Download 3. The Black-Scholes model

Financial Derivative
Reference：
1. John Hull著，张陶伟译，《期权、期货
及其它衍生产品》，第三版,华夏出版社。
2. John Hull著，张陶伟译，《期权期货入
门》
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Chapter 1
Introduction
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Outline
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1. Derivatives
2. Forward Contracts
3. Futures Contracts
4. Options
5. Types of Traders
6. Other Derivatives
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1. Derivatives
The Nature of Derivatives
A derivative is an instrument whose value
depends on the values of other more basic
underlying variables.
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Examples of Derivatives
 Forward Contracts
 Futures Contracts
 Swaps
 Options
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Derivatives Markets

Exchange-traded markets


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CBOT (Chicago Board of Trade), 1848, grain
CME (Chicago Mercantile Exchange), 1919,
futures
CBOE (Chicago Board Options Exchange), 1973,
options
Traditionally exchanges have used the openoutcry system, but increasingly they are switching
to electronic trading
Contracts are standard there is virtually no credit
risk
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
Over-the-counter (OTC)
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A computer- and telephone-linked network of
dealers at financial institutions, corporations,
and fund managers
Contracts can be non-standard and there is
some small amount of credit risk
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Ways Derivatives are Used
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To hedge risks
To speculate (take a view on the future
direction of the market)
To lock in an arbitrage profit
To change the nature of a liability
To change the nature of an investment
without incurring the costs of selling one
portfolio and buying another
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2. Forward Contracts




A forward contract is an agreement to buy or
sell an asset at a certain future time for a
certain price (the delivery price)
It can be contrasted with a spot contract
which is an agreement to buy or sell
immediately
It is traded in the OTC market
Forward contracts on foreign exchange are
very popular
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Foreign Exchange Quotes for
GBP on Aug 16, 2001
Spot
Bid
1.4452
Offer
1.4456
1-month forward
1.4435
1.4440
3-month forward
1.4402
1.4407
6-month forward
1.4353
1.4359
12-month forward
1.4262
1.4268
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Terminology
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The party that has agreed to buy
has what is termed a long position
The party that has agreed to sell
has what is termed a short position
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Example



On August 16, 2001 the treasurer of a
corporation enters into a long forward
contract to buy £1 million in six months at an
exchange rate of 1.4359
This obligates the corporation to pay
$1,435,900 for £1 million on February 16,
2002
What are the possible outcomes?
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Profit from a Long Forward
Position
Profit
K
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Price of Underlying
at Maturity, ST
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Profit from a Short Forward
Position
Profit
K
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Price of Underlying
at Maturity, ST
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Forward Price


The forward price for a contract is the
price agreed today for the delivery of the
asset at the maturity date.
When move through time the delivery
price for the forward contract does not
change, but the forward price is likely to
do so.
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1) Gold: An Arbitrage
Opportunity?

Suppose that:

The spot price of gold is US$300
The 1-year forward price of gold is US$340
The 1-year US$ interest rate is 5% per annum
Is there an arbitrage opportunity?
(We ignore storage costs and gold lease rate)?
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2) Gold: Another Arbitrage
Opportunity?
 Suppose that:
- The spot price of gold is US$300
- The 1-year forward price of gold is
-
US$300
The 1-year US$ interest rate is 5%
per annum
 Is there an arbitrage opportunity?
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The Forward Price of Gold
If the spot price of gold is S and the forward price for
a contract deliverable in T years is F, then
F = S (1+r )T
where r is the 1-year (domestic currency) risk-free
rate of interest.
In our examples, S = 300, T = 1, and r =0.05 so that
F = 300(1+0.05) = 315
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3. Futures Contracts



Agreement to buy or sell an asset for a
certain price at a certain time
Similar to forward contract
Whereas a forward contract is traded OTC, a
futures contract is traded on an exchange
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Examples of Futures Contracts
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Agreement to:
 buy 100 oz. of gold @ US$300/oz. in
December (COMEX)
 sell £62,500 @ 1.5000 US$/£ in March
(CME)
 sell 1,000 brl. of oil @ US$50/brl. in
April (NYMEX)
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4. Options
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A call option is
an option to buy
a certain asset
by a certain date
for a certain
price (the strike
price)

A put is an
option to sell a
certain asset by
a certain date for
a certain price
(the strike price)
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Terminology

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
Strike price (Exercise price)
Expiration date (maturity)
American/European option
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Exchanges Trading Options
 Chicago Board Options Exchange
 American Stock Exchange
 Philadelphia Stock Exchange
 Pacific Stock Exchange
 European Options Exchange
 Australian Options Market
 and many more (see list at end of book)
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Long Call on Microsoft
Profit from buying a European call option on Microsoft:
option price = $5, strike price = $60
30 Profit ($)
20
10
30
0
-5
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40
50
Terminal
stock price ($)
60
70
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80
90
24
Short Call on Microsoft
Profit from writing a European call option on Microsoft:
option price = $5, strike price = $60
Profit ($)
5
0
70
30
40
50 60
-10
80
90
Terminal
stock price ($)
-20
-30
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Long Put on IBM
Profit from buying a European put option on IBM:
option price = $7, strike price = $90
30 Profit ($)
20
10
0
-7
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Terminal
stock price ($)
60
70
80
90
100 110 120
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Short Put on IBM
Profit from writing a European put option on IBM:
option price = $7, strike price = $90
Profit ($)
7
0
60
70
Terminal
stock price ($)
80
90
100 110 120
-10
-20
-30
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Payoffs from Options
K = Strike price, ST = Price of asset at maturity
 Payoff from a long position in the European call:
Max(ST-K,0)
 Payoff from a short position in the European call:
-Max(ST-K,0)
 Payoff from a long position in the European putl:
Max(K-ST,0)
 Payoff from a long position in the European call:
-Max(K-ST,0)
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Payoffs from Options
Payoff
Payoff
K
K
ST
Payoff
ST
Payoff
K
K
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ST
ST
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5. Types of Derivative Traders
• Hedgers: use derivatives to reduce the risk
that they face from potential future
movements in a market variable
• Speculators: use derivatives to bet on the
future direction of a market variable
• Arbitrageurs: lock in a riskless profit by
simultaneously entering into two or more
transactions
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Hedging Examples (1)


A US company will pay £10 million for imports
from Britain in 3 months and decides to hedge
using a long position in a forward contract
The price is locked, but the outcome may be
worse
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Hedging Examples (2)


An investor owns 1,000 Microsoft shares
currently worth $73 per share. A two-month put
with a strike price of $65 costs $2.50. The
investor decides to hedge by buying 10
contracts
The difference between the use of forward and
options for hedging:
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Forward: fix the price
Option: provide insurance
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Speculation Example



An investor with $4,000 to invest feels that Cisco’s
stock price will increase over the next 2 months.
The current stock price is $20 and the price of a 2month call option with a strike of 25 is $1
Two possible alternative strategies: buy calls and
shares.
The use of futures and options for speculation:
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Both obtain leverage
The potential loss and gain are different
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Arbitrage Example




A stock price is quoted as £100 in London
and $172 in New York
The current exchange rate is 1.7500
What is the arbitrage opportunity?
Arbitrage opportunities can’t last for long.
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6. Other Derivatives

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Plain vanilla/ standard derivatives
Exotics
Credit derivatives: creditworthiness of a
company
Weather derivatives: average temperature
Insurance derivatives: dollar value of
insurance claim
Electricity derivatives: spot price of electricity
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Chapter 2
Mechanics of Futures
Markets
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Futures Contracts

CBOT, CME
Available on a wide range of underlying assets

Exchange traded



Specifications need to be defined:
 What can be delivered,
 Where it can be delivered,
 When it can be delivered
Settled daily
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Delivery





Closing out a futures position involves entering
into an offsetting trade
Most contracts are closed out before maturity
If a contract is not closed out before maturity, it
usually settled by delivering the assets underlying
the contract.
When there are alternatives about what is
delivered, where it is delivered, and when it is
delivered, the party with the short position
chooses.
A few contracts (for example, those on stock
indices and Eurodollars) are settled in cash
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Price and Position Limits



Many futures exchanges set limits on daily
price changes and holdings.
Limits are set to prevent excessive
volatility and market manipulation.
Limits are often removed in the last month
of the contract.
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Convergence of Futures to Spot
Futures
Price
Spot Price
Futures
Price
Spot Price
Time
(a)
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Time
(b)
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Margin Requirement




Initial Margin - funds deposited to provide
capital to absorb losses, generally 5%-15%.
Maintenance Margin - an established value
below which a trader’s margin may not fall.
Marking to market
When the maintenance margin is reached, the
trader will receive a margin call from her
broker to add variation margin to reach the
level of initial margin.
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Margin Requirement (cont.)




清算所(clearing house): track all the
transactions to calculate the positions
经纪人也需在清算所存入保证金(clearing
margin) 。但数额小于等于客户交给经纪人的保证
金
变动保证金必须以现金支付，初识保证金中的一
部分可以以生息债券存入。
1990年7月某经纪公司对国际货币市场合约初始保
证金和维持保证金的要求如下表。
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Margin Requirement (cont.)
合约
初始保证金
英镑
$2 800
$2 100
马克
$1 800
$1 400
瑞士法郎
$2 700
$2 000
日元
$2 700
$2 000
加拿大元
$1 000
$800
澳大利亚元
$2 000
$1 500
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Margin Calculation

An investor takes a long position in 2
December gold futures contracts on June 4



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contract size is 100 oz.
futures price is US$400
initial margin requirement is US$2,000/contract
(US$4,000 in total, 5%)
maintenance margin is US$1,500/contract
(US$3,000 in total)
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Margin Calculation (cont.)
Day
Futures
Price
(US$)
Daily
Gain
(Loss)
(US$)
Cumulative
Gain
(Loss)
(US$)
400.00
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Margin
Account Margin
Balance
Call
(US$)
(US$)
4,000
5-Jun 397.00
.
.
.
.
.
.
(600)
.
.
.
(600)
.
.
.
13-Jun 393.30
.
.
.
.
.
.
(420)
.
.
.
(1,340)
.
.
.
2,660 + 1,340 = 4,000
.
.
.
.
.
< 3,000
19-Jun 387.00
.
.
.
.
.
.
(1,140)
.
.
.
(2,600)
.
.
.
2,740 + 1,260 = 4,000
.
.
.
.
.
.
26-Jun 392.30
260
(1,540)
5,060
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3,400
.
.
.
0
.
.
.
0
45
Example
An investor enters into two long futures contracts on
frozen orange juice. Each contract is for the
delivery of 15,000 pounds. The current futures
price is 160 cents per pound, the initial margin is
$6,000 per contract, and the maintenance margin
is $4,500 per contract. What price change would
lead to a margin call? Under what circumstances
could $2,000 be withdrawn from the margin
account?
Falls by 10 cents and rises by 6.67 cents
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Newspaper quotes

Open interest: the total number of contracts
outstanding




equal to number of long positions or number of
short positions
One trading older
Settlement price: the price just before the
final bell each day
 used for the daily settlement process
Volume of trading: the number of trades in 1
day
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Patterns of Futures Prices





Normal market: price increase as the
time to maturity increase, wheat in CBT
Inverted market: Sugar-World
Mixed pattern: crude oil
Normal backwardation (现货溢价):
futures price below the expected spot
price
Contango (期货溢价)
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Orders
买卖期货合约的两种主要指令

限价指令(limit orders)：以预先讲明的价格买卖，
如，以US$0.5323/DM或更低的价格买入两份马克
期货合约

市价指令(market orders)：以交易所可得的最优价
格买卖，如，在市场上买入两份期货合约，价格
为交易所可得的最低价格
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Forward Contracts vs Futures
Contracts
FORWARDS
FUTURES
Private contract between 2 parties
Exchange traded
Non-standard contract
Standard contract
Usually 1 specified delivery date
Settled at maturity
Settled daily
No daily price change limit
交割率为90%
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Range of delivery dates
Have daily price change limit
交割率不到5%
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Foreign Exchange Quotes


Futures exchange rates are quoted as the
number of USD per unit of the foreign currency
Forward exchange rates are quoted in the same
way as spot exchange rates. This means that GBP,
EUR, AUD, and NZD are USD per unit of foreign
currency. Other currencies (e.g., CAD and JPY)
are quoted as units of the foreign currency per
USD.
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Chapter 3
Determination of
Forward and Futures
Prices
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Consumption Assets vs Investment


Investment assets are assets held by
significant numbers of people purely for
investment purposes (Examples: gold, silver)
Consumption assets are assets held
primarily for consumption (Examples:
copper, oil)
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Conversion Formulas
Define
Rc : continuously compounded rate
Rm: same rate with compounding m times per
year
Rm 

Rc  m ln 1 


m 


Rm  m e Rc / m  1
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Example
1. Consider an interest rate that is quoted
as 10% per annum with semiannual
compounding. What is the equivalent
rate with continuous compounding?
2 ln(1  0.1 / 2)  0.09758
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Example (cont.)
2. A deposit account pays 12% per annum
with continuous compounding, but
interest is actually paid quarterly. How
much interest will be paid each quarter
on a $10,000 deposit?
100000.1218/4=304.55
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Forward vs Futures Prices

Forward and futures prices are usually
assumed to be the same. When interest
rates are uncertain they are, in theory,
slightly different.
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Notation
S0 :
Spot price today
F0 :
Futures or forward price today
T:
Time until delivery date
r:
Risk-free interest rate for maturity T
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An Arbitrage Opportunity?

Suppose that:




The spot price of gold is US$300
The 1-year futures price of gold is US$340
The 1-year US$ interest rate is 5% per annum
Is there an arbitrage opportunity?
(We ignore storage costs and gold lease rate)
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Another Arbitrage Opportunity?

Suppose that:





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The spot price of gold is US$300
The 1-year futures price of gold is US$300
The 1-year US$ interest rate is 5% per
annum
Is there an arbitrage opportunity?
What if the 1-year futures price of gold
is US$315?
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The Futures Price of Gold


If the spot price of gold is S0 and the futures
price for a contract deliverable in T years is
F0, then
F0 = S0 (1+r )T
where r is the 1-year (domestic currency)
risk-free rate of interest.
In our examples, S0 = 300, T = 1, and r =0.05
so that
F0 = 300(1+0.05) = 315
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For investment asset

For any investment asset that provides no
income and has no storage costs
F0 = S0(1 + r )T

If r is compounded continuously
F0 = S0erT
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For Investment Asset Providing
Known Cash Income


stocks paying known dividends,
coupon bond
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Example: An Arbitrage Opportunity?




Consider a long forward contract to purchase a
coupon-bearing bond whose current price is $900
The forward contract matures in one year and the
bond matures in 5 years, so the forward contract
is to purchase a 4-year bond in one year
Coupon payments of $40 are expected after 6
months and 12 months
The 6-month and 1-year risk-free interest rates
(continuous compounding) are 9% per annum and
10% per annum, respectively
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An Arbitrage Opportunity? (cont.)




If the forward price $930
An arbitrageur can borrow $900 to buy the
bond and short a forward contract
Since 40e-0.090.5=$38.24, so, of the $900,
$38.24 is borrowed at 9% per annum for six
months
The remaining $861.76 is borrowed at 10% per
annum for one year
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An Arbitrage Opportunity? (cont.)



The amount owing at the end of the year is
$861.76e0.11=$952.39
The second coupon provides $40, and $930
is received from the bond selling under the
forward contract
The net profit is
$40+ $930 - $952.39=$17.61
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An Arbitrage Opportunity? (cont.)




If the forward price $905
An investor who holds the bond can sell it and
enter a forward contract
of the $900 realized from selling the bond,
$38.24 is invested at 9% per annum for 6
months so that it grows to $40
The remaining $861.76 is invested at 10% per
annum for one year and grows to $952.39
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An Arbitrage Opportunity? (cont.)

The net gain is
$952.39 -$40- $905 =$7.39

When will no arbitrage exist?
$912.39
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Generalization

When an Investment Asset Provides a
Known Dollar Income
F0 = (S0 – I )erT
where I is the present value of the income

In our example
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For investment asset (cont.)

When an Investment Asset Provides a
Known Yield
F0 = S0 e(r–q )T
where q is the average yield during the life of
the contract (expressed with continuous
compounding)
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Example
1. Consider a 10-month forward contract on a
stock with a price of $50. The risk-free
interest rate (continuous compounded) is 8%
per annum for all maturities. Assume that
dividends of $0.75 per share are expected
after three months, six months and nine
months. What is the forward price?
51.14
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Example (cont.)
2. Consider a six-month futures contract on an
asset that is expected to provide income
equal to 2% of the asset price once during
the six-month period. The risk-free rate of
interest (continuous compounded) is 10% per
annum. The asset price is $25. What is the
futures price?
25e(0.1-0.0396)/2=25.77
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For Stock Index Futures




Can be viewed as an investment asset paying a
dividend yield
The investment asset is the portfolio of stocks
underlying the index
The dividend paid are the dividends that would be
received by the holder of the portfolio
It is usually assumed that the dividends provide a
known yield rather than a known cash income
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For Stock Index Futures (cont.)

The futures price and spot price
relationship is therefore
F0 = S0 e(r–q )T
where q is the dividend yield on the
portfolio represented by the index
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For Stock Index Futures (cont.)


2005年秋
In practice, the dividend yield on the
portfolio underlying the index varies
week by week throughout the year.
The chosen value of q should represent
the average annualized dividend yield
during the life of the contract.
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Example




The risk-free interest rate is 9% per annum with
continuously compounding
The dividend on the stock index varies
throughout the year. In February, May, August
and November, dividends are paid at a rate of
5% per annum. In other months, dividends are
paid at a rate of 2% per annum.
The value of the index on July 31, 2002 is 300.
What is the futures price for a contract
deliverable on December 31, 2002?
307.34
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Index arbitrage


If F0 > S0 e(r–q )T , profits can be made by
buying the stocks underlying the index and
shorting futures contract;
If F0 < S0 e(r–q )T , profits can be made by
shorting or selling the stocks underlying the
index and taking a long position in futures
contract.
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Forward and Futures on Currencies



A foreign currency is analogous to a
security providing a dividend yield
The continuous dividend yield is the
foreign risk-free interest rate
It follows that if rf is the foreign riskfree interest rate
F0  S0e
2005年秋
( r rf ) T
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Example



Suppose that the two-year interest rate in
Australia and the United States are 5% and
7%, respectively,
The spot exchange rate between the
Australian dollar and the US dollar is
US$0.62/AUD.
What is two-year forward exchange rate?
0.6453
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Another example
You observe that British pound March 93
futures contract settled at $1.5372/pound
and the June 93 futures contract settled at
$1.5276/pound. What is the implied interest
rate difference for this period between pound
and dollar?
 FJun93 
1
ln 
r  r f  
  0.0249
 T2  T1   FMar93 
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Futures on Commodities

Storage cost for investment asset is regarded as
negative income, so
F0 = (S0+U )erT
where U is the present value of the storage costs.

Alternatively,
F0 = S0 e(r+u )T
where u is the storage cost per annum as a percent
of the spot price.
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Example

Consider a one-year futures contract on
gold. Suppose that it costs $2 per once per
year to store gold, with the payment being
made at the end of the year. Assume that
the spot price is $450, and the risk-free
rate is 7% per annum with continuous
compounding. Then the futures price is
484.63
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Consumption Assets

One keep the commodity for consumption,
so he won’t sell the commodity and buy
futures, which influence the arbitrage
argument.
F0  S0 e(r+u )T
or
F0  (S0+U )erT
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The convenience yield


The convenience yield on the consumption
asset
 ability to profit from temporary local
shortages
 ability to immediately keep a production
process running
The convenience yield, y, is defined so that
F0 eyT= S0 e(r+u )T
Or
2005年秋
F0 = S0 e(r+u-y )T
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Cost of Carry

Cost of carry refers to the cost and
benefit of holding the asset, including:
interest rate paid to finance the asset
 storage costs
 dividends or other income

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Cost-of-Carry (cont.)

Non-divident-paying stock (no storage cost and
no income): c =r

Stock index: c =r-q

Currency: c =r-rf

Commodities: c =r+u
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Cost-of-Carry and futures price

For an investment asset
F0 = S0ecT

For a consumption asset
F0 = S0e(c-y)T
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Futures Prices & Expected
Future Spot Prices

Suppose k is the expected return required
by investors on an asset

We can invest F0e–r T now to get ST back at
maturity of the futures contract

This shows that
F0 = E (ST )e(r–k )T
88
Valuing a Forward Contract



Suppose that
K is delivery price in a forward contract
F0 is forward price that would apply to the
contract today
The value of a long forward contract, ƒ, is
ƒ = (F0 – K )e–rT
Similarly, the value of a short forward
contract is
(K – F0 )e–rT
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Example

A long forward contract on a non-dividend
paying stock was entered into some time
ago. It currently has six months to
maturity. The risk-free interest rate (with
continuous compounding) is 10% per annu,
the stock price is $25 and delivery price is
$24. What is the value of the forward
contract?
$2.17
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Chapter 4
Hedging Strategies
Using Futures
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Long & Short Hedges


long futures hedge: involves a long position
in futures, appropriate when you know you
will purchase an asset in the future and
want to lock in the price
short futures hedge: involves a short
position in futures, appropriate when you
know you will sell an asset in the future &
want to lock in the price
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Example of short hedge



On May 15, X has contracted to sell 1 million
barrels of oil on August 15 at the spot price
of that day
May 15 quotes:
S1= $19.00 /barrel, F1= $18.75 /barrel
Hedging actions:
Contract size: 1000 barrels
On May 15, short 1000 August oil futures
On August 15, close out futures position
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Example (cont.)




August 15:
S2= F2=$17.50 /barrel,
X receives $17.50 per barrel per contract
Gains from futures=F1-F2
=$(18.75 - 17.50) = $1.25 per barrel
Price realized=$17.50+ $1.25
=$18.75= F1+( S2- F2)
Alternatively if S2=$19.50 /barrel
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Basis Risk


Basis is the difference between spot &
futures prices
Basis risk arises because of the uncertainty
about the basis when the hedge is closed out


2005年秋
The asset to be hedged may not be the same
as the asset underlying the futures
The hedger is uncertain about the precise
date of buying or selling the asset
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Choice of Contract


Choose a delivery month that is as close
as possible to, but later than, the end of
the life of the hedge
When there is no futures contract on
the asset being hedged, choose the
contract whose futures price is most
highly correlated with the asset price.
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Long Hedge



Suppose that
F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
b2 : Basis at time t2
You hedge the future purchase of an asset
by entering into a long futures contract
Cost of Asset=S2 –(F2 – F1) = F1 + Basis
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Example 2



It is June 8 and a company knows that it will
need to purchase 20,000 barrels of crude oil
at some time in October or November.
Oil futures contracts are currently traded
for delivery every month on NYMEX and the
contract size is 1,000 barrels.
The company therefore decides to take a long
position in 20 December contracts for
hedging (Assuming that the hedge ratio is 1).
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Example 2 (cont.)





The futures price on June 8 is F1=$18 /barrel.
The company finds that it is ready to
purchase the crude oil on November 10. It
therefore closes out its futures contract on
that date.
The pot price and futures price on November
10 are S2=$20 and F2=$ 19.10 per barrel.
The gain on the futures contract is 19.1018=$1.10 per barrel.
The effective price paid is 20-1.10=$18.90
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Short Hedge



Suppose that
F1 : Initial Futures Price
F2 : Final Futures Price
S2 : Final Asset Price
You hedge the future sale of an asset by
entering into a short futures contract
Payoff Realized=S2+ (F1 –F2) = F1 + Basis
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Basis risk



Since F1 is known at t1, hedging risk is the basis
risk b2
when asset to be hedged is different from asset
underlying futures
Effective price at t2 is
(S2 + F1 - F2) = F1 +(S2* - F2) + (S2 - S2*)
where S2* is the spot price of the asset
underlying the futures contract
The term (S2 - S2*) arises due to the difference
between the two assets
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Optimal Hedge Ratio


Hedge ratio: the ratio of the position taken in
futures contract to the size of the exposure
Optimal hedge ratio: proportion of the exposure
that should optimally be hedged is (extra1)
sS
h r
sF
*
where
 sS and sF are the standard deviations of dS and dF, the
change in the spot price and futures price during the
hedging period,
 r is the coefficient of correlation between dS and dF.
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Example 2 (cont.)


If the company decides to use a hedge
ratio of 0.8, how does the decision affect
the way in which the hedge is implemented
and the result?
If the hedge ratio is 0.8, the company
takes a long position in 16 NYM December
oil futures contracts on June 8 and closes
out its position on November 10.
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Example 2 (cont.)
The gain on the futures position is
(19.10-18)16,000=17,600
 The effective cost of the oil is therefore
20,00020-17,600=382,400
or $19.12 per barrel.
(This compares with $18.90 per barrel when
the hedge ratio is 1.)

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Hedging Using Index Futures


To hedge the risk in a portfolio the number
of index futures contracts that should be
used is
P

A
where P is the value of the portfolio,  is
its beta, and A is the value of the index
underlying one futures contract
105
Example 3
Value of S&P 500 is 1,000
Value of Portfolio is $5 million
Beta of portfolio is 1.5
What position in futures contracts on the
S&P 500 is necessary to hedge the
portfolio? (Example3)
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Chapter 6
Swaps
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Outline


A swap is an agreement to exchange
cash flows at specified future times
according to certain specified rules
Contents:




Two plain vanilla swap:


2005年秋
How swaps are defined
How they are be used
How they can be valued
Interest-rate swap,
fixed-for-fixed currency swap
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1. An Example of a “Plain Vanilla”
Interest Rate Swap


2005年秋
An agreement by Microsoft to receive
6-month LIBOR & pay a fixed rate of
5% per annum every 6 months for 3
years on a notional principal of $100
million
Next slide illustrates cash flows
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Cash Flows to Microsoft
---------Millions of Dollars--------LIBOR FLOATING
2005年秋
FIXED
Net
Date
Rate
Cash Flow Cash Flow Cash Flow
Mar.5, 2001
4.2%
Sept. 5, 2001
4.8%
+2.10
–2.50
–0.40
Mar.5, 2002
5.3%
+2.40
–2.50
–0.10
Sept. 5, 2002
5.5%
+2.65
–2.50
+0.15
Mar.5, 2003
5.6%
+2.75
–2.50
+0.25
Sept. 5, 2003
5.9%
+2.80
–2.50
+0.30
Mar.5, 2004
6.4%
+2.95
–2.50
+0.45
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Typical Uses of an
Interest Rate Swap

Converting a
liability from
 fixed rate to
floating rate
 floating rate to
fixed rate
2005年秋

Converting an
investment from
 fixed rate to
floating rate
 floating rate to
fixed rate
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Intel and Microsoft (MS)
Transform a Liability
5%
5.2%
Intel
MS
LIBOR+0.1%
LIBOR
2005年秋
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Financial Institution is Involved
4.985%
5.015%
5.2%
Intel
F.I.
MS
LIBOR+0.1%
LIBOR
2005年秋
LIBOR
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Intel and Microsoft (MS)
Transform an Asset
5%
4.7%
Intel
MS
LIBOR-0.25%
LIBOR
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Financial Institution is Involved
4.985%
5.015%
4.7%
F.I.
Intel
MS
LIBOR-0.25%
LIBOR
2005年秋
LIBOR
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The Comparative Advantage
Argument


AAACorp wants to borrow floating
BBBCorp wants to borrow fixed
Fixed
2005年秋
Floating
AAACorp
10.00%
6-month LIBOR + 0.30%
BBBCorp
11.20%
6-month LIBOR + 1.00%
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The Swap
9.95%
10%
AAA
BBB
LIBOR+1%
LIBOR
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The Swap when a Financial
Institution is Involved
9.93%
9.97%
10%
F.I.
AAA
BBB
LIBOR+1%
LIBOR
2005年秋
LIBOR
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Reason for the Comparative
Advantage



The 10.0% and 11.2% rates available to AAACorp
and BBBCorp in fixed rate markets are 5-year
rates
The LIBOR+0.3% and LIBOR+1% rates available
in the floating rate market are six-month rates
The spread reflects the probability of default
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Swaps & Forwards



A swap can be regarded as a convenient
way of packaging forward contracts
The “plain vanilla” interest rate swap in
our example consisted of 6 FRAs
The value of the swap is the sum of the
values of the forward contracts
underlying the swap
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Valuation of an Interest Rate Swap


A swap is worth zero to a company initially.
This means that it costs nothing to enter
into a swap
At a future time its value is liable to be
either positive or negative
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Valuation of an Interest Rate
Swap (cont.)


2005年秋
Interest rate swaps can be valued as
the difference between the value of a
fixed-rate bond and the value of a
floating-rate bond
Alternatively, they can be valued as a
portfolio of forward rate agreements
(FRAs)
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Valuation in Terms of Bonds



Vswap=Bfl-Bfix (or, Bfix-Bfl)
The fixed rate bond is valued in the usual
way
The floating rate bond is valued by noting
that it is worth par immediately after the
next payment date
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Example 1

Suppose that a financial institution pays 6-month
LIBOR and receives 8% per annum (with
semiannual compounding) on a swap with a notional
principle of $100 and the remaining payment
dates are in 3, 9 and 15 months. The swap has a
remaining life of 15months. The LIBOR rates with
continuous compounding for 3-month, 9-month
and 15-month maturities are 10%, 10.5% and 11%,
respectively. The 6-month LIBOR rate at the last
payment date was 10.2% (with semiannual
compounding).
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Valuation in Terms of FRAs


Each exchange of payments in an interest
rate swap is an FRA
The FRAs can be valued on the assumption
that today’s forward rates are realized
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2. An Example of a fixed-forfixed Currency Swap
An agreement to pay 11% on a sterling
principal of £10,000,000 & receive 8%
on a US$ principal of $15,000,000
every year for 5 years
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Exchange of Principal


2005年秋
In an interest rate swap the principal
is not exchanged
In a currency swap the principal is
exchanged at the beginning and the
end of the swap
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The Cash Flows
Year
2001
2002
2003
2004
2005
2006
2005年秋
Dollars Pounds
$
£
------millions-----–15.00 +10.00
+1.20 –1.10
+1.20 –1.10
+1.20 –1.10
+1.20 –1.10
+16.20 -11.10
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Typical Uses of a Currency Swap

2005年秋
Conversion from a
liability in one
currency to a
liability in another
currency

北航金融系李平
Conversion from
an investment in
one currency to
an investment in
another currency
129
Comparative Advantage
Arguments for Currency Swaps
General Motors wants to borrow AUD
Qantas wants to borrow USD
USD
AUD
General Motors 5.0%
12.6%
Qantas
13.0%
7.0%
130
Valuation of Currency Swaps


2005年秋
Like interest rate swaps, currency
swaps can be valued either as the
difference between 2 bonds or as a
portfolio of forward contracts
Valuation in Terms of Bonds:
Vswap=BD-S0 BF
(or, S0BF -BD )
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Example 2

Suppose that the term structure of interest
rates is flat in both Japan and United States.
The Japanese rate is 4% per annum and the U.S.
rate is 9% per annum (both with continuous
compounding). A financial institution enters into a
currency swap in which it receives 5% per annum
in yen and pays 8% per annum in dollars once a
year. The principles in the two currencies are $10
million and 1,200 million yen. The swap will last
for another three years and the current
exchange rate is 110yen=$1.
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Example 3
Company X wishes to borrow U.S. dollars at a fixed rate
of interest and company Y wishes to borrow Japanese
Yen at a fixed rate of interest. The companies have
been quoted the following interest rates.
Yen
Dollars
Company X
5.0%
9.6%
Company Y
6.5%
10.0%
Design a swap that will net a bank, acting as
intermediary, 50bp per annum and make the swap
equally attractive to the two companies.
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Example 4

A $100 million interest swap has a remaining
life of 10 months. Under the terms of the
swap, 6-month LIBOR is exchanged for 12%
per annum (semiannual compounding). The
average of the bid-offer rate being
exchanged for 6-month LIBOR in swaps of all
maturities is currently 10% per annum with
continuous compounding. The 6-month LIBOR
rate was 9.6% per annum two months ago.
What is the current value of the swap to the
party paying floating?
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Chapter 7
Mechanics of
Options Markets
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135
Types of Options




A call is an option to buy
A put is an option to sell
A European option can be exercised only at
the end of its life
An American option can be exercised at
any time
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Types of options (cont.)


Stock options: American in U.S.
Index options：traded on CBOE




2005年秋
An option is to buy or sell 100 times
the index value
Options on S&P500 are European
Options on S&P100 are American
Settled in cash
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Types of options (cont.)

Futures option



2005年秋
期货到期日比期权到期日稍晚
和期货合约在同一交易所交易
When a call is exercised, the holder
get a long position in the underlying
futures plus a cash amount equal to the
excess of the futures price over the
strike price
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Types of options (cont.)

Foreign currency option


2005年秋
Traded on Philadelphia Stock Exchange
以外币的本币价格表示：如英镑买入期权
的价格为$ 0.035/£
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Specification of
Exchange-Traded Options




2005年秋
Expiration date
Strike price
European or American
Call or Put (option class)
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Terminology
Moneyness :
 At-the-money option
 In-the-money option
 Out-of-the-money option
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Terminology (continued)




2005年秋
Option class (call or put)
Option series
Intrinsic value
Time value
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Dividends & Stock Splits

2005年秋
Suppose you own N options with a strike price
of K :
 No adjustments are made to the option
terms for cash dividends
 When there is an n-for-m stock split,
 the strike price is reduced to mK/n
 the no. of options is increased to nN/m
 Stock dividends are handled in a manner
similar to stock splits
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143
Dividends & Stock Splits
(continued)


2005年秋
Consider a call option to buy 100
shares for $20/share
How should terms be adjusted:
 for a 2-for-1 stock split?
 for a 25% stock dividend?

200 share, $10/share

125 share, $16/share
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Position limits and Exercise limits


2005年秋
Position limits: the maximum number of
option contracts that an investor can hold
on one side of the market (long call and
short put are considered to be on the
same side of the market)
Exercise limits: the maximum number of
contracts that can be exercised by any
investor in any period of five consecutive
trading days
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 Newspaper
quotes
 Market Makers



2005年秋
Most exchanges use market makers to
facilitate options trading
A market maker quotes both bid and ask
prices when requested
The market maker does not know whether
the individual requesting the quotes wants
to buy or sell
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146
Margins


Margins are required when options are sold
When a naked call (put) option is written the
margin is the greater of:
1
2

A total of 100% of the proceeds of the sale plus
20% of the underlying share price less the amount
(if any) by which the option is out of the money
A total of 100% of the proceeds of the sale plus
10% of the underlying share price (exercise price)
When writing covered calls, no margin is required
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Margins (cont.)
Example:
An investor writes four naked call options
on a stock. The option price is $5, the
strike price is $40, and the stock price is
$38. What is the margin requirement?

$4240
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Warrants, Executive Stock
Options and convertible bonds



Are call options that are written by a
company on its own stock
When they are exercised, the company
issues more of its own stock and sells them
to the option holder for the strike price
The exercise leads to an increase in the
number of the company’s stock outstanding
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Warrants




Warrants are call options coming into
existence as a result of a bond issue
They are added to the bond issue to
make the bond more attractive to
investors
Once they are created, they sometimes
trade separately from the bonds
宝钢权证
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150
Executive Stock Options




Call option issued by a company to
executives to motivate them to act in the
best interests of the company’s
shareholders
Usually at-the-money when issued
Can’t be traded
Often last for 10 or 15 years
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Convertible Bonds


2005年秋
Convertible bonds are regular bonds
that can be exchanged for equity at
certain times in the future according to
a predetermined exchange ratio
Is a bond with an embedded call option
on the company’s stock
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Chapter 8
Properties of
Stock Option Prices
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153
Outline


The relationship between the option price
and the underlying stock price (by
arbitrage argument)
Whether an American option should be
exercised early
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Notation






c : European call
option price
p : European put
option price
S0 : Stock price today
K : Strike price
T : Life of option
s: Volatility of stock
price
2005年秋





C : American Call option
price
P : American Put option
price
ST :Stock price at option
maturity
D : Present value of
dividends during option’s
life
r : Risk-free rate for
maturity T with cont. comp.
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Effect of Variables on Option
Pricing
Variable
S0
K
T
s
r
D
2005年秋
c
+
–
?
+
+
–
p
–
+?
+
–
+
北航金融系李平
C
+
–
+
+
+
–
P
–
+
+
+
–
+
156
American vs European Options
An American option is worth at least as much
as the corresponding European option
Cc
Pp
C  max( c, S  K )
P  max( p, K  S )
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157
Upper bounds for option prices
c  S0 , C  S0
p  Ke
2005年秋
 rT
, PK
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158
Lower bound for European calls
on non-dividend-paying stocks


Portfolio A: one European call & an amount of
cash equal to Ke-rT
Portfolio B: one share
c(t)  max(S(t) –Ke
2005年秋
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–rT,
0)
159
Calls: An Arbitrage opportunity?


Suppose that
c(t) = 3
S(t) = 20
T=1
r = 10%
K = 18
D=0
Is there an arbitrage opportunity?
S(t) –Ke –rT=3.71>3=c，buy call, short
stock. If the inflow ($17) is invested for one
year at 10% per Annum, it will be $18.79.
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160
Lower bound for European puts
on non-dividend-paying stocks


Portfolio A: one European put & one share
Portfolio B: an amount of cash equal to Ke-rt
p(t) max( Ke-rT–S(t),0)
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Puts: An Arbitrage opportunity?


Suppose that
p(t)= 1
S(t) = 37
T = 0.5
r =5%
K = 40
D =0
Is there an arbitrage opportunity?
Ke-rT–S(t)=2.01>1=p, 借$38，为期6个月，用
借款购买卖权和股票， 6个月后借款为$38.96。
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Put-call parity for nondividend-paying stocks




Portfolio A: One European call on a stock + an
amount of cash equal to Ke-rT
Portfolio B: One European put on the stock + one
share
Both are worth MAX(ST , K ) at the maturity of the
options
They must therefore be worth the same today
c(t) + Ke -rT = p(t) + S(t)
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163
Arbitrage Opportunities


2005年秋
Suppose that
c(t)= 3
S(t)= 31
T = 0.25 (3-m)
r = 10%
K =30
D=0
What are the arbitrage possibilities
when
p(t) = 2.25 ? p(t) = 1 ?
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164
When p(t)=2.25





c(t)+Ke-rt=32.26, p(t)+S(t)=33.25
Portfolio B is overpriced.
The arbitrage strategy: buy the call, short
both the stock and the put.
Generating a positive cash flow of
2.25+31-3=30.25
After three months, this amount grows to
31.02
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165
Continue




If S(T)>K, exercise the call
If S(T) K, the put is exercised
In either case, the investor ends up buying
one share for $30 to close the short position.
The net profit: 31.02-30 continue
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166
When p(t)=1




2005年秋
c(t)+Ke-rt=32.26, p(t)+S(t)=32
Portfolio A is overpriced.
The arbitrage strategy: short the call, buy
both the stock and the put.
Initial investment:
1+31-3=$29
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167
Continue



The initial investment is financed at 10%.
A repayment of $29.73 is required at the
end of three months.
Either the call or put is exercised, the
stock will be sold for $30.
The net profit: 30-29.73
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Early Exercise for American
options



2005年秋
Usually there is some chance that an
American option will be exercised early
An exception is an American call on a
non-dividend paying stock
This should never be exercised early
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169
An Extreme Situation


For an American call option:
S(t) = 50; T = 1m; K = 40; D = 0
Should you exercise immediately?
What should you do if
1.
2.
2005年秋
You want to hold the stock for one month?
You do not feel that the stock is worth
holding for the next 1 month?
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170
Reasons For Not Exercising a
Call Early--No Dividends

Case 1: should keep the option and
exercise it at the end of the month.



2005年秋
We delay paying the strike price, earn
the interest
No income is sacrificed (no dividend)
Holding the call provides insurance
against stock price falling below strike
price
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171
Reasons For Not Exercising a Call
Early--No Dividends (cont.)

Case 2:
 Better action: sell the option
 The option will be bought by another
investor who does want to hold the stock.
 Such investors must exist, otherwise the
current stock price would not be $50.
 The price obtained for the option will be
greater than its intrinsic value of $10.
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172
More formal argument
C  c  S0–Ke –rT>S0–K
 If it is optimal to exercise early,
C=S0–K

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173
Should Puts Be Exercised Early?




2005年秋
A put option should be exercised early if
it is deep in the money.
An extreme case:
S(t)= 0; K = $10; D = 0
The profit of exercise now: $10, and can
also get interest.
If wait, the profit will be less than 10.
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174
The Impact of Dividends on
Lower Bounds


Portfolio A: one European call & an
amount of cash equal to Ke-rT+D
Portfolio B: one share
c  S0  D  Ke
p  D  Ke
2005年秋
 rT
北航金融系李平
 rT
 S0
175
Impact on Put-Call Parity

European options; D > 0
c + D + Ke -rT = p + S0

American options; D = 0
S 0  K  C  P  S 0  Ke  rT

American options; D > 0
S 0  D  K  C  P  S 0  Ke
2005年秋
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 rT
176
Chapter 9
Trading Strategies
Involving Options
2005年秋
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177
Three Alternative Strategies



Take a position in the option and the
underlying
Take a position in 2 or more options of the
same type (A spread)
Combination: Take a position in a mixture
of calls & puts (A combination)
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Positions in an Option & the
Underlying




Long a stock & short a call = writing a
covered call (a)
Short a stock & long a call = reverse of a
covered call
Long a stock & long a put = protective put (b)
Short a stock & short a put= reverse of a
protective put
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179
Profit
Profit
K
K
ST
ST
(b)
(a)
Profit
Profit
K
ST
(c)
2005年秋
K
ST
(d)
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180
Bull Spread Using Calls
Profit
S
K1

2005年秋
K2
T
Buy lower & sell higher call
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181
Continue
Bull spread created from calls requires an
initial investment
 Profit from a bull spread
 Example:
K1=30, c1=3, K2=35, c2=1
Construct a bull and give the profit.

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182
Bull Spread Using Puts
Profit
K1
K2
ST
Buy lower & sell higher put
2005年秋
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183
Continue


Bull spread created from puts brings a
cash inflow to investors
A bull spread strategy limits the upside
potential as well as the downside risk
2005年秋
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184
Bear Spread Using Calls
Profi
t
K1
K2
ST
Buy higher & sell lower call
2005年秋
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185
Bear Spread Using Puts
Profit
K1
K2
ST
Buy higher & sell lower put
2005年秋
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186
Butterfly Spread Using Calls
Profit
K1
K2
K3
ST
Buy 1 high & 1 low, sell 2 middle
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187
Butterfly Spread Strategy
Generally K2 is close to the current stock
price
 When it is appropriate?
 Payoff
 Example:
K1=55, c1=10, K2=60, c2=7, K2=65, c2=5, S0=61

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188
Butterfly Spread Using Puts
Profit
K1
2005年秋
K2
北航金融系李平
K3
ST
189
Calendar Spread Using Calls
Profit
ST
K
Buy longer & sell shorter (maturity)
2005年秋
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190
Calendar Spread Using Puts
Profit
ST
K
2005年秋
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191
A Straddle Combination
Profit
K
ST
Buy a call & a put
2005年秋
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192
Continue



Payoff structure
When it is appropriate?
Example:



2005年秋
S0=69,
expect a significant move in the future,
K=70, c=4, p=3
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193
Strip & Strap
Profit
Profit
K
ST
K
Strap
Strip
Buy 1 call & 2 puts
2005年秋
ST
Buy 2 calls & 1 put
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194
A Strangle Combination
Profit
K1
K2
ST
Buy 1 call with higher strike &
1 put with lower strike
2005年秋
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195
Example 1
Suppose that put options with strike prices
$30 and $35 cost $4 and $7, respectively.
How can these two options can be used to
create
(a) a bull spread
(b) a bear spread?
Show the profit for both spreads.
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196
Example 2
Call options on a stock are available with
strike prices of $15, $17.5 and $20 and
expiration dates in three months. Their
prices are $4, $2 and $0.5 respectively.
Explain how these options can be used to
create a butterfly spread. What is the
pattern of profits from this spread?
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197
Example 3
A call with a strike price of $50 costs $2.
A put with a strike price of $45 costs $3.
Explain how a strangle can be created from
these two options. What is the pattern of
profits from the strangle?
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198
Example 4
An investor believes that there will be a big
jump in a stock price, but is uncertain to
the direction. Identify six different
strategies the investor can follow and
explain the differences between them.
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199
Chapter 10
Binomial Model
2005年秋
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200
A Simple Example


A stock price is currently $20
In three months it will be either $22 or
$18
Stock Price = $22
Stock price = $20
Stock Price = $18
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201
A Call Option
A 3-month call option on the stock has a
strike price of 21.
Stock Price = $22
Option Price = $1
Stock price = $20
Option Price=?
Stock Price = $18
Option Price = $0
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202
Setting Up a Riskless Portfolio

Consider the Portfolio:
long  shares
short 1 call option
22 – 1
20-c
18
Portfolio is riskless when 22  – 1 = 18  or
 = 0.25
2005年秋
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203
Valuing the Portfolio
(Risk-Free Rate is 12%)



The riskless portfolio is:
long 0.25 shares
short 1 call option
The value of the portfolio in 3 months is
22´0.25 – 1 = 4.50
The value of the portfolio today is (noarbitrage argument)
4.5e – 0.12´0.25 = 4.3670
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204
Valuing the Option



2005年秋
The portfolio that is
long 0.25 shares
short 1 option
is worth 4.367 today.
The value of the shares today is
5.000 (= 0.25´20 )
The value of the option is therefore
0.633 (= 5.000 – 4.367 )
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205
Generalization

A derivative lasts for time T and is
dependent on a stock
S0u
ƒu
S0
ƒ
2005年秋
S0d
ƒd
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206
Generalization (continued)

Consider the portfolio that is long  shares
and short 1 derivative
S0– f

S0 u – ƒu
S0d – ƒd
The portfolio is riskless when S0u  – ƒu =
S0d  – ƒd or
ƒu  f d

S0 u  S0 d
2005年秋
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207
Generalization (continued)




2005年秋
Value of the portfolio at time T is
S0u  – ƒu
Value of the portfolio today is
(S0u  – ƒu )e–rT
Another expression for the
portfolio value today is S0  – f
Hence
ƒ = S0  – (S0u  – ƒu )e–rT
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208
Generalization (continued)

Substituting for  we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
e d
p
ud
rT
2005年秋
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209
Risk-Neutral Valuation



ƒ = [ p ƒu + (1 – p )ƒd ]e-rT
The variables p and (1 – p ) can be interpreted as
the risk-neutral probabilities of up and down
movements
The value of a derivative is its expected payoff in a
risk-neutral world discounted at the risk-free rate
S0
ƒ
2005年秋
S 0u
ƒu
S 0d
ƒd
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210
Irrelevance of Stock’s Expected
Return
When we are valuing an option in terms of
the underlying stock the expected return
on the stock is irrelevant
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211
Original Example Revisited
S0
ƒ

S0u = 22
ƒu = 1
S0d = 18
ƒd = 0
One way is to use the formula
e rT  d e 0.120.25  0.9
p

 0.6523
ud
1.1  0.9

2005年秋
Alternatively, since p is a risk-neutral probability
20e0.12 ´0.25 = 22p + 18(1 – p ); p = 0.6523
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212
Valuing the Option
S0u = 22
ƒu = 1
S0
ƒ
S0d = 18
ƒd = 0
The value of the option is
e–0.12´0.25 [0.6523´1 + 0.3477´0]
= 0.633
2005年秋
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213
A Two-Step Example
24.2
22
19.8
20
18
16.2

2005年秋
Each time step is 3 months
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214
Valuing a Call Option
D
22
24.2
3.2
B
20
1.2823


2005年秋
2.0257
A
E
19.8
0.0
F
16.2
0.0
18
C
0.0
Value at node B
= e–0.12´0.25(0.6523´3.2 + 0.3477´0) = 2.0257
Value at node A
= e–0.12´0.25(0.6523´2.0257 + 0.3477´0)
= 1.2823
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215
A Put Option Example; K=52
60
50
4.1923
A
E
48
4
B
1.4147
40
D
72
0
C
9.4636
F
2005年秋
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32
20
216
What Happens When an Option is
American



Procedure: work back through the tree from
the end to the beginning, testing at each node
to see whether early exercise is optimal.
The value at the final nodes is the same as
for the European.
At earlier nodes the value is the greater of


2005年秋
The value given as an European;
The payoff from early exercise.
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217
An American Put Option Example;
K=52
D
60
50
5.0894
A
B
1.4147
40
48
4
E
C
12.0
F
2005年秋
72
0
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32
20
218
Examples
1. S0=$40, T=1m, ST=$42 or $38,
r=8% per annum (cont comp), what is the
value of a 1-m European call with K=$39?
Use both of the no-arbitrage argument and
the risk-neutral argument.
1.69
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219
Examples
2. S0=$100. Over each of the next two sixmonth periods it is expected to go up by
10%, or go down by 10%,
r=8% per annum (cont comp),
what are the value of a one-year
European call and a one-year European
put with K=$100? Verify the put-call
parity.
1.92, 9.61
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220
Examples
3. S0=$25, T=2m, ST=$23 or $27,
r=10% per annum (cont comp),
what is the value of a derivative that pays
off ST2 at the end of two months?
639.3
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221
Chapter 11
Model of the Behavior
of Stock Prices
2005年秋
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222
Categorization of Stochastic
Processes




2005年秋
Discrete time; discrete variable
Discrete time; continuous variable
Continuous time; discrete variable
Continuous time; continuous variable
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223
Modeling Stock Prices

2005年秋
We can use any of the four types of
stochastic processes to model stock
prices
北航金融系李平
224
Markov Processes


In a Markov process future movements in a
variable depend only on where we are, not
the history of how we got where we are
We assume that stock prices follow Markov
processes
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Weak-Form Market Efficiency


This asserts that it is impossible to produce
consistently superior returns with a trading
rule based on the past history of stock prices.
In other words technical analysis does not
work.
A Markov process for stock prices is clearly
consistent with weak-form market efficiency
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Example of a Discrete Time
Continuous Variable Model


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A stock price is currently at $40
At the end of 1 year it is considered that it
will have a probability distribution of
f(40,10) where f(m,s) is a normal
distribution with mean m and standard
deviation s.
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Questions




What is the probability distribution of the
stock price at the end of 2 years?
½ years?
¼ years?
dt years?
Taking limits we have defined a continuous
variable, continuous time process
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Variances & Standard Deviations



In Markov processes changes in successive
periods of time are independent
This means that variances are additive
Standard deviations are not additive
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Variances & Standard
Deviations (continued)


In our example it is correct to say that the
variance is 100 per year.
It is strictly speaking not correct to say
that the standard deviation is 10 per year.
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A Wiener Process



We consider a variable W whose value
changes continuously
The change in a small interval of time dt is dW
The variable W follows a Wiener process if
1. d W   d t where  is a random drawing from f (0,1)
2. The values of dW for any 2 different (nonoverlapping) periods of time are independent
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Properties of a Wiener Process



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Mean of [W(T ) – W(0)] is 0
Variance of [W(T ) – W(0)] is T
Standard deviation of [W(T ) – W(0)] is
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T
232
Taking Limits . . .


What does an expression involving dW and dt
mean?
It should be interpreted as meaning that the
corresponding expression involving dW and dt
is true in the limit as dt tends to zero
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Generalized Wiener Processes


A Wiener process has a drift rate (i.e.
average change per unit time) of 0 and a
variance rate of 1
In a generalized Wiener process the drift
rate and the variance rate can be set equal
to any chosen constants
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Generalized Wiener Processes
The variable X follows a generalized
Wiener process with a drift rate of a and
a variance rate of b2 if
dX=adt+bdW
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Generalized Wiener Processes
d X  ad t b d t

Mean change in X in time T is aT

Variance of change in X in time T is b2T

Standard deviation of change in X in time T
is b T
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The Example Revisited

A stock price starts at 40 and has a probability
distribution of f(40,10) at the end of the year

If we assume the stochastic process is Markov
with no drift then the process is
dS = 10dW

If the stock price were expected to grow by $8
on average during the year, so that the yearend distribution is f(48,10), the process is
dS = 8dt + 10dW
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Ito Process


In an Ito process the drift rate and the
variance rate are functions of time
dX=a(X,t)dt+b(X,t)dW
The discrete time equivalent
d X  a( X , t )d t  b( X , t ) d t
is only true in the limit as dt tends to zero
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Why a Generalized Wiener Process
is not Appropriate for Stocks


For a stock price we can conjecture that its
expected percentage change in a short period
of time remains constant, not its expected
absolute change in a short period of time
We can also conjecture that our uncertainty as
to the size of future stock price movements is
proportional to the level of the stock price
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An Ito Process for Stock
Prices

The well-known Geometric Brownian Motion
dS  mSdt  sSdW

where m is the expected return, s is the
volatility.
The discrete time equivalent is
dS  mSdt  sS dt
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Monte Carlo Simulation


We can sample random paths for the stock
price by sampling values for 
Suppose m= 0.14, s= 0.20, and dt = 0.01,
then
dS  0.0014S  0.02 S
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Monte Carlo Simulation –
One Path
Period
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Stock Price at
Random
Start of Period Sample for 
Change in Stock
Price, S
0
20.000
0.52
0.236
1
20.236
1.44
0.611
2
20.847
-0.86
-0.329
3
20.518
1.46
0.628
4
21.146
-0.69
-0.262
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Ito’s Lemma

If we know the stochastic process followed
by X, Ito’s lemma tells us the stochastic
process followed by some function f (X, t )

Since a derivative security is a function of
the price of the underlying and time, Ito’s
lemma plays an important part in the
analysis of derivative securities
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Taylor Series Expansion

A Taylor’s series expansion of f(X, t) gives
 f
 f
2f
2
d f 
dX 
d t ½
(d X )
2
 x
t
 x
2f
2f

d Xd t  ½ 2 (d t ) 2  
 x t
t
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Ignoring Terms of Higher
Order Than dt
In ordinary calculus we have
f
f
df 
dx
dt
x
t
In stochastic calculus this becomes
f
f
2f
df 
dX 
dt ½
(d X ) 2
x
t
 x2
because dX has a component which is
of order
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dt
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245
Substituting for dX
Suppose
dx  a ( x, t )dt  b( x, t )dz
so that
d X = ad t +b d t
Then ignoring terms of higher order than d t
 f
 f
2f 2 2
d f 
d X
d t ½
b  dt
2
 x
t
 x
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The 2dt Term
Since   f (0,1) E ( )  0
E ( 2 )  [ E ( )] 2  1
E ( 2 )  1
It follows that E ( 2dt )  dt
The variance of d t is proportional to d t 2 and can
be ignored. Hence
f
f
12f 2
d f 
d x
d t
b dt
2
x
t
2 x
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Taking Limits
Taking limits
Substituting
We obtain
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f
f
2f 2
df 
dX 
dt 
b dt
x
t
 x2
dX  a dt  b dW
2
 f


f

f
f
2


df 
a

b dt 
b dW
2
x


t
 x

x


This is Ito' s Lemma
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Application of Ito’s Lemma
to a Stock Price Process
The stock price process is
d S  m S dt  s S dW
For a function f of S and t
2
 f


f

f
f
2
2


df 
mS 
½
s S dt 
s S dW
2
S


t
S

S


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Examples
1. The forward price of a stock for a contract
maturing at time T
f  S e r (T t )
df  ( m  r ) f dt  sf dW
2. f  ln S
2

s
dt  s dW
df   m 


2


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The lognormal property
From Example 2,
ln ST  ln S 0 ~ f (( m 
s2
2
)T , s T )
or, ln ST ~ f (ln S 0  ( m 

s2
2
)T , s T )
Since the logarithm of ST is normal, ST is
lognormally distributed
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The Lognormal Distribution
(cont.)
E ( ST )  S0 e mT
2 2 mT
var ( ST )  S0 e
(e
s2T
 1)
Example


Consider a stock with an initial price of $40,
an expected return of 16% per annum, and a
volatility of 20% per annum, then the
probability distribution of the stock price, ST,
in six months’ time is given by
The confidence interval for the stock price in
six month with the probability of 95% is
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Continuously Compounded Return
ST  S0 e T
or
1 ST
 = ln
T
S0
or

s2
  f m 
,
2

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s 

T
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The Expected Return


The expected value of the stock price is
S0emT
The expected return on the stock is
m – s2/2
Eln( ST / S0 )  m  s / 2
2
ln E ( ST / S0 )  m
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The Volatility


The volatility of an asset is the standard
deviation of the continuously compounded
rate of return in 1 year
As an approximation it is the standard
deviation of the percentage change in the
asset price in 1 year
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Estimating Volatility from
Historical Data
1.
2.
Take observations S0, S1, . . . , Sn at intervals of t
years
Calculate the continuously compounded return in
each interval as:
 Si 

ui  ln
 Si 1 
3.
4.
Calculate the standard deviation, s , of the ui´s
The historical volatility estimate is: sˆ  s
t
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Chapter 12-13
Black-Scholes Model
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1. Black-Scholes Formula
The Concepts Underlying Black-Scholes:



The option price and the stock price depend
on the same underlying source of uncertainty
We can form a portfolio consisting of the stock
and the option which eliminates this source of
uncertainty
The portfolio is instantaneously riskless and
must instantaneously earn the risk-free rate
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The Derivation of the BlackScholes Differential Equation
d S  mS d t  sS d W
 f
ƒ
2 ƒ 2 2 
ƒ
d ƒ   mS   ½ 2 s S d t  sS d W
t
S
S
 S

We set up a portfolio consisting of
 1 : derivative
f
+
: shares
S
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The Derivation of the Black-Scholes
Differential Equation (cont.)
The value of the portfolio  is given by
ƒ
  ƒ 
S
S
The change in its value in time d t is given by
ƒ
d   d ƒ 
dS
S
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The Derivation of the Black-Scholes
Differential Equation (cont.)
The return on the portfolio must be the risk - free
rate. Hence
d   r d t
We substitute for d ƒ and d S in these equations
to get the Black - Scholes differenti al equation :
ƒ
ƒ
2 2  ƒ
 rS
½s S
 rƒ
2
t
S
S
2
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The Differential Equation



Any security whose price is dependent on
the stock price satisfies the differential
equation
The particular security being valued is
determined by the boundary conditions of
the differential equation
For European call option, the boundary
condition is
fT=max(0, ST-K)
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Risk-Neutral Valuation




The variable m does not appear in the
Black-Scholes equation
The equation is independent of all variables
affected by risk preference
The solution to the differential equation is
therefore the same in a risk-free world as it
is in the real world
This leads to the principle of risk-neutral
valuation
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Applying Risk-Neutral Valuation
1. Assume that the expected return from the
stock price is the risk-free rate r
2. Calculate the expected payoff from the
option
3. Discount at the risk-free rate
f0  e
 rT
Eˆ ( fT )
where Ê is the expectation under a risk-neutral
probability measure P̂
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The Black-Scholes Formulas
c  S 0 N (d1 )  K e  rT N (d 2 )
p  K e  rT N (d 2 )  S 0 N (d1 )
2
ln( S 0 / K )  (r  s / 2)T
where d1 
s T
ln( S 0 / K )  (r  s 2 / 2)T
d2 
 d1  s T
s T
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Implied Volatility



The implied volatility of an option is the
volatility for which the Black-Scholes price
equals the market price
There is a one-to-one correspondence
between prices and implied volatilities
Traders and brokers often quote implied
volatilities rather than dollar prices
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2. Dividends



European options on dividend-paying stocks
are valued by substituting the stock price less
the present value of dividends into BlackScholes
Only dividends with ex-dividend dates during
life of option should be included
The “dividend” should be the expected
reduction in the stock price expected
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European Options on Stocks
Providing Dividend Yield (cont.)


We get the same probability distribution for the stock
price at time T in each of the following cases:
1. The stock starts at price S0 and provides a
dividend yield = q
2. The stock starts at price S0e–q T and provides no
income
We can value European options by reducing the
stock price to S0e–q T and then behaving as though
there is no dividend
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Prices for European Options on
Stocks Providing Dividend Yield
c  S0e  qT N (d1 )  Ke rT N (d 2 )
p  Ke rT N (d 2 )  S0e  qT N (d1 )
ln( S0 / K )  (r  q  s 2 / 2)T
where d1 
s T
ln( S0 / K )  (r  q  s 2 / 2)T
d2 
s T
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3. Valuing European Index Options

We can use the formula for an option on a
stock paying a dividend yield


2005年秋
Set S0 = current index level
Set q = average dividend yield expected during
the life of the option
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4. The Foreign Interest Rate




We denote the foreign interest rate by rf
When a U.S. company buys one unit of the
foreign currency it has an investment of S0
dollars
The return from investing at the foreign rate
is rf S0 dollars
This shows that the foreign currency
provides a “dividend yield” at rate rf
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Valuing European Currency
Options

A foreign currency is an asset that
provides a “dividend yield” equal to rf

We can use the formula for an option
on a stock paying a dividend yield:
Set S0 = current exchange rate
Set q = rƒ
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Formulas for European Currency
Options
c  S 0e
p  Ke
r f T
 rT
N (d1 )  Ke rT N (d 2 )
N (d 2 )  S0e
where d1 
d2 
2005年秋
r f T
N (d1 )
ln( S0 / K )  (r  r f  s 2 / 2)T
s T
ln( S0 / K )  (r  r f  s 2 / 2)T
s T
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Alternative Formulas
Using
F0  S0 e
( r rf ) T
c  e  rT [ F0 N (d1 )  KN (d 2 )]
p  e  rT [ KN ( d 2 )  F0 N ( d1 )]
ln( F0 / K )  s 2T / 2
d1 
s T
d 2  d1  s T
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5. Mechanics of Call Futures
Options
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 futures price over the strike price
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Mechanics of Put Futures Option
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 futures price
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The Payoffs
If the futures position is closed out
immediately:
Payoff from call = F0 – K
Payoff from put = K – F0
where F0 is futures price at time of exercise
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Put-Call Parity for Futures
Options
Consider the following two portfolios:
1. European call on futures + Ke-rT of cash
2. European put on futures + long futures +
cash equal to F0e-rT
They must be worth the same at time T so
that
c+Ke-rT=p+F0 e-rT
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Binomial Tree Model

A derivative lasts for time T and is
dependent on a futures price
F0
ƒ
F0u
ƒu
F0d
ƒd
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Binomial Tree Model (cont.)

Consider the portfolio that is long  futures and
short 1 derivative
F0u   F0  – ƒu
F0d  F0 – ƒd

The portfolio is riskless when
ƒu  f d

F0u  F0d
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Binomial Tree Model (cont.)
2005年秋

Value of the portfolio at time T is
F0u  –F0 – ƒu

Value of portfolio today is – ƒ

Hence
ƒ = – [F0u  –F0 – ƒu]e-rT
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Binomial Tree Model (cont.)

Substituting for  we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
1 d
p
ud
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Pricing by Binomial Tree Model
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
1 d
p
ud
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Valuing European Futures
Options

We can use the formula for an option on a
stock paying a dividend yield
Set S0 = current futures price (F0)

2005年秋
Set q = domestic risk-free rate (r )
Setting q = r ensures that the expected
growth of F in a risk-neutral world is zero
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Growth Rates For Futures Prices




A futures contract requires no initial
investment
In a risk-neutral world the expected return
should be zero
The expected growth rate of the futures
price is therefore zero
The futures price can therefore be treated
like a stock paying a dividend yield of r
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Black’s Formula

The formulas for European options on
futures are known as Black’s formulas
c  e  rT F0 N (d1 )  K N (d 2 )
p  e  rT K N (d 2 )  F0 N (d1 )
ln( F0 / K )  s 2T / 2
where d1 
s T
ln( F0 / K )  s 2T / 2
d2 
 d1  s T
s T
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Summary of Key Results

We can treat stock indices, currencies,
and futures like a stock paying a dividend
yield of q
 For stock indices, q= average dividend
yield on the index over the option life
 For currencies, q= rƒ
 For futures, q= r
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Chapter 14
The Greek Letters
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


The problem to the option writer:
managing the risk
Each Greek letter measures a different
dimension to the risk in an option
position
The aim of a trader: manage the Greek
letters so that all risks are acceptable
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An Example
A bank has sold for $300,000 a European call
option on 100,000 shares of a nondividend paying
stock
 S0 = 49,
K = 50, r = 5%, s = 20%,
T = 20 weeks (0.3846y), m = 13%
 The Black-Scholes value of the option is $240,000
 The bank get $60,000 more than the theoretical
value, but it is faced the problem of hedging the
risk.

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Naked & Covered Positions



Naked position: Take no action,
works well when ST <50,
otherwise (e.g. ST=60), lose (60-50)* 100,000
Covered position
Buy 100,000 shares today
works well when exercised (ST >50),
otherwise (e.g. ST=40), lose (59-40)* 100,000
Neither strategy provides a satisfactory hedge,
most traders employ Greek letters.
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Delta ()

Delta is the rate of change of the option
price with respect to the underlying
Option
price
Slope = 
B
A
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Example




S=100, c=10,  =0.6
An investor sold 20 calls, this position could
be hedged by buying 0.6*2000=1200 shares
The gain (lose) on the option position will be
offset by the lose (gain) on the stock position
Delta of a call on a stock (0.6)
delta of the short option position (-2000*0.6)
delta of the long share position
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Delta Hedging


This involves maintaining a delta neutral
portfolio---  =0
In Black-Scholes model,
-1: option
+  : shares
set up a delta neutral portfolio
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Delta Hedging (cont.)




The delta of a European call on a nondividend-paying stock:
 =N (d 1)>0
Short position in a call should be hedged by a
long position on shares
The delta of a European put is
 = - N (-d 1) =N (d 1) – 1<0
Short position in a put should be hedged by a
short position on shares
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Delta Hedging (cont.)



The variation of delta w.r.t the stock price
The hedge position must be frequently
rebalanced
In the example, when S increase from
$100 to $110, the delta will increase from
0.6 to 0.65, then an extra 0.05*2000=100
shares should be purchased to maintain
the hedge
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Delta for other European
options



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Call on asset paying yield q
 =e-qt N (d 1)
For put
 = e-qt [N (d 1) -1]
For index option, foreign currency options
and futures options
Delta of a portfolio


  wi  i
S
i
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Gamma (G)
  c
G
 2
S  S
2

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Gamma is always positive (for buyer),
negative for writer
If gamma is large, delta is highly
sensitive to the stock price, then it will
be quite risky to leave a delta-neutral
portfolio unchanged.
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Gamma Addresses Delta Hedging
Errors Caused By Curvature
Call
price
C’’
C’
C
Stock price
S
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’
300
Making a portfolio gamma neutral


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The gamma of the underlying asset is 0, so
it can’t be used to change the gamma of a
portfolio.
What is required is an instrument such as
an option which is not linearly dependent
on the underlying asset.
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Gamma hedging


Suppose
the gamma of a delta-neutral portfolio is G,
the gamma of a traded option is GT,
then the gamma of a new portfolio with the
number of wT options added is
wT GT + G
In order that the new portfolio is gamma
neutral, the number of the options should be
wT= - G/GT
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Gamma hedging (cont.)


Including the traded option will change the
delta of the portfolio, so the position in the
underlying asset has to be changed to
maintain delta neutral.
The portfolio is gamma neutral only for a
short period of time. As time passes, gamma
neutrality can be maintained only when the
position in the option is adjusted so that it is
always equal to
- G/GT.
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An example



Suppose that a portfolio is delta neutral
and has a gamma of –3000.
The delta and gamma of a particular
traded call are 0.62 and 1.5.
The portfolio can be made gamma neutral
by including in the portfolio a long position
of 2000(=-[-3000/1.5]).
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Example (cont.)


The delta of the new portfolio will change
from 0 to 2000*0.62=1240.
A quantity of 1240 of the underlying asset
must be sold from the portfolio to keep it
delta neutral.
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Theta (Q)



Theta of a derivative is the rate of change of the
value with respect to the passage of time with all
else remain the same, often referred to as the
time decay of the option
In practice, when theta is quoted, time is
measured in days so that theta is the change in
the option value when one day passes.
Theta is usually negative for an option, since as
time passes, the option tends to be less valuable.
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Vega (n)
c
n
s


If n is the vega of a portfolio and nT is the
vega of a traded option, a position of –n/nT
in the traded option makes the portfolio
vega neutral.
If a hedger requires a portfolio to be both
gamma and vega neutral, at least two
traded derivatives dependent on the
underlying asset must be used.
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Rho (r)
c
r
r


For currency options there are 2 rhos
For a European call
r r  KTe rT N (d 2 )
r r  Te
f
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rf T
SN (d1 )
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Hedging in Practice




Traders usually ensure that their
portfolios are delta-neutral at least once a
day.
Zero gamma and zero vega are less easy to
achieve because of the difficulty of
finding suitable options.
Whenever the opportunity arises, they
improve gamma and vega
As portfolio becomes larger hedging
becomes less expensive
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Chapter 19
Exotic Options
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






Packages
Asian options
Options to
exchange
one asset for
another
Binary options
Rainbow options
Lookback options
Barrier options
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




Compound options
Nonstandard
American options
Forward start
options
Chooser options
Shout options
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Packages




Portfolios of standard options, forward
contract, cash and the underlying asset
Examples: bull spreads, bear spreads,
straddles, etc
Often structured to have zero cost
One popular package is a range forward
contract
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Range forward contract





Popular in foreign-exchange markets
Long/short-range forward = a short/long put with
the low strike price + a long/short call with the
high strike price
The prices of the call and the put are equal when
the contract is initiated
A long-range forward guarantees the underlying
asset be purchased for a price between two
strikes at the maturity
When K1 and K2 are moved closer, the price
becomes more certain
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Payoff from a long range forward
Profit
S
K1
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T
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Asian Options


Payoff related to the average price of the
underlying during some period
Payoffs:




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max(Save – K, 0) (average price call),
max(K – Save , 0) (average price put)
max(ST – Save , 0) (average strike call),
max(Save – ST , 0) (average strike put)
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Asian Options (cont.)


Average price options are less expensive
and sometimes are more appropriate than
regular options
Average strike call (put) can guarantee that
the average price paid (received) for an
asset in frequent trading over a period of
time is not greater (less) than the final
price
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Exchange Options



Option to exchange one asset for another
For example, an option to give up Japanese
yen worth UT at time T and receive in
return Australian dollars worth VT
Payoff= max(VT – UT, 0)
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Binary Options

Cash-or-nothing call:




Cash-or-nothing put:



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Pays off a fixed amount Q if ST > K,
otherwise pays off 0,
Value = e–rT Q N(d2)
Pays off a fixed amount Q if ST < K,
otherwise pays off 0,
Value = e–rT Q N(-d2)
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Binary Options (cont.)

Asset-or-nothing call:




pays off ST (an amount equal to the asset price) if
ST > K,
otherwise pays off 0.
Value = S0 N(d1)
Asset-or-nothing put:



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pays off ST (an amount equal to the asset price) if
ST < K,
otherwise pays off 0.
Value = S0 N(-d1)
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Rainbow options


Options involving two or more risky assets
The most popular rainbow option--Basket Options: whose payoff is dependent
on the value of a portfolio of assets
(stocks, indices, currencies)
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Lookback Options

Payoff from an European lookback call:
ST – Smin


Allows buyer to buy stocks at the lowest
observed price in some interval of time
Payoff from a lookback put: Smax– ST

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Allows buyer to sell stocks at the highest
observed price in some interval of time
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Barrier Options

Option comes into existence only if the
asset price hits barrier before option
maturity


‘In’ options
Option dies if the asset price hits barrier
before option maturity

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‘Out’ options
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Barrier Options (cont.)

barrier level above the asset price


barrier level below the asset price



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‘Up’ options
‘Down’ options
Option may be a put or a call
Eight possible combinations
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Barrier Options (cont.)




Up-and-in call
Up-and-in put
Down-and-in call
Down-and-in put
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


Up-and-out call
Up-and-out put
Down-and-out call
Down-and-out put
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Compound Option


Option to buy / sell an option
Two strikes and two maturities




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Call on call
Put on call
Call on put
Put on put
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Non-Standard American Options




Exercisable only on specific dates--Bermudan option
Early exercise allowed during only part of
life
Strike price changes over the life
Exm: a seven-year warrant issued by a
corporation on its own stocks
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Chooser Option




“As you like it” option
Option starts at time 0, matures at T2
At T1 (0 < T1 < T2) buyer chooses whether it
is a put or call
The value of the chooser option at time T1:
Max(c,p)
where c and p are the values of the call and
put underlying the chooser option.
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Chooser Option

If the call and the put underlying the chooser
option are both European and have the same
strike price K, then put-call parity implies that
max( c, p)  max( c, c  Ke  r (T2 T1 )  S1 )
 c  max(0, Ke  r (T2 T1 )  S1 )

Thus the chooser option is a package of


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A call option with strike price K and maturity T2
A put option with strike price Ke r (T2 T1 ) and
maturity T1
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Shout Options


A European call option where the holder can
‘shout’ to the writer once during the option life
The final payoff of a call is the maximum of




The usual European option payoff,
max(ST – K, 0), or
Intrinsic value at the time of shout, St – K
Example: K=50, St=60, when ST<60, the payoff
is 10; when ST>60, the payoff is
Similar to a lookback option, but is cheaper
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Forward Start Options


Option starting at a future time, used in
employee incentive schemes
Usually be at the money at the time they
start
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Example: Standard Oil’s Bond





It is a bond issued by Standard Oil
The holder receives no interest.
At the maturity the company promised to pay
$1000 plus an additional amount based on the
price of oil at that time.
The additional amount was equal to the product
of 170 and the excess (if any) of the price of oil
at maturity over $25.
The maximum additional amount paid was $2250
(which corresponds to a price of $40)
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Standard Oil’s Bond


Show that this bond is a combination of a
regular bond, a long position in call options on
oil with a strike price of $25 and a short
position in call options on oil with a strike
price of $40.
Relationship between a spread option
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