Assignment questions: 1.29, 1.30, 1.32, 2.28, 2.29
1.29 – St = $94, time to expiration = 3 months, K = $95, Ct = $4.70, 100 shares of equity vs. 2,000 calls.
Call options are a levered investment relative to the long position in the underlying, i.e. identical capital
invested in call options produces an exposure to 20X the number of shares of the investment in the underlying.
Payoffs ($)
Equity – 100 shares
100(St – 94)
Call options – 2,000
2,000(ST – 95 – 4.70)
-400
0
100
600
1100
1600
-9,400
-9,400
-9,400
600
10,600
20,600
ST
90
94
95
100
105
110
100(ST – 94) = 2,000(ST – 95 – 4.70)
The leverage inherent in call options magnifies the risk along several dimensions. The decision maker must be
confident concerning the direction magnitude and timing of price change before making the investment in call
options.
During the next three months a return of –100% is possible for the equity investment only if the price falls from
$94/share to $0. The investment in call options produces a –100% return for all prices from $0 - $95.
For all terminal prices less than $99.70 the call option investment will produce negative returns.
2,000(ST – 95 – 4.70) = 0
For terminal prices in the range $99.70 - $100 the rate of return of the equity investment exceeds the return
from the investment in call options.
For all terminal prices greater than $100 the return from the investment in call options exceeds the return from
the equity investment by a significant margin.
ST = $101
Equity
4(700/9400) = 0.296
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Call options
4(2600/9400) = 1.108
1.30 – zero coupon bond, long call option K = $25, short call option K = $40
long call:
short call:
Max[0, ST – 25]
Min[0, 40 – ST]
ST < 25
25 < ST < 40
ST > 40
Bond
$1,000
$1,000 + 170(ST – 25)
$1,000 + 2,550
Zero + options
$1,000 + 170( 0 + 0)
$1,000 + 170(ST – 25 + 0)
$1,000 + 170(ST –25 +40 – ST)
$1,000 + $2,550
1.32 – Difference between borrowing and lending rates – gold investment portfolio, (T – t) = 1.0, u = U = 0.
St,ask = ¢250
rborrow = 6%
St,bid =¢ 249
rlend = 5.5%
Cash and carry transactions;
Ft*  (250  U )(1.06)  265
Ft  265
Reverse cash and carry;
Ft*  (249  U )(1.055)  262 .695
Ft  262 .695
262.695  Ft  265
2.28 - U = $0.20/year, r = 5%, t = 3/15/01
Cash and carry 5/01 – 5/02
May 01’
May 02’
Ft
210.75
254
(210.75 + 20)1.05 = 242.2875
Given U(5/01– 5/02) = $0.20 and r5/01 = 5%
Long position May 01’
Short position May 02’
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Guaranteed profit 254 - 242.2875 / bushel = ¢11.71
Contango (FT2 > FT1) in forward market produced incentive to store underlying through cash and carry
transactions, thereby allocating additional units of the underlying for consumption in the delivery period. As in
this example if the size of the contango is greater than the net storage cost arbitrageurs will arrange cash and
carry transactions. The net affect is a transference (through storage) of the underlying from one time period to a
later date.
2.29 – Maintenance margin is a safety buffer for the capital of the clearing corporation. When the margin
balance for an open position reaches the maintenance level a margin call is issued. The trader has until close of
the following day to post variation margin. Variation margin is the difference required to replenish the margin
balance to the initial margin level. If the futures trader fails to post variation margin the position is closed at the
settlement price on the second day.
The clearing corporation has incentive to set required maintenance margin high enough to cover all but the most
extreme two-day price changes.
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The variance of identically and independently (iid) normal distributed price changes is directly proportional to
the time period over which the price change is measured. For instance if the variance of daily price changes is
$0.14, the variance of a 5-day price change is 5 * $0.14 = $0.70. As a result the standard deviation of iid
normal distributed price changes is proportional to the square root of elapsed time.
1-day $0.37  5-day = $0.37 5  $0.82735
Definition of a standardized normal variable;
x

z
z ~ N(0,1) Hence the cumulative probability the random variable x is less than a specific
value is equivalent to the cumulative probability z is less than a specific value.
Prob z <= 2.33 is 0.99
Prob z <= 3.09 is 0.999
Given  = 0 and
̂ estimated from a sample, the quantile of the normal distributed variable x is given by
x  z ( )ˆ 2
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Contract size: WTI oil = 1,000 bbls. Gold = 100 troy ounces .
 = 0.99
 = 0.999
WTI - oil
Gold
2.33  0.31  2  1,000
2.33  2.77  2  100
 $1,021.85
 $911.80
3.09  0.31  2  1,000
3.09  2.77  2  100
 $1,357.40
 $1,211.21
b. Long position in oil 1/10/97 @ $19.45 till 10/26/01 @ $22.03. Initial margin = $1,362 Maintenance
margin = $1,022.
Change in margin balance 1,000($22.03 - $19.45) = $2,580
N = 1,204
Assuming excess margin withdrawn and variation margin paid prior to settlement next trading date, total
withdrawls = $101,020 total variation margin = $98,140, gain/loss from return margin balance on 10/26/01
= $1,062 - $1,362 = $-300, grand total $2,580.
While position is open there were a total of 179 margin calls. On three days the margin balance from the
previous day is less than the size of variation margin on the next trading day.
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