A C T A U N I V E R S I T A T I S L O D Z I E N S I S
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 k a *
D YN AM IC SC H EM ES OF H ED G IN G
- DELTA H EDG ING AND DELTA -G AM M A H EDG IN G O N CURRENCY MARKET
Abstract
D ynam ic schemes o f hedging against currency risk are carefully stru ctu red p rocedures, which should be carried o u t im m ediately d u rin g continuous tran sactio n m onitoring in case o f changes in currency value. H ence, it can be described as selective, con tin u o u s preventing o f open positions in currency options. T h e 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 fu tu re cu rren cy m ovem ents, as well as certain readiness to b ear th a t k in d o f risk.
T h is article p resen ts tw o d ynam ic schemes o f hedging: d e lta hed g in g an d d elta-g am m a hedging w ith exam ples fro m the Polish currency m ark e t.
Key words: hedging, d e lta hedging, d elta -g am 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 ou t im m ediately d u rin g con tinu ou s tran sa ctio n m o n ito rin g 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 h e hedging techniques which reduce the loss w itho ut excluding the profits from currency m ovem ents are given preference. T h e ir application requires access to reliable forecasts o f futu re currency m ov em ents, as well as certain readiness to bear th a t kind o f risk.
N ow adays, in the financial w orld, a dynam ic schem e called d elta hedging enjoys the highest recognition as one of the m o st soph isticated m eth o d s to spread the risk. It is m o st frequently applied by in stitu tio n al investors to hedge sh o rt positions in derivative instrum ents. D e lta hedging is also used
by b anks, w hich as currency derivatives g ran to r, have to low er th e risk of cu rren t tran sactio n s on currency m arket. Only recently d elta-g am m a hedging wins its p o p u larity as a m odification o f its w ell-know n predecessor.
D elta hedging enables active hedging o f short positions in quoted derivative instru m ents, particularly in options. K now in g the function o f exercising o p tio n co n tra cts, one can calculate the risk level which is in h eren t in having sh o rt positions in these instrum ents.
Figure 1. Exercise fu n ctio n for call o p tio n
Figure 2. Exercise fun ctio n fo r p u t o p tio n
T h e functions are as follows:
Fcall = m in (E - K\ 0) F PUT = m in ( K - £ ; 0) (1)
w here К is the price o f underlying instrum en t and E is th e exercise price. It can be therefore noticed th a t, in case o f a large difference between the c u rren t prices o f underlying instrum ents and the exercise prices, huge losses can occur. It refers particularly to call o p tio n , w here the loss is theoretically infinite. In case o f ow ning such positions the ban k 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 in strum en ts are used to m an ag e open currency positions in options. A p p licatio n o f pricing o f these in stru m en ts to the risk m an agem ent m elts d ow 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 p o rtfolio. H ence, the sho rt position o f derivative instrum ents m ust be hedged by th e 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 en t 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)
w here / stands for the option price (derivative instru m en t) and К is the price o f underlying instrum ent.
A hedged portfolio based on the delta rate should be structured as follows:
fo r one und erly in g in stru m e n t th ere is ' / д o f d eriv ativ e in stru m en ts
W ith such a portfo lio stru ctu re the losses incurred by som e instrum ents shall be covered by m eans o f the profits generated by the o thers, which m eans the ban k shall achieve full hedging. O ne should how ever realise th at such a situ atio n is only possible for a very sh o rt period o f time. In order to hedge for an unlim ited period o f tim e, one should consider the necessary adju stm en t o f p ortfolio. 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 e lta hedging.
T o im plem ent the practice one should assum e the following: first, it m u st 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 rta n t because d elta rate is defined on the basis of pricing m odel. W hen currency is the underlying in stru m en t the G arm an n -K o h lh ag en 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 a ta , b u t m o re often it is implied risk calculated on the basis o f the k now n o p tio n price.
T h e follow ing exam ples based on em pirical d a ta from Polish m arket illustrate th e d elta hedging scheme for a ban k q u o tin g call o p tio n in U SD .
T h e choice o f exam ples from 1998 was d eterm inated by one o f the m ost interesting situ atio n s on the Polish currency m ark et.
Exam ple 1. O n 1st A ugust 1998 IN G Barings B ank q u o tes a E u rop ean call o p tio n for 100 000 U SD 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, th e ban k decides to hedge its p o sitio n by m ean s o f d ynam ic d e lta hed g in g w ith daily adjustm ent. In o rd e r to apply the scheme the G a rm a n -K o h lh a g e n 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 ug ust 1998 were ad opted.
E stim ated volatility as the im plied risk from G a rm an -K o h lh a g en m odel is 10.5 % . C u rre n t interest rates w ithout the risk arc as follows:
rr = 4.71% in the USA (the average re tu rn o f 3-m onth T re asu ry Bills), rd — 17.74% in P oland (the average re tu rn o f 13-week treasu ry bonds). T he ban k 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 situatio n the d elta ra te is 0.41 which m eans the necessity to buy a b o u t 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 p eratio n be covered from the b a n k ’s assets. T h e transaction cost shall rise to 141 473 P L Z , 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 a rt o f the currency assets when it falls down. In this way the ban k is entirely hedged by m eans of daily adjustm ents o f the portfolio. D etails are presented in T ab le 1.
A fter one m o n th the option expires in-the-money (which m eans the exercise price is under the curren t underlying asset price). A n optio n 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 th e bank sells th e fixed am o unt o f currency for 348 500 PLZ. H ow ever, for its p u rchase 350 679 PL Z from ban k ’s assets were spent, which results in a favourable flow o f financial means; o r 369 863 P L Z from the loan, which results in the loss. T ak in g into account the m oney gained on selling the option the b o ttom line a t th e expiration is:
B a n k ’s assets L o an
Selling the o p tio n 4 020 4 020
P ro fit fro m exercising the o p tio n 348 500 348 500 T h e c o st o f currency p u rc h a se 350 679 369 863
T o ta l 1 841 -1 7 343
D espite adverse m a rk e t condition, i.e. rise o f exchange ra te over the exercise price the bank profited on the whole o peration d ue to hedging its position from th e b a n k ’s assets. In case it was no t hedged, the b an k would have had to buy 100 000 U S D on the m ark et for 3.815 P L Z each, which should m ean the loss am o u n tin g to 28 980 PLZ. It tu rn ed o u t th a t financing the hedging schem e by m eans o f loan is disadv antag eou s, alth o u g h it is still m ore adv an tag eo u s th a n no hedging schem e at all.
O p tio n day s U S D /P L Z exchange ra te D e lta U S D p u rc h a se d U S D cu m u lated L o a n C u m u lated lo an In terests L o a n + in terests 22 3.4390 0.41 41 000.00 41 000 140 999.00 140 999 473.702 141 473 21 3.4320 0.37 - 4 000.00 37 000 - 1 3 728.00 127 271 427.582 128 172 20 3.4245 0.33 - 4 000.00 33 000 -1 3 698.00 113 573 381.562 114 856 19 3.4195 0.30 - 3 000.00 30 000 -1 0 258.50 103 315 347.097 104 944 18 3.4450 0.41 11 000.00 41 000 37 895.00 141 210 474.410 143 314 17 3.4450 0.40 -1 000.00 40 000 - 3 445.00 137 765 462.836 140 332 16 3.4460 0.40 0.00 40 000 0.00 137 765 462.836 140 795 15 3.4905 0.63 23 000.00 63 000 80 281.50 218 046 732.551 221 809 14 3.4540 0.42 -21 000.00 42 000 - 7 2 534.00 145 512 488.864 149 763 13 3.5045 0.71 29 000.00 71 000 101 630.50 247 143 830.304 252 224 12 3.4995 0.68 - 3 000.00 68 000 - 1 0 498.50 236 644 795.033 242 521 11 3.4995 0.68 0.00 68 000 0.00 236 644 795.033 243 316 10 3.5210 0.80 12 000.00 80 000 4 2 252.00 278 896 936.983 286 505 9 3.5280 0.84 4 000.00 84 000 14 112.00 293 008 984.394 301 601 8 3.6000 0.99 15 000.00 99 000 54 000.00 347 008 1 165.813 356 767 7 3.6710 1.00 1 000.00 100 000 3 671.00 350 679 1 178.147 361 616 6 3.7950 1.00 0.00 100 000 0.00 350 679 1 178.147 362 794 5 3.6650 1.00 0.00 100 000 0.00 350 679 1 178.147 363 972 4 3.6950 1.00 0.00 100 000 0.00 350 679 1 178.147 365 151 3 3.7790 1.00 0.00 100 000 0.00 350 679 1 178.147 366 329 2 3.7950 1.00 0.00 100 000 0.00 350 679 1 178.147 367 507 1 3.8150 1.00 0.00 100 000 0.00 350 679 1 178.147 368 685 0 3.8150 1.00 0.00 100 000 0.00 350 679 1 178.147 369 863 D y n a m ic S ch em es of Hedging - D el ta H e d g in g ...
Exam ple 2. O n l sl Septem ber 1998 1NG Barings Bank q uotes a European call o p tio n fo r 100 000 U SD with the exercise price 3.87 P L Z and one m o n th ex piration date. Because o f the currency m ovem ents, th e bank decides to hedge its p o sitio n by m ean s o f dy nam ic d e lta hedging with daily ad ju stm ent. In ord er to apply the scheme the G a rm an -K o h lh a g en m odel and the currency values from Septem ber 1998 were accepted as the basis o f calculation. E stim ated volatility as the implied risk is 17.8 % . C u rrent in terest rates w ith o u t the risk are as follows:
Гу = 4.69% in the U SA (the average re tu rn o f 3-m onth T reasu ry Bills), rd = 17.13% in P oland (the average re tu rn o f 13-week treasury bonds). T h e b an k opens the sh o rt position with the value o f 3.815 P L Z , gaining 7270 PL Z from selling the option. In this situation the d elta rate is 0.39 which m eans the necessity to buy a b o u t 39 000 d ollars. W ith the price o f 3.815 the b ank has to bear th e cost o f 148 785 P L Z . W ith the p o rtfo lio adjustm ents sim ilar to the first case the situ atio n 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 th an the curren t underlying asset price), so it is not exercised. H ence, on the last day the b ank sells all the dollars in question, w hich results in 5250 PL Z o f cum ulated cost o f hedging from the b a n k ’s assets. A dding to it the m oney gained on selling th e o p tio n , the bottom line at the expiration is:
By hedging its sh o rt position at the time o f q uite considerable currency m ovem ents the ban k m anaged to p rofit on the o p eratio n . It m u st be m en tio n ed how ever th a t in this p a rtic u la r case th e b a n k w ould have achieved a better result w ithou t any hedging a t all. T he p rofit would be equal to m oney gained from selling the option, i.e. 7270 P L Z . Y et, with the volatility estim ated at 17.8% it is difficult to predict w h at shall happen w ith the currency value within a one m o n th tim e and w h a t will be the c h a racter o f the changes. T he deep crisis in R ussia a t th a t tim e evoked a lot o f pessim istic forecasts for global econom y and a t the beginning of S eptem ber 1998 it was difficult to predict such a quick stabilisatio n of U S D /P L Z exchange ra te and the return to the pre-crisis value. T herefore, the decision to hedge the tran sactio n o r n o t is very com plicated and depends on individual app ro ach to the possible developm ent o f the future situation.
Selling the o p tion
T h e cost o f currency p u rch ase T o tal B an k ’s assets 7 270 - 5 250 2 020 L o an 7 270 - 6 252 1 018
O p tio n days U S D /P L Z exchange ra te D elta U SD p u rch ased U S D cu m u lated L o an C u m u lated lo an In tere sts L o an + in terests 22 3.8150 0.39 39 000.00 39 000 148 785.00 148 785 490.132 149 275 21 3.6850 0.04 - 3 5 000.00 4 000 -1 2 8 975.00 19 810 65.259 20 365 20 3.6880 0.04 0.00 4 000 0.00 19 810 65.259 20 431 19 3.6880 0.03 -1 000.00 3 000 - 3 688.00 16 122 53.110 16 796 18 3.6240 0.00 - 3 000.00 0 - 1 0 872.00 5 250 17.295 5 941 17 3.6010 0.00 0.00 0 0.00 5 250 17.295 5 958 16 3.6180 0.00 0.00 0 0.00 5 250 17.295 5 976 15 3.5650 0.00 0.00 0 0.00 5 250 17.295 5 993 14 3.6540 0.00 0.00 0 0.00 5 250 17.295 6 010 13 3.6540 0.00 0.00 0 0.00 5 250 17.295 6 028 12 3.5860 0.00 0.00 0 0.00 5 250 17.295 6 045 11 3.5860 0.00 0.00 0 0.00 5 250 17.295 6 062 10 3.5865 0.00 0.00 0 0.00 5 250 17.295 6 079 9 3.5730 0.00 0.00 0 0.00 5 250 17.295 6 097 8 3.6030 0.00 0.00 0 0.00 5 250 17.295 6 114 7 3.6030 0.00 0.00 0 0.00 5 250 17.295 6 131 6 3.5950 0.00 0.00 0 0.00 5 250 17.295 6 149 5 3.5830 0.00 0.00 0 0.00 5 250 17.295 6 166 4 3.5820 0.00 0.00 0 0.00 5 250 17.295 6 183 3 3.5490 0.00 0.00 0 0.00 5 250 17.295 6 200 2 3.5470 0.00 0.00 0 0.00 5 250 17.295 6 218 1 3.5470 0.00 0.00 0 0.00 5 250 17.295 6 235 0 3.5620 0.00 0.00 0 0.00 5 250 17.295 6 252 D y n a m ic Sc he me s of He dg ing - D elt a H e d g in g ...
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-g am m a hedging schem e is a m od ification o f d elta 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 h e gam m a rate presents to w hat extent shall d elta change when the underlying instru m en t price changes. H ence, it is second derivative o f the option price with respect to underlying instrum ent price.
r - £
G a m m a for a single o p tio n is always positive and depends on the price of underlying instru m en t K.
In d elta-g am m a hedging scheme th e flexibility o f change in the option value w ith respect to change in the underlying in stru m en t price reduced by the flexibility o f change in delta with respect to change in the prim ary in stru m en t price is used as the exact m easure o f th e n u m b er o f underlying asset. A hedged portfo lio based on the d elta and gam m a ra te should be stru ctu red as follows:
fo r one underlying in stru m e n t th ere is l/д — V r d erivative in stru m en ts
W ith such a portfo lio stru ctu re 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 b ank shall achieve full hedging.
Exam ple 3. T h e ban k q uotes a E uro p ean call o p tio n fo r 100 000 U SD with the exercise price 3.5 PLZ and one m o n th exp iratio n date. Because o f th e currency m ovem ents, the bank decides to hedge its po sitio n by m eans o f dynam ic delta-gam m a hedging with daily adjustm en t. In ord er to apply the schem e the G arm an -K o h lh ag en m odel was accepted as the basis o f calculatio n. Hence:
N' ( d . ) e ~ TfT Д = e r‘ N(d) and Г = --- -= —
K o j T
T h e rem aining rates are as above: estim ated volatility fo r one m o n th is 10.8% . C u rre n t interest rates w ithout the risk are as follows:
rf = 4.71% in the U SA (the average return o f 3-m onth T re asu ry Bills), rd = 17.74% in P oland (the average retu rn o f 13-week treasu ry bonds).
T he ban k opens the sh o rt position w ith the value o f 3.439 P L Z , gaining 4020 P L Z from selling the option. In this situ atio n th e d elta rate is 0.41 reduced by gam m a - 0,36, which m eans the necessity to buy a b o u t 36 400 dollars. W ith th e 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 a n k ’s assets. T he transaction cost shall rise to 125 240 PLZ, should the hedging be covered w ith a loan. T o hedge the p o rtfo lio efficiently the bank should buy the currency when delta-gam m a ra te goes up and sell p a rt o f the currency assets w hen it falls dow n. In this way the ban k is entirely hedged by m eans o f daily adjustm ents o f the p o rtfo lio . D etails are presented in T ab le 3.
A fter one m o n th the option expires in-the-money (which m eans the exercise price is under the cu rren t underlying asset price). A n o ptio n buyer shall w ant to exercise an optio n which m akes the bank obliged to provide 100 000 USD at the value o f 3.485 PLZ. A t the expiration the b an k sells the fixed am ou nt o f currency for 348 500 PLZ. H ow ever, for its purch ase 351 525 P L Z from ban k ’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 ak in g in to account the m oney gained on selling the option the b o tto m line at the exp iratio n is:
B a n k ’s assets L o an
Selling the o p tion 4 020 4 020
P ro fit fro m exercising the o p tion 348 500 348 500 T h e co st o f currency p u rch ase 351 525 354 145
T o ta l 995 -1 625
D espite adverse m a rk e t condition, i.e. rise o f exchange ra te over the exercise price the b a n k m anaged to p rofit on the whole o p eratio n. In case it was not hedged, the b a n k would have had to buy 100 000 U S D on th e m a rk e t for 3.815 P L Z each, which should m ean the loss am o u n tin g to 28 980 PLZ.
A s it tu rn s o u t adjusting the delta ra te with the g am m a ra te slightly reduces the o p eration profit when it is financed from b a n k ’s assets. However, in co m p ariso n to d elta scheme, it considerably reduces th e loss by financing the hedging schem e w ith a loan. In case o f a bigger fa v o u rab le currency m ovem ent the profit from delta-gam m a hedging schem e could entirely cover the interests and lead to a positive final financial result.
Example 4. O n 1st Septem ber 1998 IN G Barings B ank quotes a European call option for 100 000 U SD with the exercise price 3.87 P L Z and one m onth expiration date. In ord er to apply dynam ic delta-gam m a hedging scheme with daily adjustm ent the G arm an -K o h lh ag en m odel w as accepted as th e basis o f calculation. T h e rem aining rates are as follows: estim ated volatility as the im plied risk is 17.8% . C u rre n t interest rates w ith o u t the risk are as follows: rf = 4.69% in the USA (the average re tu rn o f 3-m onth T reasu ry Bills), rd = 17.13% in P oland (the average re tu rn o f 13-week treasu ry bonds).
O p tio n days U S D /P L Z exchange ra te D e lta G a m m a U S D p u rch ased U S D cu m u lated L o an C u m u la ted lo an In terests L o a n + interests 22 3.4390 0.41 0.05 36 400.00 36 400 125 179.60 125 180 59.915 125 240 21 3.4320 0-37 0.05 - 3 990.00 32 410 -1 3 693.68 111 486 53.361 111 599 20 3.4245 0.33 0.05 - 3 920.00 28 490 -1 3 424.04 98 062 46.935 98 222 19 3.4195 0 3 0 0.04 - 2 920.00 25 570 - 9 984.94 88 077 42.156 88 279 18 3.4450 0.41 0.05 10 360.00 35 930 35 690.20 123 767 59.239 124 029 17 3.4450 0.40 0.05 -1 120.00 34 810 - 3 858.40 119 909 57.392 120 228 16 3.4460 0.40 0.05 -1 4 0 .0 0 34 670 -482.44 119 426 57.161 119 802 15 3.4905 0.63 0.05 23 030.00 57 700 80 386.22 199 813 95.636 200 284 14 3.4540 0.42 0.06 -2 1 470.00 36 230 - 7 4 157.38 125 655 60.142 126 187 13 3.5045 0.71 0.05 29 580.00 65 810 103 663.11 229 318 109.759 229 960 12 3.4995 0.68 0.06 - 3 430.00 62 380 - 1 2 003.29 217 315 104.013 218 061 11 3.4995 0.68 0.06 -2 5 0 .0 0 62 130 -874.87 216 440 103.595 217 289 10 3.5210 0.80 0.05 13 070.00 75 200 46 019.47 262 460 125.621 263 434 9 3.5280 0.84 0.04 4 390.00 79 590 15 487.92 277 947 133.034 279 055 8 3.6000 0.99 0.00 18 980.00 98 570 68 328.00 346 275 165.738 347 549 7 3.6710 1.00 0.00 1 430.00 100 000 5 249.53 351 525 168.250 352 967 6 3.7950 1.00 0.00 0.00 100 000 0.00 351 525 168.250 353 135 5 3.6650 1.00 0.00 0.00 100 000 0.00 351 525 168.250 353 303 4 3.6950 1.00 0.00 0.00 100 000 0.00 351 525 168.250 353 472 3 3.7790 1.00 0.00 0.00 100 000 0.00 351 525 168.250 353 640 2 3.7950 1.00 0.00 0.00 100 000 0.00 351 525 168.250 353 808 1 3.8150 1.00 0.00 0.00 100 000 0.00 351 525 168.250 353 976 0 3.8150 1.00 0.00 0.00 100 000 0.00 351 525 168.250 354 145
O p tio n day's U S D /P L Z exchange ra te D elta G a m m a U S D p u rch ased U S D cu m u lated L o an C u m u lated lo an In tere sts L o a n + in terests 22 3.8150 0.39 0.0410 34 900.00 34 900 133 143.50 133 144 62.486 133 206 21 3.6850 0.04 0.0100 -3 1 900.00 3 000 -1 1 7 551.50 15 592 7.318 15 662 20 3.6880 0.04 0.0099 10.00 3 010 36.88 15 629 7.335 15 706 19 3.6880 0.03 0.0091 -9 2 0 .0 0 2 090 -3 392.96 12 236 5.743 12 319 18 3.6240 0.00 0.0014 - 2 230.00 -1 4 0 -8 081.52 4 154 1.950 4 239 17 3.6010 0.00 0.0005 90.00 -5 0 324.09 4 478 2.102 4 565 16 3.6180 0.00 0.0007 -2 0 .0 0 - 7 0 -7 2 .3 6 4 406 2.068 4 495 15 3.5650 0.00 0.0000 70.00 0 249.55 4 656 2.185 4 747 14 3.6540 0.00 0.0015 -1 5 0 .0 0 -1 5 0 -5 4 8 .1 0 4 108 1.928 4 201 13 3.6540 0.00 0.0011 40.00 -1 1 0 146.16 4 254 1.996 4 349 12 3.5860 0.00 0.0000 110.00 0 394.46 4 648 1181 4 745 11 3.5860 0.00 0.0000 0.00 0 0.00 4 648 2.181 4 748 10 3.5865 0.00 0.0000 0.00 0 0.00 4 648 2.181 4 750 9 3.5730 0.00 0.0000 0.00 0 0.00 4 648 2.181 4 752 8 3.6030 0.00 0.0000 0.00 0 0.00 4 648 2.181 4 754 7 3.6030 0.00 0.0000 0.00 0 0.00 4 648 2.181 4 756 6 3.5950 0.00 0.0000 0.00 0 0.00 4 648 2.181 4 759 5 3.5830 0.00 0.0000 0.00 0 0.00 4 648 2.181 4 761 4 3.5820 0.00 0.0000 0.00 0 0.00 4 648 2.181 4 763 3 3.5490 0.00 0.0000 0.00 0 0.00 4 648 2.181 4 765 2 3.5470 0.00 0.0000 0.00 0 0.00 4 648 2.181 4 767 1 3.5470 0.00 0.0000 0.00 0 0.00 4 648 2.181 4 769 0 3.5620 0.00 0.0000 0.00 0 0.00 4 648 2.181 4 772
T h e ban k opens the sh o rt position with the value o f 3.815 P L Z , gaining 7270 PL Z from selling the option. In this situ atio n the d e lta rate is 0.39, reduced by gam m a - 0,041 which m eans the necessity to buy a b o u t 34 900 d ollars. W ith the price o f 3.815 the bank has to bear th e cost o f 133 143.50 P L Z , should the operation be covered from the b a n k ’s assets. T h e transaction cost shall rise to 133 206 P L Z , should the hedging be covered with a loan. W ith the daily portfolio adjustm ents the situation is presented in T ab le 4. A fte r one m o n th the optio n expires out-of-the-m oney (w hich m eans the exercise price is higher th an the curren t underlying asset price), so it is not exercised. H ence, on the last day the bank sells all th e d o llars in question, which results in 4648 P L Z o f cum ulated cost o f hedging from the b a n k ’s assets o r 4772 P L Z from th e loan. A dding to it th e m oney gained on selling the o p tio n , the bottom line at the expiratio n is:
By hedging its sh o rt position in currency optio n th e b an k has profited on the o p eratio n . N aturally in case the o p tio n is n o t exercised, the unhedged po sitio n is alw ays m ore beneficial, bu t to predict th e developm ent o f such a situ atio n on q u o tin g the optio n is extrem ely difficult. O n th e o th er hand, the efficiency o f delta-gam m a hedging w ith such currency m ovem ents is slightly higher th an the one with della hedging scheme.
H edging schemes presented above appear to be only selected cases. Yet, already o n their basis it can be noticed th a t a p a rt from legal coverage o r p o rtfo lio o p tim isatio n m eth o d s there are derivative currency instrum ents which can efficiently hedge legal entities, such as b anks, ag ain st exchange risk. T h eir p rim ary advantage is a relatively sm all capital necessary to provide the hedging. M oreover, the choice and ap plication o f dynam ic schemes depends exclusively on the expectations o f the legal entity which hedges its position.
D zierżą J. (1998), O zabezpieczaniu przed ryzykiem w alutow ym , R yn e k Term inow y, N o 2, 20-25. H ull J. (1989), Options, Futures and O ther Derivative Securities, P rentice H all, N ew Y ork. H ull J. (1997), K o n tra k ty term inowe i opcje. W prowadzenie, W IG P R E S S , W arszaw a. W eron A ., W eron R . (1998), Inżynieria finansow a, W N T , W arszaw a.
Selling the optio n
T h e co st o f cu rren cy p u rc h ase T o ta l R a n k ’s assets 7 270 - 4 648 2 622 L o an 7 270 -4 772 2 498 R E F E R E N C E S
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 J Ą 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 RY N K U W A L U T O W Y M Streszczenie
D y n am iczn e strategie zabezpieczające przed ryzykiem w alutow ym to o p ra co w a n e p ro ced u ry 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, selektyw ne przeciw działanie pow staw aniu o tw arty ch pozycji w o p cjach w alutow ych. P referow ane są te techniki zab ezpieczania, k tó re o g raniczając w ysokość stra t, nie w ykluczają zysków kursow ych. Ich stosow anie w ym aga dostępu d o rzetelnych p ro g n o z przyszłych zm ian cen w alu t, a tak że o k reślo n eg o s to p n ia gotow ości d o p o n o szen ia tego typu ryzyka.
A rty k u ł p rezen tu je d w a rodzaje dynam icznych strategii hedgingow ych: d e lta hedging i d e lta -g a m m a hed g in g w raz z p rz y k ład am i d la polskiego ryn k u w alutow ego.
