Post on 25-Dec-2015
2005年秋 北航金融系李平 1
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
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 DerivativesA 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 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 open-
outcry 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) 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
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
Bid Offer
Spot 1.4452 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
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
Price of Underlying at Maturity, STK
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Profit from a Short Forward Position
Profit
Price of Underlying at Maturity, STK
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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
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 Ap
ril (NYMEX)
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4. Options
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
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 = $60
30
20
10
0-5
30 40 50 60
70 80 90
Profit ($)
Terminalstock price ($)
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Short Call on Microsoft
Profit from writing a European call option on Microsoft: option price = $5, strike price = $60
-30
-20
-10
05
30 40 50 60
70 80 90
Profit ($)
Terminalstock price ($)
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Long Put on IBM
Profit from buying a European put option on IBM: option price = $7, strike price = $90
30
20
10
0
-790807060 100 110 120
Profit ($)
Terminalstock price ($)
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Short Put on IBM
Profit from writing a European put option on IBM: option price = $7, strike price = $90
-30
-20
-10
7
090
807060
100 110 120
Profit ($)Terminal
stock price ($)
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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
ST STK
K
Payoff Payoff
ST STK
K
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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: 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 2-month 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: 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
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
Time Time
(a) (b)
FuturesPrice
FuturesPrice
Spot Price
Spot Price
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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 维持保证金 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 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$3,000 in total)
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Margin Calculation (cont.)
Daily Cumulative Margin
Futures Gain Gain Account Margin
Price (Loss) (Loss) Balance Call
Day (US$) (US$) (US$) (US$) (US$)
400.00 4,000
5-Jun 397.00 (600) (600) 3,400 0. . . . . .. . . . . .. . . . . .
13-Jun 393.30 (420) (1,340) 2,660 1,340 . . . . . .. . . . .. . . . . .
19-Jun 387.00 (1,140) (2,600) 2,740 1,260 . . . . . .. . . . . .. . . . . .
26-Jun 392.30 260 (1,540) 5,060 0
+
= 4,000
3,000
+
= 4,000
<
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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 ): futu
res 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
Private contract between 2 parties Exchange traded
Non-standard contract Standard contract
Usually 1 specified delivery date Range of delivery dates
Settled at maturity Settled daily
No daily price change limit Have daily price change limit
FORWARDS FUTURES
交割率为 90% 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
DefineRc : continuously compounded rate
Rm: same rate with compounding m times per year
R m
R
m
R m e
cm
mR mc
ln
/
1
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?
09758.0)2/1.01ln(2
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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?
100000.1218/4=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:
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 thatF0 = 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-0.090.5=$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 $861.76e0.11=$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.)
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 securi
ty providing a dividend yield The continuous dividend yield is the foreig
n risk-free interest rate It follows that if rf is the foreign risk-free int
erest rate F S e r r Tf
0 0 ( )
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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?
Jun93
f2 1 Mar93
1 Fr r ln 0.0249
T T F
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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 shor
tages ability to immediately keep a production
process running The convenience yield, y, is defined so that
F0 eyT= S0 e(r+u )T
Or 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
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Valuing a Forward Contract Suppose that K is delivery price in a forward contract F0 is forward price that would apply to the c
ontract 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
2005年秋 北航金融系李平 90
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 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
Long & Short Hedges
2005年秋 北航金融系李平 93
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
2005年秋 北航金融系李平 94
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 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 19.10-18=$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 futu
res contract to the size of the exposure Optimal hedge ratio: proportion of the exposure t
hat should optimally be hedged is (extra1)
where S and F are the standard deviations of S and F, the c
hange in the spot price and futures price during the hedging period,
is the coefficient of correlation between S and F.
h S
F
*
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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 20,00020-17,600=382,400 or $19.12 per barrel. (This compares with $18.90 per barrel when t
he hedge ratio is 1.)
北航金融系李平 105
Hedging Using Index Futures
To hedge the risk in a portfolio the number of index futures contracts that should be used is
where P is the value of the portfolio, is its beta, and A is the value of the index underlying one futures contract
P
A
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Example 3
Value of S&P 500 is 1,000Value of Portfolio is $5 millionBeta 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: How swaps are defined How they are be used How they can be valued
Two plain vanilla swap: Interest-rate swap, fixed-for-fixed currency swap
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1. An Example of a “Plain Vanilla” Interest Rate Swap
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
2005年秋 北航金融系李平 110
---------Millions of Dollars---------
LIBOR FLOATING 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
Cash Flows to Microsoft
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Typical Uses of anInterest Rate Swap
Converting a liability from fixed rate to
floating rate floating rate to
fixed rate
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
Intel MS
LIBOR
5%
LIBOR+0.1%
5.2%
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Financial Institution is Involved
F.I.
LIBOR LIBORLIBOR+0.1%
4.985% 5.015%
5.2%Intel MS
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Intel and Microsoft (MS) Transform an Asset
Intel MS
LIBOR
5%
LIBOR-0.25%
4.7%
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Financial Institution is Involved
Intel F.I. MS
LIBOR LIBOR
4.7%
5.015%4.985%
LIBOR-0.25%
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The Comparative Advantage Argument
AAACorp wants to borrow floating BBBCorp wants to borrow fixed
Fixed Floating
AAACorp 10.00% 6-month LIBOR + 0.30%
BBBCorp 11.20% 6-month LIBOR + 1.00%
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The Swap
AAA BBB
LIBOR
LIBOR+1%
9.95%
10%
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The Swap when a Financial Institution is Involved
AAA F.I. BBB10%
LIBOR LIBOR
LIBOR+1%
9.93% 9.97%
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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.)
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 w
ay The floating rate bond is valued by noting t
hat 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 $100 and the remaining payment dates are in 3, 9 and 15 months. The swap has a remaining life of 15months. The LIBOR rates with continuous compounding for 3-month, 9-month and 15-month maturities are 10%, 10.5% and 11%, respectively. The 6-month LIBOR rate at the last payment date was 10.2% (with semiannual compounding).
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Valuation in Terms of FRAs
Each exchange of payments in an interest rate swap is an FRA
The FRAs can be valued on the assumption that today’s forward rates are realized
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2. An Example of a fixed-for-fixed Currency Swap
An agreement to pay 11% on a sterling principal of £10,000,000 & receive 8% on a US$ principal of $15,000,000 every year for 5 years
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Exchange of Principal
In an interest rate swap the principal is not exchanged
In a currency swap the principal is exchanged at the beginning and the end of the swap
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The Cash Flows
Year
Dollars Pounds$
------millions------
2001 –15.00 +10.002002 +1.20 –1.10
2003 +1.20 –1.10 2004 +1.20 –1.10
2005 +1.20 –1.10 2006 +16.20 -11.10
£
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Typical Uses of a Currency Swap
Conversion from a liability in one currency to a liability in another currency
Conversion from an investment in one currency to an investment in another currency
北航金融系李平 130
Comparative Advantage Arguments for Currency Swaps
General Motors wants to borrow AUDQantas wants to borrow USD
USD AUD
General Motors 5.0% 12.6%
Qantas 7.0% 13.0%
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Valuation of Currency Swaps
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.
2005年秋 北航金融系李平 133
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 DollarsCompany 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.
Example 3
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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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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
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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Futures option 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
Types of options (cont.)
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Foreign currency option 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/£
Types of options (cont.)
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Specification ofExchange-Traded Options
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)
Option class (call or put) Option series Intrinsic value Time value
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Dividends & Stock Splits
Suppose you own N options with a strike price of K : No adjustments are made to the option ter
ms 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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Dividends & Stock Splits(continued)
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
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
2005年秋 北航金融系李平 146
Newspaper quotes Market Makers
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
2005年秋 北航金融系李平 147
Margins
Margins are required when options are sold When a naked call (put) option is written the
margin is the greater of:1 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
2 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
2005年秋 北航金融系李平 150
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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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
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 ofStock Option Prices
2005年秋 北航金融系李平 154
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 : Volatility of stock
price
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.
2005年秋 北航金融系李平 156
Effect of Variables on Option Pricing
c p C PVariable
S0
KTrD
+ + –+
? ? + ++ + + ++ – + –
–– – +
– + – +
2005年秋 北航金融系李平 157
American vs European Options
An American option is worth at least as much as the corresponding European option
C c
P p
) ,max(
) ,max(
SKpP
KScC
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Upper bounds for option prices
,
, 00
KPKep
SCSc
rT
2005年秋 北航金融系李平 159
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 –rT, 0)
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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.
2005年秋 北航金融系李平 161
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 non-dividend-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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Arbitrage Opportunities
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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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 bot
h the stock and the put. Generating a positive cash flow of
2.25+31-3=30.25 After three months, this amount grows to 31.0
2
2005年秋 北航金融系李平 166
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
Continue
2005年秋 北航金融系李平 167
When p(t)=1
c(t)+Ke-rt=32.26, p(t)+S(t)=32 Portfolio A is overpriced. The arbitrage strategy: short the call, buy b
oth the stock and the put. Initial investment:
1+31-3=$29
2005年秋 北航金融系李平 168
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
Continue
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Early Exercise for American options
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
2005年秋 北航金融系李平 170
For an American call option: S(t) = 50; T = 1m; K = 40; D = 0
Should you exercise immediately? What should you do if
1. You want to hold the stock for one month?
2. You do not feel that the stock is worth holding for the next 1 month?
An Extreme Situation
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Reasons For Not Exercising a Call Early--No Dividends
Case 1: should keep the option and exercise it at the end of the month. 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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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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More formal argument
C c S0–Ke –rT>S0–K If it is optimal to exercise early,
C=S0–K
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Should Puts Be Exercised Early?
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.
2005年秋 北航金融系李平 175
The Impact of Dividends on Lower Bounds
c S D Ke
p D Ke S
rT
rT
0
0
Portfolio A: one European call & an amount of cash equal to Ke-rT+D Portfolio B: one share
2005年秋 北航金融系李平 176
Impact on Put-Call Parity
European options; D > 0
c + D + Ke -rT = p + S0
American options; D = 0
American options; D > 0
rTKeSPCKS 00
rTKeSPCKDS 00
2005年秋 北航金融系李平 177
Chapter 9
Trading Strategies Involving Options
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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
2005年秋 北航金融系李平 180
Profit
STK
Profit
ST
K
Profit
ST
K
Profit
STK
(a) (b)
(c)
(d)
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Bull Spread Using Calls
K1 K2
S
T
Profit
Buy lower & sell higher call
2005年秋 北航金融系李平 182
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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Bull Spread Using Puts
Buy lower & sell higher put
K1 K2
Profit
ST
2005年秋 北航金融系李平 184
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 北航金融系李平 185
Bear Spread Using Calls
Profit
K1 K2 ST
Buy higher & sell lower call
2005年秋 北航金融系李平 186
Bear Spread Using Puts
K1 K2
Profit
ST
Buy higher & sell lower put
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Butterfly Spread Using Calls
K1 K3
Profit
STK2
Buy 1 high & 1 low, sell 2 middle
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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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Butterfly Spread Using Puts
K1 K3
Profit
STK2
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Calendar Spread Using Calls
Profit
ST
K
Buy longer & sell shorter (maturity)
2005年秋 北航金融系李平 191
Calendar Spread Using Puts
Profit
ST
K
2005年秋 北航金融系李平 192
A Straddle Combination
Profit
STK
Buy a call & a put
2005年秋 北航金融系李平 193
Continue
Payoff structure When it is appropriate? Example:
S0=69, expect a significant move in the future, K=70, c=4, p=3
2005年秋 北航金融系李平 194
Strip & Strap
Profit
K ST
Strip Strap
Buy 1 call & 2 puts Buy 2 calls & 1 put
K ST
Profit
2005年秋 北航金融系李平 195
A Strangle Combination
Buy 1 call with higher strike & 1 put with lower strike
K1 K2
Profit
ST
2005年秋 北航金融系李平 196
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.
2005年秋 北航金融系李平 197
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?
2005年秋 北航金融系李平 198
Example 3
A call with a strike price of $50 costs $2. A put with a strike price of $45 costs $3. Expl
ain how a strangle can be created from these two options. What is the pattern of profits from the strangle?
2005年秋 北航金融系李平 199
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.
2005年秋 北航金融系李平 200
Chapter 10
Binomial Model
2005年秋 北航金融系李平 201
A Simple Example
A stock price is currently $20 In three months it will be either $22
or $18
Stock Price = $18
Stock Price = $22
Stock price = $20
2005年秋 北航金融系李平 202
Stock Price = $22Option Price = $1
Stock Price = $18Option Price = $0
Stock price = $20Option Price=?
A Call Option
A 3-month call option on the stock has a strike price of 21.
2005年秋 北航金融系李平 203
Consider the Portfolio: long sharesshort 1 call opti
on
Portfolio is riskless when 22 – 1 = 18 or = 0.25
18
Setting Up a Riskless Portfolio
22– 1
20-c
2005年秋 北航金融系李平 204
Valuing the Portfolio(Risk-Free Rate is 12%)
The riskless portfolio is: long 0.25 sharesshort 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 (no-arbitrage argument)
4.5e – 0.12´0.25 = 4.3670
2005年秋 北航金融系李平 205
Valuing the Option
The portfolio that is long 0.25 sharesshort 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 )
2005年秋 北航金融系李平 206
Generalization
A derivative lasts for time T and is dependent on a stock
S0u ƒu
S0d ƒd
S0
ƒ
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Generalization (continued)
Consider the portfolio that is long shares and short 1 derivative
The portfolio is riskless when S0u – ƒu = S0d – ƒd or
ƒu df
S u S d0 0
S0 u– ƒu
S0d– ƒd
S0– f
2005年秋 北航金融系李平 208
Generalization (continued)
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
2005年秋 北航金融系李平 209
Generalization (continued)
Substituting for we obtain ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
pe d
u d
rT
2005年秋 北航金融系李平 210
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
S0u ƒu
S0d ƒd
S0
ƒ
p
(1– p )
2005年秋 北航金融系李平 211
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
2005年秋 北航金融系李平 212
Original Example Revisited
One way is to use the formula
Alternatively, since p is a risk-neutral probability 20e0.12 ´0.25 = 22p + 18(1 – p ); p = 0.6523
6523.09.01.1
9.00.250.12
e
du
dep
rT
S0d = 18 ƒd = 0
p
S0u = 22 ƒu = 1
S0
ƒ(1– p )
2005年秋 北航金融系李平 213
Valuing the Option
The value of the option is e–0.12´0.25 [0.6523´1 + 0.3477´0] = 0.633
S0u = 22 ƒu = 1
S0d = 18 ƒd = 0
S0
ƒ
0.6523
0.3477
2005年秋 北航金融系李平 214
A Two-Step Example
Each time step is 3 months
20
22
18
24.2
19.8
16.2
2005年秋 北航金融系李平 215
Valuing a Call Option
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
24.23.2
201.2823
22
18
19.80.0
16.20.0
2.0257
0.0
A
B
C
D
E
F
2005年秋 北航金融系李平 216
A Put Option Example; K=52
504.1923
60
40
720
484
3220
1.4147
9.4636
A
B
C
D
E
F
2005年秋 北航金融系李平 217
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 The value given as an European; The payoff from early exercise.
2005年秋 北航金融系李平 218
An American Put Option Example; K=52
505.0894
60
40
720
484
3220
1.4147
12.0
A
B
C
D
E
F
2005年秋 北航金融系李平 219
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
2005年秋 北航金融系李平 220
Examples
2. S0=$100. Over each of the next two six-month 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
2005年秋 北航金融系李平 221
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 at the end of two months?
2TS
639.3
2005年秋 北航金融系李平 222
Model of the Behavior
of Stock Prices
Chapter 11
2005年秋 北航金融系李平 223
Categorization of Stochastic Processes
Discrete time; discrete variable Discrete time; continuous variable Continuous time; discrete variable Continuous time; continuous variable
2005年秋 北航金融系李平 224
Modeling Stock Prices
We can use any of the four types of stochastic processes to model stock prices
2005年秋 北航金融系李平 225
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
2005年秋 北航金融系李平 226
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
2005年秋 北航金融系李平 227
Example of a Discrete Time Continuous Variable Model
A stock price is currently at $40 At the end of 1 year it is considered that it
will have a probability distribution of(40,10) where (,) is a normal distribution with mean and standard deviation
2005年秋 北航金融系李平 228
Questions
What is the probability distribution of the stock price at the end of 2 years?
½ years? ¼ years? t years?
Taking limits we have defined a continuous variable, continuous time process
2005年秋 北航金融系李平 229
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
2005年秋 北航金融系李平 230
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.
2005年秋 北航金融系李平 231
A Wiener Process
We consider a variable W whose value changes continuously
The change in a small interval of time t is W The variable W follows a Wiener process if
1.
2. The values of W for any 2 different (non- overlapping) periods of time are independent
(0,1) W from drawing random a is wheret
2005年秋 北航金融系李平 232
Properties of a Wiener Process
Mean of [W(T ) – W(0)] is 0 Variance of [W(T ) – W(0)] is T Standard deviation of [W(T ) – W(0)] is T
2005年秋 北航金融系李平 233
Taking Limits . . .
What does an expression involving dW and dt mean?
It should be interpreted as meaning that the corresponding expression involving W and t is true in the limit as t tends to zero
2005年秋 北航金融系李平 234
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
2005年秋 北航金融系李平 235
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
2005年秋 北航金融系李平 236
Generalized Wiener Processes
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
tbtaX
b T
2005年秋 北航金融系李平 237
The Example Revisited
A stock price starts at 40 and has a probability distribution of(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 year-end distribution is (48,10), the process is
dS = 8dt + 10dW
2005年秋 北航金融系李平 238
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
is only true in the limit as t tends to zero
ttXbttXaX ),( ),(
2005年秋 北航金融系李平 239
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
2005年秋 北航金融系李平 240
An Ito Process for Stock Prices The well-known Geometric Brownian Motion
where is the expected return, is the volatility.
The discrete time equivalent is
SdWSdtdS
tStSS
2005年秋 北航金融系李平 241
Monte Carlo Simulation
We can sample random paths for the stock price by sampling values for
Suppose = 0.14, = 0.20, and t = 0.01, then
SSS 02.00014.0
2005年秋 北航金融系李平 242
Monte Carlo Simulation – One Path
Period
Stock Price at Start of Period
Random 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
2005年秋 北航金融系李平 243
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
2005年秋 北航金融系李平 244
Taylor Series Expansion
A Taylor’s series expansion of f(X, t) gives
22
22
22
2
) (
½
) (
½
tt
ftX
tx
f
Xx
ft
t
fX
x
ff
2005年秋 北航金融系李平 245
Ignoring Terms of Higher Order Than t
t
X
Xx
ft
t
fX
x
ff
tt
fx
x
ff
)(½
22
2
order of
is whichcomponent a has because
becomes this calculus stochastic In
have wecalculusordinary In
2005年秋 北航金融系李平 246
Substituting for X
tbx
ft
t
fX
x
ff
t
tbtaX
dztxbdttxadx
½
order thanhigher of termsignoringThen
+ =
thatso
),(),(
Suppose
222
2
2005年秋 北航金融系李平 247
The 2t Term
tbx
ft
t
fx
x
ff
tt
ttE
E
EE
E
2
1
Hence ignored. be
can and toalproportion is of varianceThe
)( that followsIt
1)(
1)]([)(
0)()1,0( Since
22
2
2
2
2
22
2005年秋 北航金融系李平 248
Taking Limits
Lemma sIto' is This
obtain We
ngSubstituti
limits Taking
22
2
22
2
dWbx
fdtb
x
f
t
fa
x
fdf
dWbdtadX
dtbx
fdt
t
fdX
x
fdf
2005年秋 北航金融系李平 249
Application of Ito’s Lemmato a Stock Price Process
dWSS
fdtS
S
f
t
fS
S
fdf
tSf
WdSdtSSd
½
and of function aFor
is process pricestock The
222
2
2005年秋 北航金融系李平 250
Examples
dWdtdf
Sf
dWfdtfrdf
eSf
TtTr
2
ln 2.
)(
at time maturing
contract afor stock a of price forward The 1.
2
)(
2005年秋 北航金融系李平 251
The lognormal property
Since the logarithm of ST is normal, ST is lognormally distributed
) ,)2
((ln~ln or,
) ,)2
((~lnln
2, Example From
2
0
2
0
TTSS
TTSS
T
T
2005年秋 北航金融系李平
The Lognormal Distribution (cont.)
E S S e
S S e e
TT
TT T
( )
( ) ( )
0
02 2 2
1
var
2005年秋 北航金融系李平 253
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
2005年秋 北航金融系李平 254
Continuously Compounded Return
S S e
T
S
S
T
TT
T
0
0
1
2
or
=
or
2
ln
,
2005年秋 北航金融系李平 255
The Expected Return
The expected value of the stock price is S0eT
The expected return on the stock is
–
)/(ln
2/)/ln(
0
20
SSE
SSE
T
T
2005年秋 北航金融系李平 256
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
2005年秋 北航金融系李平 257
Estimating Volatility from Historical Data
1. Take observations S0, S1, . . . , Sn at intervals of years
2. Calculate the continuously compounded return in each interval as:
3. Calculate the standard deviation, s , of the ui´s
4. The historical volatility estimate is:
uS
Sii
i
ln1
sˆ
2005年秋 北航金融系李平 258
Chapter 12-13
Black-Scholes Model
2005年秋 北航金融系李平 259
1. Black-Scholes Formula
The Concepts Underlying Black-Scholes: The option price and the stock price depend o
n 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
2005年秋 北航金融系李平 260
The Derivation of the Black-Scholes Differential Equation
ƒ
ƒ
½ƒ
ƒ
222
2
WSS
tSSt
SS
f
WStSS
shares :+
derivative :1
of consisting portfolio a upset We
S
f
2005年秋 北航金融系李平 261
The Derivation of the Black-Scholes Differential Equation (cont.)
ƒ
ƒ
bygiven is in time valueitsin change The
ƒ
ƒ
bygiven is portfolio theof valueThe
SS
t
SS
2005年秋 北航金融系李平 262
The Derivation of the Black-Scholes Differential Equation (cont.)
ƒƒ
½ƒ
ƒ
:equation aldifferenti Scholes-Black get the to
equations in these and ƒfor substitute We
Hence rate.
free-risk thebemust portfolio on thereturn The
2
222 r
SS
SrS
t
S
tr
2005年秋 北航金融系李平 263
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)
2005年秋 北航金融系李平 264
Risk-Neutral Valuation
The variable 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
2005年秋 北航金融系李平 265
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
where Ê is the expectation under a risk-neutral probability measure
)(ˆ0 T
rT fEef
P̂
2005年秋 北航金融系李平 266
The Black-Scholes Formulas
TdT
TrKSd
T
TrKSd
dNSdNeKp
dNeKdNScrT
rT
10
2
01
102
210
)2/2()/ln(
)2/2()/ln(
)()(
)()(
where
2005年秋 北航金融系李平 267
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
2005年秋 北航金融系李平 268
2. Dividends
European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-Scholes
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
2005年秋 北航金融系李平 269
European Options on StocksProviding 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 = q2. 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
2005年秋 北航金融系李平 270
Prices for European Options on Stocks Providing Dividend Yield
T
TqrKSd
T
TqrKSd
dNeSdNKep
dNKedNeSc
qTrT
rTqT
)2/2()/ln(
)2/2()/ln( where
)()(
)()(
02
01
102
210
2005年秋 北航金融系李平 271
3. Valuing European Index Options
We can use the formula for an option on a stock paying a dividend yield Set S0 = current index level Set q = average dividend yield expected during
the life of the option
2005年秋 北航金融系李平 272
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
2005年秋 北航金融系李平 273
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
2005年秋 北航金融系李平 274
Formulas for European Currency Options
T
TfrrKSd
T
TfrrKSd
dNeSdNKep
dNKedNeSc
TrrT
rTTr
f
f
)2/2()/ln(
)2/2()/ln( where
)()(
)()(
02
01
102
210
2005年秋 北航金融系李平 275
Alternative Formulas
F S e r r Tf
0 0 ( )Using
Tdd
T
TKFd
dNFdKNep
dKNdNFec
rT
rT
12
20
1
102
210
2/)/ln(
)]()([
)]()([
2005年秋 北航金融系李平 276
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
2005年秋 北航金融系李平 277
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
2005年秋 北航金融系李平 278
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
2005年秋 北航金融系李平 279
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
2005年秋 北航金融系李平 280
Binomial Tree Model
A derivative lasts for time T and is dependent on a futures price
F0d ƒd
F0u ƒuF0
ƒ
2005年秋 北航金融系李平 281
Binomial Tree Model (cont.)
Consider the portfolio that is long futures and short 1 derivative
The portfolio is riskless when
ƒu df
F u F d0 0
F0u F0 – ƒu
F0d F0– ƒd
2005年秋 北航金融系李平 282
Binomial Tree Model (cont.)
Value of the portfolio at time T is F0u –F0 – ƒu
Value of portfolio today is – ƒ Hence
ƒ = – [F0u –F0– ƒu]e-rT
2005年秋 北航金融系李平 283
Binomial Tree Model (cont.)
Substituting for we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
pd
u d
1
2005年秋 北航金融系李平 284
Pricing by Binomial Tree Model
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
pd
u d
1
2005年秋 北航金融系李平 285
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)
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
TdT
TKFd
T
TKFd
dNFdNKep
dNKdNFecrT
rT
10
2
01
102
210
2/2)/ln(
2/2)/ln(
)()(
)()(
where
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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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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%, = 20%,
T = 20 weeks (0.3846y), = 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.
An Example
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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
A
BSlope =
Stock price
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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 non-dividend-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
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 i
iiw
S
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Gamma ()
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.
S
c
S 2
2
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Gamma Addresses Delta Hedging Errors Caused By Curvature
S
CStock price
S’
Callprice
C’C’’
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Making a portfolio gamma neutral
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 , the gamma of a traded option is T, then the gamma of a new portfolio with the num
ber of wT options added is wT T + In order that the new portfolio is gamma neutral,
the number of the options should be wT= - /T
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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
- /T.
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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 ()
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 ()
If is the vega of a portfolio and T is the vega of a traded option, a position of –/T 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.
c
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Rho ()
For currency options there are 2 rhos For a European call
r
c
)(
)(
1
2
dSNTe
dNKTeTr
r
rTr
f
f
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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 exchan
ge one asset for anoth
er Binary options Rainbow options Lookback options Barrier options
Compound options Nonstandard Am
erican 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
K1 K2
S
T
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Asian Options
Payoff related to the average price of the underlying during some period
Payoffs: 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 Japanes
e 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: Pays off a fixed amount Q if ST > K, otherwise pays off 0, Value = e–rT Q N(d2)
Cash-or-nothing put: 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: 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 Allows buyer to sell stocks at the highest obs
erved 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 ‘Out’ options
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Barrier Options (cont.)
barrier level above the asset price ‘Up’ options
barrier level below the asset price ‘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
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
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---Bermu
dan option Early exercise allowed during only part of li
fe Strike price changes over the life Exm: a seven-year warrant issued by a corp
oration 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
Thus the chooser option is a package of A call option with strike price K and maturity T2
A put option with strike price and maturity T1
) max(0,c
),max(),max(
1)(
1)(
12
12
SKe
SKeccpcTTr
TTr
)( 12 TTrKe
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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, S – K
Example: K=50, S=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