2005 年秋北航金融系李平 1 financial derivative reference ： 1. john hull...

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》


Chapter 1

Introduction


Outline

1. Derivatives 2. Forward Contracts 3. Futures Contracts 4. Options 5. Types of Traders 6. Other Derivatives


1. Derivatives

The Nature of DerivativesA derivative is an instrument whose value

depends on the values of other more basic underlying variables.


Examples of Derivatives

Forward Contracts

Futures Contracts

Swaps

Options


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


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


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


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


Foreign Exchange Quotes for GBP on Aug 16, 2001

Bid Offer

Spot 1.4452 1.4456

1-month forward 1.4435 1.4440





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


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?


Profit from a Long Forward Position

Profit

Price of Underlying at Maturity, STK


Profit from a Short Forward Position

Profit

Price of Underlying at Maturity, STK


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.


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)?


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



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


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


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)


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)


Terminology

Strike price (Exercise price) Expiration date (maturity) American/European option


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)


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 ($)


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 ($)



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 ($)



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 ($)


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)


Payoffs from Options

Payoff Payoff

ST STK

K

Payoff Payoff

ST STK

K


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


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


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


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


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.


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


Chapter 2

Mechanics of Futures Markets


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


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


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.


Convergence of Futures to Spot

Time Time

(a) (b)

FuturesPrice

FuturesPrice

Spot Price

Spot Price


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.


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。


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


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)


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

<


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


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


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 )


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


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%


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.


Chapter 3

Determination of Forward and

Futures Prices


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)


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


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


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


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.


Notation

S0: Spot price today

F0: Futures or forward price today

T: Time until delivery date

r: Risk-free interest rate for maturity T


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)


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?


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


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


For Investment Asset Providing Known Cash Income

stocks paying known dividends, coupon bond


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


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


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


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


An Arbitrage Opportunity? (cont.)

The net gain is

$952.39 -$40- $905 =$7.39 When will no arbitrage exist?

$912.39


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


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)


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


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


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


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


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.


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


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.


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 ( )


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


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


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.


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


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


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


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


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


Cost-of-Carry and futures price

For an investment asset

F0 = S0ecT

For a consumption asset

F0 = S0e(c-y)T


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


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


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


Chapter 4

Hedging Strategies Using

Futures


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


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


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


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


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.


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


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).


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


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


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


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

*


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.


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


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)


Chapter 6

Swaps


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


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


---------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


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


Intel and Microsoft (MS) Transform a Liability

Intel MS

LIBOR

5%

LIBOR+0.1%

5.2%


Financial Institution is Involved

F.I.

LIBOR LIBORLIBOR+0.1%

4.985% 5.015%

5.2%Intel MS


Intel and Microsoft (MS) Transform an Asset

Intel MS

LIBOR

5%

LIBOR-0.25%

4.7%


Financial Institution is Involved

Intel F.I. MS

LIBOR LIBOR

4.7%

5.015%4.985%

LIBOR-0.25%


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%


The Swap

AAA BBB

LIBOR

LIBOR+1%

9.95%

10%


The Swap when a Financial Institution is Involved

AAA F.I. BBB10%

LIBOR LIBOR

LIBOR+1%

9.93% 9.97%


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


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


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


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)


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


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).


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


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


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


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

£


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%


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 )


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.


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


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?


Chapter 7

Mechanics of Options Markets


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


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


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



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/£



Specification ofExchange-Traded Options

Expiration date Strike price European or American Call or Put (option class)


Terminology

Moneyness : At-the-money option In-the-money option Out-of-the-money option


Terminology (continued)

Option class (call or put) Option series Intrinsic value Time value


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


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


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


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


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


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


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


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


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


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


Chapter 8

Properties ofStock Option Prices


Outline

The relationship between the option price and the underlying stock price (by arbitrage argument)

Whether an American option should be exercised early


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.


Effect of Variables on Option Pricing

c p C PVariable

S0

KTrD

+ + –+

? ? + ++ + + ++ – + –

–– – +

– + – +


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


Upper bounds for option prices

,

, 00

KPKep

SCSc

rT


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)


Calls: An Arbitrage opportunity?

Suppose that

c(t) = 3 S(t) = 20 T = 1 r = 10% K = 18 D = 0


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.


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)


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 。


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)


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 ?


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


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


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


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


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


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


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


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.


More formal argument

C c S0–Ke –rT>S0–K If it is optimal to exercise early,

C=S0–K


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.


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


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


Chapter 9

Trading Strategies Involving Options


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)


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


Profit

STK

Profit

ST

K

Profit

ST

K

Profit

STK

(a) (b)

(c)

(d)


Bull Spread Using Calls

K1 K2

S

T

Profit

Buy lower & sell higher call


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.


Bull Spread Using Puts

Buy lower & sell higher put

K1 K2

Profit

ST


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


Bear Spread Using Calls

Profit

K1 K2 ST

Buy higher & sell lower call


Bear Spread Using Puts

K1 K2

Profit

ST

Buy higher & sell lower put


Butterfly Spread Using Calls

K1 K3

Profit

STK2

Buy 1 high & 1 low, sell 2 middle


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


Butterfly Spread Using Puts

K1 K3

Profit

STK2


Calendar Spread Using Calls

Profit

ST

K

Buy longer & sell shorter (maturity)


Calendar Spread Using Puts

Profit

ST

K


A Straddle Combination

Profit

STK

Buy a call & a put


Continue

Payoff structure When it is appropriate? Example:

S0=69, expect a significant move in the future, K=70, c=4, p=3


Strip & Strap

Profit

K ST

Strip Strap

Buy 1 call & 2 puts Buy 2 calls & 1 put

K ST

Profit


A Strangle Combination

Buy 1 call with higher strike & 1 put with lower strike

K1 K2

Profit

ST


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.


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?


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?


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.


Chapter 10

Binomial Model


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


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.


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


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


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 )


Generalization

A derivative lasts for time T and is dependent on a stock

S0u ƒu

S0d ƒd

S0

ƒ


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



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



Substituting for we obtain ƒ = [ p ƒu + (1 – p )ƒd ]e–rT

where

pe d

u d

rT


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 )


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


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 )


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


A Two-Step Example

Each time step is 3 months

20

22

18

24.2

19.8

16.2


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


A Put Option Example; K=52

504.1923

60

40

720

484

3220

1.4147

9.4636

A

B

C

D

E

F


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.


An American Put Option Example; K=52

505.0894

60

40

720

484

3220

1.4147

12.0

A

B

C

D

E

F


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


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


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


Model of the Behavior

of Stock Prices

Chapter 11


Categorization of Stochastic Processes

Discrete time; discrete variable Discrete time; continuous variable Continuous time; discrete variable Continuous time; continuous variable


Modeling Stock Prices

We can use any of the four types of stochastic processes to model stock prices


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


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


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


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


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


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.


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


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


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


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



The variable X follows a generalized Wiener process with a drift rate of a and a variance rate of b2 if

dX=adt+bdW



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


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


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 ),( ),(


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


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


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


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


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


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


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


Substituting for X

tbx

ft

t

fX

x

ff

t

tbtaX

dztxbdttxadx

½

order thanhigher of termsignoringThen

+ =

thatso

),(),(

Suppose

222

2


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


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


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


Examples

dWdtdf

Sf

dWfdtfrdf

eSf

TtTr

2

ln 2.

)(

at time maturing

contract afor stock a of price forward The 1.

2

)(


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


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


Continuously Compounded Return

S S e

T

S

S

T

TT

T

0

0

1

2

or

=

or

2

ln

,


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


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


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


Chapter 12-13

Black-Scholes Model


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


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


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


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


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)


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


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


The Black-Scholes Formulas

TdT

TrKSd

T

TrKSd

dNSdNeKp

dNeKdNScrT

rT

10

2

01

102

210

)2/2()/ln(

)2/2()/ln(

)()(

)()(

where


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


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


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


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


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


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


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


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


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(

)]()([

)]()([


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


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


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


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


Binomial Tree Model

A derivative lasts for time T and is dependent on a futures price

F0d ƒd

F0u ƒuF0

ƒ


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



Value of the portfolio at time T is F0u –F0 – ƒu

Value of portfolio today is – ƒ Hence

ƒ = – [F0u –F0– ƒu]e-rT



Substituting for we obtain

ƒ = [ p ƒu + (1 – p )ƒd ]e–rT

where

pd

u d

1


Pricing by Binomial Tree Model

ƒ = [ p ƒu + (1 – p )ƒd ]e–rT

where

pd

u d

1


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


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


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


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


Chapter 14

The Greek Letters


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


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


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.


Delta ()

Delta is the rate of change of the option price with respect to the underlying

Option

price

A

BSlope =

Stock price


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


Delta Hedging

This involves maintaining a delta neutral portfolio--- =0

In Black-Scholes model, -1: option

+ : shares set up a delta neutral portfolio


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


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


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


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


Gamma Addresses Delta Hedging Errors Caused By Curvature

S

CStock price

S’

Callprice

C’C’’


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.


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


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.


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]).


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.


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.


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


Rho ()

For currency options there are 2 rhos For a European call

r

c

)(

)(

1

2

dSNTe

dNKTeTr

r

rTr

f

f


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


Chapter 19

Exotic Options


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


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


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


Payoff from a long range forward

Profit

K1 K2

S

T


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)


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


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)


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)


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)


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)


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


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


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


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


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


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


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.


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


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


Forward Start Options

Option starting at a future time, used in employee incentive schemes

Usually be at the money at the time they start


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)


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

2005 年秋北航金融系李平 1 financial derivative reference ： 1. john hull...

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