lecture option and derivative

8/13/2019 Lecture Option and Derivative

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Fundamentals of Futures and Options Markets, 7th Ed, Ch 1, Copyright John C. Hull 2010

IntroductionChapter 1

1

Remarks

All information is available on Moodle

Lectures & Tutorials:

I present the theory, please prepare! We discuss the end-of-chapter questions &

problems, please prepare!

Textbook:

Hull, 2011, Fundamentals of Futures andOptions Markets, 7th edition, Pearson/PrenticeHall

Exam

Use one page A4 with formulas & non-progr.calculator

2

Topics

Derivatives: Introduction

Hedging, arbitrage, speculation

Futures/Forwards

Markets, history, how to use/price?

Options

Markets, hedging and pricing issues

Binomial trees vs. Black/Scholes model

Greeks: option sensitivities

Option strategies, other issues3 Fundamentals of Futures and Options Markets, 7th Ed, Ch 1, Copyright John C. Hull 2010

The Nature of Derivatives

A derivative is an instrument whose valuedepends on the values of other morebasic underlying variables

4


Examples of Derivatives

Futures Contracts

Forward Contracts

Swaps Options

5 Fundamentals of Futures and Options Markets, 7th Ed, Ch 1, Copyright John C. Hull 2010

Ways Derivatives are Used

To hedge risks

To speculate (take a view on thefuture direction of the market)

To lock in an arbitrage profit

To change the nature of a liability

To change the nature of an investmentwithout incurring the costs of sellingone portfolio and buying another

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Futures Contracts

A futures contract is an agreement to

buy or sell an asset at a certain time inthe future for a certain price

By contrast in a spot contract there isan agreement to buy or sell the assetimmediately (or within a very shortperiod of time)


Exchanges Trading Futures

CBOT and CME (now CME Group)

Intercontinental Exchange

NYSE Euronext

Eurex

BM&FBovespa (Sao Paulo, Brazil)

and many more (see list at end of book)

8


Futures Price

The futures prices for a particular contractis the price at which you agree to buy orsell

It is determined by supply and demand inthe same way as a spot price


Electronic Trading

Traditionally futures contracts have beentraded using the open outcry systemwhere traders physically meet on the floorof the exchange

Increasingly this is being replaced byelectronic trading where a computermatches buyers and sellers

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Examples of Futures Contracts

Agreement to:

buy 100 oz. of gold @ US$1050/oz. in

December sell 62,500 @ 1.5500 US$/ in

March

sell 1,000 bbl. of oil @ US$75/bbl. inApril


Terminology

The party that has agreed to buy

has a long position The party that has agreed to sell

has a short position

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Example

January: an investor enters into a long

futures contract to buy 100 oz of gold @$1050 in April

April: the price of gold $1065 per oz

What is the investors profit?


Over-the Counter Markets

The over-the counter market is an

important alternative to exchanges It is a telephone and computer-linked

network of dealers who do not physicallymeet

Trades are usually between financialinstitutions, corporate treasurers, and fundmanagers

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Size of OTC and Exchange Markets(Figure 1.2, Page 4)

Source: Bank for International Settlements. Chart shows total principal amounts

for OTC market and value of underlying assets for exchange market


Forward Contracts

Forward contracts are similar to futuresexcept that they trade in the over-the-counter market

Forward contracts are popular oncurrencies and interest rates

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Options

A call option is an option to buy acertain asset by a certain date for a

certain price (the strike price)A put option is an option to sell a

certain asset by a certain date for acertain price (the strike price)


American vs European Options

An American option can be exercised atany time during its life

A European option can be exercised onlyat maturity

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Google Option Prices (July 17,

2009; Stock Price=430.25); See page 6

Calls PutsStrikeprice Aug Sept Dec Aug Sept Dec

($) 2009 2009 2009 2009 2009 2009

380 51.55 54.60 65.00 1.52 4.40 15.00

400 34.10 38.30 51.25 4.05 8.30 21.15

420 19.60 24.80 39.05 9.55 14.70 28.70

440 9.25 14.45 28.75 19.20 24.25 38.35

460 3.55 7.45 20.40 33.50 37.20 49.90

480 1.12 3.40 13.75 51.10 53.10 63.40


Exchanges Trading Options

Chicago Board Options Exchange

International Securities Exchange

NYSE Euronext

Eurex (Europe)

and many more (see list at end of book)

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Options vs Futures/Forwards

A futures/forward contract gives the holderthe obligation to buy or sell at a certainprice

An option gives the holder the right to buyor sell at a certain price


Hedging Examples (Example 1.1 and 1.2,page 11)

A US company will pay 10 million for importsfrom Britain in 3 months and decides tohedge using a long position in a forwardcontract

An investor owns 1,000 Microsoft sharescurrently worth $28 per share. A two-monthput with a strike price of $27.50 costs $1. Theinvestor decides to hedge by buying 10contracts

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Value of Microsoft Shares with

and without Hedging (Fig 1.4, page 12)


1. Gold: An Arbitrage

Opportunity?

Suppose that:

The spot price of gold is US$1000

The quoted 1-year futures price of goldis US$1100

The 1-year US$ interest rate is 5% perannum

No income or storage costs for gold

Is there an arbitrage opportunity?

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2. Gold: Another Arbitrage

Opportunity?

Suppose that:

The spot price of gold is US$1000 The quoted 1-year futures price of

gold is US$990

The 1-year US$ interest rate is 5%per annum




The Futures Price of Gold

If the spot price of gold is S& the futures price isfor a contract deliverable in Tyears is F, then

F = S (1+r )T

where ris the 1-year (domestic currency) risk-free rate of interest.

In our examples, S=1000, T=1, and r=0.05 sothat

F = 1000(1+0.05) = 1050

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1. Oil: An Arbitrage Opportunity?

Suppose that:

The spot price of oil is US$70

The quoted 1-year futures price ofoil is US$80


The storage costs of oil are 2% perannum



2. Oil: Another Arbitrage

Opportunity?

Suppose that:

The spot price of oil is US$70

The quoted 1-year futures price ofoil is US$65


The storage costs of oil are 2% perannum


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Mechanics of Futures

Markets

Chapter 2


Futures Contracts

Available on a wide range of underlyings

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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Margins

A margin is cash or marketable securities

deposited by an investor with his or herbroker

The balance in the margin account isadjusted to reflect daily settlement

Margins minimize the possibility of a lossthrough a default on a contract


Example of a Futures Trade

An investor takes a long position in 2

December gold futures contracts onJune 5 contract size is 100 oz.

futures price is US$900

margin requirement is US$2,000/contract(US$4,000 in total)

maintenance margin is US$1,500/contract(US$3,000 in total)

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A Possible OutcomeTable 2.1, Page 27

Dai ly Cumulat ive Margin

Futures Gain Gain Account Margin

Price (Loss) (Loss) Balance Call

Day (US$) (US$) (US$) (US$) (US$)

900.00 4,000

5-Jun 897.00 (600) (600) 3,400 0. . . . . .. . . . . .. . . . . .

13-Jun 893.30 (420) (1,340) 2,660 1,340. . . . . .. . . . .. . . . . .

19-Jun 887.00 (1,140) (2,600) 2,740 1,260. . . . . .. . . . . .. . . . . .

26-Jun 892.30 260 (1,540) 5,060 0

+

= 4,000+

= 4,000


Other Key Points About Futures

They are settled daily

Closing out a futures position involvesentering into an offsetting trade

Most contracts are closed out beforematurity

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Collateralization in OTC Markets

It is becoming increasingly common forcontracts to be collateralized in OTCmarkets

Counterparties then post margin with eachother to reflect changes in the value of thecontract

Regulators are now insisting thatclearinghouses (similar to those used forfutures) be used for some OTC contracts


Delivery

If a futures contract is not closed out before maturity, it isusually settled by delivering the assets underlying thecontract. When there are alternatives about what isdelivered, where it is delivered, and when it is delivered,the party with the short position chooses.

A few contracts (for example, those on stock indicesand Eurodollars) are settled in cash

When there is cash settlement contracts are traded untila predetermined time. All are then declared to be closedout.

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Foreign Exchange Quotes

Futures exchange rates are quoted as thenumber of USD per unit of the foreign currency

Forward exchange rates are quoted in the sameway as spot exchange rates. This means thatGBP, EUR, AUD, and NZD are USD per unit offoreign currency. Other currencies (e.g., CADand JPY) are quoted as units of the foreigncurrency per USD.


Mechanics of Options

MarketsChapter 9

44


Types of Options

A call is an option to buy

A put is an option to sell

A European option can be exercised onlyat the end of its life

An American option can be exercised atany time


Option Positions

Long call

Long put

Short call

Short put

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Long Call(Figure 9.1, Page 207)

Profit from buying one European call option: option price

= $5, strike price = $100.

30

20

10

0-5

70 80 90 100

110 120 130

Profit ($)

Terminalstock price ($)


Short Call(Figure 9.3, page 208)

Profit from writing one European call option: option price

= $5, strike price = $100

-30

-20

-10

05

70 80 90 100

110 120 130

Profit ($)


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Long Put(Figure 9.2, page 208)

Profit from buying a European put option: option price =

$7, strike price = $70

30

20

10

0

-770605040 80 90 100

Profit ($)



Short Put(Figure 9.4, page 209)

Profit from writing a European put option: option price =

$7, strike price = $70

-30

-20

-10

7

070

605040

80 90 100

Profit ($)Terminal

stock price ($)

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Payoffs from Options

What is the Option Position in Each Case?

K= Strike price, ST= Price of asset at maturity

Payoff Payoff

ST STK

K

Payoff Payoff

ST STK

K


Assets Underlying

Exchange-Traded OptionsPage 210-211

Stocks

Foreign Currency

Stock Indices

Futures

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Specification of

Exchange-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

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Executive Stock Options

Option issued by a company to executives

When the option is exercised the companyissues more stock

Usually at-the-money when issued

55 Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright John C. Hull 2010

Hedging Strategies Using

FuturesChapter 3

56

Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright John C. Hull 2010

Long & Short Hedges

A long futures hedge is appropriate whenyou know you will purchase an asset inthe future and want to lock in the price

A short futures hedge is appropriatewhen you know you will sell an asset inthe future & want to lock in the price


Arguments in Favor of Hedging

Companies should focus on the mainbusiness they are in and take steps tominimize risks arising from interestrates, exchange rates, and other marketvariables

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Arguments against Hedging

Shareholders are usually well diversifiedand can make their own hedging decisions

It may increase risk to hedge whencompetitors do not

Explaining a situation where there is a losson the hedge and a gain on the underlyingcan be difficult


Convergence of Futures to Spot(Hedge initiated at time t1 and closed out at time t2)

Time

SpotPrice

Futures

Price

t1 t2

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Basis Risk

Basis is the difference between

spot & futures Basis risk arises because of the

uncertainty about the basiswhen the hedge is closed out


Long Hedge

Suppose that

F1 : Initial Futures PriceF2 : Final Futures Price

S2 : Final Asset Price

You hedge the future purchase of anasset by entering into a long futurescontract

Cost of Asset=S2(F2F1) = F1 + Basis

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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 byentering into a short futures contract

Price Realized=S2+ (F1F2) = F1 + Basis


Choice of Contract

Choose a delivery month that is as closeas possible to, but later than, the end ofthe life of the hedge

When there is no futures contract on theasset being hedged, choose the contractwhose futures price is most highlycorrelated with the asset price. There arethen 2 components to basis

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Optimal Hedge Ratio

Proportion of the exposure that should optimally behedged is

where

S is the standard deviation of S, the change in thespot price during the hedging period,

F is the standard deviation of F, the change in thefutures price during the hedging period

is the coefficient of correlation between Sand F.

F

Sh

=


Hedging Using Index Futures(Page 63)

To hedge the risk in a portfolio thenumber of contracts that should beshorted is

where VA is the current value of theportfolio, is its beta, and VF is thecurrent value of one futures (=futuresprice times contract size)

F

A

VV

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Reasons for Hedging an Equity

Portfolio

Desire to be out of the market for a short

period of time. (Hedging may be cheaperthan selling the portfolio and buying itback.)

Desire to hedge systematic risk


Example

Futures price of S&P 500 is 1,000

Size of portfolio is $5 millionBeta of portfolio is 1.5

One contract is on $250 times the index

What position in futures contracts on theS&P 500 is necessary to hedge theportfolio?

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Changing Beta

What position is necessary to reduce thebeta of the portfolio to 0.75?

What position is necessary to increase thebeta of the portfolio to 2.0?

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Stock Picking

If you think you can pick stocks that willoutperform the market, futures contractcan be used to hedge the market risk

If you are right, you will make moneywhether the market goes up or down

Fundamentals of Futures and Options Markets, 7th Ed, Ch3, Copyright John C. Hull 2010 70


Rolling The Hedge Forward

We can use a series of futurescontracts to increase the life of a

hedge Each time we switch from 1 futures

contract to another we incur a type ofbasis risk


Determination of Forward

and Futures Prices

Chapter 5

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Consumption vs Investment Assets

Investment assets are assets held by

significant numbers of people purely forinvestment purposes (Examples: gold,silver)

Consumption assets are assets heldprimarily for consumption (Examples:copper, oil)


Short Selling (Page 104-105)

Short selling involves selling

securities you do not own

Your broker borrows thesecurities from another client andsells them in the market in theusual way

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Short Selling(continued)

At some stage you must buythe securities back so theycan be replaced in theaccount of the client

You must pay dividends andother benefits the owner ofthe securities receives


Notation

S0: Spot price today

F0: Futures or forward price today

T: Time until delivery date

r: Risk-free interest rate formaturity T

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1. Gold: An Arbitrage

Opportunity?

Suppose that:


The quoted 1-year futures price of goldis US$1100

The 1-year US$ interest rate is 5% perannum




2. Gold: Another Arbitrage

Opportunity?

Suppose that:


The quoted 1-year futures price ofgold is US$990




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The Futures Price of Gold

If the spot price of gold is S& the futures price isfor a contract deliverable in Tyears is F, then

F = S (1+r )T

where ris the 1-year (domestic currency) risk-free rate of interest.

In our examples, S=1000, T=1, and r=0.05 sothat

F = 1000(1+0.05) = 1050


When Interest Rates are Measured

with Continuous Compounding

F0 = S0erT

This equation relates the forward priceand the spot price for any investmentasset that provides no income and hasno storage costs

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When an Investment Asset

Provides a Known Dollar Income

(page 110, equation 5.2)

F0 = (S0I )erT

whereI is the present value of theincome during life of forward contract


When an Investment Asset

Provides a Known Yield(Page 111, equation 5.3)

F0 = S0 e(rq )T

where q is the average yield during thelife of the contract (expressed withcontinuous compounding)

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Valuing a Forward ContractPage 112

Suppose that

Kis delivery price in a forward contract &

F0 is forward price that would apply to thecontract today

The value of a long forward contract, , is = (F0 K)e

rT

Similarly, the value of a short forward contractis

(KF0 )erT


Stock Index (Page 115)

Can be viewed as an investment assetpaying a dividend yield

The futures price and spot price

relationship is thereforeF0 = S0 e

(rq )T

where q is the dividend yield on theportfolio represented by the indexduring life of contract

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Stock Index(continued)

For the formula to be true it is

important that the index represent aninvestment asset

In other words, changes in the indexmust correspond to changes in thevalue of a tradable portfolio

The Nikkei index viewed as a dollarnumber does not represent aninvestment asset


Index Arbitrage

When F0 > S0e(r-q)Tan arbitrageur buys the

stocks underlying the index and sellsfutures

When F0 < S0e(r-q)Tan arbitrageur buys

futures and shorts or sells the stocksunderlying the index

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Index Arbitrage(continued)

Index arbitrage involves simultaneous tradesin futures and many different stocks

Very often a computer is used to generate thetrades


A foreign currency is analogous to a securityproviding a dividend yield

The continuous dividend yield is the foreignrisk-free interest rate

It follows that if rf is the foreign risk-free interestrate

Futures and Forwards on

Currencies (Page 116-120)

F S er r Tf

0 0=( )

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Futures on Consumption Assets(Page 122)

F0 S0 e(r+u )T

where u is the storage cost per unit

time as a percent of the asset value.Alternatively,

F0 (S0+U )erT

where Uis the present value of thestorage costs.

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lecture option and derivative

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