Options on
Stock Indices and Currencies
Chapter 15
1
The cash market
Stock 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.
2
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.
3
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
P

I=
;
i
N
i = 1,..., N.
Initially D = N and the index is set at some level.
To ensure continuity, the divisor is adjusted over
4
time.
EXAMPLES OF INDEX ADJUSMENTS
STOCK SPLITS: 2 for 1.
1.
(P1  P2  ... PN ) / D1  I1
2.
1
(P1  P2  ... PN ) / D 2  I1
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
5
CHANGE OF STOCKS IN THE INDEX
1.
(P1  P2 (ABC)  ... PN ) / D1  I1
2.
(P1  P2 ( XYZ)  ... PN ) / D 2  I1
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
6
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.
7
ADDITIONAL STOCKS
1.
2.
(P1  P2  ... PN ) / D1  I1
(P1P2 ,...,PN PN+1 ) / D2 I1
1. (30 + 39 + 55 + 33 + 21)/4.5035= 39.5248
2. (30 + 39 + 55 + 33 + 21 + 35)/D = 39.5248
D = 5.389
8
VALUE WEIGHTED INDEXES
S & P500, NIKKEI 225, VALUE LINE
NP
V

I

 w
V
N P
ti ti
ti
t
Bi Bi
ti
Bp
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.
9
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.
R It   w tiR ti ;
N ti Pti
Vti
w ti 

.
 N tiPti VtP
10
stock
Pti
Nti
Vti
wti
Pt+1i Rti
Federal Mogul
Martin Arietta
IBM
US West
Bausch&Lomb
First Union
Walt Disney
Delta Airlines
Total
18
73
50
45
55
50
40
55
9,000
8,000
4,000
5,000
15,000
10,000
12,000
20,000
162,000
584,000
200,000
225,000
825,000
500,000
480,000
1,100,000
4,076,000
.0397
.1432
.0491
.0552
.2024
.1227
.1178
.2699
1.000
19.8
75
48
49
52
57
46
59
.1000
.0274
-.0400
.0889
-.0545
.1400
.1500
.0727
Rp = (.0397)(.1) + (.1432(.0274) + (.0491)(-.04) + (.0552)(.0889) +
(.2024)(-.0545) + (.1227)(.14) + (.1178)(.15) + (.2699)(.0727) = 0.0543
11
or 5.43%
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%.
12
THE RATE OF RETURN ON THE INDEX
N
I t +1 I t
R It 

It
t +1i
VB
Pt +1i

N
ti
N
ti
Pti
VB
Pti
VB
N


t +1i
Pt +1i  N ti Pti
N
ti
Pti
;
but, N t +1i  N ti . Thus,
N (P


N
ti
t +1i
ti
Pti )
Pti
13
Pt 1i  Pti
 N tiPti P
ti
R It 
,
 N tiPti
N PR


N P
ti
ti
ti
ti
. Rewrite this as :
ti
N tiPti
 [
]R ti, or
 N tiPti
Vti

R ti . Finally,
VtP
R It   w tiR ti . Notice, again, that :
N ti Pti Vti
w ti 
 .
 N tiPti VtP
14
THE BETA OF A PORTFOLIO
Definitions:
COV(R i , R M )
βi 
.
VAR(R M )
COV(R P , R M )
βP 
.
VAR(R M )
R
COV(R P , R I )
βP 
.
VAR(R I )
15
THE BETA OF A PORTFOLIO
THEOREM:
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
R
market portfolio.
Let P denote any portfolio, i denote the
individual stock; i = 1, 2, …,N in the
portfolio and I denote the index.
16
By definition:
COV(R P , R I )
βP 
.
VAR(R I )
Substituti ng for R P ; R P   w i R i ,
βP 
COV([  w i R i ], R I )
VAR(R I )
.
Recall that the covariance is
a linear operator, thus :
w iCOV(R i , R I )
βP  
, or :
VAR(R I )
 COV(R i , R I 
βP   wi 
 w iβi .

 VAR(R I ) 
This concludes the proof.
17
STOCK PORTFOLIO BETA
STOCK NAME
FEDERAL MOUGUL
MARTIN ARIETTA
IBM
US WEST
BAUSCH & LOMB
FIRST UNION
WALT DISNEY
DELTA AIRLINES
PRICE
18.875
73.500
50.875
43.625
54.250
47.750
44.500
52.875
SHARES
9,000
8,000
3,500
5,400
10,500
14,400
12,500
16,600
VALUE
WEIGHT
169,875
588,000
178,063
235,575
569,625
687,600
556,250
877,725
3,862,713
BETA
.044
.152
.046
.061
.147
.178
.144
.227
1.00
.80
.50
.70
1.1
1.1
1.4
1.2
P = .044(1.00) + .152(.8) + .046(.5)
+ .061(.7) + .147(1.1) + .178(1.1)
+ .144(1.4) + .227(1.2) = 1.06
18
A STOCK PORTFOLIO BETA
STOCK NAME
BENEFICIAL CORP.
CUMMINS ENGINES
GILLETTE
KMART
BOEING
W.R.GRACE
ELI LILLY
PARKER PEN
PRICE
40.500
64.500
62.000
33.000
49.000
42.625
87.375
20.625
SHARES
11,350
10,950
12,400
5,500
4,600
6,750
11,400
7,650
VALUE
459,675
706,275
768,800
181,500
225,400
287,719
996,075
157,781
3,783,225
WEIGHT BETA
.122
.187
.203
.048
.059
.076
.263
.042
.95
1.10
.85
1.15
1.15
1.00
.85
.75
P = .122(.95) + .187(1.1) + .203(.85)
+ .048(1.15) + .059(1.15) + .076(1.0)
+ .263(.85) + .042(.75) = .95
19
Sources of calculated Betas and calculation inputs
Example: ß(GE) 6/20/00
Source
ß(GE)
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)
Quicken.Excite.com
1.23
MSN Money Central
1.20
DailyStock.com
1.21
Standard & Poors Compustat Svcs
S&P Personal Wealth
1.2287
S&P Company Report)
1.23
Index Data
S&P500I
Horizon
Monthly P ice
3 (5) yrs
S&P500I
Monthly Price
5 yrs (Monthly)
S&P500I
Daily Price
5 yrs (Daily)
S&P500I
Monthly Price
5 yrs (Monthly)
Charles Schwab Equity Report Card 1.20
S&P Stock Report
AArgus Company Report
1.23
1.12
Market Guide
YYahoo!Finance
1.23
Motley Fool
1.23
20
STOCK INDEX OPTIONS
1.
One contract = (I)($m)
(WSJ)
2.
ACCOUNTS ARE SETTLED BY CASH
21
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 riskfree rate is 8%. The SP500I dividend
payout ratio is q = 3% and its annual VOL
σ= 25%.
22
MoneyGone offer:
Deposit:
$A now.
Receive:
$AMax{0, .5RI} in 6 months.
Denote the date in six month = T.
Rewrite MoneyGone offer at T:
IT  I0
Retrurn  ($A)Max{0, (.5)(
)}
I0
$A
Return  (.5)( )Max{0, IT  I0 }
I0
23
The expression:
Max{0, IT  I0}
is equivalent to the at-expiration cash flow
of an at-the money European call option on
the index, if you notice that K = I0.
Calculate this options value based on:
S0 = K = I0; T – t = .5; r = .08; q = .03
and σ = .25. Using DerivaGem:
c = .08137.
Thus, MoneyGone’s promise is equivalent
24
to giving the client NOW, at time 0, a value
of:
(.5)(.08137)($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:
1
$A
R  ln[
]  .083
.5 $.959315A
= 8.3%
25
STOCK INDEX OPTIONS FOR
PORTFOLIO INSURANCE
Problems:
1. How many puts to buy?
2. Which exercise price will guarantee
a desired level of protection?
The answers are not easy because the
index underlying the puts is not the
portfolio to be protected.
26
The protective put with a single stock:
AT
STRATEGY
ICF
Hold the
stock
Buy put
-St
TOTAL
-St – p
-p
EXPIRATION
ST < K
ST
K - ST
K
ST ≥ K
ST
0
ST
27
The protective put consists of holding the
portfolio and purchasing n puts on an
index. Current t = 0; Expiration T = 1.
AT
STRATEGY
EXPIRATION (T = 1)
ICF (t = 0)
-V0
I1 < K
I1 ≥ K
V1
V1
Hold the
portfolio
Buy n puts
-nP($m)
TOTAL
-V0 –nP($m) V1+n($m)(K- I1)
n(K- I1)($m)
0
V1
28
WE USE
THE CAPITAL ASSET PRICING MODEL.
For any security i, the expected excess
return on the security and the expected
excess return on the market portfolio are
linearly related by their beta:
ER i  rF  βi (ER M  rF )
ER p  rF  β p (ER M  rF )
29
THE INDEX TO BE USED IN THE
STRATEGY, IS TAKEN TO BE A PROXY FOR
THE MARKET PORTFOLIO, M. FIRST,
REWRITE THE ABOVE EQUATION FOR THE
INDEX I AND ANY PORTFOLIO P :
ER p  rF  β p (ER I  rF ).
30
Second, as an approximation, rewrite the
CAPM result, with actual returns:
R p  rF  β p (R I  rF ).
In a more refined way, using V and I for the
portfolio and index market values,
respectively:
V1 - V0  D P
I1 - I 0  D I
 rF  β p [
 rF ].
V0
I0
31
NEXT, use the ratio Dp/V0 as the portfolio’s
annual dividend payout ratio qP and DI/I0
the index annual dividend payout ratio, qI.
V1 - V0
I1 - I 0
 q P  rF  β p [
 q I  rF ]
V0
I0
V1
I1
 1  q P  rF  β p [ - 1  q I - rF ]
V0
I0
The ratio V1/ V0 indicates the portfolio
required protection ratio.
32
For example:
V1
 .90,
V0
The manager wants V1, to be down to no
more than 90% of the initial portfolio
market value, V0:
V1 = (.9)V0.
We denote this desired level of hedging
by
(V1/ V0)*.
This is a decision variable.
33
1.
The number of puts is:
V0
n  βp
.
($m)I 0
34
2. The exercise price, K, is determined by
substituting I1 = K and the required level,
(V1/ V0)* into the equation:
V1
I1
 1  q P  rF  β p [ - 1  q I - rF ],
V0
I0
and solving for K:
V1
K
( ) * 1  q P  rF  β p [ - 1  q I - rF ].
V0
I0
I0 V1
K  [( ) * q p - (β p )q I  (1  rF )(β p -1)].
β p V0
35
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 riskfree rate is 10%, while the portfolio and
the index annual dividend payout ratios are
5% and 6%, respectively. The data are
36
summarized below:
V1
V0  $25,000,000; ( )*  .9; I0  1,250;
V0
$m  $100;
The annual rates are : rF  10%; q p  5%; q I  6%.
Finally, β  2.4. Period  half a year.
Solution: Purchase
V0
n  βp
($m)I 0
$25,000,000
n  2.4
 480 puts.
($100)(1,2 50)
37
The exercise price of the puts is:
I 0 V1
K  [( ) * q p - (β p )q I  (1  rF )(β p - 1)].
β p V0
1,250
K
[.9  .025  (2.4).03  (1  .05)(2.4  1)
2.4
K  1,210.
Solution: Purchase n = 480 six-months
puts with exercise price K = 1,210.
38
We rewrite the Profit/Loss table for the protective put
strategy:
AT EXPIRATION
STRATEGY
INITIAL CASH
FLOW
Hold the portfolio
-V0
I1 < K
V1
I1 ≥ K
V1
Buy n puts
-n P($m)
TOTAL
n(K - I1)($m)
-V0 - nP($m) V1+n($m)(K - I1)
0
V1
We are now ready to calculate the floor
level of the portfolio: V1+n($m)(K- I1) 39
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($m)I1
40
Substitute for n:
V0
n  βp
.
($m)I 0
V0
Min porfolio value  V1  β p
($m)K
($m)I 0
V0
 βp
($m)I 1
($m)I 0
V0
V0
Min portfolio value  V1  β p
K - βp
I1.
I0
I0
41
To substitute for V1 we solve the equation:
V1
I1
 1  q P  rF  β p [ - 1  q I - rF ]
V0
I0
I1
V1  V0 (1  q P  rF  β p [ - 1  q I - rF ])
I0
V0
V1  β p I1
I0
 V0 1  rF  q p  β p [q I  1  rF ]
42
3.
Substitution V1 into the equation for
the Min portfolio value
Min portfolio value 
V0
βp
K  V0 [β p q I  q p  (1  rF )(1  β p )].
I0
The desired level of protection is made at
time 0. This determines the exercise
price and management can also calculate
the minimum portfolio value.
43
Minimum
portfolio
value 
V0
β p K  V0 [β pq I  q p  (1  rF )(1  β p )].
I0
$25,000,000
 2.4
1,210
1,250
 $25,000,000[2.4(.03) - .025  (1  .05)(1 - 2.4)]
 $22,505,000.
44
Example (p326) protection for 3 months
V1
V0  $500,000; ( )*  .9; I 0  1,000; $m  $100;
V0
The annual rates are : rF  12%; q p  4%; q I  4%.
Finally, β  2.0
Solution: Purchase
V0
n  βp
($m)I 0
$500,000
n  2.0
 10 puts.
($100)(1,0 00)
45
The exercise price of the puts is:
I 0 V1
K  [( ) * q p - (β p )q I  (1  rF )(β p - 1)].
β p V0
1,000
K
[.9  .01  (2.0).01  (1  .03)(2.0  1)
2.0
K  960.
Solution: Purchase n = 10 three -months
puts with exercise price K = 960.
46
Min portfolio value 
V0
β p K  V0 [β pq I  q p  (1  rF )(1  β p )].
I0
$500,000
 2.0
960
1,000
 $500,000[2.0(.01) - .01  (1  .03)(1 - 2.0)]
 $450,000.
47
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 $450,000 in three
months.
48
A SPECIAL CASE: In the case that
a.
β=1
b.
qP =qI,
the portfolio is statistically similar to the
index. In this case:
V1
K  I 0 ( ) * and
V0
V0
n
($m)I 0
Min portfolio value is  V0 (V1/V0 ) .
*
49
Assume that in the above example:
βp = 1 and qP =qI, then:
V1
K  I 0 ( ) *  1,250(.9)  1,125.
V0
V0
$25,000,000
n

 200 puts and
($m)I 0
$100(1250)
Min portfolio value 
V  V0 (.9)  $25,000,000(.9)  $22,500,000.
*
1
*
50
Example: (p326-27)
βp = 1 and qP =qI, then:
V1
K  I 0 ( ) *  1,000(.9)  900.
V0
V0
$500,000
n

 5 puts and
($m)I 0
$100(1,000)
Min portfolio value 
V  V0 (.9)  $500,000(.9)  $450,000.
*
1
*
51
A Zero cost Collar
AT EXPIRATION
STRATEGY
ICF
I1 < KP
portfolio
Buy n puts
Sell n calls
TOTAL
-V0
-nP($m)
nC($m)
-V0
KP < I1 < KC
I1 ≥ KC
V1
n(KP-I1)($m)
0
V1
0
0
V1
0
n(I1-KC)($m)
V1+
n($m)(KP - I1)
V1
V1 –
n($m)(I1-KC)
52
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:
53
For I1  K P :
V0
V0 [rF  q P  β P (1  rF  q I )]  β P
KP
I0
For K P  I1  K C :
V0
V0 [rF  q P  β P (1  rF  q I )]  β P
I1
I0
For I1  K C :
V0
V0 [rF  q P  β P (1  rF  q I )]  β P
KC
I0
54
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.
55
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
56
S(USD/GBP) = 1.6821
1
1
=
=
.5945 S(GBP/USD)
S(GBP/USD)  .5945
57
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(FC1/FC3)
S(FC1/FC2) =
S(FC2/FC3)
S(FC3/FC2)
=
S(FC3/FC1)
58
CURRENCY CROSS RATES – OCT. 13, 04
USD
GBP
CAD
USD
1
1.7972
0.798212 1.2393
GBP
0.556421 1
0.444141 0.689572 0.407023
CAD
1.25279
1
1.55259
0.916425
EUR
0.806907 1.45017
.644082
1
0.590254
AUD
1.36705
1.09119
1.69418
1
2.25153
2.45686
EUR
AUD
0.731502
59
CURRENCY CROSS RATES
EXAMPLE:
FC1 = USD; FC2 = MXP; FC3 = GBP.
USD
MXP
GBP
USA
1.0000
0.0997
1.6603
MEXICO
10.0301
1.000
16.653
UK
0.6023
0.06005
1.000
S(USD/MXP)  0.0997; S(USD/GBP)  1.6603
S(MXP/USD)  10.0301; S(MXP/GBP)  16.653
S(GBP/USD)  0.6023; S(GBP/MXP)  0.06005.
60
CURRENCY CROSS RATES
Let FC1  USD;
EXAMPLE
FC2  MXP;
FC3  GBP.
S(GBP/MXP)
S(USD/GBP)
S(USD/MXP) =
=
.
S(GBP/USD)
S(MXP/GBP)
S(GBP/MXP) 0.06005

 0.0997.
S(GBP/USD)
0.6023
S(USD/GBP) 1.6603

 0.0997.
S(MXP/GBP) 16.653
61
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
S(GBP/USD)
THEORY :
= S(CHF/USD)
S(GBP/CHF)
BUT : 0.6394 = 1.8031  1.7920
0.3546
S(USD/GBP)
SIMILARLY :
= S(CHF/GBP)
S(USD/CHF)
1.5640
BUT :
= 2.8029 < 2.8200
0.5580
62
THE CASH ARBITRAGE ACTIVITIES:
USD1,000,000
USD1,006,134.26
0.6394
0.5580
GBP639,400
CHF1,803,108
2.8200
63
Forward rates, An example:
GBP
18.5.99
SPOT
USD1.6850/GBP
30 days forward
USD1.7245/GBP
60 days forward
USD1.7455/GBP
90 days forward
USD1.7978/GBP
180 days forward
USD1.8455/GBP
The existence of forward exchange rates
implies that there is a demand and supply
64
for the GBP for future dates.
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.
F(DC/FC) = S(DC/FC)e
1.8455  1.6850e
(rDOM - rFOR )(T - t)
180
(rUS rUK )
365
65
.
NO ARBITRAGE: CASH-AND-CARRY
TIME
CASH
FUTURES
t
(1) BORROW DC. rDOM
(4) SHORT FOREIGN CURRENCY
(2) BUY FOREIGN CURRENCY
FORWARD
DC/S(DC/FC) = DCS(FC/DC)]
(3) INVEST IN BONDS
DENOMINATED IN THE
Ft,T(DC/FC)
AMOUNT:
DCS(FC/DC)e
rFOR (T-t)
FOREIGN CURRENCY rFOR
T
(3) REDEEM THE BONDS EARN (4) DELIVER THE CURRENCY TO
rFOR (T-t)
CLOSE THE SHORT POSITION
DCS(FC/DC)e
(1) PAY BACK THE LOAN
DCe
DCe
rDOM (T -t)
RECEIVE:
F(DC/FC)DCS(FC/DC)e
IN THE ABSENCE OF ARBITRAGE:
rD (T t)
 F(DC/FC)DC S(FC/DC)e
Ft,T (DC/FC)  St (DC/FC)e
rFOR (T-t)
rFOR (T-t)
(rDOM - rFOR )(T-t)
66
NO ARBITRAGE: REVERSE CASH – AND - CARRY
TIME
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:
FCS(DC/FC)e
(3) INVEST IN T-BILLS
FOR RDOM
T
REDEEM THE T-BILLS EARN
rDOM (T-t)
TAKE DELIVERY TO CLOSE
FCS(DC/FC)e
THE LONG POSITION
PAY BACK THE LOAN
RECEIVE
FCe
rFOR (T - t)
IN THE ABSENCE OF ARBITRAGE:
FCe
rFOR (T - t)
R DOM (T-t)
rDOM ( T-t)
FCS(DC/FC)e
F(DC/FC)
rDOM ( T-t)
FCS(DC/FC)e

F(DC/FC)
Ft,T (DC/FC)  St (DC/FC)e
(rDOM rFOR )( T-t)
67
FROM THE CASH-AND-CARRY STRATEGY:
Ft,T (DC/FC) St (DC/FC)e (rDOM - rFOR )(T-t)
FROM THE REVERSE CASH-AND-CARRY STRATEGY:
(rDOM - rFOR )(T -t)
t
t,T
F (DC/FC)  S (DC/FC)e
THE ONLY WAY THE TWO INEQUALITIES HOLD
SIMULTANEOUSLY IS BY BEING AN EQUALITY:
Ft,T (DC/FC) = St (DC/FC)e
(rDOM - rFOR )(T - t)
68
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%
F(USD/GBP Theoretical ) = 1.5640e
(.0785 - .12)
209
365
= 1.5273
The market forward rate 1.5328 is overvalued
relative to the theoretical, no arbitrage forward
rate 1.5273.
CASH AND CARRY
69
CASH AND CARRY
TIME
CASH
MAY 25
FUTURES
(1) BORROW USD100M AT 7. 85%
FOR 209 DAYS
SHORT DEC 20
GBP68,477,215 FORWARD.
F = USD1.5328/GBP
(2) BUY GBP63,930,000
(3) INVEST THE GBP63,930,000
IN BRITISH BONDS
DEC 20 RECEIVE GBP68,477,215
209
.12
365
63,930,000e
DELIVER GBP68,477,215
FOR USD104,961,875.2
= GBP68,477, 215
REPAY YOUR LOAN:
100Me
.0785
209
365
= USD104,59 7,484.3
70
PROFIT: USD104,961,875.2 - USD104,597,484.3 = USD364,390.90
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
71
from this band of rates arbitrage exists.
Foreign Exchange Quotes for
USD/GBP on Aug 16, 2001
Spot
Bid
1.4452
Ask
1.4456
1-month forward
1.4435
1.4440
3-month forward
1.4402
1.4407
6-month forward
1.4353
1.4359
12-month forward
1.4262
1.4268
72
FOR BID AND ASK QUOTES :
1
S(EUR/FC)

ASK S(FC/EUR)
S(EUR/FC)
BID
S(USD/NZD)
S(USD/NZD)
S(NZD/USD)
S(NZD/USD)

ASK
BID
ASK
BID
1
S(FC/EUR)
BID
ASK
 USD.5NZD, buy NZD1 pay 50 cents.
 USD.480/N ZD, sell NZD1 get 48 cents.
 NZD2.083/USD, buy USD1 pay NZD2.083.
 NZD2.000/USD, sell USD1 get NZD2.
73
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.
74
NO ARBITRAGE: CASH - AND - CARRY
TIME
CASH
FUTURES
t
(1) BORROW DC. rD,ASK
(4) SHORT FOREIGN CURRENCY FORWARD
(2) BUY FOREIGN CURRENCY
DC/SASK(DC/FC)
(3) INVEST IN BONDS
DENOMINATED IN THE
FOREIGN CURRENCY rF,BID
T
REDEEM THE BONDS
EARN:
DC/SASK (DC/FC)e
rF,BID (T-t)
DELIVER THE CURRENCY TO CLOSE THE SHORT POSITION
r
DC/SASK (DC/FC)e F,BID
PAY BACK THE LOAN
DCe
FBID (DC/FC)
(T-t)
RECEIVE:
rD,ASK (T-t)
FBID (DC/FC)DC/ S(DC/FC)e rFOR (T-t)
IN THE ABSENCE OF ARBITRAGE:
DCe
rD,ASK (Tt)
 FBID (DC/FC)A/S ASK (DC/FC)e
FBID (DC/FC)  SASK (DC/FC)e
rF,BID (T-t)
(rD,ASK - rF,BID )(T-t)
75
NO ARBITRAGE:
REVERSE CASH - AND - CARRY
TIME
CASH
FUTURES
t
(1) BORROW FC .
rF,ASK
(4) LONG FOREIGN CURRENCY FORWARD FOR
FASK(DC/FC)
(2) EXCHANGE FOR
FCSBID (DC/FC)e
FCSBID (DC/FC)
(3) INVEST IN T-BILLS
rD,BID (T -t)
FOR rD,BID
T
REDEEM THE T-BILLS EARN
FCSBID (DC/FC)e
PAY BACK THE LOAN
TAKE DELIVERY TO CLOSE THE LONG POSITION
rD,BID (T -t) RECEIVE in foreign currency, the amount:
r
rF,ASK (T-t)
FCe
FCSBID (DC/FC)e D,BID
FASK (DC/FC)
IN THE ABSENCE OF ARBITRAGE:
r
D,BID
rF,ASK (T-t)  FCSBID (DC/FC)e
FCe
FASK (DC/FC)
FASK (DC/FC)  SBID (DC/FC)e
( T -t)
( T - t)
(rD,BID rF,ASK )( T-t)
76
From Cash and Carry:
(1)
FBID (DC/FC)  SASK (DC/FC)e
(rD,ASK - rF,BID )(T-t)
From reverse cash and Carry
(2)
FASK (DC/FC)  SBID (DC/FC)e
(rD,BID rF,ASK )( T-t)
(3) And FASK(DC/FC) > FBID(DC/FC)
Notice that:
RHS(1) > RHS(2)
Define: RHS(1)  BU
RHS(2)  BL
77
F($/D)
FASK
BU
FASK(DC/FC) > FBID(DC/FC).
BU
FBID (DC/FC)  BU
FASK (DC/FC)  BL
BL
BL
FBID
Arbitrage exists only if both ask and bid
futures prices are above BU,
or both are below BL.
78
A numerical example:
Given the following exchange rates:
Spot
S(USD/NZ)
Forward
F(USD/NZ)
Interest rates
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
79
Inequality (1):
FBID (USD/NZ)
 SASK (USD/NZ)e
(rUS,ASK - rNZ,BID )(T - t)
0.4450 < (0.4438)e(0.108125 – 0.05875)/12 = 0.4456 = BU
Inequality (2):
FASK (USD/NZ)  SBID (USD/NZ)e
(rUS,BID rNZ,ASK )( T- t)
0.4480 > (0.4428)e(0.106875 – 0.06000)/12 = 0.4445 = BL
No arbitrage.
Lets see the graph
80
F
FASK = 0.4480
0.4456
BU
FBID = 0.4450
BL
FBID (USD/NZ)  0.4456  BU
Clearly: FASK($/FC) > FBID($/FC).
0.4445
FASK (USD/NZ)  0.4445  BL
An example of arbitrage:
FASK = 0.4480
FBID = 0.4465
81
Currency options
Units
USD/AUD
50,000AUD
USD/GBP
31,250GBP
USD/CAD
50,000CAD
USD/EUR
62,500EUR
USD/JPY
6,250,000JPY
USD/CHF
62,500CHF
Exercise Style: American- or European
options available for physically settled
contracts; Long-term options are
82
European-style only.
Expiration/Last Trading Day
The 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.
83
With the Canadian dollar spot price currently
at a level of USD.6556/CAD, strike prices
Standardized
would be listed
in half-cent Options
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.
Exercise Prices
Exercise 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.
84
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 longterm options.
85
Premium Quotation premiums for dollarbased 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.
86
FX Options As Insurance
Options 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.
87
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
88
interest rate adjustments.
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.
89
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.
90
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.
91
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.
92
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)
93
= USD5,532.
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.
94
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.
95
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.
96
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.
97
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.
98
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.
99
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.
100
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.
101
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
102
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
103
off. This concludes Example 2.
Writing Foreign Currency Options
General 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.
104
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.
105
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.
106
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.
107
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 108
writer.
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.
109
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.
110
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:
111
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.
112
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.
113
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:
114
c(K = USD.74/CAD) = USD.01379;
c(K = USD.75/CAD) = USD.00650;
c(K = USD.76/CAD) = USD.00313.
K
Value
One call
n
.74
USD689.5
6,000
Value
Total
Total
Fees:
Premium
USD4/call
USD4,137,000 USD24,000
.75
USD325.0
6,000
USD1,950,000
USD24,000
.76
USD156.5
6,000
USD939,000
USD24,000
115
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.
116
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:
117
If K = USD.74/CAD
Strategy
Write 6,000,
.74 calls
Initial
Cash Flow
S< USD.74/CAD
S>USD.74/CAD
USD4,113,000
0
-(S-.74)CAD300M
-USD210,000
(S)CAD300M
(S)CAD300M
(S)CAD300M
USD221,780,000
(S)CAD300M
+USD4.113,000
USD225,903,000
CAD
Total
P/L
USD4,113,000
Cash flow at Expiration
118
If K = USD.75/CAD
Strategy
Write 6,000,
.75 calls
Initial
Cash Flow
S< USD.75/CAD
S>USD.75/CAD
USD1,926,000
0
-(S-.75)CAD300M
-USD210,000
(S)CAD300M
(S)CAD300M
(S)CAD300M
USD224,700,000
(S)CAD300M
+USD1.926,000
USD226,716,000
CAD
Total
P/L
USD1,926,000
Cash flow at Expiration
119
If K = USD.76/CAD
Strategy
Write 6,000,
.76 calls
Initial
Cash Flow
S< USD.76/CAD
S>USD.76/CAD
USD915,000
0
-(S-.76)CAD300M
-USD210,000
(S)CAD300M
(S)CAD300M
(S)CAD300M
USD227,790,000
(S)CAD300M
+USD915,000
USD228,705,000
CAD
Total
P/L
USD915,000
Cash flow at Expiration
120
A consolidation of the three profit profile
tables:
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
121
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.
122
One possible comparison of the three
results is to evaluate the options strategy
vis-à-vis the immediate forward exchange.
Recall that when Iris enters the options
strategy the forward exchange rate is
F = USD.7447/CAD.
Thus, Iris may exchange the CAD300M
Forward for USD223,410,000 a future
break-even Spot rate can be calculated for
Every corresponding exercise price chosen:
123
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.
124
CONCLUSION
Writing the calls will protect Iris’ flow
in USD better than purchasing the CAD
forward if the spot rate in six months
will be above the corresponding
break- even exchange rates.
125
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.
126
The terminal spot rate is the market
exchange rate when the calls expire. It
is assumed that Iris pays a brokeragefee of USD4 per option contract and an
additional fee of USD35 per option to
the Options Clearing Corporation if the
options are exercised.
127
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
.74477 < S< .75267
S < .74477
Write options
with K = .76
Write options
with K = .75
Write options
with K = .74
128
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.
129
In that event, the options would need to
be 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.
130
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:
131
Strategy Max Value
Unhedged
Long
Position Unlimited
Short
Min Value
Zero.
Forward
USD223,410,000
USD223,410,000
.76 call
USD228,705,000
Unhedged min
+ USD915,000.
.75 call
USD226,716,000
Unhedged min
+ USD1,926,000.
.74 call
USD225,903,000
Unhedged min
+ USD4,113,000.
132
Futures options
A FORWARD
IS 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.
133
Buy or sell
a forward
t
Delivery
and
payment
T
Time
BUY means OPEN A LONG POSITION
SELL means OPEN A SHORT POSITION
134
EXAMPLE:
GBP
18.5.99
SPOT
USD1,6850/GBP
30 days forward
USD1,7245/GBP
60 days forward
USD1,7455/GBP
90 days forward
USD1,7978/GBP
180 days forward
USD1,8455/GBP
The existence of forward exchange rates
implies that there is a demand and supply
for the GBP for future dates.
135
Profit from a
Long Forward Position
P/L
F
Price of Underlying
at Maturity, ST
136
Profit from a
Short Forward Position
P/L
F
Price of Underlying
at Maturity, ST
137
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
138
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.
139
NYMEX.
Light, Sweet Crude Oil
Trading Unit
Futures: 1,000 U.S. barrels (42,000 gallons).
Options: One NYMEX Division light, sweet crude oil futures contract.
Price Quotation
Futures and Options: Dollars and cents per barrel.
Trading Hours
Futures 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 Months
Futures: 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.
140
Minimum Price Fluctuation
Futures and Options: $0.01 (1¢) per barrel ($10.00 per contract).
Maximum Daily Price Fluctuation
Futures: 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 Day
Futures: 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 nonbusiness 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.
141
Exercise of Options
By 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 Prices
Twenty 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.
Delivery
F.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).
142
Delivery Period
All 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.
143
Deliverable Grades
Specific 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.
Inspection
Inspection 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.
144
Position Accountability Limits
Any 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 Requirements
Margins are required for open futures or short options positions. The
margin requirement for an options purchaser will never exceed the
premium.
Trading Symbols
Futures: CL
Options: LO
145
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
Fluctuation
Price changes are registered in multiples of five one
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.
146
Maximum Daily
Price Fluctuation
Initial price limit, based upon the preceding day's
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.
147
Delivery
Copper may be delivered against the highgrade 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.
148
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.
December, March, May, July,
September
Contract Months
Trading Hours
Last Trading Day
Deliverable Grades
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.
Seventh business day preceding the
last business day of the delivery
month.
No. 2 Yellow at par and substitution at
149
differentials established by the
exchange.
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.
150
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.
151
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.
152
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
153
$26,250.
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/oz
margin requirement is 5%.
USD2,950/contract or USD5,900 total.
Maintenance margin is 75%.
USD2,212.5/contract or USD4,425
Total.
154
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
155
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
156
overnight
Open a long position in 10 JUNE crude oil futures for:
$58.50/bbl.VALUE: (10)(1,000)($58.50) = $585,000
INITIAL MARGIN
= (.03)($585,000) = $17,550;
VAR. MARGIN = 80%
SETTLE
PRICE
VALUE
DAY 0
$58.60
$586,000 + 1,000
DAY 1
$58.42
$584,200
DAY 2
$58.75
$587,500 + $3,300
DAY 3
$ 58.32
$ 58.08
$583,200
$580,800
DAY 4
MARKET-TOMARKET
MARGIN
BALANCE
$18,550
- $1,800 $16,750
$20,050
$4,300 $15,750
-$2400 $13,350
157
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
158
Date
Settlement
price:Q
Mark-toMarket for
the long
92.23
Dollar
settlement
price = P
980,575
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
93.25
983,126
-1150
52,550
93.12
982,800
June
2
17
18
Margin
Account **
50,000
-325
52,225
•$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.
159
Delivery
If a contract is not closed out before
maturity, it is usually settled by delivering
The assets underlying the contract. When
There are alternatives about what is
delivered, where it is delivered, and when it
is delivered, the party with the short
position chooses.
Few contracts are settled in cash.
For example, those on stock indices and
Eurodollars.
160
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,
161
Mechanics of Call Futures Options
The underlying asset is
A FUTURES.
This means that when you exercise a
futures option you become committed
to BUY or SELL the asset underlying the
futures, depending on whether you
have a call or a put.
162
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.
163
The Payoff of a futures call exercise
If 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.
164
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 most
recent futures settlement price F(settle).
The put writer obtains a long futures
position and his/her margin account is
adjusted opposite to 2. above.
165
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.
166
Put-Call Parity for Futures
Options (p 329)
ct + Ke-r(T-t) = pt + Ft,Te-r(T-t)
167
Black’s Formula (P 333)
ct  e
 r(T -t)
pt  e
d1 
d2 
F N(d )  KN(d )
KN(d )  F N(d )
t,T
r(T -t)
1
2
2
t,T
1
2
ln(F t,T /K)  σ (T - t)/2
σ T-t
2
ln(F t,T /K)  σ (T - t)/2
σ T-t
168