($)
0
−10
−20
−30
38
39
40 41 42
Stock Price
43
44
64 / 74
Gamma-Neutrality (cont’d)
Delta-gamma hedging prevents the position from reacting to large
changes in the underlying stock:
Naked Position
∆-hedged
∆- and Γ-hedged
200
Daily Profit
I
100
0
−100
1
2
3
Day
4
5
65 / 74
Outline
I From Binomial Trees to the Black-Scholes Option Pricing Formula
Discrete Time vs Continuous Time
The Limiting Case of the Binomial Formula
Lognormality and the Binomial Model
Black-Scholes Assumptions
Inputs in the Binomial Model and in Black-Scholes
II Black-Scholes Formula
Black-Scholes Formula for a European Call Option
Black-Scholes Formula for a European Put Option
III Implied Volatility
IV WSJ reading
V Market-Maker Risk and Delta-Hedging
VI Option Greeks
VII Gamma-Neutrality
VIII Calendar Spreads
IX Appendix: Formulas for Option Greeks
3
4
5
11
14
15
16
17
23
27
33
38
46
60
66
70
66 / 74
Calendar Spreads
I
To protect against a stock price increase when selling a call, you can
simultaneously buy a call option with the same strike and greater time
to expiration.
I
This purchased calendar spread exploits the fact that the written
near-to-expiration option exhibits greater time decay than the
purchased far-to-expiration option, and therefore is profitable if the
stock price does not move.
67 / 74
Calendar Spreads (cont’d)
I
Suppose you sell a 40-strike call with 91 days to expiration and buy a
40-strike call with 1 year to expiration. Assume a stock price of $40,
r = 8%, σ = 30%, and δ = 0.
I
The premiums are $2.78 for the 91-day call and $6.28 for the 1-year
call.
I
Theta is more negative for the 91-day call (-0.0173) than for the
1-year call (-0.0104). Thus, if the stock price does not change over
the course of 1 day, the position will make money since the written
option loses more value than the purchased option.
68 / 74
Calendar Spreads (cont’d)
The profit diagram for this position for a holding period of 91 days is
displayed below
Profit in 91 days ($)
I
Short call
Calendar spread
0
−10
30
40
50
Stock price ($)
60
69 / 74
Outline
I From Binomial Trees to the Black-Scholes Option Pricing Formula
Discrete Time vs Continuous Time
The Limiting Case of the Binomial Formula
Lognormality and the Binomial Model
Black-Scholes Assumptions
Inputs in the Binomial Model and in Black-Scholes
II Black-Scholes Formula
Black-Scholes Formula for a European Call Option
Black-Scholes Formula for a European Put Option
III Implied Volatility
IV WSJ reading
V Market-Maker Risk and Delta-Hedging
VI Option Greeks
VII Gamma-Neutrality
VIII Calendar Spreads
IX Appendix: Formulas for Option Greeks
3
4
5
11
14
15
16
17
23
27
33
38
46
60
66
70
70 / 74
Appendix: Formulas for Option Greeks
I
I
Delta (∆) measures the change in the option price for a $1 change in
the stock price:
∂C
= e −δ(T −t) N (d1 )
∂S
∂P
=
= −e −δ(T −t) N (−d1 )
∂S
∆Call =
(11)
∆Put
(12)
Gamma (Γ) measures the change in delta when the stock price
changes:
∂2C
e −δ(T −t) N 0 (d1 )
√
=
∂S 2
Sσ T − t
∂2P
= ΓCall
=
∂S 2
ΓCall =
(13)
ΓPut
(14)
71 / 74
Appendix: Formulas for Option Greeks (cont’d)
I
I
Elasticity (Ω) measures the percentage change in the option price
relative to the percentage change in the stock price:
St ∆Call
Ct
St ∆Put
=
Pt
ΩCall =
(15)
ΩPut
(16)
Vega measures the change in the option price when volatility changes
(divide by 100 for a change per percentage point):
√
∂C
= Se −δ(T −t) N 0 (d1 ) T − t
∂σ
∂P
=
= VegaCall
∂σ
VegaCall =
(17)
VegaPut
(18)
72 / 74
Appendix: Formulas for Option Greeks (cont’d)
I
Theta (θ) measures the change in the option price with respect to
calendar time, t, holding fixed the maturity date T . To obtain
per-day theta, divide by 365.
∂C
Ke −r (T −t ) N 0 (d2 ) σ
√
= δSe −δ(T −t ) N (d1 ) − rKe −r (T −t ) N (d2 ) −
∂t
2 T −t
(19)
∂P
=
= θCall + rKe −r (T −t ) − δSe −δ(T −t )
(20)
∂t
θCall =
θPut
I
Rho (ρ) measures the change in the option price when the interest
rate changes (divide by 100 for a change per percentage point, or by
10,000 for a change per basis point):
∂C
= (T − t) Ke −r (T −t) N (d2 )
∂r
∂P
=
= − (T − t) Ke −r (T −t) N (−d2 )
∂r
ρCall =
(21)
ρPut
(22)
73 / 74
Appendix: Formulas for Option Greeks (cont’d)
I
Psi (Ψ) Measures the change in the option price when the continuous
dividend yield changes (divide by 100 for a change per percentage
point):
∂C
= − (T − t) Se −δ(T −t) N (d1 )
∂δ
∂P
=
= (T − t) Se −δ(T −t) N (−d1 )
∂δ
ΨCall =
(23)
ΨPut
(24)
74 / 74
Option Markets 232D
4. Forwards and Futures
Daniel Andrei
Fall 2012
1 / 62
Outline
I Pricing Forwards
3
II Futures Contracts
11
III Financial Forwards and Futures
Index Futures
Currency Futures
13
14
24
IV Commodity Forwards and Futures
27
V Interest Rate Forwards and Futures
41
VI Options on Futures
55
2 / 62
Outline
I Pricing Forwards
3
II Futures Contracts
11
III Financial Forwards and Futures
Index Futures
Currency Futures
13
14
24
IV Commodity Forwards and Futures
27
V Interest Rate Forwards and Futures
41
VI Options on Futures
55
3 / 62
Alternative ways to buy a stock
1. Outright Purchase:
2. Forward:
I Pay S0
I Pay F0,T =?
I Receive security
I Receive security
Time 0
Time T
I
A forward contract is an arrangement in which you both pay for the
stock and receive it at time T , with the time T price specified at
time 0.
I
What should you pay for the stock in this case?
I
Arbitrage ensures that there is a very close relationship between
prices and forward prices
4 / 62
Pricing a Forward Contract
I
Let S0 be the spot price of an asset at time 0, and r the continuously
compounded interest rate. Assume that dividends are continuous and
paid at a rate δ.
I
Then the forward price at a future time T must satisfy
F0,T = S0 e (r −δ)T
I
(1)
Suppose that F0,T > S0 e (r −δ)T . Then an investor can execute the
following trades at time 0 (buy low and sell high) and obtain an
arbitrage profit:
Transaction
Buy tailed position in stock (e −δT units)
Borrow S0 e −δT
Short forward
Total
Time 0
−S0 e −δT
+S0 e −δT
0
0
Cash Flows
Time T (expiration)
ST
−S0 e (r −δ)T
F0,T − ST
F0,T − S0 e (r −δ)T > 0
5 / 62
Pricing a Forward Contract (cont’d)
I
Suppose that F0,T < S0 e (r −δ)T . Then an investor can execute the
following trades at time 0 (buy low and sell high) and obtain once
again an arbitrage profit:
Transaction
Short tailed position in stock (e −δT units)
Lend S0 e −δT
Long forward
Total
I
Time 0
S0 e −δT
−S0 e −δT
0
0
Cash Flows
Time T (expiration)
−ST
S0 e (r −δ)T
ST − F0,T
S0 e (r −δ)T − F0,T > 0
Consequently, and assuming that the non-arbitrage condition holds,
we have
F0,T = S0 e (r −δ)T
6 / 62
Payoff Diagram for a Forward
Payoff ($) = Profit ($)
200
Long forward ST − F0,T
100
0
−100
F0,T
Short forward F0,T − ST
−200
800
900
1,000
1,100
1,200
ST
7 / 62
Risk-Neutral Valuation
I
The risk neutral technique also works for forward contracts. When the
contract is agreed to initially, its value is 0. Its payoff at maturity is
ST − F0,T
I
Therefore
0 = S0 e −δT − F0,T e −rT
which yields F0,T = S0 e (r −δ)T
8 / 62
Put-Call Parity and the Forward Price
I
Consider a call and a put having both the same maturity and the
same strike price.
I
Let the strike price be equal to the forward price with the same
maturity:
K = F0,T = S0 e (r −δ)T
I
Write the put-call parity relationship:
C0 − P0 = S0 e −δT − Ke −rT
= S0 e −δT − S0 e (r −δ)T e −rT
=0
I
The price of a call equals the price of a put with the same maturity
when the strike price is equal to the forward price.
9 / 62
Example: Let S = $100, σ = 30%, r = 8%, δ = 5%,
T = 1, and K = $100e (0.08−0.05)×1 = $103.05.
I
What is the Black-Scholes call price?
The Black-Scholes call price is
$11.34
I
What is the Black-Scholes put price?
The Black-Scholes put price is
$11.34
10 / 62
Outline
I Pricing Forwards
3
II Futures Contracts
11
III Financial Forwards and Futures
Index Futures
Currency Futures
13
14
24
IV Commodity Forwards and Futures
27
V Interest Rate Forwards and Futures
41
VI Options on Futures
55
11 / 62
Futures Contracts
I
I
Exchange-traded forward contracts
Typical features of futures contracts
I
I
I
I
Matches buy and sell orders
Keeps track of members’ obligations and payments
After matching the trades, becomes counterparty
Differences from forward contracts
I
I
I
Settled daily through the mark-to-market process ⇒ low credit risk
Highly liquid ⇒ easier to offset an existing position
Highly standardized structure ⇒ harder to customize
12 / 62
Outline
I Pricing Forwards
3
II Futures Contracts
11
III Financial Forwards and Futures
Index Futures
Currency Futures
13
14
24
IV Commodity Forwards and Futures
27
V Interest Rate Forwards and Futures
41
VI Options on Futures
55
13 / 62
Index Futures
Figure 1 : Listing of various index futures contracts from the Wall Street
Journal, October 6-7, 2012.
14 / 62
The S&P 500 Futures Contract
Specifications for the S&P 500 index futures contract
I
Underlying: S&P 500 index
I
Where traded: Chicago Mercantile Exchange
I
Size: $250 × S&P 500 index
I
Months: Mar, Jun, Sep, Dec
I
Trading ends: Business day prior to determination of settlement price
I
Settlement: Cash-settled, based upon opening price of S&P 500 on
third Friday of expiration month
I
Suppose the futures price is 1100 and you wish to enter into 8 long
futures contracts.
I
The notional value of 8 contracts is
8 × $250 × 1100 = $2, 000 × 1100 = $2.2 million
15 / 62
The S&P 500 Futures Contract (cont’d)
I
Suppose that there is 10% margin and weekly settlement (in practice
settlement is daily). The margin on futures contracts with a notional
value of $2.2 million is $220,000.
I
The margin balance today from long position in 8 S&P 500 futures
contracts is
Week
0
I
Multiplier ($)
2000.00
Futures Price
1100.00
Price Change
—
Margin Balance ($)
220,000.00
Over the first week, the futures price drops 72.01 points to 1027.99.
On a mark-to-market basis, we have lost
$2, 000 × (−72.01) = −$144, 020
I
Thus, if the continuously compounded interest rate is 6%, our margin
balance after one week is
$220, 000 × e 0.06×1/52 − $144, 020 = $76, 233.99
16 / 62
The S&P 500 Futures Contract (cont’d)
I
Because we have a 10% margin, a 6.5% decline in the futures price
results in a 65% decline in margin. The margin balance after the first
week is
Week
0
1
Multiplier ($)
2000.00
2000.00
Futures Price
1100.00
1027.99
Price Change
—
-72.01
Margin Balance ($)
220,000.00
76,233.99
I
The decline in margin balance means the broker has significantly less
protection should we default. For this reason, participants are required
to maintain the margin at a minimum level, called the maintenance
margin. This is often set at 70% to 80% of the initial margin level.
I
In this example, the broker would make a margin call, requesting
additional margin.
I
We can go on for a period of 10 weeks, assuming weekly
marking-to-market and a continuously compounded risk-free rate of
6%.
17 / 62
The S&P 500 Futures Contract (cont’d)
I
The margin balance after a period of 10 weeks is
Week
0
1
2
3
4
5
6
7
8
9
10
I
Multiplier ($)
2000.00
2000.00
2000.00
2000.00
2000.00
2000.00
2000.00
2000.00
2000.00
2000.00
2000.00
Futures Price
1100.00
1027.99
1037.88
1073.23
1048.78
1090.32
1106.94
1110.98
1024.74
1007.30
1011.65
Price Change
—
-72.01
9.89
35.35
-24.45
41.54
16.62
4.04
-86.24
-17.44
4.35
Margin Balance ($)
220,000.00
76,233.99
96,102.01
166,912.96
118,205.66
201,422.13
234,894.67
243,245.86
71,046.69
36,248.72
44,990.57
The 10-week profit on the position is obtained by subtracting from
the final margin balance the future value of the original margin
investment:
$44, 990.57 − $220, 000 × e 0.06×10/52 = −$177, 562.60
18 / 62
The S&P 500 Futures Contract (cont’d)
I
What if the position had been forwarded rather than a futures
position, but with prices the same? In that case, after 10 weeks our
profit would have been
(1011.65 − 1100) × $2, 000 = −$176, 700
I
The futures and forward profits differ because of the interest earned
on the mark-to-market proceeds (in the present cases, we have
founded losses as they occurred and not at expiration, which explains
the loss).
19 / 62
Quanto Index Contracts
Specifications for the Nikkei 225 index futures contract
I
Underlying: Nikkei 225 stock index
I
Where traded: Chicago Mercantile Exchange
I
Size: $5 × Nikkei 225 index
I
Months: Mar, Jun, Sep, Dec
I
Trading ends: Business day prior to determination of settlement price
I
Settlement: Cash-settled, based upon opening Osaka quotation of the
Nikkei 225 index on the second Friday of expiration month
I
There is one very important difference with the S&P 500 contract:
Settlement of the contract is in a different currency (dollars) than the
currency of denomination for the index (yen)
I
Consequently, the contract insulates investors from currency risk,
permitting them to speculate solely on whether the index rises or falls
20 / 62
Uses Of Index Futures
I
Why buy an index futures contract instead of synthesizing it using the
stocks in the index? — Lower transaction costs
I
Asset allocation: switching investments among asset classes
Example: invested in the S&P 500 index and wish to temporarily
invest in bonds instead of the index. What to do?
I
I
I
Alternative #1: sell all 500 stocks and invest in bonds
Alternative #2: take a short forward position in S&P 500 index
21 / 62
Uses Of Index Futures (cont’d)
I
Suppose that the current index price, S0 , is $100, and the effective
1-year risk-free rate is 10%. The forward price is therefore $110. The
effect of owning the stock and selling forward is
Transaction
Own stock @ $100
Short forward @ $110
Total
Today
-$100
0
-$100
Cash Flows
1 year, S1 = $80 1 year, S1 = $130
$80
$130
$110 - $80
$110 - $130
$110
$110
22 / 62
Uses Of Index Futures (cont’d)
I
I
General asset allocation: futures overlay, alpha-porting
Cross-hedging: hedge portfolios that are not exactly the index
I
I
I
Cross-hedging with perfect correlation
Cross-hedging with imperfect correlation
Risk management for stock-pickers
23 / 62
Currency Futures
Figure 2 : Listing of various currency futures contracts from the Wall Street
Journal, October 6-7, 2012.
24 / 62
Currency Contracts
I
Currency forwards and futures are widely used to manage foreign
exchange risk
I
Suppose that T years from today you want to buy U1 with dollars.
Denote the yen-denominated interest rate by ry , the
dollar-denominated interest rate by r , and the exchange rate today
($/U) by x0
I
The forward price for a yen is
F0,T = x0 e (r −ry )T
(2)
I
The forward currency rate will exceed the current exchange rate when
the domestic risk-free rate is higher than the foreign risk-free rate
I
Notice that equation (2) is just like equation (1), for stock index
futures, with the foreign interest rate equal to the dividend yield.
25 / 62
Currency Contracts (cont’d)
I
We can prove equation (2) by absence of arbitrage and by building a
synthetic currency forward: borrowing in one currency and lending in
another, which creates the same cash-flow as the forward contract:
Cash Flows
Transaction
Borrow x0 e −ry T dollar at r
Convert to yen @ x0
Invest in yen-denominated bill at ry
Total
I
Year 0
$
+x0 e −ry T
−x0 e −ry T
—
0
U
—
+e −ry T
−e −ry T
0
Year T
$
−x0 e (r −ry )T
—
—
−x0 e (r −ry )T
U
—
—
1
1
Example: Suppose that the yen-denominated interest rate is ry = 2%
and the dollar-denominated rate is r = 6%. The current exchange
rate rate is x0 = 0.009 dollars per yen. The 1-year forward rate is
0.009e 0.06−0.02 = 0.009367
26 / 62
Outline
I Pricing Forwards
3
II Futures Contracts
11
III Financial Forwards and Futures
Index Futures
Currency Futures
13
14
24
IV Commodity Forwards and Futures
27
V Interest Rate Forwards and Futures
41
VI Options on Futures
55
27 / 62
Commodity Forwards and Futures
Figure 3 : Listing of various currency futures contracts from the Wall Street
Journal, October 6-7, 2012.
28 / 62
Introduction to Commodity Forwards and Futures
I
Financial forward prices are described by the general formula
F0,T = S0 e (r −δ)T
I
At a general level, commodity forward prices can be described by the
same formula. There are, however, important differences:
I
I
I
For financial assets δ is the dividend yield, whereas for commodities δ is
the commodity lease rate
While the dividend yield for a financial asset can typically be observed
directly, the lease rate for a commodity can be estimated only by
observing the forward price
The formula for a commodity forward price is
F0,T = S0 e (r −δl )T
(3)
29 / 62
Forward Prices and The Lease Rate
I
When we observe the forward price, we can infer the lease rate.
Specifically, if the forward price is F0,T , the annualized lease rate is
F0,T
1
δl = r − ln
T
S0
I
(4)
If instead we use an effective annual interest rate, the effective annual
lease rate is
δl =
1+r
(F0,T /S0 )1/T
−1
(5)
30 / 62
The Forward Curve
I
Commodities are complex because every commodity market differs in
the details. For example:
I
I
I
I
Storage is not possible for electricity
Gold is durable and relatively inexpensive to store (compared to its
value)
Some commodities feature seasonality in production (for example, corn
in the United States is harvested primarily in the fall)
One way to observe and understand this heterogeneity is to build the
forward curve (or the forward strip), i.e., the set of prices for
different expiration dates for a given commodity
I
I
I
If on a given date the forward curve is upward-sloping, then the market
is in contango.
If the forward curve is downward sloping, the market is in
backwardation
Forward curves can have portions in backwardation and portions in
contango
31 / 62
Forward Curves for Various Commodities, May 5, 2004
Corn Futures Price (cents per bushel)
Gasoline Futures Price (cents per gallon)
130
319
125
318
Backwardation
120
317
Backwardation
Contango
115
316
110
315
1
2
3
4
Months to Maturity
5
2
3
4
5
6
Months to Maturity
7
Gold Futures Price ($ per ounce)
Crude Oil Futures Price ($ per barrel)
39.5
395.5
39
395
38.5
Contango
Backwardation
394.5
38
394
37.5
393.5
2
3
4
5
6
Months to Maturity
7
1
2
3
4
5
Months to Maturity
6
32 / 62
Futures Prices for Corn for the first Wednesday in June,
1995-2004
Futures Prices (cents per bushel)
450
400
350
300
250
200
Jul Sep
Dec
Mar May Jul Sep
Dec
Expiration Month
33 / 62
Oil Prices: Contango vs. Backwardation
I
NPR news, Dec 17, 2008: Contango in Oil Markets Explained
I
I
The supply glut in the oil market has led to a contango price structure,
in which oil futures are priced higher than their spot price... Usually,
the market goes into contango when there is a supply glut, when
there’s too much oil on the market. And this is what we’ve seen
happen just in the last few months as the recession fully set in and
people realized the days of easy credit and high oil prices are gone...
There’s just a lot of oil floating around with no demand to take it up.
Wall Street Journal, Oct 28, 2011: Oil Futures Return to
Backwardation
I
Crude oil traded on the New York Mercantile Exchange has been
trading in contango, a commodity price structure in which futures
contracts for near-term delivery are cheaper than further-out contracts.
But that’s changed in the last few days... Currently, we do have rising
production from Libya, which does suggest more generous supply in the
months ahead... Aside from that, we’re also at a time of year when
global demand is strong seasonally and low seasonal demand in the
second quarter points to a potential surplus at that time—a reason for
prices to be softer then.
34 / 62
Nymex Crude Falls as Brent Divergence Widens
From the Wall Street Journal,
October 8, 2012:
I They’re really in two different
universes... In terms of supply and
demand, they’re just two different
markets.
I Declining output in the North Sea,
tension in the Middle East, and the
Iranian oil embargo are having a
disproportionate impact on the Brent
market.
I In the U.S., the Energy Information
Administration reported domestic
crude-oil output rose to its highest
level since 1996 during the prior
week. The surge in production has
been fueled by rising flows from U.S.
shale-oil fields.
35 / 62
Gold Futures
Specifications for the NYMEX gold futures contract
I
Underlying: Refined gold bearing approved refiner stamp
I
Where traded: New York Mercantile Exchange
I
Size: 100 troy ounces
I
Months: Feb, Apr, Aug, Oct, out two years. Jun, Dec, out 5 years
I
Trading ends: Third-to-last business day of maturity month
I
Delivery: Any business day of the delivery month
36 / 62
Suppose that on June 6, 2001, the gold spot price is
$265.7 and the gold future price with maturity in
December is $269. The June to December interest rate
(annualized, effective) is 3.9917%. What is the annualized
6-month gold lease rate?
I
Using equation (5), the annualized 6-month lease rate is
6-month lease rate =
1 + 0.039917
(269/265.7)1/0.5
− 1 = 1.456%
37 / 62
Gold Lease Rate Slides to Lowest on Record as European
Banks Seek Dollars
I
Bloomberg, December 8, 2011:
I
I
The interest rate for lending gold in exchange for dollars plunged to the
lowest on record this week as European banks sought ways to secure
the U.S. currency amid the region’s debt crisis
European banks especially are having liquidity funding problems, which
does see a lot of lending of gold and that’s putting downward pressure
on lease rates
38 / 62
Coffee Standoff Tests Growers’ Grit
Source: Wall Street Journal, October 8, 2012
39 / 62
Gas Futures Tumble After Power Outages Sap Demand
From the Wall Street Journal, October 31, 2012:
I
Natural-gas futures tumbled 3% on Tuesday as traders bet on falling
demand from gas-fired power plants after Storm Sandy cut power to
millions of people along the East Coast.
I
“We’re faced with the loss of demand in a well-supplied market,” said
Teri Viswanath, an energy analyst at BNP Paribas.
I
Futures prices could see further declines if data later this week and
next show falling demand and rising stockpiles.
I
Natural gas for December delivery settled 11.2 cents lower at $3.691
per million British thermal units on New York Mercantile Exchange
electronic trading.
I
After tumbling to decade lows in April under $2/MMBtu, gas futures
have made a steady climb amid rising demand from utilities. Many
power producers switched to burning gas from coal to take advantage
of low gas prices.
40 / 62
Outline
I Pricing Forwards
3
II Futures Contracts
11
III Financial Forwards and Futures
Index Futures
Currency Futures
13
14
24
IV Commodity Forwards and Futures
27
V Interest Rate Forwards and Futures
41
VI Options on Futures
55
41 / 62
Bond Basics
1
[1+r (0,t2 )]t2
P(0, t2 )
P(0, t1 )
0
1
[1+r (0,t1 )]t1
$1
t1 years
P0 (t1 , t2 )
I
$1
t2 years
1
[1+r0 (t1 ,t2 )]t2 −t1
$1
Notation:
I
I
I
I
P(0, t1 ) and P(0, t2 ): zero coupon bond
r (0, t1 ) and r (0, t2 ): effective annual interest rate (yield to maturity)
P0 (t1 , t2 ): implied forward zero-coupon bond price (quoted at time 0)
r0 (t1 , t2 ): implied forward rate (prevailing at time 0)
42 / 62
Implied Forward Rates and Zero Coupon Bonds
I
Suppose that today is 0 and we plan to borrow $1 at t1 and pay back
at t2
I
We can synthetically create this by trading zero-coupon bonds
Transaction
Sell P(0, t1 )/P(0, t2 ) z-c bonds maturing @ t2
Buy 1 z-c bond maturing @ t1
Total
I
Time 0
+P(0, t1 )
−P(0, t1 )
0
Cash Flows
Time t1 Time t2
P(0,t1 )
— − P(0,t
2)
$1
—
1)
$1 − P(0,t
P(0,t2 )
It follows that the implied forward zero-coupon bond price must be
consistent with the implied forward interest rate and the zero-coupon
bond prices with maturities t1 and t2 :
P0 (t1 , t2 ) =
P(0, t2 )
1
t2 −t1 =
P(0, t1 )
[1 + r0 (t1 , t2 )]
43 / 62
Forward Rate Agreements
I
Consider a firm expecting to borrow $100m for 91 days (3 months),
beginning 120 days from today, in June (using our previous notation,
t1 = 120/360 is the borrowing date and t2 = 211/360 is the loan
repayment date).
I
In June, the effective quarterly interest rate can be either 1.5% or 2%.
The implied June 91-day forward rate (the rate from June to
September) is 1.8%.
I
Here is the risk faced by the borrower, assuming no hedging:
Transaction
June (t1 = 120/360)
Borrow $100m
+$100m
September (t2 = 211/360)
rquarterly = 1.5% rquarterly = 2%
-$101.5m
-$102.0m
44 / 62
Forward Rate Agreements (cont’d)
I
I
A forward rate agreement (FRA) is an over-the-counter contract
that guarantees a borrowing or lending rate on a given notional
principal amount.
FRAs can be settled at maturity (in arrears) or at the initiation of
the borrowing or lending transaction
I
FRA settlement in arrears:
Payment to the borrower = rquarterly − rFRA × notional principal
I
FRA settlement at the borrowing date:
Payment to the borrower =
rquarterly − rFRA
× notional principal
1 + rquarterly
45 / 62
Forward Rate Agreements (cont’d)
I
I
FRA settlement in arrears:
Transaction
June (t1 = 120/360)
Borrow $100m
FRA payment if rquarterly = 1.5%
FRA payment if rquarterly = 2%
Total
+$100m
—
—
+$100m
September (t2 = 211/360)
rquarterly = 1.5% rquarterly = 2%
-$101.5m
-$102.0m
-$0.3m
—
—
+$0.2m
-$101.8m
-$101.8m
FRA settlement at the time of borrowing:
Transaction
June (t1 = 120/360)
Borrow $100m
FRA payment if rquarterly = 1.5%
FRA payment if rquarterly = 2%
Total
+$100m
−$295, 566.50
+$196, 078.43
+$100m
September (t2 = 211/360)
rquarterly = 1.5% rquarterly = 2%
-$101.5m
-$102.0m
-$0.3m
+$0.2m
-$101.8m
-$101.8m
46 / 62
Eurodollar Futures
Specifications for the Eurodollar futures contract
I
Where traded: Chicago Mercantile Exchange
I
Size: 3-month Eurodollar time deposit, $1 million principal
I
Months: Mar, Jun, Sep, Dec, out 10 years, plus 2 serial months and
spot month
I
Trading ends: 5 A.M. (11 A.M. London) on the second London bank
business day immediately preceding the third Wednesday of the
current month
I
Delivery: Cash settlement
I
Settlement: 100 - British Banker’s Association Futures Interest
Settlement Rate for 3-Month Eurodollar Interbank Time Deposits.
(This is a 3-month rate annualized by multiplying by 360/90.)
47 / 62
Eurodollar Futures (cont’d)
Figure 4 : Listing for the 3-month Eurodollar futures contract from the Wall
Street Journal, October 17, 2012.
48 / 62
Eurodollar Futures (cont’d)
I
Since the notional principal is $1 million, a change in annualized
LIBOR of 1 basis point would change the borrowing cost by
0.0001
× $1million = $25
4
I
The payoff at expiration of a long Eurodollar futures contract is
[(100 − rLIBOR ) − Futures Price] × 100 × $25
Convert the change in the
futures price to basis points
Change in borrowing cost
per basis point
49 / 62
Eurodollar Futures: Example
I
Let’s consider again the example in which we wish to guarantee a
borrowing rate for a $100m loan from June to September
I
Suppose the June Eurodollar futures price is 92.8. Implied 3-month
LIBOR is
100 − 92.8
= 1.8% over 3 months
4
I
As before, consider 2 possible 3-month borrowing rates in June:
I
I
1.5%, which correspond to a Eurodollar futures price in June of 94
2%, which correspond to a Eurodollar futures price in June of 92
50 / 62
Eurodollar Futures: Example (cont’d)
I
We expect to borrow $100m for 91 days, beginning 120 days from
today. Thus, we are short the interest rate.
I
We wish to hedge this exposure ⇒ we short 100 Eurodollar futures
contracts
I
The futures contract settles in June:
[(92.8 − 94) × 100 × $25] × 100
Transaction
June (t1 = 120/360)
Borrow $100m
Eurodollar payment if rquarterly = 1.5%
Eurodollar payment if rquarterly = 2%
+$100m
−$300, 000
+$200, 000
September (t2 = 211/360)
rquarterly = 1.5% rquarterly = 2%
-$101.5m
-$102.0m
[(92.8 − 92) × 100 × $25] × 100
51 / 62
Eurodollar Futures: Example (cont’d)
I
This is like the payment on an FRA paid in arrears, except that the
futures contract settles in June, but our interest expense is not paid
until September.
I
We need to tail the position in order to earn or pay interest on our
Eurodollar gain or loss before we actually have to make the interest
payment:
Number of Eurodolar contracts = −
100
= −98.2318
1 + 0.018
Transaction
June (t1 = 120/360)
Borrow $100m
Eurodollar payment if rquarterly = 1.5%
Eurodollar payment if rquarterly = 2%
Total
+$100m
−$294, 695
+$196, 464
+$100m
September (t2 = 211/360)
rquarterly = 1.5% rquarterly = 2%
-$101.5m
-$102.0m
-$299,116
+$200,393
-$101.799m
-$101.799m
52 / 62
Convexity Bias
I
I
The net borrowing cost appears to be a little less than 1.8%
This is not due to rounding error. Actually, the settlement structure
of the Eurodollar contract works systematically in favor of the
borrower:
I
I
When the interest rate turns out to be high (2%), the short Eurodollar
contract has a positive payoff and the proceeds can be reinvested at
the high realized rate
When the interest rate turns out to be low (1.5%), the short Eurodollar
contract has a negative payoff and we can found this loss until the loan
payment date by borrowing at a low rate
I
In order for the futures price to be fair to both the borrower and
lender, the rate implicit in the Eurodollar futures price must be higher
than a comparable FRA rate.
I
This difference between the FRA rate and the Eurodollar rate is called
convexity bias.
53 / 62
Suppose that on June 6, 2001, the gold spot price is
$265.7 and the gold future price with maturity in
December is $269. The June and September Eurodollar
futures prices on this date are 96.09 and 96.13. What is
the annualized 6-month gold lease rate?
54 / 62
Suppose that on June 6, 2001, the gold spot price is
$265.7 and the gold future price with maturity in
December is $269. The June and September Eurodollar
futures prices on this date are 96.09 and 96.13. What is
the annualized 6-month gold lease rate?
I
The 3-month LIBOR from June to September is
100−96.09
× 91
400
90 = 0.988%. The 3-month LIBOR from September to
× 91
December is 100−96.13
400
90 = 0.978%
I
The June to December interest rate is therefore
1.00988 × 1.00978 − 1 = 1.9763%, or 3.9917% annualized.
I
Using equation (5), the annualized 6-month gold lease rate is
6-month lease rate =
1 + 0.039917
(269/265.7)1/0.5
− 1 = 1.456%
54 / 62
Outline
I Pricing Forwards
3
II Futures Contracts
11
III Financial Forwards and Futures
Index Futures
Currency Futures
13
14
24
IV Commodity Forwards and Futures
27
V Interest Rate Forwards and Futures
41
VI Options on Futures
55
55 / 62
Options on Futures
I
The options we have considered so far, spot options, provide the
holder with the right to buy or sell a certain asset by a certain date
for a certain price
I
For futures options, the exercise of the option gives the holder a
position in the futures contract.
Why trade option on futures rather than options on the underlying
asset?
I
I
I
I
I
Futures contracts are, in many circumstances, more liquid than the
underlying asset
A futures price is known immediately from trading on the futures
exchange, whereas the spot price of the underlying may not be so
readily available
Futures on commodities are often easier to trade than the commodities
themselves
Futures options entail lower transactions costs than spot options in
many situations
56 / 62
Options on Futures: Mechanics
I
Call Futures Options (American Style)
I
When a call futures option is exercised, the holder acquires
1. A long position in the futures
2. A cash amount equal to the excess of the most recent settlement
futures price over the strike price
I
Put Futures Options (American Style)
I
When a put futures option is exercised, the holder acquires
1. A short position in the futures
2. A cash amount equal to the excess of the strike price over the most
recent settlement futures price
57 / 62
Options on Futures: Mechanics (cont’d)
I
I
If we compare how the spot and futures price evolve on the tree, we
observe a different solution for u and d in the case of future prices
The solution is exactly what we would get for an option on a stock
index if δ, the dividend yield, were equal to the risk-free rate:
St+h = St e (r −δ)h+σ
Ft+h,T = Ft,T e σ
St
Ft,T = St e (r −δ)nh
√
I
h
h
St+h = St e (r −δ)h−σ
Ft+h,T = Ft,T e −σ
√
√
√
h
h
Thus, the nodes are constructed as
u = eσ
√
d = e −σ
h
√
h
58 / 62
Options on Futures Contracts: Example
I
An option has a gold futures contract as the underlying asset.
Assume S = $300, K = $300, σ = 0.1, r = 0.05, T = 1 year, δ = 0.02,
and h = 1/3.
I
The following figure shows the tree for pricing an American call
option on a gold futures contract.
59 / 62
Options on Futures Contracts: Example (cont’d)
S3uuu = F3uuu = $367.6
F2uu = $347.0
S2uu = $343.5
F1u = $327.5
F0 = $309.1
S1u = $321.0
S3uud = F3uud = $327.5
F2ud = $309.1
S2ud = $306.1
S0 = $300.0
F1d = $291.8
S1d = $286.0
S3udd = F3udd = $291.8
F2dd = $272.7
S2dd = $275.4
S3ddd = F3ddd = $260.0
60 / 62
Options on Futures Contracts: Example (cont’d)
I
Binomial tree for pricing an
American call option on a gold
futures contract
F1u = $327.5
F0 = $309.1
C0 = $17.1
C0,NO = $17.1
C0,EX = $9.1
C1u
u
C1,NO
u
C1,EX
F1d
C1d
d
C1,NO
= $29.1
= $29.1
= $27.5
F3uuu = $367.6
F2uu = $347.0
C2uu
uu
C2,NO
uu
C2,EX
= $47.0
= $46.2
= $47.0
F2ud = $309.1
= $13.1
= $291.8
C2ud
ud
C2,NO
= $6.3
ud
C2,EX
= $9.1
= $6.3
d
C1,EX
= $0
C3uuu = $67.6
F3uud = $327.5
C3uud = $27.5
= $13.1
F2dd = $275.4
F3udd = $291.8
C3udd = $0
C2dd = $0
dd
C2,NO
= $0
dd
C2,EX
= $0
C·,NO = value of call if not exercised
C·,EX = value of call if exercised
F3ddd = $260.0
C3ddd = $0
61 / 62
Options on Futures Contracts: Example (cont’d)
I
Early exercise is optimal when the futures price is $347.0 (which
corresponds to a $343.5 spot price for the gold).
I
The intuition for early exercise is that when an option on a futures
contract is exercised, the option holder pays nothing, is entered into a
futures contract, and receives mark-to-market proceeds of the
difference between the strike price and the futures price.
I
The motive for exercise is the ability to earn interest on the
mark-to-market proceeds.
62 / 62
Option Markets 232D
5. Other Topics
Daniel Andrei
Fall 2012
1 / 67
Outline
I Swaps
Commodity Swaps
Interest Rate Swaps
Variance and Volatility Swaps
3
5
11
16
II Measurement and Behavior of Volatility
26
III Extending the Black-Scholes Model
Option Pricing When the Stock Price Can Jump
Stochastic Volatility
35
37
42
IV Revision
49
V About My Current Research
60
2 / 67
Outline
I Swaps
Commodity Swaps
Interest Rate Swaps
Variance and Volatility Swaps
3
5
11
16
II Measurement and Behavior of Volatility
26
III Extending the Black-Scholes Model
Option Pricing When the Stock Price Can Jump
Stochastic Volatility
35
37
42
IV Revision
49
V About My Current Research
60
3 / 67
Introduction to Swaps
I
A swap is a contract calling for an exchange of payments, on one or
more dates, determined by the difference in two prices
I
A swap provides a means to hedge a stream of risky payments
I
A single-payment swap is the same thing as a cash-settled forward
contract
4 / 67
An Example of a Commodity Swap
I
An industrial producer, IP Inc., needs to buy 100,000 barrels of oil 1
year from today and 2 years from today
I
The forward prices for delivery in 1 year and 2 years are $20 and
$21/barrel
I
The 1- and 2-year zero-coupon bond yields are 6% and 6.5%
(annualized, effective)
5 / 67
An Example of a Commodity Swap (cont’d)
I
IP can guarantee the cost of buying oil for the next 2 years by
entering into long forward contracts for 100,000 barrels in each of the
next 2 years. The present value of this cost per barrel is
$21
$20
+
= $37.383
1.06 1.0652
I
Thus, IP could pay an oil supplier any payment stream with a present
value of $37.383. Typically, a swap will call for equal payments in
each year
I
For example, the payment per year per barrel, x , will have to satisfy
the following equation
x
x
+
= $37.383
1.06 1.0652
I
We find x = $20.483 and we say that the 2-year swap price is $20.483
6 / 67
Computing the Commodity Swap Rate
I
Suppose there are n swap settlements, occurring on dates ti ,
i = 1, ..., n
I
The price of a zero-coupon bond maturing on date ti is P(0, ti )
I
The fixed payment on a commodity swap is
Pn
i=1 P(0, ti )F0,ti
F̄ = P
n
i=1 P(0, ti )
(1)
where F0,ti is the forward price with maturity ti
I
The commodity swap price is a weighted average of commodity
forward prices, where zero-coupon bond prices are used to determine
the weights
7 / 67
Computing the Commodity Swap Rate (cont’d)
I
A buyer with seasonally varying demand (e.g., someone buying gas for
heating) might enter into a swap in which quantities vary over time
I
The swap price with seasonally-varying quantities is
Pn
i=1 Qti P(0, ti )F0,ti
F̄ = P
n
i=1 Qti P(0, ti )
(2)
where Qti is the quantity of gas purchased at time ti
I
when Qt = 1, the formula is the same as equation (1), when the
quantity is not varying
I
It is also possible for prices to be time-varying (e.g., a gas buyer who
needs gas for heating can enter into a swap in which the summer
price is fixed at a low value, and then winter price is then determined
by the zero present value condition)
8 / 67
Physical Versus Financial Settlement
$20.483
Oil Buyer
Oil
Swap Counterparty
Physical settlement
Spot – $20.483
Oil Buyer
Swap Counterparty
Spot
Oil
Oil Seller
Financial settlement
9 / 67
Swaps are nothing more than forward contracts coupled
with borrowing and lending money
I
Consider the swap price of $20.483/barrel. Relative to the forward
curve price of $20 in 1 year and $21 in 2 years, we are overpaying by
$0.483 in the first year, and we are underpaying by $0.517 in the
second year
I
Thus, by entering into the swap, we are lending the counterparty
money for 1 year. The interest rate on this loan is
0.517
− 1 = 7%
0.483
I
Given 1- and 2- year zero-coupon bond yields of 6% and 6.5%, the
1-year implied forward yield from year 1 to year 2 is
r0 (1, 2) =
P(0, 1)
− 1 = 7%
P(0, 2)
10 / 67
Interest Rate Swaps
I
The notional principal of the swap is the amount on which the
interest payments are based
I
The life of the swap is the swap term or swap tenor
I
If swap payments are made at the end of the period (when interest is
due), the swap is said to be settled in arrears
11 / 67
An Example of an Interest Rate Swap
I
XYZ Corp. has $200M of floating-rate debt at LIBOR, i.e., every year
it pays that year’s current LIBOR
I
XYZ would prefer to have fixed-rate debt with 3 years to maturity
I
XYZ could enter a swap, in which they receive a floating rate and pay
the fixed rate, which is 6.9548%
12 / 67
An Example of an Interest Rate Swap (cont’d)
Pay LIBOR
Lender
Borrower (XYZ)
Pay 6.9548%
Receive LIBOR
I
Swap Counterparty
On net, XYZ pays 6.9548%
XYZ net payment = −LIBOR + LIBOR − 6.9548% = −6.9548%
13 / 67
Computing the Swap Rate
I
Suppose there are n swap settlements, occurring on dates ti ,
i = 1, ..., n
I
The implied forward interest rate from date ti−1 to date ti , known at
date 0, is r0 (ti−1 , ti )
I
The price of a zero-coupon bond maturing on date ti is P(0, ti )
I
The fixed swap rate, R, is
Pn
R=
i=1 P(0, ti )r (ti−1 , ti )
Pn
i=1 P(0, ti )
(3)
where ni=1 P(0, ti )r (ti−1 , ti ) is the present value of interest payments
P
implied by the strip of forward rates, and ni=1 P(0, ti ) is the present
value of a $1 annuity when interest rates vary over time
P
14 / 67
Computing the Swap Rate (cont’d)
I
We can rewrite equation (3) to make it easier to interpret
R=
n
X
i=1
I
"
#
P(0, ti )
Pn
r (ti−1 , ti )
j=1 P(0, tj )
(4)
Thus, the fixed swap rate is a weighted average of the implied forward
rates, where zero-coupon bond prices are used to determine the
weights
15 / 67
Variance and Volatility Swaps: Why Trade Volatility?
I
Just as stock investors think they know something about the direction
of the stock market, or bond investors think they can foresee the
probable direction of interest rates, so you may think you have insight
into the level of future volatility
I
What do you do if you simply want exposure to a stock’s volatility?
I
Stock options are impure: they provide exposure to both the direction
of the stock price and its volatility
I
The easy way to trade volatility is to use volatility swaps, sometimes
called realized volatility forward contracts
I
These products provide pure exposure to volatility (and only to
volatility)
16 / 67
Volatility Swaps
I
A stock volatility swap is a forward contract on annualized volatility.
Its payoff at expiration is equal to
(σR − Kvol ) × N
(5)
where
I
I
I
σR is the realized stock volatility (quoted in annual terms) over the life
of the contract
Kvol is the annualized volatility delivery price, typically quoted as a
volatility, for example 30%
N is the notional amount of the swap in dollars per annualized
volatility point, for example N = $250, 000/(volatility point)
17 / 67
Volatility Swaps (cont’d)
I
The holder of a volatility swap at expiration receives N dollars for
every point by which the stock’s realized volatility σR has exceeded
the volatility delivery price Kvol
I
He or she is swapping a fixed volatility Kvol for the actual (floating)
future volatility σR
The procedure for calculating the realized volatility should be clearly
specified with respect to the following aspects:
I
I
I
I
I
I
How frequently the return is measured
Whether returns are continuously compounded or arithmetic
Whether the variance is measured by subtracting the mean or by simply
squaring the returns
The period of time over which variance is measured
How to handle days on which trading does not occur
18 / 67
Uses for Volatility Swaps
I
Directional trading of volatility levels. Clients who want to
speculate on the future levels of stock or index volatility can go long
or short realized volatility with a swap
I
Hedging implicit volatility exposure. Risk arbitrageurs, investors
following active benchmarking stragegies, portfolio managers, equity
funds are implicitly short volatility. They can hedge this exposure by
going long realized volatility with a swap
19 / 67
Variance Swaps
I
A variance swap is a forward contract on annualized variance, the
square of the realized volatility. Its payoff at expiration is equal to
σR2 − Kvar × N
(6)
where
I
I
I
σR2 is the realized stock variance (quoted in annual terms) over the life
of the contract
Kvar is the delivery price for variance, for example (30%)2
N is the notional amount of the swap in dollars per annualized
volatility point squared, for example N = $100, 000/(volatility point)2
20 / 67
Variance vs. Volatility Contracts
10
Variance Swap
Payoff
5
0
Kvol
−5
−10
Volatility Swap
−15
15
20
25
30
35
40
σR
21 / 67
Replicating Variance Swaps
I
If you own a portfolio of options of all strikes, weighted in inverse
proportion to the square of the strike level, you will obtain an
exposure to variance that is independent of stock price, just what is
needed to trade variance.
I
There is no simple replication strategy for synthesizing a volatility
swap.
I
We can approximate a volatility swap by statically holding a suitably
chosen variance contract.
22 / 67
The CBOE Volatility Index – VIX
I
In 1993, the Chicago Board Options Exchange (CBOE) introduced
the CBOE Volatility Index, VIX, which was originally designed to
measure the market’s expectation of 30-day volatility implied by
at-the-money S&P 100 Index option prices
I
Ten years later in 2003, CBOE and Goldman Sachs updated the VIX
to reflect a new way to measure expected volatility. The new VIX is
based on the S&P 500 Index, and estimates expected volatility by
averaging the weighted prices of puts and calls over a wide range of
strike prices.
23 / 67
The VIX Calculation
I
I
The CBOE utilizes a wide variety of strike prices for SPX puts and
calls to calculate the VIX
The generalized formula used in the VIX calculation is
2
2 X ∆Ki rT
1 F0,T
2 X ∆Ki rT
e Put(Ki ) +
e Call(Ki ) −
−1
σ̂ =
T
T
T K0
Ki2
Ki2
2
Ki ≤K0
Ki >K0
where
I
I
I
I
I
σ̂ is VIX/100
T is time to expiration (in order to arrive at a 30 day implied volatility
value, the calculation blends options expiring on two different dates,
with the result being an interpolated implied volatility number)
F0,T is the forward index price
Ki is the strike price of i th out-of-the-money option
r is the risk-free interest rate to expiration
24 / 67
Why should we care about VIX?
I
I
VIX provides important information about investor sentiment. Since
volatility often signifies financial turmoil, the VIX is often referrred to
as the investor fear gauge.
Investors can use VIX options and VIX futures to hedge their
portfolios. The VIX is a good hedging tool because it has a strong
negative correlation to the S&P 500
80
60
40
20
Jan 2004
Jan 2006
Jan 2008
Jan 2010
Jan 2012
25 / 67
Outline
I Swaps
Commodity Swaps
Interest Rate Swaps
Variance and Volatility Swaps
3
5
11
16
II Measurement and Behavior of Volatility
26
III Extending the Black-Scholes Model
Option Pricing When the Stock Price Can Jump
Stochastic Volatility
35
37
42
IV Revision
49
V About My Current Research
60
26 / 67
Volatility is Crucial in Finance
I
for forecasting return
I
for the pricing of derivatives
I
for asset allocation (trade-off between return and risk)
I
for risk management (evaluation of the risk of a portfolio)
27 / 67
The major problem with volatility is that it is not directly
observable from returns
I
Unconditional volatility is estimated as the sample standard
deviation
v
u
T
u1 X
σ=t
(rt − µ)2
T
(7)
t=1
where rt is the log return on period t and µ is the sample mean over
T periods
I
However, volatility is actually not constant through time. Therefore,
conditional volatility σt is a more relevant measure of risk at time t.
28 / 67
Volatility is not constant through time (volatility
clustering)
SP500 return
0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
−0.25
0
200
400
600
800
1000
1200
1400
29 / 67
Using squared daily returns to proxy volatility will
produce a very noisy volatility estimator
SP500 squared return
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0
200
400
600
800
1000
1200
1400
30 / 67
An alternative measure of volatility is the absolute return
SP500 absolute return
0.25
0.2
0.15
0.1
0.05
0
0
200
400
600
800
1000
1200
1400
31 / 67
The GARCH Model
I
The ARCH (AutoRegressive Conditional Heteroskedasticity)
model has been introduced by Engle (1982)
I
Basic idea: unexpected returns εt are serially uncorrelated but
dependant. The dependency in εt is described by a quadratic function
of its lagged values:
rt = µ + εt
εt = σt zt
σt2 = ω + α1 ε2t−1 + ... + αp ε2t−p
zt ∼ iid N(0, 1)
I
Due to the large persistence in volatility, the ARCH model often
requires a large p to fit the data
32 / 67
The GARCH Model (cont’d)
I
In such cases, it is more parsimonious to use the GARCH
(Generalized ARCH) model proposed by Bollerslev (1986). The
GARCH(1,1) model is defined as
rt = µ + εt
εt = σt zt
2
σt2 = ω + αε2t−1 + βσt−1
zt ∼ iid N(0, 1)
33 / 67
GARCH(1,1) volatility for the S&P 500 (weekly data)
Volatility estimated using a GARCH(1,1) model
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
200
400
600
800
1000
1200
1400
34 / 67
Outline
I Swaps
Commodity Swaps
Interest Rate Swaps
Variance and Volatility Swaps
3
5
11
16
II Measurement and Behavior of Volatility
26
III Extending the Black-Scholes Model
Option Pricing When the Stock Price Can Jump
Stochastic Volatility
35
37
42
IV Revision
49
V About My Current Research
60
35 / 67
Extensions of the Black-Scholes Model
I
Three extensions
1. The Merton jump diffusion model
2. The Constant Elasticity of Variance model (CEV)
3. The Heston model
36 / 67
Valuation Formula when the Underlying Can Jump
I
Stock prices sometimes move more than would be expected from a
lognormal distribution:
I
I
I
I
If market volatility is 20% and the expected return is 15%, a one-day
5% drop in the market occurs about once every 2.5 million days
A 20% drop (as in October 1987) is virtually impossible if prices are
lognormally distributed
Merton (1976) used the Poisson distribution to count the number of
price jumps that occur over a period of time.
The Poisson distribution is summarized by the parameter λ:
I
I
λh is the probability that one event occurs over the short interval h.
Thus, λ is like an annualized probability of the event occurring over a
short interval.
37 / 67
Valuation Formula when the Underlying Can Jump (cont’d)
I
Let the jump magnitude be Y . That is, if S is the pre-jump price,
Y × S is the post-jump price
I
Let us assume that Y is lognormally distributed (hence, this is called
the Poisson-lognormal model):
ln(Y ) ∼ N αJ , σJ2
I
You can interpret ln(Y ) as the continuously compounded stock return
if there is a jump; ln(Y ) can go from −∞ to +∞
I
You can interpret Y − 1 as the effective stock return if there is a
jump; Y − 1 can go from −1 to ∞, or Y can go from 0 to ∞
Thus, the Merton formula for pricing with jumps requires 3 extra
parameters:
I
1. The jump probability, λ
2. The expected size of a jump when one does occur, αJ
3. The volatility of the jump magnitude, σJ
38 / 67
Valuation Formula when the Underlying Can Jump (cont’d)
I
The resulting process for the stock is
(
0,
dSt
= (α − λk) dt + σdZ +
St
Y − 1,
I
if there is no jump
if there is a jump
(8)
Merton (1976) shows that with the stock following equation (8), and
with jumps diversifiable, the price of an European call is
∞ −λ0 T
X
e
(λ0 T )i
i=0
i!

BSCall S, K ,
s

iσ 2
iαJ
σ 2 − J , r − λk +
, T , δ  (9)
T
T
39 / 67
Jump Risk and Implied Volatility
I
Suppose that the stock can jump to zero (i.e., Y = 0) with λ = 0.5%
probability per year
I
In this case, the value of a European call becomes
BSCall (S, K , σ, r + λ, T , δ)
I
Suppose that other parameters are: S = $40, σ = 30%, r = 8%,
T = 0.25, and δ = 0. The jump parameters are: λ = 0.005,
αJ = ln(0) = −∞, and σJ = 0
I
We do the following experiment: generate correct option prices—i.e.,
prices properly accounting for the jump—for a variety of strikes
I
We then ask what implied volatility we would compute for these
options using the ordinary Black-Scholes formula
40 / 67
Jump Risk and Implied Volatility (cont’d)
0.34
CallJump = 10.67, CallStd = 10.63
Implied Volatility
0.33
0.32
0.31
0.3
30
35
40
45
50
Strike Price
55
60
41 / 67
Constant Elasticity of Variance (CEV) Model
I
Cox (1975) proposed the constant elasticity of variance (CEV) model
I
Volatility varies with the level of the stock price
β−2
dS
= (α − δ)dt + σ̄S 2 dZ
S
I
The instantaneous standard deviation of the stock return is
σ(S) = σ̄S
I
I
I
β−2
2
If β < 2 volatility decreases with the stock price
If β > 2 volatility increases with the stock price
If β = 2 the CEV model reduces to the lognormal process
42 / 67
The CEV Call Price
I
For the case β < 2, the CEV call price is
Se
−δT
2
, 2x
1 − Q 2y , 2 +
2−β
− Ke
−rT
2
Q 2x ,
, 2y
2−β
(10)
where
k=
2(r − δ)
σ̄ 2 (2 − β)(e (r −δ)(2−β)T
x = ks
y = kK
I
− 1)
2−β (r −δ)(2−β)T
e
2−β
Q(a, b, c) dentoes the noncentral Chi-squared distribution function
with b degrees of freedom and noncentrality parameter c, evaluated
at a
43 / 67
The CEV Call Price (cont’d)
I
For the case β > 2, the CEV call price is
Se
−δT
2
, 2y
1 − Q 2x ,
β −2
− Ke
−rT
2
Q 2y , 2 +
, 2x
β −2
I
When β < 2, the CEV model generates a Black-Scholes implied
volatility skew
I
Implied volatility decreases with the option strike price
(11)
44 / 67
Implied Volatility in the CEV Model
I
I
Suppose the usual parameters: S = $40, σ = 30%, r = 8%, T = 0.25,
and δ = 0.
√
Additionally, β = 1 and σ̄ = 0.3 40 = 1.897
I
Now we do the following experiment: generate correct option
prices—i.e., prices properly accounting for time-varying volatility—for
a variety of strikes
I
We then ask what implied volatility we would compute for these
options using the ordinary Black-Scholes formula
45 / 67
Implied Volatility in the CEV Model (cont’d)
0.34
Implied Volatility
CallJump = 10.67, CallStd = 10.63
CallCEV = 10.65, CallStd = 10.63
0.32
0.3
0.28
30
35
40
45
50
Strike Price
55
60
46 / 67
The Heston Stochastic Volatility Model
I
The model allows volatility to vary stochastically but still be
correlated with the stock price
q
dS
= (α − δ)dt + v (t)dZ1
S
q
dv (t) = k(v̄ − v (t))dt + σv
v (t)dZ2
(12)
(13)
p
I
v (t) is the volatility (instantaneous standard deviation of the stock
return)
I
The brownians Z1 and Z2 can be correlated:
E (dZ1 dZ2 ) = ρdt
I
Heston’s stochastic volatility model has an integral solution that can
be evaluated numerically
47 / 67
Implied Volatility in the Heston Model
48 / 67
Outline
I Swaps
Commodity Swaps
Interest Rate Swaps
Variance and Volatility Swaps
3
5
11
16
II Measurement and Behavior of Volatility
26
III Extending the Black-Scholes Model
Option Pricing When the Stock Price Can Jump
Stochastic Volatility
35
37
42
IV Revision
49
V About My Current Research
60
49 / 67
Below is a profit diagram for a position. In the table below,
record option quantities which construct the diagram
100
Slope=1
Profit
50
0
−50
Slope=1
−100
Strike
Calls
80
120
0
50
100
150
Stock Price in one year
200
Shares
50 / 67
Below is a profit diagram for a position. In the table below,
record option quantities which construct the diagram
100
Slope=1
Profit
50
0
−50
Slope=1
−100
Strike
Puts
80
120
0
50
100
150
Stock Price in one year
200
Shares
51 / 67
Which of the following options will NOT be exercised
early?
(a) Put on a dividend paying stock
(b) Call on a dividend paying stock
(c) Put on a non-dividend paying stock
(d) Call on a non-dividend paying stock
52 / 67
Why is an American call option rarely exercised early, and
thus priced similar to European options?
(a) Early exercise requires purchasing the stock
(b) Most option sellers cannot deliver the stock
(c) Option prices usually exceed intrinsic values
(d) Short sellers disrupt delivery
53 / 67
Assume that an investor is currently holding a reverse
straddle position (i.e. a short put and short call), which is
currently a profitable investment. All else being equal,
what would this investor like to happen to vega?
(a) Decrease
(b) Increase
(c) Stay constant
(d) Indifferent
54 / 67
Suppose the delta of an option is 5.15
(a) It is a call option
(b) It is a put option
(c) Could be either call or put
55 / 67
The price of an option is decreasing with the stock price
(a) It is a call option
(b) It is a put option
(c) Could be either call or put
56 / 67
The price of an option is decreasing with the strike price
(a) It is a call option
(b) It is a put option
(c) Could be either call or put
57 / 67
What actions are required to both delta-hedge and
gamma-hedge a written option position?
(a) No action is required
(b) We must buy more bonds
(c) We must buy more shares of the underlying
(d) We must long a different option so as to offset the short call gamma.
58 / 67
During periods when measured volatility is high, the typical
day tends to exhibit high volatility. This behavior is
referred to as volatility ______
(a) Clustering
(b) Smile
(c) Skew
(d) Stochasticity
59 / 67
Outline
I Swaps
Commodity Swaps
Interest Rate Swaps
Variance and Volatility Swaps
3
5
11
16
II Measurement and Behavior of Volatility
26
III Extending the Black-Scholes Model
Option Pricing When the Stock Price Can Jump
Stochastic Volatility
35
37
42
IV Revision
49
V About My Current Research
60
60 / 67
February 2010: Wall Street takes break for Tiger Woods
apology
Source: Bloomberg, By Michael Patterson and Eric Martin - February 19, 2010 14:16 EST
61 / 67
We explain the strong positive relationship between
investors’ attention and stock market volatility
S&P500 Volatility
0.4
0.3
0.2
0.1
2004
2005
2006
2007
2008
2009
2010
2011
2009
2010
2011
Focus on Economic News
3
2
1
2004
2005
2006
2007
2008
62 / 67
Bottom fishing: The quest for the bottoming-out point
that suggests a return to more bullish days
Source: Yu (2009)
63 / 67
Disagreement over the speed of the reversion of the
economy drives volatility
0.3
0.2
Stock’s Diffusion
0.1
0
−0.1
−0.2
−0.3
σω
−0.4
0
σF
10
σG
20
σδ
30
σ
40
50
Time
64 / 67
Volatility is clustered
Returns
0.06
0.04
0.02
10
20
30
40
50
Time
-0.02
-0.04
-0.06
65 / 67
May 1995: IBM secretary Lorraine Cassano was asked to
photocopy some papers...
May 1995
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
June 1995
1 2 3 4
5 6 7 8 9 10 11
12 13 14 15 16 17 18
19 20 21 22 23 24 25
26 27 28 29 30
66 / 67
May 1995: IBM secretary Lorraine Cassano was asked to
photocopy some papers...
May 1995
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
June 1995
1 2 3 4
5 6 7 8 9 10 11
12 13 14 15 16 17 18
19 20 21 22 23 24 25
26 27 28 29 30
66 / 67
May 1995: IBM secretary Lorraine Cassano was asked to
photocopy some papers...
May 1995
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
June 1995
Takeover of Lotus by IBM
(to be announced June 5)
1 2 3 4
5 6 7 8 9 10 11
12 13 14 15 16 17 18
19 20 21 22 23 24 25
26 27 28 29 30
66 / 67
May 1995: IBM secretary Lorraine Cassano was asked to
photocopy some papers...
May 1995
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
June 1995
Takeover of Lotus by IBM
(to be announced June 5)
The information
percolated during 6 hours
1 2 3 4
5 6 7 8 9 10 11
12 13 14 15 16 17 18
19 20 21 22 23 24 25
26 27 28 29 30
66 / 67
May 1995: IBM secretary Lorraine Cassano was asked to
photocopy some papers...
May 1995
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
June 1995
Takeover of Lotus by IBM
(to be announced June 5)
1 2 3 4
5 6 7 8 9 10 11
12 13 14 15 16 17 18
19 20 21 22 23 24 25
26 27 28 29 30
The information
percolated during 6 hours
Investors: 25
Investment: $5000 000
Trading profits: $10 3000 000
66 / 67
May 1995: IBM secretary Lorraine Cassano was asked to
photocopy some papers...
May 1995
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
June 1995
Takeover of Lotus by IBM
(to be announced June 5)
1 2 3 4
5 6 7 8 9 10 11
12 13 14 15 16 17 18
19 20 21 22 23 24 25
26 27 28 29 30
The information
percolated during 6 hours
Investors: 25
Investment: $5000 000
Trading profits: $10 3000 000
Source: New York Daily News, p. 5, Thursday, May 27th 1999.
U.S. SECURITIES AND EXCHANGE COMMISSION Litigation Release No. 16161, MAY 26, 1999.
66 / 67
The data are never for the whole society given to a single
mind (Hayek, 1945, p. 519)
Prices aggregate knowledge
Language aggregates knowledge
67 / 67
The data are never for the whole society given to a single
mind (Hayek, 1945, p. 519)
Prices aggregate knowledge
Language aggregates knowledge
I
Word-of-mouth communication generates momentum
and reversal in asset returns
I
Word-of-mouth communication generates persistent
volatility
67 / 67