0 Introduction Chapter 1. 1 DERIVATIVES : FORWARDS FUTURES OPTIONS SWAPS

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Introduction

Chapter 1

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DERIVATIVES:

FORWARDS

FUTURES

OPTIONS

SWAPS

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DERIVATIVES ARE CONTRACTS:

Two parties

Agreement, or Contract

Underlying security

Contract termination date

4

The Difference between Derivatives:

The contract determines the rights and/or obligations of

thetwo parties

in relation to the sale/purchase

of the underlying asset.

5

The Nature of Derivatives

A derivative is a contract whose value depends on the values of other more basic underlying variables

In particular, it depends on the market price of the so called:

underlying asset.

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Underlying assets:StocksInterest bearing securities: BondsForeign currencies: Euro, GBP, AUD,…Metals: Gold, Silver…Energy commodities: Crude oil, Natural

gas, Gasoline, heating oil…

Agricultural commodities: Wheat, corn, rice, grain feed, soy beans, pork bellies…

Stock indexes

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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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Types of risk:Price riskCredit risk

Operational riskCompletion risk

Human riskRegulatory risk

Tax risk

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IN THIS CLASS WE WILL ONLY ANALYZE THE

RISK ASSOCIATED WITH THESPOT MARKET PRICE

OFTHE UNDERLYING ASSET

10t

Probability distribution

St

T time

ST

PRICE RISK: At time t, the asset’s price at time T is not

known.

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FORWARDS

A FORWARDIS A CONTRACT IN WHICH:

ONE PARTY COMMITS TO BUY AND THE OTHER PARTY COMMITS TO

SELL A SPECIFIED AMOUNTOF AN AGREED UPON COMMODITY

OF A SPESIFIC QUALITYFOR A PREDETERMINED PRICE

ON AN AGREED UPON FUTURE DATE AT A GIVEN LOCATION

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BUY = OPEN A LONG POSITIONSELL = OPEN A SHORT POSITION

t

Buy or sell a forward T Tim

e

Delivery and payment

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Forward Price• The forward price for a contract

is the delivery price that would be applicable to the contract if were negotiated today (i.e., it is the delivery price that would make the contract worth exactly zero)

• The forward price may be different for contracts of different maturities

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Foreign Exchange Quotes for USD/GBP on DEC 19, 2007

Bid Ask

Spot 1.99480 1.99720

1-month forward 1.99283 1.99530

3-month forward 1.99127 1.99376

6-month forward 198353 1.98650

12-month forward 1.96990 1.972663

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Profit from aLong 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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A FUTURESis

A STANDARDIZED FORWARD TRADED ON AN ORGANIZED EXCHANGE.

STANDARDIZATION THE COMMODITY

THE QUANTITY

THE QUALITY

PRICE QUOTES

DELIVERY DATES

DELIVERY PROCEDURES

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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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AN OPTION IS A CONTRACT IN WHICH ONE PARTY HAS THE RIGHT,

BUT NOT THE OBLIGATION, TO BUY OR SELL A SPECIFIED AMOUNT OF AN AGREED UPON COMMODITY

FOR A PREDETERMINED PRICE BEFORE OR ON A SPECIFIC DATE IN THE FUTURE. THE OTHER PARTY HAS

THE OBLIGATION TO DO WHAT THE FIRST PARTY WISHES TO DO. THE FIRST PARTY, HOWEVER, MAY

CHOOSE NOT TO EXERCISE ITS RIGHT AND LET THE OPTION EXPIRE WORTHLESS.

A CALL = A RIGHT TO BUY THE UNDERLYING ASSET

A PUT = A RIGHT TO SELL THE UNDERLYING ASSET

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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 ($)


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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 ($)


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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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A SWAPIS A CONTRACT IN WHICH THE TWO PARTIES COMMIT TO EXCHANGE A

SERIES OF CASH FLOWS. THE CASH FLOWS ARE BASED ON AN AGREED UPON PRINCIPAL AMOUNT. NORMALLY, ONLY

THE NET FLOW EXCHANGES HANDS.Principal amount = EUR100,000,000; semiannual payments. Tenure: three

years.

Party A Party B7%

6-months LIBOR

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EXAMPLE: A RISK MANAGEMENT SWAP

BONDS MARKET

BANK

12%

10%

FIRM A BORROWS AT A FIXED RATE FOR 5 YEARS

FL1

SWAP DEALER A

FL2

LOAN

LOAN

FL1 = 6-MONTH BANK RATE.

FL2 = 6-MONTH LIBOR.

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THE BANK’S CASH FLOW:

12% - FLOATING1 + FLOATING2 – 10% = 2% + SPREAD

Where the

SPREAD = FLOATING2 - FLOATING1

RESULTS THE BANK EXCHANGES THE RISK ASSOCIATED WITH THE DIFFERENCE BETWEEN FLOATING1

and 12% WITH THE RISK ASSOCIATED WITH THE

SPREAD = FLOATING2 - FLOATING1.

The bank may decide to swap the SPREAD for fixed, risk-free cash flows.

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EXAMPLE: A RISK MANAGEMENT SWAP

BOND MARKET

BANK

12%

10% FL1

SWAP DEALER A

FL2

SWAP DEALER BFL1

FL2

FIRM A

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THE BANK’S CASH FLOW:

12% - FL1 + FL2 – 10% + (FL1 - FL2 ) = 2%

RESULTS THE BANK EXCHANGES THE RISK ASSOCIATED WITH

THE SPREAD = FL2 - FL1

WITH A FIXED RATE OF 2%.

THIS RATE IS A FIXED RATE!

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WHY TRADE DERIVATIVES?

THE FUNDAMENTALREASON FOR TRADING

DERIVATIVES IS TO HEDGE: THE PRICE RISK

Exhibited by the Underlying commodity’s spot price volatility

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PRICE RISK IS THE

VOLATILITYASSOCIATED WITH THE COMMODITY’S

PRICE IN THE CASH MARKET

REMEMBER THAT THE CASH MARKET IS WHERE FIRMS DO THEIR BUSINESS. I.E., BUY

AND SELL THE COMMODITY.

ZERO PRICE VOLATILITYNO DERIVATIVES!!!!

31t

Probability distribution

St

T time

ST

PRICE RISK: At time t, the asset’s price at time T is not

known.

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Derivatives Traders

• Speculators

• Hedgers

•Arbitrageurs

Some of the large trading losses in derivatives occurred because

individuals who had a mandate to hedge risks switched to being

speculators

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THE ECONOMIC PURPOSES OF DERIVATIVE MARKETS

HEDGINGPRICE DISCOVERYSAVING

HEDGING IS THE ACTIVITY OF MANAGING PRICE RISK EXPOSURE

PRICE DISCOVERY IS THE REVEALING OF INFORMSTION ABOUT THE FUTURE CASH MARKET PRICE OF A PRODUCT.

SAVING IS THE COST SAVING ASSOCIATED WITH SWAPING CASH FLOWS

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A Review of Some Financial Economics Principles

Arbitrage: A market situation whereby an investor can make a profit with:

no equity and no risk.

Efficiency: A market is said to be efficient if prices are such that there exist no arbitrage opportunities.

Alternatively, a market is said to be inefficient if prices present arbitrage opportunities for investors in this market.

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Valuation: The current market value (price) of any project or investment is the net present value of all the future expected cash flows from the project.

One-Price Law: Any two projects whose cash flows are equal in every possible state of the world have the same market value.

Domination: Let two projects have equal cash flows in all possible states of the world but one. The project with the higher cash flow in that particular state of the world has a higher current market value and thus, is said to dominate the other project.

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The Holding Period Rate of Return (HPRR):Buy shares of a stock on date t and sellthem later on date T. While holding theshares, the stock has paid a cash dividendIn the amount of $D/share.The Holding Period Rate of Return HPRR is:

t

ttTTtT S

SDSR

tTannual RtT

365R

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Example:

St = $50/share

ST = $51.5/share

DT-t = $1/share

T = t + 73days.

25.]05[.73

365R

05.50

50- 1 51.5R

annual

73days

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A proof by contradiction:

is a method of proving that an assumption, or a set of assumptions, is incorrect by showing that the implication of the assumptions contradicts these very same assumptions.

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Risk-Free Asset: is a security of investment whose return carries no risk. Thus, the return on this security is known and guaranteed in advance.

Risk-Free Borrowing And Landing: By purchasing the risk-free asset, investors lend their capital and by selling the risk-free asset, investors borrow capita at the risk-free rate.

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The One-Price Law:There exists only one risk-free rate in an efficient economy.

Proof: By contradiction. Suppose two risk-freerates exist in a market and R > r. Since both arefree of risk, ALL investors will try to borrow at rand invest the money borrowed in R, thusassuring themselves the difference. BUT, theexcess demand for borrowing at r and excesssupply of lending (investing) at R will changethem. Supply = demand only when R = r.

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Compounded Interest (p. 76) Any principal amount, P, invested at

an annual interest rate, R, compounded annually, for n years would grow to:

An = P(1 + R)n.

If compounded Quarterly:

An = P(1 +R/4)4n.

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In general: Invest P dollars in an account which paysAn annual interest rate R with mCompounding periods every year. The rate in every period is R/m.The number of compounding periods is

nm.

Thus, P grows to:

An = P(1 +R/m)mn.

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An = P(1 +R/m)mn.

Monthly compounding becomes:

An = P(1 +R/12)12n

and daily compounding yields:

An = P(1 +R/365)365n.

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EXAMPLES:

n =10 years; R =12%; P = $1001.Simple compounding, m = 1, yields:

A10 = $100(1+ .12)10 = $310.5848

2.Monthly compounding, m = 12, yields:

A10 = $100(1 + .12/12)120 = $330.0387

3.Daily compounding, m = 365, yields:

A10 = $100(1 + .12/365)3,650 = $331.9462.

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DISCOUNTING

The Present Value of a future income, FVT,

on date T hence, is given byDISCOUNTING:

TT

t R][1

FVPV

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DISCOUNTINGLet cj, j = 1,2,3,….,; be a sequence of

payments paid out m times a year over thenext n years. Let R be the annual rateDuring these years. DISCOUNTING this cash flow yields the

Present Value:

mn

1j j

jt

]mR

[1

cPV

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CONTINUOUS COMPOUNDINGIn the early 1970s, banks came up withThe following economic reasoning: Since The bank has depositors money all the time, this money should be working for the depositor all the time! This idea, of course, leads to the concept of

continuous compounding. We want to apply this idea to the formula:

.m

R1PA

mn

n

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CONTINUOUS COMPOUNDINGAs m increases the time span of

every compounding period diminishes

Compounding

m Time span

Yearly 1 1 yearDaily 365 1 day

Hourly 8760 1 hourEvery

second3,153,60

0One second

Continuously

∞ Infinitesimally small

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CONTINUOUS COMPOUNDINGThis reasoning implies that in order to impose the concept of continuous time on the above compounding expression, we need to solve:

}m

R1{PLimitA

mn

mn

This expression may be rewritten as:

Rnn PeA

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Recall that the number “e” is:

}x

11{Limite

x

x

X e

1 2

100 2.70481382

10,000 2.71814592

1,000,000 2.71828046

In the limit 2.718281828…..

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Recall that in our example:n = 10 years.R = 12%P=$100. So, P = $100 invested at a 12% annual

rate, continuously compounded for ten years will grow to:

0117.332$

$100e

PeA(.12)(10)

Rnn

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Continuous compounding yields the

highest return:

Compounding m Factor

Simple 1 3.105848208

Quarterly 43.262037792

Monthly 12 3.300386895

Daily 365 3.319462164

Continuously ∞3.320116923

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Rn -n

n

Rnn

eAP

n, and R ,Agiven Thus,

PeA

Continuous Discounting

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Rt -Tt

T

eCF PV

: tme,present ti for the

discountedly continuous becan

,CF general,In

Continuous Discounting

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Recall that in our example:P = $100; n = 10 years and R = 12% Thus, $100 invested at an annual rate of 12% , continuously compounded for ten years will grow to: $332.0117.

Therefore, we can write the continuously discounted value of $320.0117:

.100$

$332.0117e

eAA(.12)(10) -

-Rnn0

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Equivalent Interest Rates (p.77)

Rm = The annual rate with m compounding periods

every year.

mnmn ]

m

RP[1A

57


rc = The annual rate with continuous compounding

nrn

cPeB

58


Rm = The annual rate with m

compounding periods every year.rc = The annual rate with continuous

compounding.

Definition: Rm and rc are said to be equivalent

if:

nn AB

59


1]m[eR

]m

Rmln[1r

]m

RP[1Pe

AB

mrm

mc

mnmnr

nn

c

c

60

Risk-free lending and borrowingTreasury bills: are zero-coupon bonds, or

pure discount bonds, issued by the Treasury. A T-bill is a promissory paper which promises its

holder the payment of the bond’s Face Value (Par- Value) on a specific future maturity date.

The purchase of a T-bill is, therefore, an investment that pays no cash flow between the purchase date and the bill’s maturity. Hence, its current market price is the NPV of the bill’s Face Value:

Pt = NPV{the T-bill Face-Value}

61

Risk-free lending and borrowing

Risk-Free Asset: is a security whose return is a known constant and it carries no risk.

T-bills are risk-free LENDING assets. Investors lend money to the Government by purchasing T-bills (and other Treasury notes and bonds)

We will assume that investors also can borrow money at the risk-free rate.

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Risk-free lending and borrowing

LENDING:By purchasing the risk-free asset,

investors lend capital. BORROWING:

By selling the risk-free asset, investors borrow capital.

Both activities are at the risk-free rate.

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We are now ready to calculate the current value of a T-Bill.

Pt = NPV{the T-bill Face-Value}.

Thus:the current time, t, T-bill price, Pt , which pays FV upon its maturity on date T, is:

Pt = [FV]e-r(T-t)

r is the risk-free rate in the economy.

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EXAMPLE: Consider a T-bill that promises its holder FV = $1,000 when it matures in 276 days, with a risk-free yield of 5%: Inputs for the formula:

FV = $1,000; r = .05; T-t = 276/365yrs

Pt = [FV]e-r(T-t)

Pt = [$1,000]e-(.05)276/365

Pt = $962.90.

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EXAMPLE: Calculate the yield-to -maturity of a bond which sells for $965 and matures in 100 days, with FV = $1,000.

Pt = $965; FV = $1,000; T-t= 100/365yrs.Solving for r:

Pt = [FV]e-r(T-t)

13%]965

1,000ln[

365100

1 r

]P

FVln[

t-T

1 r

t

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SHORT SELLING STOCKS (p. 97)An Investor may call a broker and ask to “sell a

particular stock short.”This means that the investor does not own shares

of the stock, but wishes to sell it anyway. The investor speculates that the stock’s share price will fall and money will be made upon buying the shares back at a

lower price. Alas, the investor does not own shares of the stock. The broker will lend the investor shares from the broker’s or a client’s account and sell it

in the investor’s name. The investor’s obligation is to hand over the shares some time in the future, or upon the broker’s request.

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SHORT SELLING STOCKSOther conditions:The proceeds from the short sale cannot be

used by the short seller. Instead, they are deposited in an escrow account in the investor’s name until the investor makes

good on the promise to bring the shares back. Moreover, the investor must deposit an additional amount of at least 50% of the short sale’s proceeds in the escrow account. This additional amount guarantees that there

is enough capital to buy back the borrowed shares and hand them over back to the broker, in case the shares price increases.

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SHORT SELLING STOCKSThere are more details associated with short

selling stocks. For example, if the stock pays dividend, the short seller must pay the dividend to the broker. Moreover, the short seller does not gain interest on the amount deposited in the escrow account, etc.

We will use stock short sales in many of strategies associated with options trading. In all of these strategies, we will assume that no cash flow occurs from the time the strategy is opened with the stock short sale until the time the strategy terminates and the stock is repurchased.

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SHORT SELLING STOCKSIn terms of cash flows per share: St is the cash flow/share from

selling the stock short thereby, opening a SHORT POSITION on date t.

-ST is the cash flow from purchasing

the stock back on date T (and delivering it to the lender thereby, closing the SHORT POSITION.)

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0 Introduction Chapter 1. 1 DERIVATIVES : FORWARDS FUTURES OPTIONS SWAPS