1 options on stock indices and currencies chapter 15

1

Options onStock Indices and Currencies

Chapter 15

2

The cash marketStock indexes are not traded per se.

Several mutual funds trade portfolio that are the index portfolio, or a portfolio that closely mimic the index.

The market values of all stock indexes are calculated virtually continuously.

3

STOCK INDEXES (INDICES)

A STOCK INDEX IS A SINGLE NUMBER BASED ON INFORMATION ASSOCIATED WITH A SET OF

STOCK PRICES AND QUANTITIES.

A STOCK INDEX IS SOME KIND OF AN AVERAGE OF THE PRICES AND THE QUANTITIES OF THE STOCKS THAT ARE

INCLUDED IN A GIVE PORTFOLIO.

THE MOST USED INDEXES ARE

A SIMPLE PRICE AVERAGE

AND

A VALUE WEIGHTED AVERAGE.

4

STOCK INDEXES - THE CASH MARKET

AVERAGE PRICE INDEXES: DJIA, MMI:

N = The number of stocks in the portfolio.

Pi = The i-th stock market price

D = Divisor

Initially D = N and the index is set at some level.

To ensure continuity, the divisor is adjusted over time.

N.1,..., = i ;N

P = I i

5

EXAMPLES OF INDEX ADJUSMENTS

STOCK SPLITS: 2 for 1.

1.

2.

1. (30 + 40 + 50 + 60 + 20) /5 = 40

I = 40 and D = 5.

2. (30 + 20 + 50 + 60 + 20)/D = 40

The new divisor is D = 4.5

(P P P D I1 2 N 1 1 ... ) /

(P P P D I1 2 N 2 1 1

2... ) /

6

CHANGE OF STOCKS IN THE INDEX

1.

2.

1. (32 + 18 + 55 + 56 + 19)/4.5 = 40

I = 40and D =4.5.

2. (32 + 118 + 55 + 56 + 19)/D = 40

The new divisor is D = 7.00

(P P ABC) P D I1 2 N 1 1 ( ... ) /

(P P XYZ) P D I1 2 N 2 1 ( ... ) /

7

STOCK #4 DISTRIBUTED 66 2/3% STOCK DIVIDEND

(22 + 103 + 44 + 58 + 25)/7.00 = 36

D = 7.00.

Next, (22 + 103 + 44 + 34.8 + 25)/D = 36

The new divisor is D = 6.355.

STOCK # 2 SPLIT 3 for 1.

(31 + 111 + 54 + 35 + 23)/6.355 = 39.9685

(31 + 37 + 54 + 35 + 23)/D = 39.9685

The new Divisor is D = 4.5035.

8

ADDITIONAL STOCKS

1.

2.

1. (30 + 39 + 55 + 33 + 21)/4.5035= 39.5248

2. (30 + 39 + 55 + 33 + 21 + 35)/D = 39.5248

D = 5.389

(P P P D I1 2 N 1 1 ... ) /

121+NN21 ID/)PP,...,P(P

9

VALUE WEIGHTED INDEXES

S & P500, NIKKEI 225, VALUE LINE

B = SOME BASIS TIME PERIOD

INITIALLY t = B THUS, THE INITIAL INDEX VALUE IS SOME

ARBITRARILY CHOSEN VALUE: M. Examples:

The S&P500 index base period was 1941-1943 and its initial value was set at M = 10.

The NYSE index base period was Dec. 31, 1965 and its initial value was set at M = 50.

ti

Bp

ti

BiBi

titit w

V

V

PN

PNI

10

The rate of return on the index:

The HPRR on a value weighted index in any period t, is the

weighted average of the individual stock returns; the weights are the

dollar value of the stock as a proportion of the entire portfolio

value. ;Rw R titiIt

.V

V

PN

PN w

tP

ti

titi

tititi

11

stock Pti Nti Vti wti Pt+1i Rti

Federal Mogul 18 9,000 162,000 .0397 19.8 .1000

Martin Arietta 73 8,000 584,000 .1432 75 .0274

IBM 50 4,000 200,000 .0491 48 -.0400

US West 45 5,000 225,000 .0552 49 .0889

Bausch&Lomb 55 15,000 825,000 .2024 52 -.0545

First Union 50 10,000 500,000 .1227 57 .1400

Walt Disney 40 12,000 480,000 .1178 46 .1500

Delta Airlines 55 20,000 1,100,000 .2699 59 .0727

Total 4,076,000 1.000

Rp = (.0397)(.1) + (.1432(.0274) + (.0491)(-.04) + (.0552)(.0889) +

(.2024)(-.0545) + (.1227)(.14) + (.1178)(.15) + (.2699)(.0727) = 0.0543

or 5.43%

12

Of course, the HPRR on the portfolio may be calculated directly.

With the end-of-period prices – Pt+1i we calculate the end-of-period portfolio value: 4,297,200.

Thus, the portfolio’s HPRR is:

= [4,297,200 – 4,076,000]/4,076,000

= .0543

Or 5.43%.

13

THE RATE OF RETURN ON THE INDEX

titi

ti1i+tti

ti1i+t

titi

titi1i+t1i+t

titi

titi1i+t1i+t

t

t1+tIt

PN

)P(PN

Thus, .N N but,

;PN

PNPN

VB

PNVB

PN

VB

PN

I

II R

14

: thatagain, Notice, .Rw R

Finally, .RV

V

or ,R]PN

PN[

:as thisRewrite .PN

RPN

,PN

PPP

PN R

titiIt

titP

ti

tititi

titi

titi

tititi

titi

ti

ti1ittiti

It

.V

V

PN

PNw

tP

ti

titi

tititi

15

THE BETA OF A PORTFOLIO

Definitions:

R

.)VAR(R

)R,COV(R β

.)VAR(R

)R,COV(R β

.)VAR(R

)R,COV(R β

I

IPP

M

MPP

M

Mii

16

THE BETA OF A PORTFOLIOTHEOREM:

A PORTFOLIO’S BETA IS THE WEIGHTED AVERAGE OF THE BETAS OF THE STOCKS THAT COMPRISE THE PORTFOLIO. THE WEIGHTS ARE THE DOLLAR VALUE

WEIGHTS OF THE STOCKS IN THE PORTFOLIO.

Proof: Assume that the index is a well diversified portfolio, I.e., the index represents the market portfolio.

Let P denote any portfolio, i denote the individual stock; i = 1, 2, …,N in the

portfolio and I denote the index.

R

17proof. theconcludes This

.β w )VAR(R

R,COV(Rwβ

:or ,)VAR(R

)R,COV(Rwβ

: thusoperator,linear a

is covariance that theRecall

.)VAR(R

)R,]RwCOV([ β

,Rw R;for R ngSubstituti

.)VAR(R

)R,COV(R β

iiI

IiiP

I

IiiP

I

IiiP

iiPP

I

IPP

By definition:

18

STOCK PORTFOLIO BETA

FEDERAL MOUGUL 18.875 9,000 169,875 .044 1.00MARTIN ARIETTA 73.500 8,000 588,000 .152 .80IBM 50.875 3,500 178,063 .046 .50US WEST 43.625 5,400 235,575 .061 .70BAUSCH & LOMB 54.250 10,500 569,625 .147 1.1FIRST UNION 47.750 14,400 687,600 .178 1.1WALT DISNEY 44.500 12,500 556,250 .144 1.4DELTA AIRLINES 52.875 16,600 877,725 .227 1.2

3,862,713

P = .044(1.00) + .152(.8) + .046(.5)

+ .061(.7) + .147(1.1) + .178(1.1)

+ .144(1.4) + .227(1.2) = 1.06

STOCK NAME PRICE SHARES VALUE WEIGHT BETA

19

BENEFICIAL CORP. 40.500 11,350 459,675 .122 .95CUMMINS ENGINES 64.500 10,950 706,275 .187 1.10GILLETTE 62.000 12,400 768,800 .203 .85KMART 33.000 5,500 181,500 .048 1.15BOEING 49.000 4,600 225,400 .059 1.15W.R.GRACE 42.625 6,750 287,719 .076 1.00ELI LILLY 87.375 11,400 996,075 .263 .85PARKER PEN 20.625 7,650 157,781 .042 .75

3,783,225

A STOCK PORTFOLIO BETA

STOCK NAME PRICE SHARES VALUE WEIGHT BETA

P = .122(.95) + .187(1.1) + .203(.85)

+ .048(1.15) + .059(1.15) + .076(1.0)

+ .263(.85) + .042(.75) = .95

20

Sources of calculated Betas and calculation inputs

Example: ß(GE) 6/20/00

Source ß(GE) Index Data Horizon Value Line Investment Survey 1.25 NYSECI Weekly Price 5 yrs (Monthly)

Bloomberg 1.21 S&P500I Weekly Price 2 yrs (Weekly)

Bridge Information Systems 1.13 S&P500I Daily Price 2 yrs (daily)

Nasdaq Stock Exchange 1.14

Media General Fin. Svcs. (MGFS) S&P500I Monthly P ice3 (5) yrs Quicken.Excite.com 1.23

MSN Money Central 1.20

DailyStock.com 1.21

Standard & Poors Compustat Svcs S&P500I Monthly Price 5 yrs (Monthly)

S&P Personal Wealth 1.2287

S&P Company Report) 1.23

Charles Schwab Equity Report Card 1.20

S&P Stock Report 1.23

AArgus Company Report 1.12 S&P500I Daily Price 5 yrs (Daily)

Market Guide S&P500I Monthly Price 5 yrs (Monthly)

YYahoo!Finance 1.23

Motley Fool 1.23

21

STOCK INDEX OPTIONS

1. One contract = (I)($m)

(WSJ)2. ACCOUNTS ARE SETTLED BY CASH

22

EXAMPLE: Options on a stock index

MoneyGone, a financial institution, offers its clients the following deal:

Invest $A ≥ $1,000,000 for 6 months. In 6 months you receive a guaranteed return:

The Greater of {0%, or 50% of the return on the SP500I during these 6 months.}

For comparison purposes: The annual risk-free rate is 8%. The SP500I dividend payout ratio is q = 3% and its annual VOL σ= 25%.

23

}II)Max{0,I

$A(.5)( Return

)}I

II(.5)(($A)Max{0, Retrurn

0T0

0

0T

MoneyGone offer:

Deposit: $A now.

Receive: $AMax{0, .5RI} in 6 months. Denote the date in six month = T.

Rewrite MoneyGone offer at T:

24

}IIMax{0, 0T The expression:

is equivalent to the at-expiration cash flow of an at-the money European call option on the index, if you notice that K = I

0.

Calculate this options value based on:

S0 = K = I0; T – t = .5; r = .08; q = .03 and σ = .25. Using DerivaGem:

c = .08137.

Thus, MoneyGone’s promise is equivalent

25

to giving the client NOW, at time 0, a value of: (.5)(.08137)($A) = $.040685A.

Therefore, the investor’s initial deposit is only 95.9315% of A.

Investing $.959315A and receiving $A in six months, yields a guaranteed return of:

= 8.3%

.083]$.959315A

$Aln[

.5

1R

26

STOCK INDEX OPTIONS FOR

PORTFOLIO INSURANCE

Problems:

1.How many puts to buy?

2.Which exercise price will guarantee a desired level of protection?

The answers are not easy because the

index underlying the puts is not the

portfolio to be protected.

27

STRATEGY

ICF AT EXPIRATION

ST < K ST ≥ K

Hold the stockBuy put

-St

-p

ST

K - ST

ST

0

TOTAL -St – p K ST

The protective put with a single stock:

28

STRATEGY ICF (t = 0)

AT EXPIRATION (T = 1)

I1 < K I1 ≥ K

Hold the portfolioBuy n puts

-V0

-nP($m)

V1

n(K- I1)($m)

V1

0

TOTAL -V0 –nP($m)

V1+n($m)(K- I1) V1

The protective put consists of holding the portfolio and purchasing n puts on an index. Current t = 0; Expiration T = 1.

29

WE USE

THE CAPITAL ASSET PRICING MODEL.

For any security i, the expected excess return on the security and the expected excess return on the market portfolio are linearly related by their

beta:

)r(ERβrER

)r(ERβrER

FMpFp

FMiFi

30

THE INDEX TO BE USED IN THE STRATEGY, IS TAKEN TO BE A PROXY

FOR THE MARKET PORTFOLIO, M. FIRST, REWRITE THE ABOVE

EQUATION FOR THE INDEX I AND ANY PORTFOLIO P :

).r(ERβrER FIpFp

31

].rI

D I - I[βr

V

D V - VF

0

I01pF

0

P01

Second, as an approximation, rewrite the CAPM result, with actual returns:

).r(RβrR FIpFp In a more refined way, using V and I for the portfolio and index market values, respectively:

32

NEXT, use the ratio Dp/V0 as the portfolio’s annual dividend payout ratio qP and DI/I0 the index annual dividend payout ratio, qI. ]rq

I

I - I[βrq

V

V - VFI

0

01pFP

0

01

]r - q 1 - I

I[βr q1

V

VFI

0

1pFP

0

1

The ratio V1/ V0 indicates the portfolio required protection ratio.

33

For example:

,90.V

V

0

1

The manager wants V1, to be down to no

more than 90% of the initial portfolio

market value, V0: V1 = (.9)V0.

We denote this desired level of hedging by

(V1/ V0)*.

This is a decision variable.

34

.($m)I

Vβn

0

0p

1. The number of puts is:

35

2. The exercise price, K, is determined by substituting I1 = K and the required level, (V1/ V0)* into the equation:

].r - q 1 - I

K[βr q1*)

V

V( FI

0pFP

0

1

1)].-)(βr(1 )q(β - q*)V

V[(

β

I K pFIpp

0

1

p

0

],r - q 1 - I

I[βr q1

V

VFI

0

1pFP

0

1

and solving for K:

36

EXAMPLE: A portfolio manager expects the market to fall by 25% in the next six months. The current portfolio value is $25M. The manager decides on a 90% hedge by purchasing 6-month puts on the S&P500 index. The portfolio’s beta with the S&P500 index is 2.4. The S&P500 index stands at a level of 1,250 points and its dollar multiplier is $100. The annual risk-free rate is 10%, while the portfolio and the index annual dividend payout ratios are 5% and 6%, respectively. The data are summarized below:

37

year. a half Period 2.4.β Finally,

6%.q 5%;q 10%;r :are rates annual The

$100;$m 1,250;I .9;)*V

V( 0;$25,000,00V

IpF

00

10

Solution: Purchase

puts. 48050)($100)(1,2

0$25,000,002.4n

($m)I

Vβn

0

0p

38

The exercise price of the puts is:

1,210.K

1).05)(2.4(1(2.4).03.025[.92.4

1,250K

1)].-)(βr(1 )q(β - q*)V

V[(

β

I K pFIpp

0

1

p

0

Solution: Purchase n = 480 six-months puts with exercise price K = 1,210.

39

STRATEGY

INITIAL CASH FLOW

AT EXPIRATION

I1 < K I1 ≥ K

Hold the portfolio

Buy n puts

-V0

-n P($m)

V1

n(K - I1)($m)

V1

0

TOTAL -V0 - nP($m) V1+n($m)(K - I1)

V1

We are now ready to calculate the floor level of the portfolio: V1+n($m)

(K- I1)

We rewrite the Profit/Loss table for the protective put strategy:

40

We are now ready to calculate the floor level of the portfolio:

Min portfolio value = V1+n($m)(K- I1)

This is the lowest level that the portfolio value can attain. If the index falls below the exercise price and the portfolio value declines too, the protective puts will be exercised and the money gained may be invested in the portfolio and bring it to the value of:

V1+n($m)K- n($m)I1

41

Substitute for n:

.II

Vβ-K

I

VβV valueportfolioMin

($m)I($m)I

Vβ

($m)K($m)I

VβV valueporfolioMin

10

0p

0

0p1

10

0p

0

0p1

.($m)I

Vβn

0

0p

42

To substitute for V1 we solve the equation:

]r - q 1 - I

I[βr q1

V

VFI

0

1pFP

0

1

]r1[qβqr1V

II

VβV

])r - q 1 - I

I[βrq(1VV

FIppF0

10

0p1

FI0

1pFP01

43

3. Substitution V1 into the equation for the Min portfolio value

.)]β)(1r(1qq[βVKI

Vβ

valueportfolioMin

pFpIp00

0p

The desired level of protection is made at time 0. This determines the exercise price and management can also calculate the minimum portfolio

value.

44

0.$22,505,00

2.4)]-.05)(1(1.025-0[2.4(.03)$25,000,00

210,1250,1

0$25,000,002.4

.)]β)(1r(1qq[βVKI

Vβ

valueportfolio Minimum

pFpIp00

0p

45

2.0β Finally,

4%.q 4%;q 12%;r :are rates annual The

$100;$m 1,000;I .9;)*V

V( $500,000;V

IpF

00

10

Solution: Purchase

puts. 1000)($100)(1,0

$500,0002.0n

($m)I

Vβn

0

0p

Example (p326) protection for 3 months

46

The exercise price of the puts is:

960.K

1).03)(2.0(1(2.0).01.01[.92.0

1,000K

1)].-)(βr(1 )q(β - q*)V

V[(

β

I K pFIpp

0

1

p

0

Solution: Purchase n = 10 three -months puts with exercise price K = 960.

47

$450,000.

2.0)]-.03)(1(1.01-.0(.01)$500,000[2

960000,1

$500,0002.0

.)]β)(1r(1qq[βVKI

Vβ

valueportfolioMin

pFpIp00

0p

48

CONCLUSION:

Holding the portfolio and purchasing 10, 3-months protective puts on the S&P500 index, with the exercise price K = 960, guarantees that the portfolio value, currently $500,000 will not fall below $450,000 in three months.

49

A SPECIAL CASE: In the case that

a. β = 1

b. qP =qI,

the portfolio is statistically similar to the

index. In this case: 0

0

($m)I

Vn

.)/V(VV is valueportfolioMin

and *)V

V(IK

*010

0

10

50

.000,500,22$)9(.000,000,25$)9(.VV

valueportfolioMin

and puts 200$100(1250)

000,000,25$

($m)I

Vn

.125,1)9(.250,1 *)V

V(IK

*0

*1

0

0

0

10

Assume that in the above example:

βp = 1 and qP =qI, then:

51

.000,450$)9(.000,500$)9(.VV

valueportfolioMin

and puts 5)$100(1,000

000,500$

($m)I

Vn

.900)9(.000,1 *)V

V(IK

*0

*1

0

0

0

10

Example: (p326-27)

βp = 1 and qP =qI, then:

52

STRATEGY

ICF

AT EXPIRATION

I1 < KP KP < I1 < KC I1 ≥ KC

portfolio

Buy n puts

Sell n calls

-V0

-nP($m) nC($m)

V1 V1

n(KP-I1)($m) 0

0 0

V1

0

n(I1-KC)($m)

TOTAL -V0 V1+ V1

n($m)(KP - I1)

V1 –

n($m)(I1-KC)

A Zero cost Collar

53

A zero cost Collar

If the Collar is to be zero cost that the cost of the puts is equal to the revenue from the calls, given that: n(p) = n(c).

Using the same relationship between the portfolio value and the index value, i.e., the CAPM the solution for the P/L profile of the Collar is given by:

54

C0

0PIFPPF0

C1

10

0PIFPPF0

C1P

P0

0PIFPPF0

P1

KI

Vβ)]qr(1βq[rV

:KIFor

II

Vβ)]qr(1βq[rV

:KIKFor

KI

Vβ)]qr(1βq[rV

:KIFor

55

FOREIGN CURRENCY (FORX) OPTIONS(p.321)

FORX options are traded all over the world. The main exchange in the U.S. is the Philadelphia exchange (PHLX).

First we describe several characteristics of the spot market for FORX.

56

FOREIGN CURRENCY: THE SPOT MARKET

EXCHANGE RATES: The value of one currency in one unit of another currency is the EXCHANGE RATE between the two currencies.There are two quote formats: 1. S(USD/FC); The number of USD in one unit of the foreign currency.

2. S(FC/USD); The number of the foreign currency in one USD.

www.x-rates.com

57

.5945 S(GBP/USD)

S(GBP/USD)1 =

.59451 =

1.6821 = S(USD/GBP)

58

CURRENCY CROSS RATES

Let FC1, FC2 AND FC3 denote 3 different currencies. Then, in the absence of arbitrage, the following relationship must hold for their spot exchange rate:

S(FC3/FC1)S(FC3/FC2) =

S(FC2/FC3)S(FC1/FC3) = S(FC1/FC2)

59

CURRENCY CROSS RATES – OCT. 13, 04

USD GBP CAD EUR AUD

USD 1 1.7972 0.798212

1.2393 0.731502

GBP 0.556421

1 0.444141

0.689572

0.407023

CAD 1.25279 2.25153 1 1.55259 0.916425

EUR 0.806907

1.45017 .644082 1 0.590254

AUD 1.36705 2.45686 1.09119 1.69418 1

60

CURRENCY CROSS RATES

EXAMPLE: FC1 = USD; FC2 = MXP;FC3 = GBP.

USD MXP GBP

USA 1.0000 0.0997 1.6603

MEXICO 10.0301 1.00016.653

UK 0.6023 0.06005 1.000

0.06005. S(GBP/MXP) 0.6023; S(GBP/USD) 16.653 S(MXP/GBP) 10.0301; S(MXP/USD) 1.6603 S(USD/GBP) 0.0997; S(USD/MXP)

61

CURRENCY CROSS RATESEXAMPLE

0.0997. 16.6531.6603

S(MXP/GBP)S(USD/GBP)

0.0997. 0.6023

0.06005 S(GBP/USD)S(GBP/MXP)

.S(MXP/GBP)S(USD/GBP) =

S(GBP/USD) S(GBP/MXP) = S(USD/MXP)

GBP. FC3 MXP; FC2 USD; FC1Let

62

AN EXAMPLE OF CROSS SPOT RATES ARBITRAGE

COUNTRY USD GBP CHF

SWITZERLAND 1.7920 2.8200 1.0000

U.K 0.6394 1.0000 0.3546

U.S.A 1.0000 1.5640 0.5580

2.8200 < 2.8029 = 0.5580

1.5640 :BUT

S(CHF/GBP) = S(USD/CHF)

S(USD/GBP) :SIMILARLY

1.7920 1.8031 = 0.35460.6394 :BUT

S(CHF/USD) = S(GBP/CHF)

S(GBP/USD) :THEORY

63

THE CASH ARBITRAGE ACTIVITIES:

USD1,000,000 USD1,006,134.26

0.6394 0.5580

GBP639,400 CHF1,803,108

2.8200

64

Forward rates, An example:

GBP 18.5.99

SPOT USD1.6850/GBP

30 days forwardUSD1.7245/GBP




The existence of forward exchange rates implies that there is a demand and supply for the GBP for future dates.

65

THE INTEREST RATES PARITY

Wherever financial flows are unrestricted, the exchange rates, the forward rates and the interest rates in

any two countries must maintain a NO- ARBITRAGE relationship:

Interest Rates Parity.

365

180)r(r

t)- )(Tr - (r

UKUS

FORDOM

1.6850e1.8455

.S(DC/FC)e = F(DC/FC)

66

NO ARBITRAGE: CASH-AND-CARRYTIME CASH FUTURES

t (1) BORROW DC. rDOM (4) SHORT FOREIGN CURRENCY

(2) BUY FOREIGN CURRENCY FORWARD Ft,T(DC/FC)

DC/S(DC/FC) = DCS(FC/DC)] AMOUNT:

(3) INVEST IN BONDS

DENOMINATED IN THE

FOREIGN CURRENCY rFOR

T (3) REDEEM THE BONDS EARN (4) DELIVER THE CURRENCY TO

CLOSE THE SHORT POSITION

(1) PAY BACK THE LOAN RECEIVE:

IN THE ABSENCE OF ARBITRAGE:

t)-(TrFOReDCS(FC/DC)

t)-(TrFOReDCS(FC/DC)

t)-(TrFORS(FC/DC)eF(DC/FC)DCt)-(TrDOMDCe

t)-(Trt)(Tr FORD S(FC/DC)eF(DC/FC)DC DCe

t)-)(Tr - (rtt,T

FORDOM(DC/FC)eS (DC/FC)F

67

NO ARBITRAGE: REVERSE CASH – AND - CARRYTIME CASH FUTURES

t (1) BORROW FC . rFOR (4) LONG FOREIGN CURRENCY (2) BUY DOLLARS FORWARD Ft,T(DC/FC)

FCS(DC/FC) AMOUNT IN DOLLARS:

(3) INVEST IN T-BILLS

FOR RDOM

T REDEEM THE T-BILLS EARN TAKE DELIVERY TO CLOSE

THE LONG POSITION

PAY BACK THE LOAN RECEIVE


t)-(TRDOMeFCS(DC/FC)

t)-(TrDOMeFCS(DC/FC)

F(DC/FC)

eFCS(DC/FC) t)-T(rDOMt)-(TrFORFCe

t)-(TrFORFCeF(DC/FC)

eFCS(DC/FC) t)-T(rDOM

t)-T)(r(rtt,T

FORDOM(DC/FC)eS (DC/FC)F

68

t)- )(Tr - (rtt,T

FORDOM(DC/FC)eS = (DC/FC)F

FROM THE CASH-AND-CARRY STRATEGY:

(DC/FC)Ft,T

FROM THE REVERSE CASH-AND-CARRY STRATEGY:

t)-)(Tr - (rt

FORDOM(DC/FC)eS (DC/FC)Ft,T

THE ONLY WAY THE TWO INEQUALITIES HOLD SIMULTANEOUSLY IS BY BEING AN EQUALITY:

t)-)(Tr - (rt

FORDOM(DC/FC)eS

69

ON MAY 25 AN ARBITRAGER OBSERVES THE FOLLOWING MARKET PRICES:

S(USD/GBP) = 1.5640 <=> S(GBP/USD) = .6393

F(USD/GBP) = 1.5328 <=> F(GBP/USD) = .6524

rUS = 7.85% ; rGB = 12%

The market forward rate 1.5328 is overvalued relative to the theoretical, no arbitrage forward rate 1.5273.

CASH AND CARRY

1.5273 = 1.5640e = )F(USD/GBP 365

209.12) - (.0785

lTheoretica

70

CASH AND CARRY

TIME CASH FUTURES

MAY 25 (1) BORROW USD100M AT 7. 85% SHORT DEC 20

FOR 209 DAYS GBP68,477,215 FORWARD.

F = USD1.5328/GBP

(2) BUY GBP63,930,000

(3) INVEST THE GBP63,930,000

IN BRITISH BONDS

DEC 20RECEIVE GBP68,477,215 DELIVER GBP68,477,215

FOR USD104,961,875.2

REPAY YOUR LOAN:

PROFIT: USD104,961,875.2 - USD104,597,484.3 = USD364,390.90

7,484.3 USD104,59= 100Me 365

209.0785

215GBP68,477, = e63,930,000 365

209.12

71

Example 2: THE INTEREST RATES PARITY

In the real markets, buyers pay the ask price while sellers receive the bid price. Moreover, borrowers pay the ask interest rate while lenders only

receive the bid interest rate.

Therefore, in the real markets, it is possible for the forward exchange rate to fluctuate within a band of

rates without presenting arbitrage opportunities.Only when the market forward exchange rate diverges from this band of rates arbitrage exists.

72

Foreign Exchange Quotes for USD/GBP on Aug 16, 2001

Bid Ask

Spot 1.4452 1.4456

1-month forward 1.4435 1.4440




73

NZD2.get USD1sell SD,NZD2.000/U BIDS(NZD/USD)

NZD2.083.pay buy USD1 SD,NZD2.083/U ASKS(NZD/USD)

cents. 48get NZD1 sell ZD, USD.480/N BIDS(USD/NZD)

cents. 50pay NZD1buy USD.5NZD, ASKS(USD/NZD)

ASKS(FC/EUR)

1 BIDS(EUR/FC)

BIDS(FC/EUR)

1 ASKS(EUR/FC)

:QUOTESASK AND BID FOR

74

Example 2: THE INTEREST RATES PARITY

We now show that in the real markets it is possible for the forward

exchange rate to fluctuate within a band of rates without presenting

arbitrage opportunities.Only when the market forward exchange rate diverges from this band of rates

arbitrage exists.

Given are: Bid and Ask domestic and foreign spot rates; forward rates

and interest rates.

75

NO ARBITRAGE: CASH - AND - CARRYTIME CASH FUTURES

t (1) BORROW DC. rD,ASK (4) SHORT FOREIGN CURRENCY FORWARD

(2) BUY FOREIGN CURRENCY

DC/SASK(DC/FC) FBID (DC/FC)

(3) INVEST IN BONDS

DENOMINATED IN THE

FOREIGN CURRENCY rF,BID

T REDEEM THE BONDS DELIVER THE CURRENCY TO CLOSE THE SHORT POSITION

EARN:

PAY BACK THE LOAN RECEIVE:


t)-(TrASK

BIDF,(DC/FC)eDC/S

t)-(TrBID

FORS(DC/FC)e(DC/FC)DC/Ft)-(Tr ASKD,DCe

t)-(TrASKBID

t)(Tr BIDF,ASKD, (DC/FC)e(DC/FC)A/SF DCe

t)-)(Tr - (rASKBID

BIDF,ASKD,(DC/FC)eS (DC/FC)F

t)-(TrASK

BIDF,(DC/FC)eDC/S

76

NO ARBITRAGE:

REVERSE CASH - AND - CARRYTIME CASH FUTURES

t (1) BORROW FC . rF,ASK (4) LONG FOREIGN CURRENCY FORWARD FOR FASK(DC/FC)

(2) EXCHANGE FOR

FCSBID (DC/FC)

(3) INVEST IN T-BILLS

FOR rD,BID

T REDEEM THE T-BILLS EARN TAKE DELIVERY TO CLOSE THE LONG POSITION

RECEIVE in foreign currency, the amount:

PAY BACK THE LOAN


t)-(TrBID

BIDD,(DC/FC)eFCS

(DC/FC)F

(DC/FC)eFCS

ASK

t)-T(rBID

BIDD,

t)-(Tr ASKF,FCe

t)-T)(r(rBIDASK

ASKF,BIDD,(DC/FC)eS (DC/FC)F

t)-(TrBID

BIDD,(DC/FC)eFCS

t)-(Tr ASKF,FCe (DC/FC)F

(DC/FC)eFCS

ASK

t)-T(rBID

BIDD,

77

t)-T)(r(rBIDASK

ASKF,BIDD,(DC/FC)eS (DC/FC)F (2)

t)-)(Tr - (rASK

BIDF,ASKD,(DC/FC)eS

From Cash and Carry:

(DC/FC)F (1) BID

From reverse cash and Carry

Notice that:

RHS(1) > RHS(2)

Define: RHS(1) BU RHS(2) BL

(3) And FASK(DC/FC) > FBID(DC/FC)

78

BU

BL

FASKFASK(DC/FC) > FBID(DC/FC).

Arbitrage exists only if both ask and bid futures prices are above BU,

or both are below BL.

FBID

LASK B(DC/FC)F

UBID B(DC/FC)F

F($/D)

BU

BL

79

A numerical example:

Given the following exchange rates:

Spot Forward Interest rates

S(USD/NZ) F(USD/NZ) r(NZ) r(US)

ASK 0.4438 0.4480 6.000% 10.8125%

BID 0.4428 0.4450 5.875% 10.6875%

Clearly, F(ask) > F(bid). (USD0.4480NZ > USD0.4450/NZ)

We will now check whether or not there exists an opportunity for arbitrage profits. This will require comparing these

forward exchange rates to: BU and BL

80

t)-T)(r(rBIDASK

ASKNZ,BIDUS,(USD/NZ)eS (USD/NZ)F

t)-)(Tr - (rASK

BIDNZ,ASKUS,(USD/NZ)eS Inequality (1):

(USD/NZ)FBID

0.4450 < (0.4438)e(0.108125 – 0.05875)/12 = 0.4456 = BU

0.4480 > (0.4428)e(0.106875 – 0.06000)/12 = 0.4445 = BL

No arbitrage.

Lets see the graph

Inequality (2):

81

BU

BL

Clearly: FASK($/FC) > FBID($/FC).

An example of arbitrage:

FASK = 0.4480

FBID = 0.4465

FBID = 0.4450

LASK B4445.0 (USD/NZ)F

UBID B0.4456(USD/NZ)F

FFASK = 0.4480

0.4445

0.4456

82

Currency options UnitsUSD/AUD 50,000AUDUSD/GBP 31,250GBPUSD/CAD 50,000CADUSD/EUR 62,500EURUSD/JPY 6,250,000JPYUSD/CHF 62,500CHFExercise Style: American- or Europeanoptions available for physically settled contracts; Long-term options are European-style only.

83

Expiration/Last Trading DayThe PHLX offers a variety of expirations

in its physically settled currency options contracts, including Mid-month, Month-end and Long-term expirations. Expiration, which is also the last day of trading, occurs on both a quarterly and consecutive monthly cycle. That is, currency options are available for trading with fixed quarterly months of March, June, September and December and two additional near-term months. For example, after December expiration, trading is available in options which expire in January, February, March, June, September, and December. Month-end option expirations are available in the three nearest months.

84

Standardized Options

Exercise PricesExercise prices are expressed in terms of U.S. cents per unit of foreign currency. Thus, a call option on EUR with an exercise price of 120 would give the option buyer the right to buy Euros at 120 cents per EUR.

With the Canadian dollar spot price currently at a level of USD.6556/CAD, strike prices would be listed in half-cent intervals ranging from 60 to 70. i.e., 60, 60.5, 61, …, 69, 69.5, 70. If the Canadian dollar spot rate should move to say USD.7060/CAD, additional strikes would be listed. E.G, 70, 70.5, 71, …, 75.

With the Canadian dollar spot price currently at a level of USD.6556/CAD, strike prices would be listed in half-cent intervals ranging from 60 to 70. i.e., 60, 60.5, 61, …, 69, 69.5, 70. If the Canadian dollar spot rate should move to say USD.7060/CAD, additional strikes would be listed. E.G, 70, 70.5, 71, …, 75.

85

It is important that available exercise prices relate closely to prevailing currency values. Therefore, exercise prices are set at certain intervals surrounding the current spot or market price for a particular currency. When significant price changes take place, additional options with new exercise prices are listed and commence trading.Strike price intervals vary for the different expiration time frames. They are narrower for the near-term and wider for the long-term options.

86

Premium Quotation premiums for dollar-based options are quoted in U.S. cents per unit of the underlying currency with the exception of Japanese yen which are quoted in hundredths of a cent.Example: A premium of 1.00 for a given EUR option is one cent (USD.01) per EUR. Since each option is for 62,500 EURs, the total option premium would be[62,500EUR][USD.01/EUR] = USD625.

87

FX Options As InsuranceOptions on spot represent insurance bought or written on the spot rate. An individual with foreign currency to sell can use put options on spot to establish a floor price on the domestic currency value of the foreign currency. For example, a put on EUR with an exercise price of USD1.180/EUR ensures that, if the value of the EUR falls below USD1.180/EUR, the EUR can be sold for USD1.180/EUR.

88

If the put option costs USD.03/EUR, the floor price can be roughly approximated as:USD1.180/EUR - USD.O3/EUR =

USD1.15/EUR.

That is, if the PUT is used, the put holder will be able to sell the EUR for the USD1.180/EUR strike price, but in the meantime, have paid a premium of USD.03/EUR. Deducting the cost of the premium leaves USD1.15/EUR as the floor price established by the purchase of the put. This calculation ignores fees and interest rate adjustments.

89

Similarly, an individual who must buy foreign currency at some point in the future can use CALLS on spot to establish a ceiling price on the domestic currency amount that will have to be paid to purchase the foreign exchange.

90

For example, a call on EUR with an exercise price of USD1.23/EUR will ensure that, in the event that the value of the EURO rises above USD1.23/EUR, the call will be exercised and the EUR bought for USD1.23/EUR. If the call costs USD.02/EUR, this ceiling price can be approximated: USD1.23/EUR + USD.02/EUR = USD1.25/EUR

or the strike price plus the premium.

91

Several real world considerations:The calculations so far are only approximate for essentially two reasons. First, the exercise price and the premium of the option on spot cannot be added directly without an interest rate adjustment. The premium will be paid now, up front, but the exercise price (if the option is eventually exercised) will be paid later. The time difference involved in the two payment amounts implies that one of the two should be adjusted by an interest rate factor.

92

Second, there may be brokerage or other expenses associated with the purchase of an option, and there may be an additional fee if the option is exercised. The following two examples illustrate the insurance feature of FX options on spot and show how to calculate floor and ceiling values when some additional transactions costs are included.

93

Example 1: An American importer will have a net cash out flow of GBP250,000 in payment for goods bought in Great Britain. The payment date is not known with certainty, but should occur in late November. On September 16 the importer locks in a ceiling purchase for pounds by buying 8 PHLX calls [GBP250,000/GBP31,250 = 8] on the pound, K = USD1.90/GBP and a December expiration. The call premium on September 16 is USD.0220/GBP. With a brokerage commission of USD4/call, the total cost of the eight calls is:

8(GBP3l,250)(USD.0220/GBP) + 8(USD4)= USD5,532.

94

Measured from today's viewpoint, the importer has essentially assured that the purchased exchange rate will not be greater than:

USD5,532/GBP250,000 + USD1.90/GBP= USD.02213/GBP + USD1.90/GBP =USD1.92213/GBP.

Notice here that the add factor USD.02213/GBP is larger than the call premium of USD.0220/GBP by USD.00013/GBP, which represents the dollar brokerage cost per pound. The number USD1.92213/GBP is the importer's ceiling price. The importer is assured he will not pay more than this, but he could pay less.

95

Case A. The spot rate on the November payment date is USD1.86/GBP. The importer would not exercise the call but would buy pounds spot at the rate of USD1.86/GBP. The importer then sell the eight calls for whatever market value they had remaining. Assuming, a brokerage fee of USD4 per contract for the sale, the options would be sold as long as their remaining market value was greater than USD4 per option. The total cost will have turned out to be:

USD1.96/GBP+USD.02213/GBP - (sale value of options- USD32)/GBP250,000.

96

If the resale value is not greater than USD32, then the total cost per pound is

USD1.86 + USD.02213 = USD1.88213.The USD.02213 that was the original cost of the premium and brokerage fee turned out in this case to be an unnecessary expense.

97

Now, to be strictly correct, a further adjustment to the calculation should be made. Namely, the USD1.86 and USD.02213 represent cash flows at two different times. Thus, if R is the amount of interest paid per dollar over the September 16 to November time period, the proper calculation is the cost per pound:

USD1.86+USD.02213(l+R)- (sale value of options-USD32)/250,000.

98

Case B. The spot rate on the November payment date is USD1.95/GBP. The importer can either exercise the calls or sell them for their market value. Assume the importer sells them at a current market value of USD.055 and pays USD32 total in brokerage commissions on the sale of eight option contracts. The importer then buys the pounds in the spot market for USD1.95/GBP. The total cost is, before adding the premium and commission costs paid in September:

(USD1.95/GBP)(GBP250,000) – (USD.055/GBP))( GBP250,000) + 8(USD4)

= USD473,718.This amount implies an exchange rate of:USD473,718/GBP250,000 = USD1.89487/GBP.

99

Adding in the premium and commission costs paid back in September, the exchange rate is:

USD1.89487/GBP + USD.02213(l +R)/GBP. If the importer chooses instead to exercise the call, the calculations will be similar except that the brokerage fee will be replaced by an exercise fee.This concludes Example 1.

100

Example 2 A Japanese company must exchange USD50M into JPY and wishes to lock in a minimum yen value. The USD50M, is to be sold between July1 and December 31. Since the company will sell USD and receive JPY, the company will buy a put option on USD, with an exercise price stated in terms of JPY. The company buys an American put on USD50M with a strike price of JPY130/USD from a financial institution. The premium is JPY4/USD. Clearly, this is an OTC transaction.

101

The put was purchased directly from the bank thus, there is no resale value to the put. Assume there are no additional fees. Then, the Japanese firm has established a floor value for its USD, approximately at:

JPY130/USD - JPY4/USD = JPY126/USD. Again, we can consider two scenarios, one in which the yen falls in value to JPY145/USD and the other in which the yen rises in value to JPY115/USD.

102

Case A. The yen falls to JPY145/USD. In this case the company will not exercise the option to sell dollars for yen at JPY13O/USD, since the company can do better than this in the exchange market. The company will have obtained a net value of

JPY145/USD - JPY4/USD = JPY141/USD.In total:[JPY141/USD][USD50M] = JPY7.050B

103

Case B. The JPY rises to JPY115/USD. The company will exercise the put and sell each U.S. dollar for JPY130/USD. The company will obtain, net,

JPY130/USD - JPY4/USD = JPY126/USD.In total[JPY126/USD][USD50M] = JPY6.3B

This is JPY11 better than would have been available in the FX market and reflects a case where the “insurance” paid off. This concludes Example 2.

104

Writing Foreign Currency OptionsGeneral considerations. The writer of a foreign currency option on spot or futures is in a different position from the buyer of these options. The buyer pays the premium up front and then can choose to exercise the option or not. The buyer is not a source of credit risk once the premium has been paid. The writer is a source of credit risk, however, because the writer has promised either to sell or to buy foreign currency if the buyer exercises his option. The writer could default on the promise to sell foreign currency if the writer did not have sufficient foreign currency available, or could default on the promise to buy foreign currency if the writer did not have sufficient domestic currency available.

105

If the option is written by a bank, this risk of default may be small. But if the option is written by a company, the bank may require the company to post margin or other security as a hedge against default risk. For exchange-traded options, as noted previously, the relevant clearinghouse guarantees fulfillment of both sides of the option contract. The clearinghouse covers its own risk, however, by requiring- the writer of an option to post margin. At the PHLX, for example, the Options Clearing Corporation will allow a writer to meet margin requirements by having the actual foreign currency or U.S. dollars on deposit, by obtaining an irrevocable letter of credit from a suitable bank, or by posting cash margin.

106

If cash margin is posted, the required deposit is the current market value of the option plus 4 percent of the value of the underlying foreign currency. This requirement is reduced by any amount the option is out of the money, to a minimum requirement of the premium plus .75 percent of the value of the underlying foreign currency. These percentages can be changed by the exchanges based on currency volatility. Thus, as the market value of the option changes, the margin requirement will change. So an option writer faces daily cash flows associated with changing margin requirements.

107

Other exchanges have similar requirements for option writers. The CME allows margins to be calculated on a net basis for accounts holding both CME FX futures options and IMM FX futures. That is, the amount of margin is based on one's total futures and futures options portfolio. The risk of an option writer at the CME is the risk of being exercised and consequently the risk of acquiring a short position (for call writers) or a long position (for put writers) in IMM futures. Hence the amount of margin the writer is required to post is related to the amount of margin required on an IMM FX futures contract. The exact calculation of margins at the CME relies on the concept of an option delta.

108

From the point of view of a company or individual, writing options is a form of risk-exposure management of importance equal to that of buying options. It may make perfectly good sense for a company to sell foreign currency insurance in the form of writing FX calls or puts. The choice of strike price on a written option reflects a straightforward trade-off. FX call options with a lower strike price will be more valuable than those with a higher strike price. Hence the premiums the option writer will receive are correspondingly larger. However, the probability that the written calls will be exercised by the buyer is also higher for calls with a lower strike price than for those with a higher strike. Hence the larger premiums received reflect greater risk taking on the part of the insurance seller, ie., the option writer.

109

Writing Foreign Currency Options:

a detailed example.

The following example illustrates the risk/return trade-off for the case of an oil company with an exchange rate risk, that chooses to become an option writer.

110

Example 3 Iris Oil Inc., a Houston-based energy company, has a large foreign currency exposure in the form of a CAD cash flow from its Canadian operations. The exchange rate risk to Iris is that the CAD may depreciate against the USD. In this case, Iris’ CAD revenues, transferred to its USD account will diminish and its total USD revenues will fall. Iris chooses to reduce its long position in CAD by writing CAD calls with a USD strike price.

111

By writing the options, Iris receives an immediate USD cash flow representing the premiums. This cash flow will increase Iris' total USD return in the event the CAD depreciates against the USD or, remains unchanged against the USD, or appreciates only slightly against the USD. Clearly, the calls might expire worthless or they might be exercised. In either case, however, Iris walks away with the full amount of the options premiums:

112

1. If the USD value of the CAD remains unchanged, the option premium received is simply an additional profit.

2. If the value of the CAD falls, the premium received on the written option will offset part or all of the opportunity loss on the underlying CAD position.

3. If the value of the CAD rises sharply, Iris will only participate in this increased value up to a ceiling level, where the ceiling level is a function of the exercise price of the written option.

113

In sum, the payoff to Iris' strategy will depend both on exchange rate movements and on the selection of the strike price of the written calls. To illustrate Iris' strategy, consider an anticipated cash flow of CAD300M over the next 180 days.

With hedge ratio of 1:1*, Iris sells CAD300,000,000/CAD50,000

= 6,000 PHLX calls.

*every CAD option is for CAD50,000.

114

Assume: Iris writes 6,000 PHLX calls with a 6-month expiration; the current spot rate is S = USD.75/CAD and the 6-month forward rate is:

F = USD.7447/CAD. For the current level of spot rate, logical strike price choices for the calls might be K = USD.74, or USD.75, or USD.76 per CAD, of course. For the illustration, assume that Iris’ brokerage fee is USD4 per written call and let the hypothetical market values of the options be as follows:

115

KValue

One call

nValueTotal

Premium

TotalFees:

USD4/call

.74 USD689.5 6,000 USD4,137,000 USD24,000

.75 USD325.0 6,000 USD1,950,000 USD24,000

.76 USD156.5 6,000 USD939,000 USD24,000

c(K = USD.74/CAD) = USD.01379;

c(K = USD.75/CAD) = USD.00650;

c(K = USD.76/CAD) = USD.00313.

116

We now introduce an additional cost that is associated with the exercise fee, which exists in the real markets. If the calls are exercised, an additional OCC fee of USD35/call is assumed. In our example then, an exercise of the calls requires a total OCC fee of:

USD35(6,000) = USD210,000

for the 6,000 written calls.

117

In six months Iris will receive a cash flow of CAD300M. At that time, the total value of the long CAD position of Iris, plus the short calls position will depend on the strike price chosen. Let S = the spot exchange rate at expiration.The next three tables show the possible values for Iris:

118

Strategy Initial Cash Flow

Cash flow at Expiration

S< USD.74/CAD

S>USD.74/CAD

Write 6,000, .74

calls

USD4,113,000

0 -(S-.74)CAD300M -USD210,000

CAD (S)CAD300M (S)CAD300M

Total

P/L

USD4,113,000

(S)CAD300M

(S)CAD300M+USD4.113,00

0

USD221,780,000

USD225,903,000

If K = USD.74/CAD

119



S< USD.75/CAD

S>USD.75/CAD

Write 6,000, .75

calls

USD1,926,000

0 -(S-.75)CAD300M -USD210,000


Total

P/L

USD1,926,000

(S)CAD300M

(S)CAD300M+USD1.926,00

0

USD224,700,000

USD226,716,000

If K = USD.75/CAD

120



S< USD.76/CAD

S>USD.76/CAD

Write 6,000, .76

calls

USD915,000 0 -(S-.76)CAD300M -USD210,000


Total

P/L

USD915,000 (S)CAD300M

(S)CAD300M+USD915,000

USD227,790,000

USD228,705,000

If K = USD.76/CAD

121

SPOT RATE USD/CAD

STRIKE PRICE

USD.74/CAD USD.75/CAD USD.76/CAD

S<.74 S(CAD300M) +

USD4,113,000

S(CAD300M) +

USD1,926,000

S(CAD300M) +

USD915,000

.74<S<.75 USD225,903,000

S(CAD300M) +

USD1,926,000

S(CAD300M) +

USD915,000

.75<S<.76 USD225,903,000

USD226,716,000

S(CAD300M) +

USD915,000

.76<S USD225,903,000

USD226,716,000

USD228,705,000

A consolidation of the three profit profile tables:

122

As illustrated by the consolidated table and the three separate profit profile tables, the lower the strike price chosen, the better the protection against a depreciating CAD. On the other hand, a lower strike price limits the corresponding profitability of the strategy if the CAD happens to appreciate against the USD in six months. The optimal decision of which strategy to take is a function of the spot exchange rate at expiration.

123

One possible comparison of the threeresults is to evaluate the options strategyvis-à-vis the immediate forward exchange.Recall that when Iris enters the optionsstrategy the forward exchange rate is

F = USD.7447/CAD. Thus, Iris may exchange the CAD300MForward for USD223,410,000 a futurebreak-even Spot rate can be calculated forEvery corresponding exercise price chosen:

124

F =.7447. Iris may exchange today, CAD300M forward for:

CAD300,000,000(USD.7447/CAD)= USD223,410,000.

IF: K =.74,S(CAD300M) + 4,113,000 = USD223,410,000

SBE = USD.7310/CAD.IF K= .75,

S(CAD300M) + 1,926,000 = USD223,410,000 SBE = USD.7383/CAD.

IF K= .76,S(CAD300M) + 915,000 = USD223,410,000

SBE = USD.7416/CAD.

125

CONCLUSIONWriting the calls will protect Iris’

flow in USD better than purchasing the

CADforward if the spot rate in six

months will be above the correspondingbreak- even exchange rates.

126

A second possible analysis of the optimal decision depends on all possible values of the spot exchange rate, given our assumptions. Recall that the assumptions are:Iris maintains an open long position of CAD300M un hedged. Alternatively, Iris writes 6,000 PHLX calls with 180-day expiration period. Possible strike prices are USD.76/CAD, USD.75/CAD, USD.74/CAD. Current spot and forward exchange rates are USD.75/CAD and USD.7447/CAD, respectively.

127

The terminal spot rate is the market exchange rate when the calls expire. It is assumed that Iris pays a brokerage-fee of USD4 per option contract and an additional fee of USD35 per option to the Options Clearing Corporation if the options are exercised.

128

Optimal decision as a function of the unknown terminal spot rate

Terminal Spot rate Optimal Decision

S >.76235 Hold long currency only

.75267 < S< .76235 Write options with K = .76

.74477 < S< .75267 Write options with K =

.75S < .74477 Write options

with K = .74

129

Final comments on Example 3.In the example, the OCC charges a USD35 per exercised call. Thus, it might be cheaper for Iris to buy back the calls and pay the brokerage fee of USD4 per call in the event the options were in danger of being exercised. In addition, it is assumed that Iris will have the CAD300M on hand if the options are exercised. This would not be the case if actual Canadian dollar revenues were less than anticipated.

130

In that event, the options would need tobe repurchased prior to expiration. Each of the three choices of strike price will have a different payoff, depending

on the movement in the exchange rate. But Iris' expectation regarding the exchange rate is not the only relevant criterion for choosing a risk-management strategy.The possible variation in the underlying position should also be considered.

131

Here are the maximal and minimal payoffs for each of the call-writing choices, compared to the un

hedged position and a forward market

hedge:

132

Strategy Max Value Min ValueUnhedgedLong Position Unlimited Zero.

Short Forward USD223,410,000 USD223,410,000

.76 callUSD228,705,000 Unhedged min + USD915,000.

.75 callUSD226,716,000 Unhedged min + USD1,926,000.

.74 callUSD225,903,000 Unhedged min + USD4,113,000.

133

Futures options

A FORWARDIS A CONTRACT IN WHICH ONE PARTY COMMITS TO BUY AND THE OTHER PARTY COMMITS TO SELL A PRESPECIFIED AMOUNT OF AN AGREED UPON COMMODITY FOR A PREDETERMINED PRICE ON A SPECIFIC DATE IN THE FUTURE.

134

BUY means OPEN A LONG POSITIONSELL means OPEN A SHORT POSITION

t

Buy or sell a forward T Tim

e

Delivery and payment

135

EXAMPLE:

GBP 18.5.99

SPOT USD1,6850/GBP

30 days forward USD1,7245/GBP




The existence of forward exchange rates implies that there is a demand and supply for the GBP for future dates.

136

Profit from aLong Forward Position

P/L

Price of Underlying at Maturity, STF

137

Profit from a Short Forward Position

P/L

Price of Underlying at Maturity, STF

138

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, futures contracts are traded on organized exchanges

139

A FUTURES

Is a STANDARDIZED FORWARD traded on an organized exchange.

STANDARDIZATION

THE COMMODITY, TYPE AND QUALITY,THE QUANTITY , PRICE QUOTES,DELIVERY DATES and PROCEDURES,MARGIN ACCOUNTS,The MARKING TO MARKET process.

140

NYMEX. Light, Sweet Crude Oil

Trading UnitFutures: 1,000 U.S. barrels (42,000 gallons).Options: One NYMEX Division light, sweet crude oil futures contract.Price QuotationFutures and Options: Dollars and cents per barrel.Trading HoursFutures and Options: Open outcry trading is conducted from 10:00 A.M. until 2:30 P.M.After hours futures trading is conducted via the NYMEX ACCESS® internet-based trading platform beginning at 3:15 P.M. on Mondays through Thursdays and concluding at 9:30 A.M. the following day. On Sundays, the session begins at 7:00 P.M. All times are New York time. Trading MonthsFutures: 30 consecutive months plus long-dated futures initially listed 36, 48, 60, 72, and 84 months prior to delivery.Additionally, trading can be executed at an average differential to the previous day's settlement prices for periods of two to 30 consecutive months in a single transaction. These calendar strips are executed during open outcry trading hours.Options: 12 consecutive months, plus three long-dated options at 18, 24, and 36 months out on a June/December cycle.

141

Minimum Price FluctuationFutures and Options: $0.01 (1¢) per barrel ($10.00 per contract). Maximum Daily Price FluctuationFutures: Initial limits of $3.00 per barrel are in place in all but the first two months and rise to $6.00 per barrel if the previous day's settlement price in any back month is at the $3.00 limit. In the event of a $7.50 per barrel move in either of the first two contract months, limits on all months become $7.50 per barrel from the limit in place in the direction of the move following a one-hour trading halt.Options: No price limits.Last Trading DayFutures: Trading terminates at the close of business on the third business day prior to the 25th calendar day of the month preceding the delivery month. If the 25th calendar day of the month is a non-business day, trading shall cease on the third business day prior to the last business day preceding the 25th calendar day.Options: Trading ends three business days before the underlying futures contract.

142

Exercise of OptionsBy a clearing member to the Exchange clearinghouse not later than 5:30 P.M., or 45 minutes after the underlying futures settlement price is posted, whichever is later, on any day up to and including the option's expiration.Options Strike PricesTwenty strike prices in increments of $0.50 (50¢) per barrel above and below the at-the-money strike price, and the next ten strike prices in increments of $2.50 above the highest and below the lowest existing strike prices for a total of at least 61 strike prices. The at-the-money strike price is nearest to the previous day's close of the underlying futures contract. Strike price boundaries are adjusted according to the futures price movements.DeliveryF.O.B. seller's facility, Cushing, Oklahoma, at any pipeline or storage facility with pipeline access to TEPPCO, Cushing storage, or Equilon Pipeline Co., by in-tank transfer, in-line transfer, book-out, or inter-facility transfer (pumpover).

143

Delivery PeriodAll deliveries are rateable over the course of the month and must be initiated on or after the first calendar day and completed by the last calendar day of the delivery month.Alternate Delivery Procedure (ADP)An alternate delivery procedure is available to buyers and sellers who have been matched by the Exchange subsequent to the termination of trading in the spot month contract. If buyer and seller agree to consummate delivery under terms different from those prescribed in the contract specifications, they may proceed on that basis after submitting a notice of their intention to the Exchange.Exchange of Futures for, or in Connection with, Physicals (EFP)The commercial buyer or seller may exchange a futures position for a physical position of equal quantity by submitting a notice to the exchange. EFPs may be used to either initiate or liquidate a futures position.

144

Deliverable GradesSpecific domestic crudes with 0.42% sulfur by weight or less, not less than 37° API gravity nor more than 42° API gravity. The following domestic crude streams are deliverable: West Texas Intermediate, Low Sweet Mix, New Mexican Sweet, North Texas Sweet, Oklahoma Sweet, South Texas Sweet.Specific foreign crudes of not less than 34° API nor more than 42° API. The following foreign streams are deliverable: U.K. Brent and Forties, and Norwegian Oseberg Blend, for which the seller shall receive a 30¢-per-barrel discount below the final settlement price; Nigerian Bonny Light and Colombian Cusiana are delivered at 15¢ premiums; and Nigerian Qua Iboe is delivered at a 5¢ premium.InspectionInspection shall be conducted in accordance with pipeline practices. A buyer or seller may appoint an inspector to inspect the quality of oil delivered. However, the buyer or seller who requests the inspection will bear its costs and will notify the other party of the transaction that the inspection will occur.

145

Position Accountability LimitsAny one month/all months: 20,000 net futures, but not to exceed 1,000 in the last three days of trading in the spot month.Margin RequirementsMargins are required for open futures or short options positions. The margin requirement for an options purchaser will never exceed the premium.Trading SymbolsFutures: CLOptions: LO

146

NYMEX Copper Futures

Trading Unit 25,000 pounds.

Price Quotation Cents per pound. For example, 75.80¢ per pound.

Trading Hours Open outcry trading is conducted from 8:10 A.M. until 1:00 P.M. After-hours futures trading is conducted via the NYMEX ACCESS®

Trading Months Trading is conducted for delivery during the current

calendar month and the next 23 consecutive calendar months.

Minimum Price Price changes are registered in multiples of five one Fluctuation hundredths of one cent ($0.0005, or 0.05¢) per pound,

equal to $12.50 per contract. A fluctuation of one cent ($0.01 or 1¢) is equal to $250.00 per contract.

147

Maximum Daily Initial price limit, based upon the preceding day'sPrice Fluctuation settlement price is $0.20 (20¢) per pound. Two

minutes after either of the two most active months trades at the limit, trading in all months of futures and options will cease for a 15-minute period. Trading will also cease if either of the two active months is bid at the upper limit or offered at the lower limit for two minutes without trading. Trading will not cease if the limit is reached during the final 20 minutes of a day's trading. If the limit is reached during the final half hour of trading, trading will resume no later than 10 minutes before the normal closing time. When trading resumes after a cessation of trading, the price limits will be expanded by increments of 100%.

Last Trading Day Trading terminates at the close of business on the third to last business day of the maturing delivery month.

148

Delivery Copper may be delivered against the high-grade copper contract only from a warehouse in the United States licensed or designated by the Exchange. Delivery must be made upon a domestic basis; import duties or import taxes, if any, must be paid by the seller, and shall be

made without any allowance for freight.

Delivery Period The first delivery day is the first business day of the delivery month; the last delivery day is the last business day of the delivery month.

Margin Requirements Margins are required for open futures and

short options positions. The margin requirement for an options purchaser

will never exceed the premium paid.

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CBOT Corn Futures Trading Unit 5,000 bushels

Tick Size ¼ cent per bushel ($12.50 per contract)

Daily Price Limit 12 cents per bushel ($600 per contract) above or below the previous day’s settlement price (expandable to 18 cents per bushel). No limit in the spot month.

Contract Months December, March, May, July, September

Trading Hours 9:30 a.m. to 1:15 p.m. (Chicago time), Monday through Friday. Trading in expiring contracts closes at noon on the last trading day.

Last Trading Day Seventh business day preceding the last business day of the delivery month.

Deliverable Grades No. 2 Yellow at par and substitution at differentials established by the exchange.

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MARGIN ACCOUNTS

A MARGIN is an amount of money that must be deposited in a margin

account in order to open any futures position, long or short. It is a “good

will” deposit. The clearinghouse maintains a system of margin

requirements from all traders, brokers and futures commercial merchants.

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MARGIN ACCOUNTS.

There are two types of margins:

The initial margin: The amount that every trader must deposit with the broker upon opening a futures account; short or long.

The initial deposit is the investor EQUITY. This equity changes on a daily

basis because: all profits and losses must be realized by the

end of every trading day.

152

MARGIN ACCOUNTS.

The maintenance (variable) margin:

This is a minimum level of the trader’s equity in the margin account.

If the trader’s equity falls below this level, the trader will receive a margin

call requiring the trader to deposit more funds and bring the account to

its initial level. Otherwise, the account will be closed.

153

Most of the time, Initial margins are between 2% to 10% of the position

value. Maintenance (variable) margin is usually around 70 - 80% of the

initial margin.

Example: a position of 10 CBT treasury bonds futures ($100,000 face value each) at a price of $75,000 each.

The initial margin deposit of 5% of $750,000 is: $37,500.

If the variable margin is 75% Margin call if the amount in the margin account falls to $26,250.

154

Example of a Futures Trade

On JUN 5 an investor takes a long position in 2 NYMEX DEC gold futures.

contract size is 100 oz.futures price is USD590/ozmargin requirement is 5%.USD2,950/contract or USD5,900

total.Maintenance margin is 75%.USD2,212.5/contract or USD4,425Total.

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Daily equity changes in the margin account:

MARKING TO MARKET

Every day, upon the market close, all profits and losses for that day must be

realized. I.e.,

SETTLED.

The benchmark prices for this process are:

SETTLEMENT PRICES

156

A SETTLEMENT PRICE IS

the average price of trades during the last several minutes of the trading

day.

Every day, when the markets close,

SETTLEMENT PRICES

for the futures of all products and for all months of delivery are set. They

are then compared with the previous day settlement prices and to the

trading prices on that day and the difference must be settled overnight

157

Open a long position in 10 JUNE crude oil futures for: $58.50/bbl. VALUE: (10)(1,000)($58.50) = $585,000INITIAL MARGIN

= (.03)($585,000) = $17,550; VAR. MARGIN = 80%

SETTLE PRICE

VALUE

MARKET-TO-MARKET

MARGIN BALANCE

DAY 0 $58.60 $586,000 + 1,000 $18,550 DAY 1 $58.42 $584,200 - $1,800 $16,750

DAY 2 $58.75 $587,500 + $3,300 $20,050

DAY 3 DAY 4

$ 58.32 $ 58.08

$583,200 $580,800

$4,300 -$2400

$15,750 $13,350

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13,350/17,550 = .761 < .8

MARGIN CALL

SEND $4,200 TO MARGIN ACCOUNT

TO BRING IT UP TO $17,550

DAY 5 $58.27 $582,700 + $1,900 $19,450

159•$1M face value of 90-day T-bills. P = 1,000,000[1 - (1 – Q/100)(90/360)]. ** Initial Margin is assumed to be 5% of contract fee.

Date Settlement price:Q

Dollar settlement price = P

Mark-to-Market for the long

Margin Account **

June 2 92.23 980,575 50,000

3 92.73 981,825 $1250 51,250

4 92.83 982,075 250 51,500

5 93.06 982,650 575 52,075

6 93.07 982,675 25 52,100

9 93.48 983,700 1025 53,125

10 93.18 982,850 -750 52,375

11 93.32 983,300 350 52,725

12 93.59 983,975 675 53,400

13 93.84 984,600 625 54,025

16 93.71 984,275 -325 53,700

17 93.25 983,126 -1150 52,550

18 93.12 982,800 -325 52,225

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DeliveryIf a contract is not closed out beforematurity, it is usually settled by deliveringThe assets underlying the contract. WhenThere are alternatives about what isdelivered, where it is delivered, and when

itis delivered, the party with the short position chooses.Few contracts are settled in cash.For example, those on stock indices andEurodollars.

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A futures markets statistic:97-98% of all the futures for all

delivery months and for all underlying commodities do not get

to delivery!!This means that:1. Only 2-3% do reach delivery.2. Most traders close their positions

before they get to delivery.3. Most traders do not open futures

positions for business.4. Most futures are traded for Risk

Management reasons,

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Mechanics of Call Futures Options

The underlying asset isA FUTURES.

This means that when you exercise afutures option you become committedto BUY or SELL the asset underlying

thefutures, depending on whether youhave a call or a put.

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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 most recent settlement futures price, F(settle) over K.

The writer obtains short position in the futures and the cash amount in his/her margin account is adjusted opposite to 2. above.

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The Payoff of a futures call exerciseIf the futures position is closed out on date j, which is immediately upon the call exercise:Payoff:

F(settle) – K + Fj,T – F(settle)

= Fj,T – K,

where Fj,T is futures price at time the futures is closed.

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Mechanics of Put Futures Option

When a put futures option is exercised the holder acquires

1. A short position in the futures 2. A cash amount equal to the excess of

the put strike price, K, over the mostrecent futures settlement price F(settle).

The put writer obtains a long futures position and his/her margin account is adjusted opposite to 2. above.

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The Payoff of a futures put exercise

Payoff from put exercise:

K – F(settle) + F(settle) – Fj,T

= K – Fj,T

where Fj,T is futures price at time the put is exercised and the futures is closed.

167

Put-Call Parity for Futures Options (p 329)

ct + Ke-r(T-t) = pt + Ft,Te-r(T-t)

168

Black’s Formula (P 333)

t-Tσ

t)/2-(T2σ/K)ln(Fd

t-Tσ

t)/2-(T2σ/K)ln(Fd

)dN(F)dKN(ep

)KN(d)N(dFec

t,T2

t,T1

1t,T2t)-r(T

t

21t,Tt)-r(T

t

1 options on stock indices and currencies chapter 15

Documents

index portfolio

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period portfolio value

entire portfolio value

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