ACTA
UNIVERSITATIS
LODZIENSIS
F O L IA O E C O N O M IC A 194, 2005
M o n i k a Z i e l iń s ka *
-
DYNAM IC SCH EM ES OF H EDG ING
DELTA HEDGING AND DELTA-GAM M A HEDGING
O N CURRENCY MARKET
Abstract
D ynam ic schemes o f hedging against currency risk are carefully structured procedures, which
should be carried out imm ediately during continuous transaction m onitoring in case o f changes in
currency value. H ence, it can be described as selective, continuous preventing o f open positions in
currency options. T he hedging techniques which reduce the loss w ithout excluding the profits from
currency m ovem ents are given preference. T heir application requires access to reliable forecasts of
fu tu re currency m ovem ents, as well as certain readiness to bear th a t kind o f risk.
T h is article presents tw o dynam ic schemes o f hedging: d elta hedging and delta-gam m a
hedging w ith exam ples from the Polish currency m arket.
Key words: hedging, d elta hedging, delta-gam m a hedging.
I. IN T R O D U C T IO N
D ynam ic schemes o f hedging against currency risk are carefully structured
procedures, which should be carried out im m ediately during continuous
tran sactio n m o n ito rin g in case o f changes in currency value. Hence, it can
be described as selective, continuous preventing o f open positions in currency
options. T he hedging techniques which reduce the loss w ithout excluding
the profits from currency m ovem ents are given preference. T h eir application
requires access to reliable forecasts o f future currency m ovem ents, as well
as certain readiness to bear th at kind o f risk.
N ow adays, in the financial world, a dynam ic scheme called delta hedging
enjoys the highest recognition as one of the m o st sophisticated m ethods to
spread the risk. It is m ost frequently applied by institutional investors to
hedge sh o rt positions in derivative instrum ents. D elta hedging is also used
by banks, which as currency derivatives grantor, have to low er the risk of
current transactions on currency m arket. Only recently delta-gam m a hedging
wins its p o pularity as a m odification of its well-know n predecessor.
Delta hedging enables active hedging o f short positions in quoted derivative
instrum ents, particularly in options. K now ing the function o f exercising
optio n co n tracts, one can calculate the risk level which is inherent in having
sh o rt positions in these instrum ents.
Figure 1. Exercise function for call option
Figure 2. Exercise function for p u t option
T h e functions are as follows:
F call = m in (E - K\ 0)
F PUT = m in ( K - £ ; 0)
(1)
where К is the price o f underlying instrum ent and E is the exercise price.
It can be therefore noticed th at, in case o f a large difference between
the cu rren t prices o f underlying instrum ents and the exercise prices, huge
losses can occur. It refers particularly to call option, where the loss is
theoretically infinite. In case of owning such positions the bank has a natural
need to hedge them .
II. D E L T A H E D G IN G S C H E M E
M ethods based solely on the idea o f pricing derivative instrum ents are
used to m anage open currency positions in options. A pplication o f pricing
o f these instrum ents to the risk m anagem ent m elts dow n to observing the
value changes o f underlying instrum ents and adjusting the respective num ber
o f underlying instrum ents, which shall be included in portfolio. Hence, the
short position o f derivative instrum ents m ust be hedged by the long position
o f the assets they refer to.
T h e flexibility o f change in the option value with respect to change in
the underlying in stru m ent price is used as the exact m easure o f the num ber
o f underlying asset. T he rate is defined as delta:
Д= 1
(2)
where / stands for the option price (derivative instrum ent) and К is the
price o f underlying instrum ent.
A hedged portfolio based on the delta rate should be structured as follows:
for one underlying instrum ent there is ' / д o f derivative instrum ents
W ith such a portfolio structure the losses incurred by som e instrum ents
shall be covered by m eans o f the profits generated by the others, which
m eans the bank shall achieve full hedging. One should however realise that
such a situ atio n is only possible for a very short period o f time. In order
to hedge for an unlim ited period o f time, one should consider the necessary
adjustm ent o f portfolio. Periodical (e.g. weekly) adjusting o f the hedge is
called rebalancing and the whole scheme conducted in this way is defined
as dynam ic d elta hedging.
T o im plem ent the practice one should assum e the following: first, it
m ust be decided at w hat tim e intervals the portfolio will be adjusted. Next,
which pricing m odel will be adopted by the bank. It is im p o rtan t because
delta rate is defined on the basis of pricing m odel. W hen currency is the
underlying instrum ent the G arm ann-K ohlhagen m odel is used. Finally, the
correct volatility o f underlying instrum ents should be accepted. It m ay be
the value estim ated on the basis o f historical d ata, but m ore often it is
implied risk calculated on the basis o f the know n option price.
T h e follow ing examples based on em pirical d a ta from Polish m arket
illustrate the delta hedging scheme for a bank quo tin g call option in USD.
T he choice o f examples from 1998 was determ inated by one o f the m ost
interesting situations on the Polish currency m arket.
Example 1. O n 1st A ugust 1998 IN G Barings Bank quotes a E uropean
call op tio n for 100 000 USD with the exercise price 3.485 P L Z and one
m o n th expiration date. Because o f the currency m ovem ents, the bank decides
to hedge its po sitio n by m eans o f dynam ic d elta hedging w ith daily
adjustm ent. In order to apply the scheme the G arm an -K o h lh ag en m odel
was accepted as the basis o f calculation.
Д = e - r' TN ( d x).
F o r the needs o f the exam ple the currency values from A ugust 1998 were
adopted.
E stim ated volatility as the implied risk from G arm an -K o h lh ag en m odel
is 10.5 % . C u rren t interest rates w ithout the risk arc as follows:
rr = 4.71% in the USA (the average return o f 3-m onth T reasury Bills),
rd — 17.74% in Poland (the average return o f 13-week treasury bonds).
T he bank opens the short position with the value o f 3.439 PLZ, gaining 4020
P L Z from selling the option. In this situation the d elta rate is 0.41 which
m eans the necessity to buy about 41 000 dollars. W ith the price o f 3.439 the
bank has to bear the cost o f 140 999 PLZ, should the o peration be covered
from the b an k ’s assets. T he transaction cost shall rise to 141 473 PLZ, should
the hedging be covered with a loan. T o hedge the portfolio efficiently the bank
should buy the currency when delta rate goes up and sell p art o f the currency
assets when it falls down. In this way the bank is entirely hedged by m eans of
daily adjustm ents o f the portfolio. D etails are presented in T able 1.
A fter one m onth the option expires in-the-money (which m eans the exercise
price is under the current underlying asset price). A n option buyer shall w ant
to exercise an option which m akes the bank obliged to provide 100 000 USD
at the value o f 3.485 PLZ. A t the expiration the bank sells the fixed am ount
o f currency for 348 500 PLZ. However, for its purchase 350 679 PLZ from
bank’s assets were spent, which results in a favourable flow o f financial means;
or 369 863 PL Z from the loan, which results in the loss. T aking into account
the m oney gained on selling the option the bottom line at the expiration is:
Selling the o ption
P rofit from exercising the o ption
T h e co st o f currency p urchase
T o ta l
B an k ’s
4
348
350
1
assets
020
500
679
841
L oan
4 020
348 500
369 863
-1 7 343
D espite adverse m ark et condition, i.e. rise o f exchange rate over the exercise
price the bank profited on the whole operation due to hedging its position
from the b an k ’s assets. In case it was not hedged, the bank would have
had to buy 100 000 U SD on the m arket for 3.815 PL Z each, which should
m ean the loss am ounting to 28 980 PLZ. It turned out th at financing the
hedging scheme by m eans o f loan is disadvantageous, although it is still
m ore advantageous th a n no hedging scheme at all.
0.41
0.37
0.33
0.30
0.41
0.40
0.40
0.63
0.42
0.71
0.68
0.68
0.80
0.84
0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
41
-4
-4
-3
11
-1
000.00
000.00
000.00
000.00
000.00
000.00
0.00
23 000.00
-21 000.00
29 000.00
- 3 000.00
0.00
12 000.00
4 000.00
15 000.00
1 000.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
41
37
33
30
41
40
40
63
42
71
68
68
80
84
99
100
100
100
100
100
100
100
100
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
L o an
140
-1 3
-1 3
-1 0
37
-3
80
-7 2
101
-1 0
42
14
54
3
999.00
728.00
698.00
258.50
895.00
445.00
0.00
281.50
534.00
630.50
498.50
0.00
252.00
112.00
000.00
671.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
C um ulated
loan
140
127
113
103
141
137
137
218
145
247
236
236
278
293
347
350
350
350
350
350
350
350
350
999
271
573
315
210
765
765
046
512
143
644
644
896
008
008
679
679
679
679
679
679
679
679
Interests
473.702
427.582
381.562
347.097
474.410
462.836
462.836
732.551
488.864
830.304
795.033
795.033
936.983
984.394
1 165.813
1 178.147
1 178.147
1 178.147
1 178.147
1 178.147
1 178.147
1 178.147
1 178.147
L oan +
interests
141
128
114
104
143
140
140
221
149
252
242
243
286
301
356
361
362
363
365
366
367
368
369
473
172
856
944
314
332
795
809
763
224
521
316
505
601
767
616
794
972
151
329
507
685
863
H edging...
3.4390
3.4320
3.4245
3.4195
3.4450
3.4450
3.4460
3.4905
3.4540
3.5045
3.4995
3.4995
3.5210
3.5280
3.6000
3.6710
3.7950
3.6650
3.6950
3.7790
3.7950
3.8150
3.8150
USD
cum ulated
- Delta
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
U SD
pu rch ased
of Hedging
D elta
Schemes
U S D /P L Z
exchange ra te
Dynamic
O ption
days
Exam ple 2. On l sl Septem ber 1998 1NG Barings Bank quotes a European
call o p tio n for 100 000 USD with the exercise price 3.87 PL Z and one
m o n th expiration date. Because o f the currency m ovem ents, the bank decides
to hedge its position by m eans o f dynam ic d elta hedging with daily
adjustm ent. In order to apply the scheme the G arm an-K ohlhagen m odel
and the currency values from Septem ber 1998 were accepted as the basis
o f calculation. Estim ated volatility as the implied risk is 17.8 % . C urrent
interest rates w ithout the risk are as follows:
Гу = 4.69% in the USA (the average return o f 3-m onth T reasury Bills),
rd = 17.13% in P oland (the average return o f 13-week treasury bonds).
T he ban k opens the short position with the value o f 3.815 PL Z , gaining
7270 PLZ from selling the option. In this situation the delta rate is 0.39
which m eans the necessity to buy ab o u t 39 000 dollars. W ith the price
o f 3.815 the bank has to bear the cost o f 148 785 PLZ. W ith the
p ortfolio adjustm ents sim ilar to the first case the situation is presented
in T ab le 2.
A fter one m o n th the option expires out-of-the-m oney (which m eans the
exercise price is higher than the current underlying asset price), so it is not
exercised. H ence, on the last day the bank sells all the dollars in question,
which results in 5250 PLZ o f cum ulated cost o f hedging from the b an k ’s
assets. A dding to it the m oney gained on selling the option, the bottom
line at the expiration is:
Selling the option
T h e cost o f currency purchase
T otal
B ank’s assets
7 270
- 5 250
2 020
L oan
7 270
- 6 252
1 018
By hedging its short position at the time o f quite considerable currency
m ovem ents the bank m anaged to profit on the operation. It m ust be
m en tio n ed how ever th a t in this p artic u la r case the b an k w ould have
achieved a better result w ithout any hedging a t all. T he profit would be
equal to m oney gained from selling the option, i.e. 7270 PLZ. Y et, with
the volatility estim ated at 17.8% it is difficult to predict w hat shall happen
with the currency value within a one m onth tim e and w hat will be the
ch aracter o f the changes. T he deep crisis in R ussia at th a t time evoked
a lot o f pessim istic forecasts for global econom y and at the beginning of
Septem ber 1998 it was difficult to predict such a quick stabilisation of
U S D /P L Z exchange rate and the return to the pre-crisis value. Therefore,
the decision to hedge the transaction or no t is very com plicated and
depends on individual approach to the possible developm ent o f the future
situation.
3.8150
3.6850
3.6880
3.6880
3.6240
3.6010
3.6180
3.5650
3.6540
3.6540
3.5860
3.5860
3.5865
3.5730
3.6030
3.6030
3.5950
3.5830
3.5820
3.5490
3.5470
3.5470
3.5620
0.39
0.04
0.04
0.03
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
39 000.00
-3 5 000.00
0.00
-1 000.00
- 3 000.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
39
4
4
3
000
000
000
000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
L oan
148 785.00
-1 2 8 975.00
0.00
- 3 688.00
-1 0 872.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
C um ulated
lo an
148
19
19
16
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
785
810
810
122
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
Interests
490.132
65.259
65.259
53.110
17.295
17.295
17.295
17.295
17.295
17.295
17.295
17.295
17.295
17.295
17.295
17.295
17.295
17.295
17.295
17.295
17.295
17.295
17.295
L o an +
interests
149
20
20
16
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
275
365
431
796
941
958
976
993
010
028
045
062
079
097
114
131
149
166
183
200
218
235
252
H edging...
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
U SD
cum ulated
- Delta
U SD
purchased
of Hedging
D elta
Schemes
U S D /P L Z
exchange ra te
Dynamic
O ption
days
III. D E L T A -G A M M A H E D G IN G S C H E M E
D elta-gam m a hedging scheme is a m odification o f delta hedging sche­
m e, in which an additional sensitivity rate is calculated to adjust the
selling or purchase value o f underlying instrum ents to hedge the position
in derivative instrum ents. T he gam m a rate presents to w hat extent shall
d elta change when the underlying instrum ent price changes. H ence, it is
second derivative o f the option price with respect to underlying instrum ent
price.
r - £
G am m a for a single option is always positive and depends on the price of
underlying instrum ent K.
In delta-gam m a hedging scheme the flexibility o f change in the option
value with respect to change in the underlying instrum ent price reduced by
the flexibility o f change in delta with respect to change in the prim ary
in strum ent price is used as the exact m easure o f the num ber o f underlying
asset. A hedged portfolio based on the delta and gam m a rate should be
structured as follows:
fo r one underlying instrum ent there is
l/д —V r
derivative instrum ents
W ith such a portfolio structure the losses incurred by som e instrum ents
shall be covered by m eans o f the profits generated by the others, which
m eans the bank shall achieve full hedging.
Exam ple 3. T he bank quotes a E uropean call optio n for 100 000 USD
with the exercise price 3.5 PLZ and one m o n th expiration date. Because
o f the currency m ovem ents, the bank decides to hedge its position by
m eans o f dynam ic delta-gam m a hedging with daily adjustm ent. In order to
apply the scheme the G arm an-K ohlhagen m odel was accepted as the basis
o f calculation. Hence:
Д = e r‘ N(d)
and
N' ( d . ) e ~ TfT
Г = --------- -= —
K o jT
T he rem aining rates are as above: estim ated
10.8% . C u rren t interest rates w ithout the risk
rf = 4.71% in the USA (the average return
rd = 17.74% in Poland (the average return
volatility for one m o n th is
are as follows:
o f 3-m onth T reasury Bills),
o f 13-week treasury bonds).
T he bank opens the short position with the value o f 3.439 P L Z , gaining
4020 P L Z from selling the option. In this situation the delta rate is 0.41
reduced by gam m a - 0,36, which m eans the necessity to buy ab o u t 36 400
dollars. W ith the price o f 3.439 the bank has to bear the cost o f 125 179.60
PLZ, should the operation be covered from the b an k ’s assets. T he transaction
cost shall rise to 125 240 PLZ, should the hedging be covered with a loan.
T o hedge the p ortfolio efficiently the bank should buy the currency when
delta-gam m a rate goes up and sell p art o f the currency assets when it falls
dow n. In this way the bank is entirely hedged by m eans o f daily adjustm ents
o f the p o rtfolio. D etails are presented in T able 3.
After one m o n th the option expires in-the-money (which m eans the exercise
price is under the current underlying asset price). An option buyer shall want
to exercise an option which m akes the bank obliged to provide 100 000 USD
at the value o f 3.485 PLZ. A t the expiration the bank sells the fixed am ount
of currency for 348 500 PLZ. However, for its purchase 351 525 PL Z from
bank’s assets were spent, which results in a favourable flow o f financial means;
or 354 145 P L Z from the loan, which results in the loss. T aking into account
the m oney gained on selling the option the bottom line at the expiration is:
Selling the option
P rofit from exercising the option
T h e cost o f currency purchase
T o tal
B an k ’s assets
4 020
348 500
351 525
995
L oan
4 020
348 500
354 145
-1 625
D espite adverse m ark e t condition, i.e. rise o f exchange rate over the exercise
price the b an k m anaged to profit on the whole operation. In case it was not
hedged, the b an k would have had to buy 100 000 U SD on the m ark et for
3.815 P L Z each, which should m ean the loss am ounting to 28 980 PLZ.
As it tu rn s o u t adjusting the delta rate with the gam m a rate slightly
reduces the operation profit when it is financed from b an k ’s assets. However,
in co m parison to d elta scheme, it considerably reduces the loss by financing
the hedging schem e with a loan. In case o f a bigger favourable currency
m ovem ent the profit from delta-gam m a hedging scheme could entirely cover
the interests and lead to a positive final financial result.
Example 4. On 1st September 1998 IN G Barings B ank quotes a European
call option for 100 000 USD with the exercise price 3.87 PLZ and one m onth
expiration date. In order to apply dynam ic delta-gam m a hedging scheme with
daily adjustm ent the G arm an-K ohlhagen m odel was accepted as the basis o f
calculation. T h e rem aining rates are as follows: estim ated volatility as the
implied risk is 17.8%. C urrent interest rates w ithout the risk are as follows:
rf = 4.69% in the USA (the average return o f 3-m onth T reasury Bills),
rd = 17.13% in Poland (the average return o f 13-week treasury bonds).
O ptio n
days
U S D /P L Z
exchange ra te
D e lta
G am m a
U SD
purchased
U SD
cum ulated
L o an
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
3.4390
3.4320
3.4245
3.4195
3.4450
3.4450
3.4460
3.4905
3.4540
3.5045
3.4995
3.4995
3.5210
3.5280
3.6000
3.6710
3.7950
3.6650
3.6950
3.7790
3.7950
3.8150
3.8150
0.41
0-37
0.33
030
0.41
0.40
0.40
0.63
0.42
0.71
0.68
0.68
0.80
0.84
0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.05
0.05
0.05
0.04
0.05
0.05
0.05
0.05
0.06
0.05
0.06
0.06
0.05
0.04
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
36 400.00
-3 990.00
- 3 920.00
- 2 920.00
10 360.00
-1 120.00
-140.00
23 030.00
-21 470.00
29 580.00
- 3 430.00
-250.00
13 070.00
4 390.00
18 980.00
1 430.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
36 400
32 410
28 490
25 570
35 930
34 810
34 670
57 700
36 230
65 810
62 380
62 130
75 200
79 590
98 570
100 000
100 000
100 000
100 000
100 000
100 000
100 000
100 000
125 179.60
-1 3 693.68
-1 3 424.04
- 9 984.94
35 690.20
-3 858.40
-482.44
80 386.22
-7 4 157.38
103 663.11
-1 2 003.29
-874.87
46 019.47
15 487.92
68 328.00
5 249.53
0.00
0.00
0.00
0.00
0.00
0.00
0.00
C um ulated
lo an
125
111
98
88
123
119
119
199
125
229
217
216
262
277
346
351
351
351
351
351
351
351
351
180
486
062
077
767
909
426
813
655
318
315
440
460
947
275
525
525
525
525
525
525
525
525
Interests
L o an +
interests
59.915
53.361
46.935
42.156
59.239
57.392
57.161
95.636
60.142
109.759
104.013
103.595
125.621
133.034
165.738
168.250
168.250
168.250
168.250
168.250
168.250
168.250
168.250
125
111
98
88
124
120
119
200
126
229
218
217
263
279
347
352
353
353
353
353
353
353
354
240
599
222
279
029
228
802
284
187
960
061
289
434
055
549
967
135
303
472
640
808
976
145
O ptio n
day's
U S D /P L Z
exchange ra te
D elta
G am m a
USD
p urchased
U SD
cum ulated
L oan
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
3.8150
3.6850
3.6880
3.6880
3.6240
3.6010
3.6180
3.5650
3.6540
3.6540
3.5860
3.5860
3.5865
3.5730
3.6030
3.6030
3.5950
3.5830
3.5820
3.5490
3.5470
3.5470
3.5620
0.39
0.04
0.04
0.03
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.0410
0.0100
0.0099
0.0091
0.0014
0.0005
0.0007
0.0000
0.0015
0.0011
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
34 900.00
-31 900.00
10.00
-920.00
- 2 230.00
90.00
-20.00
70.00
-150.00
40.00
110.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
34 900
3 000
3 010
2 090
-1 4 0
-5 0
-7 0
0
-1 5 0
-1 1 0
0
0
0
0
0
0
0
0
0
0
0
0
0
133 143.50
-1 1 7 551.50
36.88
-3 392.96
-8 081.52
324.09
-7 2 .3 6
249.55
-548.10
146.16
394.46
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
C um ulated
loan
133
15
15
12
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
144
592
629
236
154
478
406
656
108
254
648
648
648
648
648
648
648
648
648
648
648
648
648
In terests
L o an +
interests
62.486
7.318
7.335
5.743
1.950
2.102
2.068
2.185
1.928
1.996
1181
2.181
2.181
2.181
2.181
2.181
2.181
2.181
2.181
2.181
2.181
2.181
2.181
133 206
15 662
15 706
12 319
4 239
4 565
4 495
4 747
4 201
4 349
4 745
4 748
4 750
4 752
4 754
4 756
4 759
4 761
4 763
4 765
4 767
4 769
4 772
T he bank opens the short position with the value o f 3.815 PL Z , gaining
7270 PLZ from selling the option. In this situation the delta rate is 0.39,
reduced by gam m a - 0,041 which m eans the necessity to buy about 34 900
dollars. W ith the price o f 3.815 the bank has to bear the cost o f 133 143.50
PLZ, should the operation be covered from the ban k ’s assets. T he transaction
cost shall rise to 133 206 PLZ, should the hedging be covered with a loan.
W ith the daily portfolio adjustm ents the situation is presented in T able 4.
A fter one m o n th the option expires out-of-the-m oney (which m eans the
exercise price is higher than the current underlying asset price), so it is not
exercised. H ence, on the last day the bank sells all the dollars in question,
which results in 4648 PLZ o f cum ulated cost o f hedging from the b an k ’s
assets o r 4772 PL Z from the loan. A dding to it the m oney gained on
selling the o p tion, the bottom line at the expiration is:
Selling the option
T h e cost o f currency purchase
T o tal
R an k ’s assets
7 270
- 4 648
2 622
L oan
7 270
-4 772
2 498
By hedging its short position in currency option the bank has profited on
the o peration. N aturally in case the option is no t exercised, the unhedged
position is always m ore beneficial, but to predict the developm ent o f such
a situation on quoting the option is extrem ely difficult. On the other hand,
the efficiency o f delta-gam m a hedging with such currency m ovem ents is
slightly higher than the one with della hedging scheme.
H edging schemes presented above appear to be only selected cases. Yet,
already on their basis it can be noticed th at ap a rt from legal coverage or
portfolio o p tim isation m ethods there are derivative currency instrum ents
which can efficiently hedge legal entities, such as banks, against exchange
risk. T heir p rim ary advantage is a relatively small capital necessary to
provide the hedging. M oreover, the choice and application o f dynam ic
schemes depends exclusively on the expectations o f the legal entity which
hedges its position.
REFEREN CES
D zierżą J. (1998), O zabezpieczaniu przed ryzykiem walutow ym , R ynek Term inowy, N o 2, 20-25.
H ull J. (1989), Options, Futures and Other Derivative Securities, Prentice H all, New Y ork.
H ull J. (1997), K on tra kty terminowe i opcje. Wprowadzenie, W IG PR E S S, W arszaw a.
W eron A ., W eron R. (1998), Inżynieria finansow a, W N T , W arszaw a.
Monika Zielińska
D Y N A M IC Z N E S T R A T E G IE Z A B E Z P IE C Z A JĄ C E
- D E L T A H E D G IN G I D E L T A -G A M M A H E D G IN G
NA RYN KU W A L U T O W Y M
Streszczenie
D ynam iczne strategie zabezpieczające przed ryzykiem w alutow ym to o pracow ane procedury
czynności, k tó re należy w ykonyw ać n a bieżąco, w sytuacjach zm ian k u rsu w aluty, w czasie
ciągłego m o n ito ro w a n ia transakcji. Jest to zatem stałe, selektywne przeciw działanie pow staw aniu
otw artych pozycji w opcjach w alutow ych. Preferow ane są te techniki zabezpieczania, które
ograniczając w ysokość strat, nie wykluczają zysków kursow ych. Ich stosow anie w ym aga dostępu
d o rzetelnych pro g n o z przyszłych zm ian cen w alut, a także określonego sto p n ia gotow ości d o
p o noszenia tego typu ryzyka.
A rty k u ł prezentuje d w a rodzaje dynam icznych strategii hedgingow ych: d elta hedging
i d elta-g am m a hedging w raz z przykładam i d la polskiego rynku w alutow ego.